MULTIPLE BOREL CANTELLI LEMMA IN DYNAMICS AND …

MULTIPLE BOREL CANTELLI LEMMA IN DYNAMICS ANDMULTILOG LAW FOR RECURRENCE.

DMITRY DOLGOPYAT, BASSAM FAYAD AND SIXU LIU

Contents

1. Introduction. 22. Multiple Borel Cantelli Lemma. 62.1. The result. 62.2. Estimates on a fixed scale. 82.3. Convergent case. Proof of Theorem 2.2 (a). 102.4. Divergent case. Proof of Theorem 2.2 (b). 102.5. Prescribing some details. 112.6. Poisson regime 122.7. Notes. 133. Multiple Borel Cantelli Lemma for exponentially mixing dynamical systems. 133.1. Good maps, good targets. 133.2. Multiple Borel-Cantelli Lemma for admissible targets. 153.3. Notes. 184. MultiLog Laws for recurrence and hitting times 184.1. Results. 194.2. Weak non-recurrence and proof of Theorem 4.2. 204.3. Generic failure of the MultiLog Law. Proof of Theorem 4.3. 234.4. The case of flows. Proof of Theorem 4.5. 244.5. Notes. 265. Poisson Law for near returns. 265.1. Introduction. 265.2. Poisson Law for Returns. 275.3. Notes. 286. Gibbs measures on the circle: Law of iterated logarithm for hitting times 296.1. Gibbs measures. 296.2. Good targets for Gibbs measures. 306.3. Law of iterated logarithm for hitting times. 316.4. Law of iterated logarithm for return times. Lower bound 326.5. Law of iterated logarithm for return times. Upper bound. 336.6. Notes. 347. Geodesic excursions. 347.1. Excursions in finite volume hyperbolic manifolds. 347.2. Multilog law for geodesic excursions. Proof of Theorem 7.1 367.3. Poisson Law for excursions. Proof of Corollary 7.2 38

1

2 DMITRY DOLGOPYAT, BASSAM FAYAD AND SIXU LIU

7.4. Notes. 398. Multiple Khintchine-Groshev Theorem. 398.1. Statements. 398.2. Reduction to a problem on the space of lattices 408.3. Modifying the initial distribution: homogeneous case. 418.4. Rogers identities 438.5. Proof of Proposition 8.10 438.6. Proof of Theorem 8.11 448.7. The argument in the inhomogeneous case. 458.8. Modifying the initial distribution: inhomogeneous case. 468.9. Proofs of Proposition 8.16 and Theorem 8.17. 468.10. Multiple recurrence for toral translations. 478.11. Notes. 519. Recurrence in configuration space. 519.1. The results. 519.2. MultiLog Law. 529.3. Poisson regime. 559.4. Notes. 5510. Extreme values. 5610.1. From hitting balls to extreme values. 5610.2. Notes. 58Appendix A. Multiple exponentially mixing 58A.1. Basic properties 58A.2. Mixing for Gibbs measures. 60A.3. Examples of exponentially mixing systems 60Appendix B. Fluctuations of local dimension. Proof of Lemma 6.6. 61References 63

1. Introduction.

The study of rare events constitutes an important subject in probability theory. Onone hand, in many applications there are significant costs associated to certain rareevents, so one needs to know how often whose events occur. On the other hand, thereare many phenomena in science which are driven by rare events including metastability,anomalous diffusion (Levy flights) and traps for motion in random media, to mentionjust a few examples.

In the independent setting there are three classical regimes. For the first two consideran array tΩk,nu

nk“1 of independent events such that pn “ PpΩk,nq does not depend on

k. Let Nn be the number of events from the n-th array which have occurred. The firsttwo regimes are:

(i) CLT regime: npn Ñ 8. In this case Nn is asymptotically normal.(ii) Poisson regime: npn Ñ λ. In this case Nn is asymptotically Poisson with param-

eter λ.

MULTIPLE BOREL CANTELLI LEMMA IN DYNAMICS 3

For the third, Borel Cantelli regime we consider a sequence tΩnu of independentevents with different probabilities. In this case the classical Borel Cantelli Lemma says

that infinitely many Ans occur iffÿ

n

PpΩnq “ 8.

A vast literature is devoted to extending the above classical results to the case inde-pendence is replaced by weak dependence. In particular, there are convenient momentconditions which imply similar results in for (weakly) dependent events. One impor-tant distinction between the Poisson Law and the other two results is that the Poissonregime additional geometric conditions on close-by events are required to extend thestatement to dependent case. Without such conditions one can have clusters of rareevents there the number of clusters has Poisson distribution but there may be severalevents inside each cluster. We refer the reader to [4] for the comprehensive discussionof Poisson clustering.

In the present paper we consider a regime which is intermediate between the Pois-son and Borel Cantelli. Namely we consider a family of events Ωn

ρ which are nested:Ωnpρ1q Ă Ωnpρ2q for ρ1 ă ρ2 and such that σpρq “ PpΩρq does not depend on n. LetNnpρq be a number of Ωkpρq, k ď n which has occurred. We fix a sequence ρn such thatnσpρnq Ñ 0 as nÑ 8 and r P N, and ask if infinitely many events

Nnpρnq “ r

occurs. Even if the events Ωnρ are independent for different ρ, the variables Nn1pρn1q and

Nn2pρn2q are strongly dependent if n1 and n2 are of the same order. On the other handif n2 " n1 then those variables are weakly dependent since conditioned on Nn2pρn2q ‰ 0it is very likely that all the events Ωkpρn2q occur for k ą n1. Using this, one can showunder appropriate monotonicity assumptions (see [114]) that Nnpρnq “ r infinitely ofteniff

ÿ

M

PpN2M pρ2M q “ rq “ 8.

Under the condition nρn Ñ 0 it follows that in the independent case

P pNnpρnq “ rq «pnσpρnqq

r

r!.

Therefore, under independence, infinitely many Nnpρnq “ r occur iffÿ

M

2Mrσr pρ2M q “ 8.

The multiple Borel Cantelli Lemma was extended to the dependent setting in [1]. How-ever, the mixing assumptions made in [1] are quite strong requiring good symbolicdynamics which limits greatly the applicability of that result. In the present paper wepresent more flexible mixing conditions for the multiple Borel Cantelli Lemma. Ourconditions are similar to the assumptions typically used to prove Poisson limit theoremsfor dynamical systems. The precise statements of our abstract results will be given inthe Sections 2 and 3 here we describe sample applications to dynamics, geometry, andnumber theory.

MultiLog Law for recurrence. Let f be a map preserving a measure µ. Given

two points x, y let dprqn px, yq be the r closest distance among dpfky, xq for 0 ď k ă n. In


particular, dp1qn px, yq is the closest distance the orbit of y comes to x up to time n. It is

shown in [61] that for systems with superpolynomial decay, for all x and µ-almost all y

limnÑ8

| ln dp1qn px, yq|

lnn“

1

dwhere d is the local dimension of µ at x provided that it exists.

Under some additional assumptions one can prove dynamical Borel Cantelli Lemmawhich implies in particular that, if µ is smooth then for all x and almost all y we have

lim supnÑ8

| ln dp1qn px, yq| ´ 1

dlnn

ln lnn“

1

d.

In Section 4 we extend this result to r ą 1. For example, if f is an expanding map ofthe circle, we shall show that for Lebesgue almost all x and y we have

lim supnÑ8

| ln dprqn px, yq| ´ lnn

ln lnn“

1

r.

Both the smoothness assumption on the invariant measure, Lebesgue typicality assump-tion on x and hyperbolicity assumption on f are essential. Namely if µ is an invariantGibbs measure which is not conformal, then we show in Section 6 that that there is aconstant c “ cpµq ‰ 0 such that for µ almost all x and y and for all r P N

lim supnÑ8


a

plnnq ln ln lnn“ c.

We shall also show that there is Gδ–dense set B such that for all x P B and Lebesguealmost all y we have

lim supnÑ8


ln lnn“ 1.

Finally if the expanding map is replaced by a rotation Tα then we have that for almostall px, y, αq it holds that

lim supnÑ8


ln lnn“

#

1 if r “ 112

if r ą 1.

Records of geodesic excursions. Consider a hyperbolic manifold Q of dimen-sion d`1 which is not compact but has finite volume. Such manifold admits a thick-thindecomposition. Namely Q is a union of compact part and several cusps. A cusp excur-sion is a maximal time segment such that the geodesic stays in a cusp for the wholesegment. Let

Hp1qpT q ě Hp2q

pT q ě . . . HprqpT q ě . . .

be the maximal heights achieved during the excursions which occur before time T placedin the decreasing order. Sullivan Logarithm Law is equivalent to saying that for almostevery geodesic

(1.1) lim supTÑ8

Hp1qpT q

lnT“

1

d.


The proof of (1.1) relies on Sullivan’s Borel-Cantelli Lemma and it also shows that foralmost every geodesic

lim supTÑ8

Hp1q ´ lnTd

ln lnT“

1

d.

We obtain a multiple version of this result by showing that for almost every geodesic

lim supTÑ8

Hprq ´ lnTd

ln lnT“

1

rd.

Multiple Khinchine Groshev Theorem. Let ψ : RÑ R be a positive function(in dimension 1 we also assume that ψ is monotone). The classical Khinchine GroshevTheorem ([91, 69, 126]) says that for almost all α P Rd there are are infinitely manysolutions to

(1.2) |xk, αy `m| ď ψpk8q with k P Zd,m P Ziff

(1.3)8ÿ

r“1

rd´1ψprq “ 8.

In particular the inequality

||k||d|xk, αy `m| ď1

ln kpln ln kqs

has infinitely many solutions for almost every α iff s ď 1. We now replace the aboveinequality by

(1.4) ||k||d|xk, αy `m| ď1

lnNpln lnNqs

and say that α is pr, sq approximable if there are infinitely N for which (1.4) has rpositive solutions (that is, solutions with k1 ą 0) such that1 gcdpk1, . . . kd,mq “ 1. (Ourinterest in smallness of

(1.5) ||k||d|xk, αy `m|

is motivated by [44] where the discrepancy of Kronecker sequences with respect toconvex sets is studied. Indeed ks where (1.5) is small are small denominators of thediscrepancy and they determine its growth rate.) We show in Section 8 that almostevery α P Rd is pr, sq approximable iff s ď 1

r.

The layout of the paper is the following. In Section 2 we describe an abstract result onan array of rare events in a probability space which ensures that for a given r, r events inthe same row happen for infinitely many (respectively, finitely many) rows. In Section 3this abstract criterion is applied for in the case the rare events in question involve visitsto a sublevel set of a Lipshitz function by an orbit of a smooth exponentially mixing

1The last conditions is needed to disregard trivial solutions plk, lmq where 1 ă l is not to big while

||k||d|xk, αy `m| !1

lnNpln lnNqs.


dynamical systems. The results of Section 3 are used to obtain MultiLog Laws in varioussettings. Namely, Section 4 studies hitting and return times in the phase space, whileSection 9 treats similar problems in the configuration space when the system at hand is ageodesic flow on compact negatively curved manifold. Geodesic excursion are discussedin Section 7, and Diophantine approximations are treated in Section 8. The MultiLogLaw for non-conformal measures is discussed in Section 6. As it was mentioned, theregime we consider is intermediate between the Poisson and Borel-Cantelli. Section 5contains an application of our results to the Poisson regime. Namely we derive Poissondistribution for hits and mixed Poisson distribution for returns for exponentially mixingsystems on smooth manifolds. Section 10 describes the application of our results to theextreme value theory for dynamical systems.

Some auxiliary are collected in the appendices.

2. Multiple Borel Cantelli Lemma.

2.1. The result. The classical Borel Cantelli Lemma is a standard tool for decidingwhen an infinite number of rare events occur with probability one. However in casean infinite number of events do occur, the Borel Cantelli Lemma does not give aninformation about how well separated in time those occurrences are. In this sectionwe present a criterion which allows to decide when several rare events occur on thesame time scale. The criterion is based on various mixing or independence conditionsbetween the rare events.

Namely, consider a probability space pΩ,Pq and a family of events Ωnρ such that

(2.1) Ωnρ1Ă Ωn

ρ2if ρ1 ď ρ2

Let Nnρ be the number of times k ď n such that Ωk

ρ occurs. For r P N and tρnu adecreasing sequence, we would like to give a criterion that allows to tell when Nn

ρn ě rwill almost surely hold for infinitely many n.

We introduce several conditions quantifying asymptotic independence between theevents Ωn

ρ . The conditions require the existence of:

‚ an increasing function σ : R` Ñ R`,‚ a sequence εn Ñ 0,‚ a function s : N ý such that spnq ď plnnq2,‚ a function s : N ý such that εn ď spnq ă np1´qqp2pr´1qq for some 0 ă q ă 1,

and some 0 ă ε ă p1´ qqp2pr ´ 1qq,

for which the following holds.For an arbitrary r-tuple 0 ď k1 ă k2 ¨ ¨ ¨ ă kr ď n we consider the separation indices

Sepnpk1, . . . , krq “ Card tj P t0, . . . r ´ 1u : kj`1 ´ kj ě spnqu , k0 :“ 0,

ySepnpk1, . . . , krq “ Card tj P t0, . . . r ´ 1u : kj`1 ´ kj ě spnqu , k0 :“ 0.

pM1qr If 0 ď k1 ă k2 ă . . . kr ď n are such that Sepnpk1, . . . , krq “ r then

σpρnqrp1´ εnq ď P

˜

rč

j“1

Ωkjρn

¸

ď σpρnqrp1` εnq.


pM2qr There exists K ą 0 such that if 0 ď k1 ă k2 ă . . . kr ď n are such thatSepnpk1, . . . , krq “ m ă r, then

P

˜

rč

j“1

Ωkjρn

¸

ďK

σpρnqmplnnq100r.

pM3qr If 0 ă k1 ă k2 ă ¨ ¨ ¨ ă kr ă l1 ă l2 ă ¨ ¨ ¨ ă lr, are such that 2i ă kα ď 2i`1, 2j ălβ ď 2j`1, for 1 ď α, β ď r, and such that

ySep2i`1pk1, . . . , krq “ r, ySep2j`1pl1, . . . , lrq “ r, l1 ´ kr ě sp2j`1q,

then

P

˜«

rč

α“1

Ωkαρ2i

ff

č

«

rč

β“1

Ωlβρ2j

ff¸

ď σpρ2iqrσpρ2jq

rp1` εiq.

Fix r P N.

Definition 2.1. Ωnρ are r–almost independent at a fixed scale if pM1qr and pM2qr

are satisfied for r ď r. Ωnρ are r–almost independent at all scales if pM1qr, pM2qr and

pM3qr are satisfied for r ď r.

Theorem 2.2. Fix r P N˚. Define

Sr “8ÿ

j“1

`

2jσpρ2jq˘r.

(a) If Sr ă 8, and Ωrn are 2r–almost independent at a fixed scale then with probability

1, we have that for large n Nnρn ă r.

(b) If Sr “ 8, and Ωrn are 2r–almost independent at all scales then with probability

1, there are infinitely many n such that Nnρn ě r.

Observe that since ρn is decreasing and σ is an increasing function we have that

2j`1´1ÿ

n“2j

σrpρnqnr´1

ď`

2j`1σpρ2jq˘rď 22r

2j´1ÿ

n“2j´1

σrpρnqnr´1

the convergence of Sr is equivalent to the convergence of8ÿ

n“1

σrpρnqnr´1.

Remark 2.3. An analogous statement has been obtained in [1] under different mixingconditions 2

2We refer the readers to the notes at the end of each section for a detailed discussion of the relatedliterature.


2.2. Estimates on a fixed scale. For m P N let

Um “ tpk1, . . . , krq such that 2m ă k1 ă k2 ă ¨ ¨ ¨ ă kr ď 2m`1 and ySep2m`1pk1, . . . krq “ ru.

Am :“ tD 0 ă k1 ă ¨ ¨ ¨ ă kr ď 2m`1 s.t. Ωkαρ2m

happens for all α P r1, rsu

Dm :“ tD pk1, . . . , krq P Um s.t. Ωkαρ2m`1

happens for all α P r1, rsu.

The goal of this section is to prove the following estimates from which it will be easyto derive Theorem 2.2.

Proposition 2.4. There exists constants c, C ą 0 and a sequence θm Ñ 0 such that if

(2.2) nσpρnq Ñ 0 as nÑ 8

then for m1 ą m` 1 we have3

PpAmq ď C`

2rmσpρ2m`1qr`m´10

˘

(2.3)

PpDmq ě cp2rmσpρ2m`1qr´m´10

q(2.4)

PpDm XDm1q ď pPpDmq `m´10qpPpDm1q `m

1´10qp1` θmq(2.5)

We start with some notations and a lemma. For n P N˚, for k1, . . . , kr ď n, define

Ak1,...,krρn :“rč

j“1

Ωkjρn .

With these notations

Am “ď

0ăk1ăk2ă¨¨¨ăkrď2m`1

Ak1,...,krρ2m(2.6)

Dm “ď

pk1,...,krqPUm

Ak1,...,krρ2m`1.(2.7)

Lemma 2.5. Fix 0 ă a1 ă a2 ď 2. If pM1qr and pM2qr hold then there exists twosequences δn Ñ 0, ηn Ñ 0 such that

(2.8)ÿ

a1năk1ăk2ă¨¨¨ăkrďa2n

PpAk1,...,krρn q “ppa2 ´ a1qnσpρnqq

r

r!p1` δnq ` ηn plnnq

´10 .

For a2 ´ a1 ě12, there exists constant cr such that

(2.9)ÿ


ySepnpk1,...krq“r

PpAk1,...,krρn q ě crppa2 ´ a1qnσpρnqqr.

Proof. For m ď r, denote

Sm :“ÿ


Sepnpk1,...krq“m

PpAk1,...,krρn q.

3Note that Proposition 2.4 does not provide a bound for P pDm XDm`1q .


Note that Sr includes ppa2á1qnqr

r!p1` δ1nq terms for some sequence δ1n Ñ 0, hence pM1qr

yields

(2.10)ÿ


Sepnpk1,...krq“r

PpAk1,...,krρn q “ppa2 ´ a1qnσpρnqq

r

r!p1` δnq.

where δn Ñ 0 as nÑ 8.For m ă r, Sm includes O pnmsr´mpnqq terms. Hence pM2qr gives

(2.11) Sm ď Cnmsr´mpmqKσpρnq

m

plnnq100r“ ηn plnnq

´10

for some sequence ηn Ñ 0. Combining (2.10) with (2.11) we obtain (2.8). The proofof (2.9) is similar to that of (2.10), except that the number of terms is not anymoreequivalent to 1

r!ppa2á1qnq

rp1`δ1nq but just larger than 1r!ppa2á1qn´pr´1qspnqqrp1`δ2nq

for some sequence δ2n Ñ 0, which is larger than qr

r!ppa2 ´ a1qnq

rp1` δ2nq. ˝

Proof of Proposition 2.4. First, (2.3) follows directly from (2.6) and (2.8). Next, define

Im “ÿ

pk1,...krqPUm

PpAk1,...,krρ2m`1q

Jm “ÿ

pk1,...,krqPUmpk11,...,k

1rqPUm

tk1,...,kru‰tk11,...,k1ru

P´

Ak1,...,krρ2m`1

č

Ak11,...,k

1r

ρ2m`1

¯

.

From (2.7) and Bonferroni inequalities we get that

(2.12) Im ´ Jm ď PpDmq ď Im

Now, (2.9) implies that

(2.13) Im ě cr2rmσpρ2m`1q

r.

On the other hand, since

Ak1,...,krρ2m`1

č

Ak11,...,k

1r

ρ2m`1“ Atk1,...,kruYtk

11,...,k

1ru

ρ2m`1,

we get that

Jm ď Cr

2rÿ

l“r`1

ÿ

k1ă¨¨¨ăkl

PpAk1,...,klρ2m`1q,

and (2.8) then implies that

(2.14) Jm ď Crp2pr`1qmσpρ2m`1q

r`1`m´10

q.

Combining (2.12), (2.13) and (2.14), and using the assumption (2.2) we obtain (2.4).Finally, observe that

PpDm XDm1q ďÿ

pk1,...,krqPUm,pl1,...,lrqPUm1

PpAk1,...,kr2m`1 X Al1,...,lr2m1`1 q.

But pM3qr implies that

PpAk1,...,kr2m`1 X Al1,...,lr2m1`1 q ď PpAk1,...,kr2m`1 qPpAl1,...,lr

2m1`1 qp1` εmq,


so that summing over all pk1, . . . , krq P Um, pl1, . . . , lrq P Um1 we get that

PpDm XDm1q ď ImIm1p1` εmq

and (2.5) then follows from (2.12), (2.13) and (2.14). ˝

2.3. Convergent case. Proof of Theorem 2.2 (a). Suppose that Sr ă 8. Thennσpρnq Ñ 0. By (2.3) of Proposition 2.4 we have that

ř

PpAmq ă 8. By Borel-Cantelli Lemma, Am happen only finitely many times with probability 1. Observe thattNn

ρn ě ru Ă Am for 2m ă n ď 2m`1.Hence with probability one Nn

ρn ě r happen only finitely many times. ˝

2.4. Divergent case. Proof of Theorem 2.2 (b). Suppose that Sr “ 8. We give aproof under the assumption (2.2). The case where (2.2) does not hold requires minimalmodifications which will be explained at the end of this section.

Claim 2.6. Let Zn “řnm“1 1Dm . There exists a subsequence tZnku such that

ZnkEpZnkq

Ñ 1, a.s..

Since EpZnq Ñ 8, the fact that Zn Ñ 8 almost surely immediately follows. That

is, with probability one infinitely many of Dm happen. Note that Dm Ă tN2m`1

ρ2m`1ě ru,

which completes the proof of Theorem 2.2 (b) in case (2.2) holds.

Proof of Claim 2.6. We first prove that (2.4) and (2.5) imply that

ZnEpZnq

Ñ 1 in L2,

or equivalently that

(2.15)VarpZnq

E2pZnqÑ 0.

Note that

(2.16) VarpZnq “nÿ

m“1

PpDmq ` 2ÿ

iăj

rPpDi XDjq ´ PpDiqPpDjqs .

By (2.5) for each δ there exists mpδq such that if i ě mpδq, j ´ i ě mpδq

(2.17) PpDi XDjq ´ PpDiqPpDjq ď δPpDiqPpDjq ` i´10PpDjq ` j

´10PpDiq ` pijq´10.

Split (2.16) into two parts:(a) Due to (2.17), the terms where i ě mpδq, j ´ i ě mpδq contribute at most

δÿ

i,j

PpDjqPpDjq ď δpEpZnqq2 ` EpZnq ` 1

(b) The terms where i ď m or j ´ i ď mpδq (including i “ j) contribute at most

r2mpδq ` 1snÿ

j“1

PpDjq “ r2mpδq ` 1sEpZnq.


Since EpZnq Ñ 8, the case (a) dominates for large n giving

lim supnÑ8

VarpZnq

pEpZnqq2ď δ.

Since δ is arbitrary, (2.15) follows.Let nk “ inftn : pEpZnqq2 ě k2VarpZnqu.Then by Chebyshev inequality

Pp|Znk ´ EpZnkq| ą δEpZnkqq ď1

δ2k2.

Thus8ÿ

k“1

Pp|Znk ´ EpZnkq| ą δEpZnkqq ď8ÿ

k“1

1

δ2k2ă 8. Therefore, by Borel-Cantelli

Lemma, with probability 1 for large k |ZnkÉpZnkq| ă δEpZnkq. HenceZnk

EpZnk qÑ 1 a.s.,

as claimed. ˝

It remains to consider the case where (2.2) fails. That is, there exists δ0 ą 0 anda sequence jl such that σpρ2jl q2

jl ě δ0. In this case we consider the sets Ωkρ2l

defined

as follows. If Ωkρ2l

does not occur then Ωkρ2l

does not occur. Conditionally on Ωkρ2jl

occurring, Ωkρ2l

occurs with probability

γl “ min

ˆ

1,1

2jlσpρ2jl q ln l

˙

independently of the other events. In other words, each time Ωkρ2jl

occurs, we keep it

with probability γl and discard it with probability 1 ´ γl. Then the sets tΩkρjlukď2jl ,lPN

satisfy pM1qr, pM2qr, and pM3qr and

8ÿ

l“1

´

PpΩρjlq2jl

¯r

ě

8ÿ

l“l0

ˆ

δ0

ln l

˙r

“ 8

where l0 is the number such that γl “1

2jlσpρ2jl q ln lfor l ě l0. Hence more than r

events among the events tΩkρjlukď2jl occurs for infinitely many l with probability 1.

Since ΩkρjlĂ Ωk

ρjlit follows that more than r events among the events tΩk

ρjlukď2jl occurs

for infinitely many l with probability 1. The proof of Theorem 2.2 (b) is thus completed.˝

2.5. Prescribing some details. In the remaining part of Section 2 we describe someextensions of Theorem 2.2(b).

Namely, we assume that Ωnρ “ Y

pi“1Ωn,i

ρ and there exists a constant ε ą 0 such that

for each i, PpΩn,iρ q ě εPpΩn

ρq. We also assume the following extension of pM1qr: for eachpk1, k2, . . . , krq with Sepnpk1, k2, . . . , krq “ r and each pi1, i2, . . . , irq P t1, . . . , pu

r

ĆpM1qr

«

rź

j“1

PpΩkj ,ijρn q

ff

p1´ εnq ď P

˜

rč

j“1

Ωkj ,ijρn

¸

ď

«

rź

j“1

PpΩkj ,ijρn q

ff

p1` εnq;


and the following extension of pM3qr: for each i1, i2 . . . ir, j1, j2 . . . jr

ĆpM3qr P

˜«

rč

α“1

Ωkα,iαρ2i

ff

č

«

rč

β“1

Ωlβ ,jβρ2j

ff¸

ď

«

rź

α“1

PpΩkα,iαρ2i

q

ff«

rź

β“1

PpΩlβ ,jβρ2jq

ff

p1` εiq.

Theorem 2.7. If Sr “ 8, and ĆpM1qk, pM2qk as well as ĆpM3qk for k “ 1, . . . , 2r aresatisfied then for any i1, i2 . . . ir for any intervals I1, I2 . . . Ir Ă r0, 1s with probability 1

there are infinitely many n such that for some k1pnq, k2pnq . . . krpnq withkjpnq

nP Ij Ω

kj ,ijρn

occur.

The proof of Theorem 2.7 is similar to the proof of Theorem 2.2(b). Without the lossof generality we may assume that Ij does not contain 0. Then we consider the followingmodification of Dm

Dm :“ tD2m ă k1 ă ¨ ¨ ¨ ă kr ď 2m`1 such thatkα2m

P Iα,Ωkα,iαρ2m`1

happens and kα`1 ´ kα ě sp2m`1q, 0 ď α ď r ´ 1u.

Arguing as in Proposition 2.4 we conclude that Dm1 and Dm2 are asymptotically in-dependent (in the sense of (2.5)) if m2 ą m1 ` p and p is so large that 2´p R Iα forα “ 1, 2 . . . r. The rest of the proof is identical to the proof of Theorem 2.2(b).

2.6. Poisson regime.

Theorem 2.8. Suppose that pM1qr and pM2qr hold for all r and that limnÑ8

nσpρnq “ λ.

Then Nnρn converges in law as nÑ 8 to the Poisson distribution with parameter λ.

Proof. We compute all (factorial) moments of the limiting distribution. Let X denote

the Poisson random variable with parameter λ. Below

ˆ

mr

˙

denotes the binomial

coefficientm!

r!pm´ rq!. Since

(2.18) Eˆˆ

Nnρn

r

˙˙

“ÿ

k1ăk2ă¨¨¨ăkrďn

PpAk1,...,krρn q,

Lemma 2.5 implies for each r

(2.19) limnÑ8

Eˆˆ

Nnρn

r

˙˙

“λr

r!“ E

ˆˆ

Xr

˙˙

.

Since this holds for all r we also have that for all r, limnÑ8

EppNnρnq

rq “ EpX r

q. Since the

Poisson distribution is uniquely determined by its moments the result follows. ˝

Similarly to Borel-Cantelli Lemma, we also have the following extension of Theo-rem 2.8 in the setting of §2.5. Denote Nn,i

I the number of times event Ωk,iρn occurs with

kn P I. Write Nn,i :“ Nn,ir0,1s.


Theorem 2.9. Suppose that ĆpM1qr and pM2qr hold for all r and that

limnÑ8

nPpΩn,iρn q “ λi.

Then tNn,iρn u

pi“1 converge in law as nÑ 8 to the independent Poisson random variables

with parameter λi.Moreover if I1, I2, . . . Is are disjoint intervals then tNn,i

Iju, i “ 1 . . . p, j “ 1 . . . s

converge in law as nÑ 8 to the independent Poisson random variables with parameterλi|Ij|.

Proof. It suffices to prove the second statement. The proof is similar to the proof ofTheorem 2.8. Namely, similarly to (2.19) we show that for each set rij P N we have

(2.20) limnÑ8

E

˜

ź

i,j

ˆ

Nn,iIj

rij

˙

¸

“ź

ij

pλi|Ij|qrij

prijq!“ź

ij

Eˆ

Xij

rij

˙

where Xij are independent Poisson random variables with parameters λi|Ij|. ˝

2.7. Notes. The usual Borel Cantelli Lemma is a classical subject in probability. Thereare many extensions to weakly dependent random variables, see e.g. [134, §12.15],[129, §1]. The connection between Borel-Cantelli Lemma and Poisson Limit Theoremis discussed in [49, 55] There is also a vast literature on Borel-Cantelli Lemmas fordynamical systems starting with [118]. Some representative examples dealing withhyperbolic systems are [5, 34, 54, 68, 71, 79, 73, 74, 87, 100] while [28, 29, 87, 93, 94,95, 104, 130] deal with systems of zero entropy. The later cases is more complicated ascounterexamples in [53, 64] show. Survey [6] reviews the results obtained up to 2009and contains many applications, some of which parallel the results obtained in Sections4–8 of the present paper. The multiple Borel Cantelli Lemma for independent eventsis proven in [114]. [1] obtains multiple Borel Cantelli Lemma for systems admittinggood symbolic dynamics. Extending multiple Borel Cantelli Lemma for more generalsequences allows to obtain many new applications, see Sections 4–10.

3. Multiple Borel Cantelli Lemma for exponentially mixing dynamicalsystems.

3.1. Good maps, good targets. Let f be a map of metric space X preserving ameasure µ. Given a family of sets Ωρ Ă X, we will, in a slight abuse of notations,sometimes call Ωρ the event 1Ωρ and Ωn

ρ the event 1Ωρ ˝ fn. Recall the notation σpρq “

µpΩρq.To deal with recurrence, we need to consider slightly more complicated events.Given a family of events Ωρ in X ˆX let Ωn

ρ Ă X be the event

Ωnρ “ tx : px, fnxq P Ωρu.

Let

Sr “8ÿ

j“1

`

2jvj˘r

where vj “ σpρ2jq for targets Ωnρ and vj “ σpρ2jq “ µˆ µpAρq for the targets Ωn

ρ .


Let Nnρ be the number of times k ď n such that Ωk

ρ (or Ωkρ) occurs. We want to

give conditions on the system pf,X, µq and on a sequence of targets Ωkρn (or Ωk

ρn), thatimply the validity of the dichotomy of Theorem 2.2 for the number of hits Nn

ρn .

Definition 3.1 (r-fold exponentially mixing systems). Let B be a space of real val-ued functions defined over Xr`1, with a norm ¨ B.We say that pf,X, µ,Bq is r-foldexponentially mixing, if there exists constants C ą 0, L ą 0 and θ ă 1 such that

(Prod) A1A2B ď CA1BA2B,

(Gr) A ˝ pfk1 , . . . , fkrqB ď CLřri“1 kiAB,

If 0 “ k0 ď k1 ď . . . ď kr are such that @j P r0, r ´ 1s, kj`1 ´ kj ě m, then

pEMqr

ˇ

ˇ

ˇ

ˇ

ż

X

Apx, fk1x, ¨ ¨ ¨ , fkrxqdµpxq ´

ż

Xr`1

Apx0, ¨ ¨ ¨ , xrqdµpx0q ¨ ¨ ¨ dµpxrq

ˇ

ˇ

ˇ

ˇ

ď Cθm AB .

Given a system pf,X, µ,Bq, we now define the notion of simple admissible targetsfor f .

Definition 3.2 (Simple admissible targets). Let Ωρ, ρ P R˚`, be a collection of sets inX there are positive constants C, η, τ such that for all sufficiently small ρ ą 0

(Appr) There are functions A´ρ , A`ρ : X Ñ R such that A˘ρ P B and

(i) A˘ρ 8 ď 2 and A˘ρ B ď Cρ´τ ;(ii) A´ρ ď 1Ωρ ď A`ρ ;

(iii) µpA`ρ q ´ µpA´ρ q ď Cσpρq1`η.

Let tρnu be a decreasing sequence of positive numbers. We say that the sequencetΩρnu is a simple admissible sequence of targets for f if there exists u ą 0 such that forsome u ą 0

(Poly) ρn ě nú, σpρnq ě nú,

and

(Mov) @R DC : @k ď R lnn µpΩρn X f´kΩρnq ď Cσpρnqplnnq

´1000r.

Remark 3.3. Note that properties (Appr)(ii) and (iii) imply that

µpA`ρ q ´ µpΩρq ď CµpΩρq1`η, µpΩρq ´ µpA

´ρ q ď CµpΩρq

1`η.

Remark 3.4. A typical situation where one wants to verify these properties is thefollowing. Suppose that f is Lipshitz and B is the space of Lipshitz functions. Then(Prod) is clear and (Gr) holds with L being the Lipshitz constant of f. Moreover incase

Ωρ “ tΦpxq P ra1pρq, a2pρqsu, and µpΩρq ě ρá, for some a ą 0,

and Φ is some Lipschitz function, one can often verify (Appr) by taking A`ρ “ φ`pΦq,A´ρ “ φ´pΦq where φ˘ : RÑ R are Lipshitz functions such that

1ra1`ρs,a2´ρss ď φ´ ď 1ra1,a2s ď φ` ď 1ra1´ρs,a2`ρss

for some s ą 0.

To deal with recurrence, the following definition is useful.


Definition 3.5 (Composite admissible targets). Let Ωρ and be a collection of sets inX ˆX satisfying the following conditions for some positive constants C, η, τ .

For any ρ ą 0

pApprq There are functions A´ρ , A`ρ : X ˆX Ñ R such that A˘ρ P B and

(i) A˘ρ 8 ď 2 and A˘ρ B ď Cρ´τ ;

(ii) A´ρ ď 1Ωρ ď A`ρ ;(iii) There is a function σpρq such that for any fixed x,

σpρq ´ Cσpρq1`η ď

ż

A´ρ px, yqdµpyq ď

ż

A`ρ px, yqdµpyq ď σpρq ` Cσpρq1`η,

(iv) For any fixed y,

ż

A`ρ px, yqdµpxq ď Cσpρq.

The sequence Ωρn is said to be composite admissible if

(Poly) ρn ě nú, σpρnq ě nú,

and there is a constant a such that

(Mov) @k ‰ 0, µpΩkaρnq ď Cplnnq´1000r

and

(Sub) Ωk1ρ X Ωk2

ρ Ă f´k1Ωk2´k12ρ for k1 ă k2.

Observe that integrating condition Apprpiiiq with respect to x we obtain for eachn ‰ 0,

C´1µ`

Ωnρ

˘

ď µ`

A´ρ px, fnxq

˘

ď µ`

A`ρ px, fnxq

˘

ď Cµ`

Ωnρ

˘

.

3.2. Multiple Borel-Cantelli Lemma for admissible targets. The goal of thissection is to establish the following.

Theorem 3.6. Assume pf,X, µq is p2rq-fold exponentially mixing. Thena) If Ωρn are as in Definition 3.2 then pM1qr, pM2qr and pM3qr hold for the sequenceΩρn and the probability measure µ.b) If Ωρn are as in Definition 3.5 then pM1qr, pM2qr and pM3qr hold for the sequenceΩρn and the probability measure µ.

Hence, Theorem 2.2 implies

Corollary 3.7. If pf,X, µq is p2rq-fold exponentially mixing, and if Ωρn (or Ωρn) areas in Definition 3.2 (or Definition 3.5), then

(a) If Sr ă 8, then with probability 1, we have that for large n Nnρn ă r.

(b) If Sr “ 8, then with probability 1, there are infinitely many n such that Nnρn ě r.

In fact Theorem 3.6 is a direct consequence of the following Proposition.

Proposition 3.8. Given a dynamical system pf,X, µq and a sequence of decreasingsets Ωρn such that (Poly) holds, then:

(i) If pProdq, pEMqr and pApprq hold then pM1qr is satisfied, with the function s :N ý: spnq “ R lnn, where R is sufficiently large depending on r, θ, C, C, σ and η.

(ii) If pProdq, pGrq, pEMqr, pApprq and pMovq hold, then pM2qr is satisfied.


(iii) If pProdq, pGrq, pEMq2r, and pApprq hold, then for arbitrary ε ą 0 pM3qr issatisfied with spnq “ εn.

Similarly, given a dynamical system pf,X, µq and a sequence of decreasing sets Ωρn

such that (Poly) holds, then:(i) If pProdq, pEMqr and pApprq hold then pM1qr is satisfied, with the function s :

N ý: spnq “ R lnn, where R is sufficiently large depending on r, θ, C, C, σ and η.(ii) If pProdq, pGrq, pEMqr, pApprq, pMovq and pSubq hold, then pM2qr is satisfied.(iii) If pProdq, pGrq, pEMq2r, and pApprq hold, then for arbitrary ε ą 0 pM3qr is

satisfied with spnq “ εn.

Proof of Proposition 3.8. We use C to denote a constant that may change from line toline but that will not depend on ρn, Ωρn , Ωρn , the order of iteration of f , etc.

(i) For Ωρn , we prove (M1) in case ki`1 ´ ki ě?R lnn where R is a sufficently large

constant.

µ

˜

rź

i“1

1Ωρn pfkixq

¸

ď µ

˜

rź

i“1

A`ρnpfkixq

¸

ď

rź

i“1

µ`

A`ρn˘

` Cρ´rσn n´?R ln θ

ď`

µpΩρnq ` CµpΩρnq1`η

˘r` Cρ´rσn n´

?R ln θ.

Likewise

µ

˜

rź

i“1

1Ωρn pfkixq

¸

ě`

µpΩρnq ´ CµpΩρnq1`η

˘r´ Cρ´rσn n´

?R ln θ.

This proves (M1), since ρn ě nú, µpΩρnq ě nú, and R is sufficiently large.

For Ωρn , we approximate 1Ωρnby A˘ρn , apply pApprq, pEMqr to the functions

B`ρnpx0, ¨ ¨ ¨ , xrq “ A`ρnpx0, x1q ¨ ¨ ¨ A`ρnpx0, xrq,

B´ρnpx0, ¨ ¨ ¨ , xrq “ A´ρnpx0, x1q ¨ ¨ ¨ A´ρnpx0, xrq,

and get

µ

˜

rč

i“1

Ωkiρn

¸

ď µ`

B`ρnpx0, fk1x0, ¨ ¨ ¨ , f

krx0q˘

ď µ`

B`ρnpx0, ¨ ¨ ¨ , xrq˘

` Cρn´rσn

?R ln θ

ď`

σpρnq ` Cσpρnq1`η

˘r` Cρn

´rσn?R ln θ,

Likewise,

µ

˜

rč

i“1

Ωkiρn

¸

ě`

σpρnq ´ Cσpρnq1`η

˘r´ Cρn

´rσn?R ln θ

This proves pM1qr for the composite admissible sequence of targets Ωρn by taking Rlarge.

(ii) For Ωρn , it is enough to consider the case Seppk1, . . . , krq “ r ´ 1 otherwise wecan estimate all 1Ωρn ˝ f

ki with ki ´ ki´1 ă spnq, except the first, by 1.


So we assume that 0 ă kj ´ kj´1 ă R lnn and ki ´ ki´1 ě R lnn for i ‰ j. Since

pM1qr was proven under the assumption that minipki´ki´1q ą?R lnn we may assume

that kj ´ kj´1 ă?R lnn. Note that by (Appr) and Remark 3.3

µ`

A`ρn`

A`ρn ˝ fk˘˘

´ µ`

1Ωρn

`

1Ωρn ˝ fk˘˘

ď 4µ`

A`ρn ´ 1Ωρn

˘

ď 4CµpΩρnq1`η.

Therefore (Mov) implies :

µ`

A`ρn`

A`ρn ˝ fkj´kj´1

˘˘

ď CµpΩρnqplnnq´1000r.

TakeB “ A`ρn`

A`ρn ˝ fkj´kj´1

˘

, we get using pEMqr and the fact that ρn ě nú, µpΩρnq ě

nú,

µ

˜

rź

i“1

1Ωρn

`

fkix˘

¸

ď µ

˜

rź

i“1

A`ρn`

fkix˘

¸

“ µ

˜

ź

i‰j´1,j

A`ρn`

fkix˘

Bpfkj´1xq

¸

ď µ

˜

ź

i‰j´1,j

A`ρn`

fkix˘

¸

µpBq ` Cρn´σrL

?R lnnθR lnn

ď CµpΩρnqr´1plnnq´1000r

proving pM2qr.For Ωρn , as the proof of (i), we approximate 1Ωρn

by A`ρn . Consider

Brpx0, ¨ ¨ ¨ , xj´1, xj`1, ¨ ¨ ¨ , xrq

“ 1Ωρnpx0, x1q ¨ ¨ ¨ 1Ωρn

px0, xj´1q1Ωkj´kj´12ρn

pxj´1q1Ωρnpx0, xj`1q ¨ ¨ ¨ 1Ωρn

px0, xrq.

Brpx0, ¨ ¨ ¨ , xj´1, xj`1, ¨ ¨ ¨ , xrq

“ A`ρnpx0, x1q ¨ ¨ Ä`ρnpx0, xj´1qA

`2ρnpxj´1, f

kj´kj´1xj´1qA`ρnpx0, xj`1q ¨ ¨ Ä

`ρnpx0, xrq,

Since pApprq, pMovq and pSubq hold, we obtain

µ

˜

rč

j“1

Ωkjρn

¸

ď µ´

Brpx, ¨ ¨ ¨ , fkj´1x, fkj`1x, ¨ ¨ ¨ , fkrxq

¯

ď µ´

Brpx, ¨ ¨ ¨ , fkj´1x, fkj`1x, ¨ ¨ ¨ , fkrxq

¯

ď µr`1pBrq ` Cρn

´σrL?R lnnθR lnn.

Integrating with respect to all all variables except for xj´1 we get

(3.1) µr`1pBrq ď

`

σpρnq ` σpρnq1`η

˘rCµpΩ

kj´kj´1

2ρn q.

Therefore, pM2qr follows when pMovq holds.(iii) Choose a large constant b and consider two cases.

Case 1 : j ď i` b. then the proof of pM3qr is the same as the proof of pM1qr exceptthat we need to use pEMq2r instead of pEMqr.Case 2 : j ą i`b. Consider first simple targets Ωρn , Denoting Bpxq “

śrα“1A

`ρ2ipfkαxq

we obtain from (Prod), (Gr), (Appr) and (Poly) so that

BB ď CL2i`1

.


µ

˜˜

rź

α“1

1Ωρ2ipfkαxq

¸˜

rź

β“1

1Ωρ2jpf lβxq

¸¸

ď µ

˜˜

rź

α“1

A`ρ2i pfkαxq

¸˜

rź

β“1

A`ρ2jpf lβxq

¸¸

ď µ

˜

Bpxq

˜

rź

β“1

A`ρ2jpf lβxq

¸¸

ď µpBqrź

j“1

µ´

A`ρ2j

¯

` CL2i`1

ρ´σ2iρ´rσ

2jθ2jε.

Applying already established pM1qr to estimate µpBq, and observing that the second

term is smaller than CpL12b´1

q2j2σrjθ2jε, which is thus much smaller than the first, we

see that if b is sufficiently large pM3q follows.Next, we analyze Case 2 for Ωρn . Consider

B˚px, x1, x2 . . . xrq “

˜

rź

α“1

1Ωkαρ2ipxq

¸˜

rź

β“1

1Ωρ2jpx, xβq

¸

.

By pApprq and pEMqr, we get

µ´

č

1ďα, βďr

`

Ωkαρ2i

č

Ωlβρ2j

˘

¯

ď µ`

B˚px, f l1x, . . . , f lrxq˘

ď µ

˜

rź

α“1

A`ρ2i px, fkαxq

¸

`

σpρ2jq ` σpρ2jq1`η

˘r` CL2i`1

ρ´σ2iρ´rσ

2jθ2jε.

Using pM1qr we conclude that

µ

˜

rź

α“1

A`ρ2i px, fkαxq

¸

ď C`

σpρ2iq ` σpρ2iq1`η

˘r.

This completes the proof of pM3qr. ˝

3.3. Notes. We refer the reader to Appendix A for more background on multiple ex-ponential mixing as we for some examples. We note that limit theorems for smoothsystems which are only assumed to be multiply exponentially mixing (but without anyadditional assumptions) are considered in [20, 32, 127]. [61] obtains Logarithm Law forhitting times under an assumption of superpolynomial mixing which is weaker than ourexponentially mixing assumption.

4. MultiLog Laws for recurrence and hitting times

In this section we apply the results of Section 3 to obtain Multilog laws for multipleexponentially mixing diffeomorphisms and flows.


4.1. Results. Let f be a map of a compact d´dimensional Riemannian manifold M

preserving a smooth measure µ. Let dprqn px, yq be the r-th minimum of

dpx, fyq, ¨ ¨ ¨ , dpx, fnyq.

The following result was obtained for a large class of weakly hyperbolic systems as aconsequence of dynamical Borel-Cantelli Lemmas

For almost all x, lim supnÑ8

| ln dp1qn px, xq| ´ 1

dlnn

ln lnn“

1

d,(4.1)

For all x and almost all y, lim supnÑ8


dlnn

ln lnn“

1

d.(4.2)

In particular, the following result is a special case of [61, Corollary 9].

Theorem 4.1. Suppose that f is smooth, preserves a smooth measure µ and has su-perpolynomial decay of correlations. That is,

|µpApxqBpfnxqq ´ µpAqµpBq| ď apnqALipBLip where @s limnÑ8

nsapnq “ 0

Then (4.1) and (4.2) hold.

MultiLog Law for recurrence and for hitting times. The goal of this

section is to obtain an analogue of (4.1) (4.2) for dprqn with r ą 1, for exponentially

mixing systems as in Definition 3.1.Given a system pf,M, µq, define

Gr “

#

x : for a.e. y, lim supnÑ8

| ln dprqn px, yq| ´ 1

dlnn

ln lnn“

1

rd

+

,

Gr “

#

x : lim supnÑ8

| ln dprqn px, xq| ´ 1

dlnn

ln lnn“

1

rd

+

.

Theorem 4.2. Suppose that pf,M, µ,Bq is an r-fold exponentially mixing system. Then(a) µpGrq “ 1; (b) µpGrq “ 1.

Failure of the Multilog laws for generic points. Naturally, one can ask ifin fact, Gr equals to M. If r “ 1 the answer is often positive (see Theorem 4.1). It turnsout that for larger r the answer is often negative. Define

H “

#

x : for a.e. y, for all r ě 1 : lim supnÑ8


dlnn

ln lnn“

1

d

+

,

H “

#

x : for all r ě 1 : lim supnÑ8

| ln dprqn px, xq|

en“ 8

+

.

Theorem 4.3. Suppose that pf,M, µ,Bq is a 1-fold exponentially mixing system andthat the periodic points of f are dense. Then for any r P N, we havea) H contains a Gδ dense set.b) H contains a Gδ dense set.


Thus for r ě 2 topologically typical points do not belong to Gr or Gr.Failure of the Multilog laws for non mixing systems. The case of toraltranslations.

Theorem 4.3 emphasizes the necessity of a restriction on x in Theorem 4.2.In a similar spirit, we show that strong mixing assumptions made in this paper are

essential. To this end we consider the case when the dynamical system is pTα,Td, λqwhere Tα is the translation of vector α and λ is the Haar measure on Td.

Define

Er “

#

x : for a.e. y, lim supnÑ8


dlnn

ln lnn“

1

2d

+

,

Er “

#

x : lim supnÑ8


dlnn

ln lnn“

1

d

+

.

Theorem 4.4. For λ-a.e. α P Td, the system pTα,Td, λq, satisfiesa) λpG1q “ 1 and λpErq “ 1 for r ě 2;b) Er “M for all r ě 1.

The proof requires different techniques, so it will be given in Section 8.

The case of flows. Here we describe the analogue results of Theorems 4.2 and4.3 for flows. Let φ be a piecewise smooth flow on a pd ` 1q dimensional Riemannianmanifold M preserving a smooth measure µ.

Observe that if φtpyq is close to x for some t, then the same is true for φtpyq witht close to t. Thus we would like to count only one return for the whole connectedcomponent lying in the ball. Namely, for some fixed ρ ą 0 let rtí , t

ì s be consecutive

connected components such that φtx P Bpx, rhoq for t P rtí , tì s. Let ti be the argmin

of dpφty, xq for t P rt´t , tì s. Let d

prqn px, yq be the r ´ th minimum of

(4.3) dpx, φt1pyqq, . . . , dpx, φtkpyqq, tk ď n ă tk`1.

Theorem 4.5. Suppose that for each a ą 0, pφa,M, µ,Bq is an r-fold exponentiallymixing system. Thena) µpGrq “ 1;b) µpGrq “ 1.

If, in addition, periodic points of φ are dense thenc) H contains a Gδ dense set;d) H contains a Gδ dense set.

4.2. Weak non-recurrence and proof of Theorem 4.2. In the following statementr ě 2 is fixed, B is supposed to be equal to the space of C1 functions on M r`1. Sinceµ is a smooth measure there is a smooth function γpxq such that

(4.4) µ pBpx, ρqq “ γpxqρd Ò`

ρd`1˘

.

Given x PM, c P R` let

(4.5) Ωρ,x “ ty : dpx, yq ď ρu


and

(4.6) Ωρ “

"

px, yq : dpx, yq ďcρ

pγpxqq1d

*

Proposition 4.6. For each x, the targets Ωρ,x satisfy pApprq and the targets Ωρ satisfypApprq

Proof. For targets Ωρ,x the statement follows from Remark 3.4.

To handle the composite targets Denote Φpx, yq “ dpx,yqγpxq1d

c. For s ą 1 let φ˘px, yq

be Lipschitz functions such that

1Bp0,ρ´ρsq ď φ´ ď 1Bp0,ρq ď φ` ď 1Bp0,ρ`ρsq.

Take A˘ρ “ φ˘pΦq. Then

pρ´ ρsqd ď

ż

1Bp0,ρ´ρsqpΦpx, yqqdµpyq ď

ż

A´ρ px, yqdµpyq

ď

ż

A`ρ px, yqdµpyq ď

ż

1Bp0,ρ`ρsqpΦpx, yqqdµpyq ď pρ` ρsqd.

Hence pApprqpiíiiq follow.Denote C0 “ supxPM γpxqγpyq, we get

ż

A`ρ px, yqdµpxq ď

ż

1Bp0,ρ`ρsqpΦpx, yqqdµpxq ď C0

ż

1Bp0,ρ`ρsqpΦpx, yqqdµpyq ď C0pρ`ρsqd

which gives pApprqpivq. ˝

To check (Mov) and pMovq we need the following definitions.

Definition 4.7 (Weakly non-recurrent points). Call x weakly non-recurrent if for eachA,K ą 0, there Dρ0 such that for all ρ ă ρ0 for all n ď K| ln ρ| we have

(4.7) µpBpx, ρq X fńBpx, ρqq ď µpBpx, ρqq| ln ρ|Á.

Definition 4.8 (Weakly non-recurrent system). Call the system pf,M, µ,Bq weaklynon-recurrent if for each A ą 0 Dρ0 such that for all ρ ă ρ0 for all n P N˚ we have

(4.8) µ ptx : dpx, fnxq ă ρuq ď | ln ρ|Á.

Proposition 4.9. Suppose that pf,M, µ,Bq is an r-fold exponentially mixing system.Then

iq pf,M, µ,Bq is weakly non-recurrent.iiq Almost every point is weakly non-recurrent.

Proof. Take B :“ A2. If k ě B ln | ln ρ|, take ρ “ | ln ρ|Á.By exponential mixing, we get

(4.9) µpx : dpx, fkxq ď ρq ď µpx : dpx, fkxq ď ρq

ď µ`

A`ρ px, fkxq

˘

ď C

ˆ

ρd ` ρd`dη `1

ρσθk˙

ď | ln ρ|´2A,

provided ρ is sufficiently small.


Now fix any k ď B ln | ln ρ|. Assume that x satisfies dpx, fkxq ď ρ, then for any l wehave that

dpf pl´1qkpxq, f lkxq ď f

pl´1qk1 ρ

If we take L “ r4B ln | ln ρ|ks ` 1 we find that

dpx, fLkxq ďÿ

lďL´1

flk1 ρ ď?ρ,

provided ρ is sufficiently small. But kL ě B ln | ln?ρ, hence (4.9) applies and we get

µpx : dpx, fkxq ď ρq ď µpx : dpx, fLkxq ď?ρq ď | ln ρ|Á,

proving iq.We proceed now to the proof of iiq. Define for j, k P N˚

Hj,kpxq :“ µpBpx, 12jq X f´kBpx, 12jqq.

Note thatż

Hj,kpxqdµpxq “

ĳ

112jpdpx, yqq112jpdpx, fkyqqdµpxqdµpyq

ď

ĳ

112jpdpx, yqq112j´1pdpy, fkyqqdµpxqdµpyq

ď CµpBpx, 12jqq

ż

112j´1pdpy, fkyqqdµpyq

where we used that µpBpy, 12jqq ď CµpBpx, 12jqq for any x, y P M . Part iq thenimplies that for sufficiently large j it holds that

ż

Hj,kpxqdµpxq ď µpBpx, 12jqqjÁ´3.

For such j we get from Markov inequality

µ`

x : Dk P p0, Kjs : Hj,kpxq ą µpBpx, 12jqqjÁ˘

ď Kj´2.

Hence Borel Cantelli Lemma implies that for almost every x there exists j such thatHj,kpxq ď µpBpx, 12jqqjÁ for every j ě j and every k P p0, Kjs, which implies iiq. ˝

Proof of Theorem 4.2. a). If we take ρn “ n´1d ln´s n, we see that for almost every x,

tΩρn,xu is a simple admissible sequence of targets as in Definition 3.5 ((Appr) followsfrom Proposition 4.6(a) and (Mov) follows from Proposition 4.9iiq). Moreover, withthe notation vj “ σpρ2jq and Sr “

ř8

j“1 p2jvjq

r, we see that Sr “ 8 iff s ě rd. Hence,

Theorem 4.2 a) follows from Corollary 3.7.

b). Take ρn “ n´1d ln´s n. Note that when x P Ωk1

ρ X Ωk2ρ , we have

dpfk1x, fk2xq ď dpx, fk1xq ` dpx, fk2xq ď2cρ

pγpxqq1dď

c1ρ

pγpfk1xqq1d.

Hence pSubq is satisfied. Combining this with Propositions 4.6(b) and 4.9(i), we knowthat tΩρnu defined by (4.6) is a composite admissible sequence of targets. Thus we getb) by a similar argument to a). ˝


4.3. Generic failure of the MultiLog Law. Proof of Theorem 4.3.

Proof. To prove part a), we first prove that periodic points belong to Hr. We knowfrom Theorem 4.1 that for any x PM and almost every y,

(4.10) lim supnÑ8


dlnn

ln lnn“

1

d.

Since dprqn px, yq ě d

p1qn px, yq, it follows that for any x PM , any r ě 1, and almost every

y

(4.11) lim supnÑ8


dlnn

ln lnnď

1

d.

To prove the opposite inequality let

Hm,l,r “

#

x : DY-open, µpYq ą 1´1

l: @y P Y ,

| ln dprqm px, yq| ´ 1

dlnm

ln lnmą

1

d´

1

l

+

We have that#

x : for a.e. y, for all r ě 1 : lim supnÑ8


dlnn

ln lnně

1

d

+

“č

lě1,rě1

ď

mě1

Hm,l,r.

But Hl,m is an open set. Hence we finish if we show that for any fixed r and l,Ť

mHm,l,r

contains the dense set of periodic points.Let x be a periodic point of period p. Take U to be some small neighbourhood of x

and denote by Λ the Lipschitz constant of fp in U .By (4.10), there exists n ě exp ˝ exppΛ` lq and Y such that µpYq ą 1´ 1

l, such that

for every y P Y , there exists k P r1, ns satisfying

dpx, fkyq ď

ˆ

1

n

˙1dˆ

1

lnn

˙1d´ 1

2l

.

Then

dpx, fk`pjyq “ dpfpjx, fk`pjyq ď Λr

ˆ

1

n

˙1dˆ

1

lnn

˙1d´ 1

2l

, 0 ď j ď r ´ 1.

Hence for y P Y and m “ n` ppr ´ 1q, we have that

dprqm px, yq ď Λr

ˆ

1

n

˙1dˆ

1

lnn

˙1d´ 1

2l

ă

ˆ

1

m

˙1dˆ

1

lnm

˙1d´ 1l

,

because we took n ě exp ˝ exppΛ` rq. Hence x P Hm,l and the proof of paq is finished.We now turn to the proof of pbq. Define

Am,l “

x : | ln dplqm px, xq| ą e2m(

.

Observe that H ĂŞ

l

Ť

mAm,l. But Am,l is open andŤ

mAm,l clearly contains theperiodic points. Part pbq is thus proved. ˝


4.4. The case of flows. Proof of Theorem 4.5. The proof of Theorem 4.5 for flowsproceed in the same way as for maps with minimal modifications. Namely, let

Ωρ,x “ ty : D s P r0, 1s, dpx, φsyq ď ρu,

and

Ωρ “

"

px, yq : D s P r0, 1s, dpx, φsyq ďcρ

γpxq

*

where γpxq “ limρÑ0

µpΩρ,xqρd. Consider the targets

Ωnρ,x “ φńΩρ,x, Ωn

ρ “ tx : px, φnxq P Ωρu

and let σpρq “ µpΩρ,xq, σpρq “ pµˆ µqpΩρq.Fix A ąą 1 arbitrarily large.

Definition 4.10 (Weakly non-recurrent points for flow). Call x weakly non-recurrentif for each A,K ą 0, there exists ρ0 ą 0 such that for all ρ ă ρ0 for all n P N and0 ă n ă K| ln ρ|,

(4.12) µ`

Ωρ,x X Ωnρ,x

˘

ď µpΩρ,xq| ln ρ|Á.

Definition 4.11 (Weakly non-recurrent flow). Call the system pφ,M, µ,Bq weakly non-recurrent if for each A ą 0 Dρ0 ą 0 such that for all ρ ă ρ0 and for all n P N˚ wehave

(4.13) µ`

Ωnρ

˘

ď | ln ρ|Á.

We proceed analogously to the proof of flow version of Proposition 4.9.

Proposition 4.12. Suppose that for any a ą 0, pφa,M, µ,Bq is an exponentially mixingsystem. Then

iq pφa,M, µ,Bq is weakly non-recurrent.iiq Almost every point is weakly non-recurrent.

Proof of Proposition 4.12. Take B “ A2. If k ě B ln | ln ρ|, the same argument as inthe proof of Proposition 4.9 gives that

(4.14) µ`

Ωkρ

˘

ď | ln ρ|Á.

Now fix any k ă B ln | ln ρ|. Denote φ1 “ maxtPr1,2s Dφt. Assume that x satisfies

dpx, φk`txq ď ρ for some 0 ď t ď 1, then for any l we have that

dpφpl´1qpk`tqpxq, φlpk`tqxq ď φ

pl´1qk1 ρ

If we take L “ r4B ln | ln ρ|ks ` 1 we find that

dpx, φLpk`tqxq ďÿ

lďL´1

φlk1 ρ ď?ρ,

provided ρ is sufficiently small. But kL ě B ln | ln ρ|, hence (4.14) applies and we get

µ`

Ωkρ

˘

ď

L´1ÿ

m“0

µ´

ΩLk`m?ρ

¯

ď 4Bpln | ln ρ|q| ln?ρ|Á ď | ln ρ|Á2,

proving iq.


We proceed now to prove iiq. Denote M0 “ maxtPr´1,1s

Dφt,

Ωρ “

"

px, yq : Ds P r´1, 1s, dpx, φsyq ďcρ

γpxq

*

where γpxq “ limρÑ0

µpΩρq

ρdand Ωn

ρ “ tx : px, φnpxqq P Ωρu. Define for j, k P N˚,

Hj,kpxq “ µ´

Ω12j ,x X Ωk12j ,x

¯

.

Note thatż

Hj,kpxqdµpxq “

ż

1Ωγpxq2j

px, yq1Ωγpxq2j

px, fkyqdµpxqdµpyq

ď M0C1

ż

1Ωγpxq2j

px, yq1Ωk22jpyqdµpxqdµpyq

ď M0C2

ż

µ`

Ωγpxq2j ,y

˘

1Ωk22jpyqdµpyq

ď M0C3µpΩ12j ,xqµ´

Ωk22j

¯

where we used that C´1 ď γpxq ď C for any x P M and µpBpy, ρqq ď CµpBpx, ρqq forany x, y PM.

Part iq then implies that for sufficiently large j it holds thatż

Hj,kpxqdµpxq ď µpΩ12j ,xqjÁ´3.

For such j we get from Markov inequality

µ`

x : Dk P p0, Kjs : Hj,kpxq ą µpΩ12j ,xqjÁ

˘

ď Kj´2.

Hence Borel Cantelli Lemma implies that for almost every x there exists j such thatHj,kpxq ď µpΩ12j ,xqj

Á for every j ě j and every k P p0, Kjs, which is iiq. ˝

Proof of Theorem 4.5. If the point x is weakly non-recurrent, we see that Ωρn,x is an

simple admissible sequence as in Definition 3.2 by taking ρn “ n´1d ln´s n. Moreover,

with the notation vj “ σpρ2jq and Sr “ř8

j“1 p2jvjq

r, we see that Sr “ 8 iff s ě rd.

Hence, x P Gr.The proof of part b) is similar when we replace the targets Ωρn,x by Ωρn and apply

the same argument. It remains to verify pSubq. Note that when x P Ωt1ρ XΩt2

ρ for t1 ă t2,we have some s1, s2 P r0, 1s such that

dpx, φt1`s1xq ďcρ

pγpxqq1d, dpx, φt2`s2xq ď

cρ

pγpxqq1d.

Hence

dpφt1x, φt2`s2´s1xq ďM0dpφt1`s1x, φt2`s2xq ď

2M0cρ

pγpxqq1dď

c1ρ

pγpφt1xqq1d.

It follows that Ωt1ρ X Ωt2

ρ P φ´t1Ωt2´t1ρ , which gives pM2qr by a similar argument of

Proposition 3.8 (ii).


The proof of part c) and part d) also proceeds in the same way as for maps. Namelywe first see that periodic orbits of the flow belong to Hr and Hr and then use thegenerality argument. ˝

4.5. Notes. Many authors obtain Logarithm Law (4.2) for hitting times as a conse-quence of dynamical Borel-Cantelli Lemmas. See [34, 42, 60, 79] and references wherein.[61] studies also return times. We note that [61] obtains (4.1) and (4.2) under muchweaker conditions than those imposed in the present paper, however, his results arevalid only for r “ 1 (the first visit).

[87, 92, 97] study the recurrence problem in the case in the case lim sup in (4.2) isreplaced by lim inf. In particular, [97] proves that for several expanding maps the

lim infnÑ8

n dp1qn px, yq

ln lnn

exists for almost all y.

5. Poisson Law for near returns.

5.1. Introduction. In this section we suppose that µ is a smooth measure and thatpf,M, µ, C1q is an r-fold exponentially mixing system for all r. In the previous sectionwe verified properties pM1qr and pM2qr for targets Ωρ,x given by (4.5), for almost everyx, and for targets Ωρ given by (4.6). Accordingly Theorem 2.8 gives

Theorem 5.1. (a) For almost all x the following holds. Let y be uniformly distributedwith respect to µ the number of visits to Bpx, ρq up to time τρ´d converges to a Poissondistribution with parameter τγpxq as ρ Ñ 0. Moreover letting n “ τρ´d we have hesequence

(5.1)dp1qn px, yq

ρ,dp2qn px, yq

ρ, . . . ,

dprqn px, yq

ρ, . . .

converges to the Poisson process with intensityγpxqτtd´1

d.

(b) Let x be chosen uniformly with respect to µ. Then the number of visits to B

ˆ

x,ρ

γ1dpxq

˙

up to time τρ´d converges to a Poisson distribution with parameter τ as ρÑ 0.

Proof. All the results except for Poisson limit for (5.1) follows from Theorem 2.8. Toprove the Poisson limit for (5.1) we need to check that for each r´1 ă r`1 ă r´2 ă

r`2 ă ¨ ¨ ¨ ă r´s ă r`s the number of times where dpfky, xq P“

r´j ρ, r`j ρ

‰

are independentPoisson random variables with parameters

γpxq

ż r`j

r´j

τtd´1dt

d“ γpxqτ

“

pr`j qd´ pr´j q

d‰

.

but this follows from Theorem 2.9. This theorem can be applied since ĆpM1qr followsfrom condition (Appr) for targets

Ωk,iρ “ ty : dpfky, xq P rrí ρ, r

ì ρsu


and the (Appr) holds due to Remark 3.4. ˝

There are two natural questions dealing with improving this result. In part (a) wewould like to specify more precisely the set of x where the Poisson law for hits hold.In part (b) we would to remove an annoying factor γ1dpxq from the denominator.Regarding the first question we have

Conjecture 5.2. If f is exponentially mixing then the conclusion of Proposition 5.1(a)holds for all non-periodic points.

Regarding the second question we have the following.

Theorem 5.3. Let x be chosen uniformly with respect to µ. Then the number of visitsto Bpx, ρq up to time τρ´d converges to a mixture of Poisson distributions. Namely,for each l

(5.2) limρÑ0

µpCardpn ď τρ´d : dpfnx, xq ď ρq “ lq “

ż

M

e´γpzqτpγpzqτql

l!dµpzq.

In other words to obtain the limiting distribution in Theorem 5.3 we first samplez P M according to the measure µ and then consider Poisson random variable withparameter τγpzq.

Corollary 5.4. If f preserves a smooth measure and is r fold exponentially mixing onC1 for all r then

(a) For almost all x we have that if τεpyq is an the first time an orbit of y entersBpx, εq then for each t

limµpy : τεpyqεdą tq “ e´γpxqt

(b) If Tεpxq is the first time the orbit of x returns to Bpx, εq then

limµpx : Tεpxqεdą tq “

ż

M

e´γpzqtdµpzq.

Proof. This is a direct consequence of Theorems 5.1(a) and 5.3. For example to getpart (b), take l “ 0 in (5.2). ˝

5.2. Poisson Law for Returns.

Proof of Theorem 5.3. Consider the targets

Ωρpx, yq “ tpx, yq PM ˆM : dpx, yq ď ρu

and let Ωkρ “ tx : px, fkxq P Ωρu. Note that pM2qr for Ωk

ρ implies pM2qr for Ωkρ. However,

pM1qr is false for targets Ωkρ. We now argue similarly to the proof of Theorem 3.6 to

obtain that for separated tuples k1, k2, . . . , kr,

(5.3) µ

˜

rč

j“1

Ωkjρ

¸

“ ρrdż

M

γrpzqdµpzqp1` op1qq.

Namely, note thatż

1Ωρpx0, x1q . . . 1Ωρ

px0, xrqdµpx0qdµpx1q . . . dµpxrq


“

ż

µrpBpx0, ρqqdµpx0q “ ρrdp1Òpρqq

ż

M

γrpx0qdµpx0q.

Thus approximating 1Ωρby A˘ρ satisfying pApprq, and applying pEMqr to the functions

B`ρ px0, ¨ ¨ ¨ , xrq “ A`ρ px0, x1q ¨ ¨ ¨ A`ρ px0, xrq,

B´ρ px0, ¨ ¨ ¨ , xrq “ A´ρ px0, x1q ¨ ¨ ¨ A´ρ px0, xrq,

we get that if kj`1 ´ kj ą R| ln ρ| for all 0 ď j ď r ´ 1, then

µ

˜

rč

j“1

Ωkjρ

¸

ď µ´

B`ρ px0, fk1x0, ¨ ¨ ¨ , f

krx0q

¯

ď µ´

B`ρ px0, ¨ ¨ ¨ , xrq¯

` Cρ´rσθR| ln ρ|

ď`

ρd ` Cρdp1`ηq˘rż

M

γrpzqdµpzq ` Cρ´rσθR| ln ρ|,

and, likewise,

µ

˜

rč

j“1

Ωkjρ

¸

ě`

ρd ´ Cρdp1`ηq˘rż

M

γrpzqdµpzq ´ Cρ´rσθR| ln ρ|.

Taking R large we obtain (5.3).Summing (5.3) over all well separated couples with nj ď τρ´d and using that the

contribution of non-separated couples is negligible due to pM2qr we obtain

(5.4) limρÑ0

ż

M

ˆ

N rρ,τ,x

r

˙

dµpxq “

ż

M

λrpzq

r!dµpzq

where

Nρ,τ,x “ Cardpk ď τρ´d : dpfkx, xq ď ρq.

Since the RHS coincides with factorial moments of the Poisson mixture from (5.2), theresult follows. ˝

5.3. Notes. Early works on Poisson Limit Theorems for dynamical systems include[35, 40, 80, 81, 82, 119]. [30, 75, 78, 120] prove Poisson law for visits to balls centeredat a good point for nonuniformly hyperbolic dynamical systems and show that the setof good points has a full measure. [42] obtains Poisson Limit Theorem for partiallyhyperbolic systems. Some of those papers, including [27, 42, 75] show that in varioussettings is the hitting time distributions are Poisson for all non-periodic points (cf.our Conjecture 5.2). The rates of convergence under appropriate mixing conditions arediscussed in [2, 3, 76]. The Poisson limit theorems for flows are obtained in [112, 116].Convergence of the level of random measures where one records some extra informationabout the close encounters, such as for example, the distance of approach is discussed in[42, 56, 57]. A mixed exponential distribution for a return time for dynamical systemssimilar to Corollary 5.4 has been obtained in [37] in a symbolic setting. For morediscussion of the distribution of the entry times to small measure sets we refer thereaders to [36, 86, 123, 137] and references wherein.


6. Gibbs measures on the circle: Law of iterated logarithm forhitting times

6.1. Gibbs measures. The goal of this section is to show how absence of the hy-pothesis of smoothness on the invariant measure µ may also alter the law of multiplerecurrence.

For simplicity we consider the case where f is an expanding map of the circle T andµ is a Gibbs measure with Lipschitz potential g. Adding a constant to g if necessarywe may and will assume in all the sequel that the topological pressure of g is 0, that isP pgq “ µpgq ` hµpfq “ 0.

This means (see [125] for background on Gibbs measures) that for each ε ą 0 thereis a constant Kε such that if Bnpx, εq is the Bowen ball

Bnpx, εq “ ty : dpfky, fkxq ď ε for k “ 0, . . . , n´ 1u

then1

Kε

ďµpBnpx, εqq

exp“`řn´1k“0 gpf

kxq˘‰ ď Kε

Denote by λ “ λpµq the Lyapunov exponent of µ

λ “ limnÑ8

log |pfnq1pxq|

n“

ż

| log f 1|dµ ą 0,

and by d the dimension of the point x,

d “ limδÑ0

lnµpBpx, δqq

ln δ.

We know from [106] that the limit exists for µ-a.e. x and d “ hµpfq

λ“ ´

µpgqµpfuq

with

the notation fu “ ln |f 1|.We say that µ is conformal if there is a constant K such that for each x and each

r ď 1

(6.1)1

KďµpBpx, rqq

rdď K.

It is known (see e.g. [117]) that µ is conformal iff g can be represented in the form

g “ tfu ´ P ptfuq ` g ´ g ˝ f

for some Holder function g and t P R.

Theorem 6.1. (a) If µ is conformal then Theorems 4.2 and 4.3 remain valid with dreplaced by d.

(b) If µ is not conformal then for µ almost every x and µˆ µ almost every px, yq, itholds that

lim supnÑ8


dlnn

a

2plnnqpln ln lnnq“

σ

d?

dλ,(6.2)

lim supnÑ8


dlnn

a

2plnnqpln ln lnnq“

σ

d?

dλ.(6.3)


6.2. Good targets for Gibbs measures. Here we prepare for the proof of Theo-rem 6.1 by showing that certain targets are good for expanding maps equipped with aGibbs measure. We caution the reader that the targets Ωρ from Section 4 are not goodtargets in the non-conformal case so we need to use a roundabout approach to proving(6.2). We collect several useful facts.

Proposition 6.2. For each pf, µq enjoys r fold exponential mixing with respect to Lip-schitz functions.

The proof is given in §A.2.In the rest of the argument it will be important that if µ is a Gibbs measure then

there are constants a, b such that for all sufficiently all r and for all x

(6.4) ra ď µpBpx, rqq ď rb.

We also need the fact that Gibbs measures are Alhfors regular, that is there is aconstant R such that for each x, ρ we have

(6.5) µpBpx, 4ρq ď Rµpx, ρq.

Proposition 6.3. For any Gibbs measure µ for µ almost all x the targets Bpx, ρnq areadmissible provided that ρn ą nú for some u.

Proof. pPolyq follows from (6.4).Next take a large s and consider A˘ρ such that

1Bpx,ρ´ρsq ď A´ρ ď A`ρ ď 1Bpx,ρ`ρsq.

Then A`ρ ´ A´ρ is supported on two segments of length 2ρs showing that

µpA`ρ ´ A´ρ q ď ρbs ď ρa ď µpBpx, ρqq

if s is large proving pApprq.It remains to show that pMovq holds for a. e. x. Namely we show that for a.e. x and

all k

(6.6) µpBpx, ρq X f´kBpx, ρqq ď µpBpx, ρqq1`η.

We consider two cases.(I) k ą ε| ln ρ| where ε is sufficiently small (see case (II) for precise bound on ε). Take

A`ρ such that A`ρ “ 0 on Bpx, ρq, µpA`ρ q´µpBpx, ρqq ď 2µpBpx, ρqq and A`ρ Lip ď Cρ´τ

for some τ “ τpµq. Let ρ “ ρσ where σ is a small constant. Then (A.3) gives

µpBpx, ρq X f´kBpx, ρqq ď µpA`ρ pxqpA`ρ ˝ f

kqq

ď 4µpBpx, ρqqµpBpx, ρqq ` Cθkρ´τµpBpx, ρqq “ CµpBpx, ρqq“

µpρσb ` ρερ´τσ‰

.

Taking σ small we can make the second term smaller than ρε2 which is enough forpMovq in view of already established pPolyq. Note that no restrictions on x are imposedin case (I).

(II) k ą ε| ln ρ|. In this case for a.e. x the intersection Bpx, ρq Y f´kBpx, ρq is emptyfor small ρ due to the Proposition 6.4 below. ˝


Proposition 6.4. ([14, Lemma 5] ) Let T : X Ñ X be a Lipschitz map with Lipschitzconstant L ą 1 on a compact metric space X. If µ is an ergodic measure with hµpT q ą 0.Then for almost every x, there exists n0pxq such that for all n ě n0pxq, and all 0 ă k ď1

2L| ln ρ|, we have T´kB px, ρq XB px, ρq “ H.

Proposition 6.5. If µ is conformal, that is Bpx, ρq “ γpxqρdp1 ` oρÑ0p1qq then thetargets Ωρn defined by (4.6) are admissible provided that ρn ą nú for some u.

The proof of this fact is the same as in the smooth measure case, so we leave it tothe reader.

Proof of Theorem 6.1(a). Given Propositions 6.2, 6.3 and 6.5 the proof is exactly thesame as in Section 4.

Consider for example, the MultiLog Law for hitting times. Taking ρn “ n´1d ln´s n,

we see that for almost every x, Ωρn,x “ ty : dpx, yq ď ρnu is a simple admissiblesequence of targets as in Definition 3.2. Moreover, with the notation vj “ σpρ2jq andSr “

ř8

j“1 p2jvjq

r, we see from (6.1), that Sr “ 8 iff s ě rd. Hence the MultiLog Law

for hitting times follows from Corollary 3.7.The proof of other results are similar. ˝

6.3. Law of iterated logarithm for hitting times. The proof of part pbq of Theo-rem 6.1 relies on the following lemma on the fluctuations of local dimension of Gibbsmeasures of expanding circle maps. We leave the proof of the lemma to Appendix B.

Denote ψpxq “ gpxq`dfupxq, then we haveş

ψdµ “ 0 under the assumption P pgq “ 0.Define σ “ σpµq by the relation

σ2“

ż

ψ2dµ` 28ÿ

n“1

ż

fupxqfupfnxqdµpxq.

Lemma 6.6.(a) σpµq “ 0 iff µ is conformal.(b) For µ almost every x

lim supδÑ0

| lnµ pBpx, δqq | ´ d| ln δ|a

2| ln δ|pln ln | ln δ|q“

σ?λ, lim inf

δÑ0

| lnµ pBpx, δqq | ´ d| ln δ|a

2| ln δ|pln ln | ln δ|q“ ´

σ?λ.

We proceed to the proof of part pbq of Theorem 6.1, that uses only the lim inf of partpbq of Lemma 6.6.

Take an arbitrary ε and let

ϑ˘pδq “ δd exp

ˆ

p1˘ εqσ?

dλ

a

2| ln δ| pln ln | ln δ|q

˙

.

Take

ρn “1

n1dexp

!

ća

2plnnqpln ln lnnq)

.

Denote ϑ˘pnq “ ϑ˘pρnq, then

ϑ˘pnq “1

nexp

"ˆ

ćd` p1˘ εqσ?

dλ` ηn

˙

?2 lnn

*


for some ηn Ñ 0 as nÑ 8.The lim inf in Lemma 6.6, has the following straightforward consequences, for any

ε ą 0 and for µ almost every x:

(i) There exists npxq such that for n ě npxq, we have

µ pBpx, ρnqq ď ϑ`pnq

(ii) For a subsequence nl Ñ 8 we have

µ pBpx, ρnlqq ě ϑ´pnlq

From there it follows that for any r ě 1, Sr “ř8

k“1p2kµBpx, ρ2kqq

r is finite if c ąp1 ` εq σ

d?dλ

and is infinite if c ă p1 ´ εq σd?dλ. Thus the result follows from Theorems

2.2 and 3.6 since the assumptions of Section 3 have been verified in §6.2.

6.4. Law of iterated logarithm for return times. Lower bound. Here we provethat

(6.7) lim supnÑ8


dlnn

a

2plnnqpln ln lnnqě

σ

d?

dλ.

Suppose p to be a fixed point of f. Take the Markov partition Pn of T such that ifPn P Pn, then fnpBPnq “ p. Denote Pnpxq “ tPn P Pn : x P Pnu and a sequence njpxq,j P N such that n0pxq “ 0,

njpxq “ min

n ą kj´1pxq2 : µ pPnpxqq ě ϑ´ p|Pnpxq|q

(

where

kjpxq “2

µpPnpxqq.

Let

Aj “

Card`

kj´1pxq ď k ď kjpxq : fkx P Pnjpxq˘

ě r(

and denote Fj “ B`

Pk1 , ¨ ¨ ¨ ,Pkj˘

. Then

lim supnÑ8


dlnn

a

2plnnqpln ln lnnqě

σ

d?

dλp1´ εq

if x belongs to infinitely many Ajs.We will use Levy’s extension of the Borel-Cantelli Lemma.

Theorem 6.7. ([134, §12.15]) Ifÿ

j

P pAj`1|Fjq “ 8 a.s.

then Aj happen infinitely many times almost surely.

Hence (6.7) follows from the Lemma below.

Lemma 6.8. There exists c˚ ą 0, such that for almost all x there is j0 “ j0pxq suchthat P pAj`1|Fjq ě c˚ for all j ě j0.


Proof. For any Ω Ă T, Pk P Pk,

µ`

fkpΩX Pkq˘

“µ`

fkpΩX Pkq˘

µ pfkpPkqqď C

µpΩX Pkq

µpPkq

by bounded distortion property. Note that

E`

1Aj`1|Fj

˘

pxq “µ`

Aj`1 X Pkjpxqpxq˘

µ`

Pkjpxqpxq˘ ě

µ`

fkjpxq`

Aj`1 X Pkjpxqpxq˘˘

C.

By construction

fkjpxq`

Aj`1 X Pkjpxqpxq˘

is the set of points y P T which visit Pnj`1pxq at least r times before time

kj`1pxq “ kj`1pxq ´ kjpxq.

By Proposition 6.3 for almost all x the targets Pnj`1pxq satisfy pM1qr and pM2qr for

all r. Since by construction limjÑ8

µpPnjpxqqkjpxq “ 2 we can apply Theorem 2.8 to get

P pAj`1|Fjq pxq ě C´1µ`

fnjpAj`1 X Pnjq˘

ě C0

8ÿ

k“r

e´2 2k

k!:“ c˚.

proving the lemma. ˝

6.5. Law of iterated logarithm for return times. Upper bound. Now we turnto the proof of

(6.8) lim supnÑ8


dlnn

a

2plnnqpln ln lnnqď

σ

d?

dλ.

Since dprqn px, xq ě d

p1qn px, xq, we only need to show (6.8) for r “ 1.

Denote

rn “1

n1dexp

"

´p1` εqσ

d?

dλ

a

2plnnqpln ln lnnq

*

.

Let Nk “ 2k. Similarly to Section 2 it is enough to show that for almost all x, for allsufficiently large k we have that

dpfmx, xq ě rNk for m “ 1, . . . Nk.

Proposition 6.4 allows us to further restrict the range of m by assuming m ě ε lnNk

there ε is sufficiently small.We say x P T is n´good if µ pBpx, rnqq ď ϑ`prnq. Fix k0 and let

Ak “ tx : x is n´good for n ě Nk0 but dpfmx, xq ď rNk for m “ ε lnNk, . . . Nk.u

Let Xk “ txj,kulkj“1 to be a maximal rNk separated set of Nk´good points. Thus if x

is Nk good then there is j such that x P Bpxj,k, rNkq Therefore if fmx P Bpx, rnkq thenfmx P Bpxj,k, 2rNkq. Fix a large K then for m ď K lnNk we use (6.6) telling us that

µpBpxj,k, 2rNkq X f´mBpxj,k, 2rNkq ď KµpBpxj,k, 2rNkqq

1`η.


while for m ą K lnNk we get by exponential mixing that tells us that

µpBpxj,k, 2rNkq X f´mBpxj,k, 2rNkq ď KµpBpxj,k, 2rNkqq

2.

Summing those estimate for we obtain

Nkÿ

m“ε lnNk

µpBpxj,k, 2rNkq X f´mBpxj,k, 2rNkq ď KµpBpxj,k, 2rNkqqe

´κ?k

for some κ “ κpεq ą 0. Since Bpxj,k, rNk2q are disjoint for different j we conclude using(6.5) that

ÿ

j

µpBpx, 2rNkq ď Rÿ

j

µpBpx, rNk2q ď R.

It follows thatµpAkq ď KRe´κ

?k.

Now the result follows from Borel–Cantelli Lemma.

6.6. Notes. The fact that return times for the non-conformal Gibbs measures are dom-inated by fluctuations of measures of the balls has been explored in various settings[23, 24, 31, 37, 77, 84, 115, 121, 131]. In particular, [72] obtains a result similar toour Lemma 6.6 in the context of symbolic systems. The papers mentioned above dealwith either one dimensional or symbolic systems. In higher dimensions even the leadingterm of lnµpBpx, rqq is rather non-trivial and is analyzed in [13], while fluctuations aredetermined only for a limited class of systems [107]. Thus extending the results of thissection to higher dimension is an interesting open problem.

7. Geodesic excursions.

7.1. Excursions in finite volume hyperbolic manifolds. Let Q be a finite volumenon compact manifold of curvature -1 and dimension d ` 1. For q P Q, v P SQ letγpt, q, vq be the geodesic such that γp0q “ q 9γp0q “ v. We call gt the correspondingflow, that is gtpγp0q, 9γp0qq “ pγptq, 9γptqq. Fix a reference point O P Q and letDpq, v, tq “distpγptq, Oq, According to Sullivan’s Logarithm Law for excursions [129] for almost allq and v

(7.1) lim supTÑ8

Dpq, v, T q

lnT“

1

d.

In fact, the argument used to prove (7.1) also shows using the Borel–Cantelli Lemmaof [129] that

(7.2) lim supTÑ8

Dpq, v, T q ´ 1d

lnT

ln lnT“

1

d.

Here we present a multiple version of (7.1). Recall ([16, Proposition D.3.12]) that Qadmits a decomposition Q “ KY

`

Ypj“1Cj

˘

where K is a compact set and Cj are cusps.Moreover each cusp is isometric to Vi ˆ rLj,8q endowed with the metric

ds2“dx2 ` dy2

y2where x P Vj, y P rLj,8q


where Vi is a compact flat manifold and dx is the Euclidean metric on Vj. Cusp aredisjoint, so that a geodesic can not pass between different cusps without visiting thethick part K in between. We note that for each px0, y0q P Ci there is a unique ge-odesic (tx “ x0u) which remains in the cusp for all positive time. We will call thisgeodesic escaping geodesic passing through px0, y0q. Let hpq, v, tq “ 0 if γpq, v, tq P Kand hpq, v, tq “ ln yptq if γpq, v, tq P Ci and has coordinates pxptq, yptqq where. It is easyto see using the triangle inequality that there exists a constant C such that

|Dpq, v, tq ´ hpq, v, tq| ď C.

A geodesic excursion is a maximal interval I such that γptq belong to some cusp Ci forall t P I. hpIq “ max

tPIhpq, v, tq is called the height of the excursion I. Let

Hp1qpq, v, T q ě Hp2q

pq, v, T q ě ¨ ¨ ¨ ě Hprqpq, v, T q . . .

be the heights of excursions finished up to time T ordered in a decreasing order. Bythe foregoing discussion (7.2) can be restated as

(7.3) lim supTÑ8

Hp1qpq, v, T q ´ 1d

lnT

ln lnT“

1

d.

We have the following extension of this result

Theorem 7.1. For a.e. pq, vq and all r we have

(7.4) lim supTÑ8

Hprqpq, v, T q ´ 1d

lnT

ln lnT“

1

rd.

We also have the following byproduct of our analysis

Corollary 7.2. There is a constant ai such that for each h the following holds. Supposethat pq, vq is uniformly distributed on SQ. Then the number of excursions in the cuspCi which finished before time T and reached the height lnT

d`h is asymptotically Poisson

with parameter aie´dh.

In other words, for every r ě 1, we have

(7.5) limTÑ8

Pˆ

Hprqi pq, v, T q ă

lnT

d` h

˙

“

r´1ÿ

l“0

paie´dhql

l!exp

`

áie´dh

˘

.

In particular, taking r “ 1 in (7.5) we obtain

Corollary 7.3. (Gumbel distribution for the maximal excursion) If pq, vq is uniformly

distributed on SQ. Let Hp1qi pq, v, T q denote the maximal height reached by γpt, q, vq up

to time T inside cusp Ci. Then

limTÑ8

Pˆ

Hp1qi pq, v, T q ´

lnT

dă h

˙

“ exp`

áie´dh

˘

.


7.2. Multilog law for geodesic excursions. Proof of Theorem 7.1. To prove thisresult we need to discuss the probability of having an excursion reaching a given level.To this end let Π be the plane passing through γ and the escaping geodesic. In thisplane the geodesics are half circles centered at the absolute ty “ 0u. The geodesic givenby px´ x0q

2 ` y2 “ R2 reaches the maximum height of lnRÒp1q. Let n˚ be the firstinteger moment of time after the beginning of the excursion. Then the y coordinate ofγpn˚q is uniformly bounded from above and below so the radius of the circle defining

the geodesic is given by R “ypn˚q

sin θwhere θ is the angle with the escaping geodesic.

It follows that the condition R ě R0 is equivalent to the condition sin θ ď ypn˚qR. Thus

the probability that a geodesic which enters the cusp at time n˚ will reach height H issandwiched between

e´dH

Cand Ce´dH

for some constant C.Thus given H we consider the set AH which consists of points pq, vq PM such that(i) The first backward time such that γp´t, q, vq exits the cusp is between 0 and 1;(ii) The angle v makes with the exiting geodesic is less than e´H .By the foregoing discussion

(7.6)e´dH

Cď µpAHq ď Ce´dH .

In fact, it is not difficult to see that for each cusp Ci there is a constant ai such that foreach n P Z

(7.7) µpx enters Ci at time n and reaches height Hq “ aie´dH

p1` op1qq

but this will be needed in the proof of Corollary 7.2 and not in the proof of Theorem7.1.

To prove Theorem 7.1 we verify the conditions of Corollary 3.7. Let B “ C1pSQq.Then the system pg1,M, µq is exponentially mixing in the sense of Definition 3.1. Indeed,pEMqr follows from [100] and (Prod) and (Gr) are clear.

Next, for ρ ą 0, we consider the following functions defined in a neighborhood ofA´ ln ρ : tpxq-time since entering the cusp, and ψpxq the angle with escaping geodesics.We then define the following sequence of targets.

(7.8) Ωρ “ A´ ln ρ “

"

x PM : tpxq P r0, 1s, ψpxq P

„

0,1

ρ

*

.

We need to check the conditions of Definition 3.2 for these targets.Since both t and ψ are Lipshitz (Appr) follows as in Remark 3.4.As for (Mov), it is a direct consequence of the following.

Lemma 7.4. There is a constant K such that for each n1, n2

µ pAHpn1qAHpn2qq ď Kµ pAHpn1qqµ pAHpn2qq .


Before we prove Lemma 7.4, we finish the proof of Theorem 7.1.We let

(7.9) ρn :“1

n1d lns n

.

By (7.6), we have that

µpΩρnq P

„

C´1 1

n lnsd n,C

1

n lnsd n

hence (Poly) is satisfied.Finally, we have that p2jµpΩρ

2jqqr “ 1

jsdr, hence by Corollary 3.7 we get that if s ą 1

dr,

then with probability one, we have that for large n, Nnpρnq ă r. This implies that foralmost every pq, vq PM , it holds that for n sufficiently large

Hprqpq, v, nq ă ´ ln ρn “

lnn

d` sln lnn.

Conversely, if s ď 1dr

, then with probability one, we have that for large n, Nnpρnq ě r.This implies that for almost every pq, vq PM , it holds that for n sufficiently large

Hprqpq, v, nq ě ´ ln ρn “

lnn

d` sln lnn.

Thus (7.4) is proved.To finish we still need to give the

Proof of Lemma 7.4. Let AH “ IAH where I denotes the involution Ipq, vq “ pq,´vq.Given n1, n2 define

BH,n1,n2 “ tx : gn1x P AH , gn2x P AH , g

nx R K for n1 ă n ă n2u.

Thus BH,n1,n2 consists of points which enter a cusp at time n1, reach the height H, and

then exit the cusp at time n2. Fix a small δ and let BH,n1,n2 “ YxPBH,n1,n2Wu px, δeń2q .

Note that if y P BH,n1,n2 then gn1y P AH´1 for some n1 P rn1´1, n1`1s and gn2y P AH´1

for some n2 P rn2 ´ 1, n2 ` 1s. In particular for each n1 the sets tBH,n1,n2un2ěn1 have atleast 3 intersection multiplicity and hence

(7.10)ÿ

n2

µpBH,n1,n2q ď 3µpAH´1q ď Ce´dH .

Also for n ą n1

µpAHpn1qAHpnqq ďÿ

n2

µpBH,n1,n2AHpnqq.

We claim that for each n1, n2, n

(7.11) µpAHpnqBH,n1,n2q ď Ce´dHµpBH,n1,n2q

(7.11) implies that

µpAHpn1qAHpnqq ď Ce´dHÿ

n2

µpBH,n1,n2q ď Ce´dHPpAHq

where the last step relies on (7.10).


It remains to establish (7.11). To this end fix a large H and partition a smallneighborhood U of AH into unstable cubes of size δ. Let

BH,n1,n2 “ď

xPBH,n1,n2

Wuδ pg

n2xq

where Wuδ pyq is the element of the above partition containing y. Note that

BH,n1,n2 Ă gń2BH,n1,n2 Ă BH,n1,n2 .

ThusµpBH,n1,n2AHpnqq ď µppgń2BH,n1,n2qAHpnqq “ µpBH,n1,n2AHpn

˚qq

where n˚ “ n´ n2. We claim that

(7.12) µpBH,n1,n2AHpn˚qq ď Cµpgn2BH,n1,n2qe

´dH .

Since µpgn2BH,n1,n2q “ µpBH,n1,n2q (7.12) implies (7.11). By construction BH,n1,n2 ispartitioned into unstable cubes of size δ, it suffices to show that for any such cube Wwhich is contained in U we have

(7.13) µpAHpn˚q|Wq ď Ce´dH

where µp¨|¨q denotes the conditional expectation. Let Q “ YxPW Y|t|ăδ Wsδ pg

txq. Notethat if δ is sufficiently small then due to the local product structure, for each pointy P Q there is unique x P W and t P r´δ, δs such that y P W s

δ pgtxq. In addition if

gn˚

x P AH then gn˚

y P AH´1. Since the measure of Q is bounded from below uniformlyin W Ă U , it follows that

µpAHpn˚q|Wq ď µpAH´1pn

˚q|Qq “

µpAH´1pn˚qQq

µpQqď cµpAH´1pn

˚qq “ cµpAH´1q ď ce´dH .

This establishes (7.13) and, hence (7.12) completing the proof of Lemma 7.4. ˝

With the proof of Lemma 7.4, we finished the proof of Theorem 7.1. l

7.3. Poisson Law for excursions. Proof of Corollary 7.2. We fix h P R and fix acusp index i. We change the targets Ωρn where Ωρ is defined by (7.8) and ρn is definedby (7.9) to

(7.14) Ωpiqρ “ A´ ln ρ “

"

x PM : tipxq P r0, 1s, ψipxq P r0,1

hρs

*

.

and

(7.15) ρn :“ n´1d .

where tipxq is the time since entering the cusp Ci, and ψipxq the angle with escapinggeodesics.

As in the proof of Theorem 7.1, we have that tΩpiqρnu satisfies the assumptions pM1qr

and pM2qr for all r. Moreover, by (7.7)

limnÑ8

nµpΩpiqρnq “ aie´dh.

Therefore Corollary 7.2 follows from Theorem 2.8. l


7.4. Notes. The logarithm law for the highest excursion was proven in [129]. Theextensions for infinite volume hyperbolic manifolds is studied in [128]. Corollary 7.3for surfaces is obtained in [85] where the authors also consider infinite volume surfaces.Papers [11, 50] obtain stable laws for geodesic windings on hyperbolic manifolds. Indeedthe main contribution to windings comes from long excursions, so the proofs of stablelaws and of the Poisson laws for excursions are closely related, see e.g. [46, 48]. Incase the hyperbolic manifold under consideration is the modular surface, the length ofthe n-th geodesic excursion is approximately equal to the size of the n-th convergent ofthe continued fraction expansion of the geodesic endpoint [70], therefore the multipleBorel-Cantelli Lemma in that case follows from the results of [1].

Several authors discussed extended Logarithm Law for excursion to other homoge-neous spaces. Namely, [100] studies partially hyperbolic flows on homogenous spacesand presents applications to metric number theory, cf. Section 8 of the present paper.Logarithm Law for unipotent flows is considered in [8, 9, 65, 87]. In the next sectionwe obtain MultiLog Law for certain diagonal flows on the space of lattices.

8. Multiple Khintchine-Groshev Theorem.

8.1. Statements.

Homogenous approximations.

Definition 8.1 (pr, sq-approximable vectors). Given α “ pα1, . . . αdq P Rd, s ě 0,c ą 0, let DNpα, s, cq be the set of k “ pk1, . . . , kdq P Zd such that

|k| ď N and Dm P Z : gcdpk1, . . . kd,mq “ 1 and |k|d |xk, αy `m| ďc

lnNpln lnNqs.

Call α pr, sq-approximable if for any c ą 0, CardpDNpα, s, cqq ě 2r for infinitely manyNs.

For x P Rd, we used the notation |x| “a

ř

x2i .

Theorem 8.2. If s ď 1r then the set of pr, sq-approximable vectors α P Td has fullmeasure. If s ą 1r then the set of pr, sq-approximable numbers has zero measure.

Remark 8.3. Observe that an equivalent statement of Theorem 8.2 is to replace 2rwith r in the definition of pr, sq approximable vectors provided we restrict to k P Zdsuch that k1 ą 0. This will be the version that we will prove in the sequel.

Inhomogeneous approximations.

Definition 8.4 (pr, sq-approximable couples). Given α “ pα1, . . . αdq P Rd and z P R,s ě 0 and c ą 0, let DNpα, z, s, cq be the set of k “ pk1, . . . , kdq P Zd such that

|k| ď N and Dm P Z : and |k|d |z ` xk, αy `m| ďc

lnNpln lnNqs.

Call the couple pα, zq pr, sq-approximable if for any c ą 0, CardpDNpα, z, s, cqq ě r forinfinitely many Ns.


Theorem 8.5. If s ď 1r then the set of pr, sq-approximable couples pα, zq P Rd ˆ Rhas full measure. If s ą 1r then the set of pr, sq-approximable couples pα, zq P Rd ˆ Rhas zero measure.

Extensions. One can extend the above results to general Kintchine Groshev 0 ´ 1laws for Diophantine approximations of linear forms. For example

Definition 8.6 (pr, sq-simultaneously approximable vectors). Given α “ pα1, . . . αdq PRd, s ě 0, c ą 0, let DNpα, s, cq be the set of k P Z˚ such that

k ď N and Dm P Zd : gcdpk,m1, . . . ,mdq “ 1

and for all i “ 1, . . . , d, k1d |kαi `mi| ď

c

plnNq1d pln lnNq

sd

.

Call α pr, sq-simultaneously approximable if for any c ą 0, CardpDNpα, s, cqq ě r forinfinitely many Ns.

Theorem 8.7. If s ď 1r then the set of pr, sq-simultaneously approximable vectorsα P Td has full measure. If s ą 1r then the set of pr, sq-simultaneously approximablenumbers has zero measure.

We omit the of Theorem 8.7 since it is obtained by routine modification of the proofof Theorem 8.2.

8.2. Reduction to a problem on the space of lattices. Let M be the space ofd` 1 dimensional unimodular lattices. We identify M with SLd`1pRqSLd`1pZq.

Λα “

ˆ

Idd 0α 1

˙

.

For t P R, we consider gt P SLd`1pRq

gt “

¨

˚

˚

˝

2´t

. . .2´t

2dt

˛

‹

‹

‚

(8.1)

For a lattice L Ă Md`1, we say that a vector in L is prime if it is not an integermultiple of another vector in L.

Given a function f on Rd`1 we consider its Siegel transform Spfq : M Ñ R definedby

(8.2) SpfqpLq “ÿ

ePL, e prime

fpeq.

For a ą 0, let φa be the indicator of the set4

Ea :“

px, yq P Rdˆ R | x1 ą 0, |x| P r1, 2s, |x|d|y| P r0, as

(

.

4We added x1 ą 0 in the definition of Ea since we will restrict to vectors k P Zd with k1 ě 0.


Fix s ě 0, c ą 0. For M P N˚, define

(8.3) ν :“c

M lnM s, Φν :“ Spφνq.

For t ě 0, we then define

AtpMq :“ tα P Td : ΦνpgtΛαq ě 1u

It is readily checked that α P AtpMq if and only if there exists k “ pk1, . . . , kdq withk1 ě 0, and 2t ă |k| ď 2t`1 such that

(8.4) .Dm, gcdpk1, . . . kd,mq “ 1, |k|d|xk, αy `m| ďc

M lnsM.

If α is such that ΦνpgtΛαq ď 1 for every t P N, then we get that α is pr, sq-approximable if and only if there exists infinitely many M for which there exists0 ă t1 ă t2 ă . . . ă tr ďM satisfying α P

Şrj“1AtjpMq.

But in general, for α and t ďM such that α P AtpMq, there may be multiple solutionsk such that 2t ă |k| ď 2t`1 for the same t. Since in Theorem 8.2 we are counting allsolutions we have to deal with this issue.

The following proposition shows that almost surely on α, multiple solutions do notoccur. Its proof is based on Rogers identity for the second moment of the Siegel trans-forms.

Proposition 8.8. For almost every α, we have that for every M sufficiently large, forevery t P r0,M s, it holds that ΦνpgtΛαq ď 1

Hence, Theorem 8.2 is equivalent to the following.

Theorem 8.9. If rs ď 1, then for almost every α P Td, there exists infinitely many Mfor which there exists 0 ă t1 ă t2 ă . . . ă tr ďM satisfying

α Prč

j“1

AtjpMq.

If rs ą 1, then for almost every α P Td, there exists at most finitely many M forwhich there exists 0 ă t1 ă t2 ă . . . ă tr ďM satisfying

α Prč

j“1

AtjpMq.

8.3. Modifying the initial distribution: homogeneous case. We transformed ourproblem into a problem of multiple recurrence of the diagonal action gt when appliedto a piece of horocycle in the direction of Λα : α P Td. But this horocycle is exactlythe full strong unstable direction of the rapidly mixing partially hyperbolic action gt.Due to the equidistribution of the strong unstable horocycles, it is thus possible andmuch more convenient to work with Haar measure on M instead of Haar measure onΛα : α P Td.

Hence, we define

BtpMq :“ tL PM : ΦνpgtLq ě 1u,


where we recall that

ν :“c

M lnM s, Φν :“ Spφνq,

and φν is the indicator of the setEν “

px, yq P Rd ˆ R | x1 ą 0, |x| P r1, 2s, |x|d|y| P r0, νs(

.Our goal becomes to prove the following.

Proposition 8.10. For µ-almost every L PM, we have that for every M sufficientlylarge, for every t P r0,M s, it holds that ΦνpgtLq ď 1

Theorem 8.11. If rs ď 1, then for µ-almost every L PM, there exists infinitely manyM for which there exists 0 ă t1 ă t2 ă . . . ă tr ďM satisfying

L Prč

j“1

BtjpMq.

If rs ą 1, then for µ-almost every L PM, there exists at most finitely many M forwhich there exists 0 ă t1 ă t2 ă . . . ă tr ďM satisfying

L Prč

j“1

BtjpMq.

Proof that Proposition 8.10 and Theorem 8.11 imply Proposition 8.8 and Theorem 8.9.Recall that for M P N we defined ν “ c

M lnMs . Fix η ą 0 and define Φ˘ν as in (8.7)

but with p1` ηqc and p1´ ηqc instead of c. Next, define for β P Rd

Λ´β “

ˆ

Idd β0 1

˙

,

and for B P SLdpRq we define

DB “

ˆ

B 00 1

˙

,

and finally

Λα,β,B “ DBΛ´β Λα.

The idea now is that if 0 ă ε ! η is fixed, and if B is distributed according to asmooth measure with respect to Haar measure of SLdpRq in an ε neighborhood of theIdentity, and if β is distributed in some ε neighborhood of 0 in Rd with a smooth densityaccording to Haar measure of Td, and α is distributed according to any measure withsmooth density with respect to Haar measure on Td, we have that the lattice Λα,β,B isdistributed according to a smooth distribution in M according to the Haar measure µ.

Moreover, because Λ´β forms the stable direction of gt and because UB forms thecentralizer of gt, we have that if M is sufficiently large, then

Φ´ν pgtΛα,β,Bq ě 1 ùñ ΦνpgtΛαq ě 1 ùñ Φ`ν pgtΛα,β,Bq ě 1,

hence, the implication of Proposition 8.8 and Theorem 8.9 from Proposition 8.10 andTheorem 8.11. l


8.4. Rogers identities. The following identities (see [110, 133]) play an importantrole in our argument. Denote

c1 “ ζpd` 1q´1, c2 “ ζpd` 1q´2, where ζpd` 1q “8ÿ

n“1

n´pd`1q

is the Riemann zeta function.Let f, f1, f2 be piecewise smooth functions with compact support on Rd`1.Let

F pLq “ÿ

ePL, prime

fpeq, F pLq “ÿ

e1‰˘e2PL, prime

f1pe1qf2pe2q.

F is the Siegel transform of f that we denoted Spfq.

Lemma 8.12. We have

paq

ż

MF pLqdµpLq “ c1

ż

Rd`1

fpxqdx,

pbq

ż

MF pLqdµpLq “ c2

ż

Rd`1

f1pxqdx

ż

Rd`1

f2pxqdx.

8.5. Proof of Proposition 8.10. Recall that ν “ cM lnMs

Lemma 8.13. There exists a constant C ą 0, such that for every M , for every t P R,it holds that

PHaar pΦνpgtLq ą 1q ď Cc2M´2 ln´2sM.

For K ě 0, apply the lemma for M “ 2K and sum over all t P r0,M s, then

PHaar

`

Dt ď 2K ,Φν4cpgtLq ą 1˘

ď 16Cc22´K ln´2sp2Kq.

The straightforward side of Borel Cantelli lemma gives that for almost every L, for Ksufficiently large, for any t ď 2K ,Φν4cpgtLq ď 1. For the same L, it then holds that forM sufficiently large, for any t ďM,ΦνcpgtLq ď 1.

To finish the proof of Proposition 8.10 we give

Proof of Lemma 8.13. Since gt preserves Haar measure on M it suffices to prove thelemma for t “ 0. But the condition k1 ě 0 implies that

Φ2νpLq ´ ΦνpLq “

ÿ

e1‰e2PL prime

φνpe1qφνpe2q “ÿ

e1‰˘e2PL prime

fpe1qfpe2q.

It then follows from Rogers identity (b) of Lemma 8.12 that

PHaar pΦνcpLq ą 1q ď E`

Φ2νcpLq ´ ΦνcpLq

˘

ď c2

ˆż

Rd`1

φνcpuqdu

˙2

ď Cc2M´2 ln´2sM.

˝


8.6. Proof of Theorem 8.11.We want to apply Corollary 3.7. For the system pf,X, µq we take pg1,M, µq, where

µ is the Haar measure on M. For the targets, we take Ωρ “ tL : ΦρpLq ě 1u. Notethat by the invariance of the Haar measure by gt we have that µpΩt

ρq “ µpΩρq for any t.For s P N, we define the sequence ρM :“ c

M lnsM. The conclusions of Theorem 8.11

will then follow from the conclusion of Corollary 3.7 applied to NMpρMq, where Nnpρqis the number of times t ď n such that Ωtpρq occurs.

Indeed, if we recall the definition of

Sr “8ÿ

j“1

`

2jvj˘r, vj “ σpρ2jq, σpρq “ µpΩρq

we see that Sr “ 8 iff rs ă 1.This being said, to be able to apply Corollary 3.7 and finish, we still need to check

the conditions of Theorem 3.6 for the system pg1,M, µq and for the family of targetsgiven by Ωρ and the sequence ρM “ c

M lnsM.

Fix s P N, s ě 2. We define Csc pMq the space of smooth functions defined over M

that are constant outside a compact set.Next we let B “ Cs

c pMq. For this norm, it is known that the system pg1,M, µq isexponentially mixing as in Definition 3.1 [18].

We need to check the conditions of Definition 3.2. The approximation condition(Appr) can be checked as follows.

Claim. There exists σ ą 0 such that, for every ρ ą 0 sufficiently small, there existsA´ρ , A

`ρ P C

sc pMq such that

(i) A˘ρ 8 ď 2 and A˘ρ B ď ρ´σ;(ii) A´ρ ď 1Ωρ ď A`ρ ;

(iii) µpA`ρ q ´ µpA´ρ q ď ρ2

Clearly the claim implies (Appr) since µpΩρq “ Opρq. The proof of the claim issimilar to the corresponding argument in [47].

Next we show now how Rogers identity of Lemma 8.12(b) implies (Mov). Define

Eτν “

px, yq P Rdˆ R | x1 ą 0, 2´τ |x| P r1, 2s, |x|d|y| P r0, νs

(

and let φτν be the indicator function of Eτν . Then

µpΩρ X f´τΩρq ď EpΦρΦρ ˝ gτ q “

ż

M

ÿ

e2‰˘e1PL prime

φρpe1qφτρpe2qdµpLq

where the contribution of e2 “ é1 vanishes because the contribution of any pair pe1, e2q

where not both e1,1 and e2,1 are positive is zero. Applying Lemma 8.12 (b) we get that

µpΩρ X f´τΩρq ď CµpΩρq

2

which is stronger than the required (Mov).Finally, the condition (Poly) clearly holds for the sequence ρM “ c

M lnsMdue to

Lemma 8.12 (a). l


8.7. The argument in the inhomogeneous case. The proof of Theorem 8.5 is verysimilar to that of Theorem 8.2, and below we only outline the main differences.

Let ĂM be the space of d` 1 dimensional unimodular affine lattices. We identify ĂMSLd`1pRq ˙ Rd`1SLd`1pZq ˙ Zd`1, where the multiplication rule in SLd`1pRq ˙ Rd`1

is defined as pA, aqpB, bq “ pAB, a` Abq. We denote by µ the Haar measure on ĂM.For α P Rd and z P R, we define

(8.5) Λα,z “ pΛα, p0, . . . , 0, zqq

Given a function f on Rd`1 we consider its Siegel transform Spfq : ĂM Ñ R definedby

(8.6) SpfqpLq “ÿ

ePL

fpeq.

Note that, unlike our definition of the Siegel transform in the case of regular lat-tices, we do not require in this affine setting that the vectors e in the summation beprime. This is because in this affine setting, when a vector k P Zd contributes to theDiophantine approximation counting problem there is no reason for the multiples of kto contribute.

For a ą 0, let φa be the indicator of the set5

Ea :“

px, yq P Rdˆ R | |x| P r1, 2s, |x|d|y| P r0, as

(

Fix s ě 0, c ą 0. For M P N˚, define

(8.7) ν :“c

M lnsM, Φν :“ Spφνq.

For t ě 0, we then define

AtpMq :“ tpα, zq P Rdˆ R : ΦνpgtΛα,zq ě 1u

It is readily checked that pα, zq P AtpMq if and only if there exists k “ pk1, . . . , kdqsuch that 2t ă |k| ď 2t`1 and that

(8.8) Dm, |k|d|z ` xk, αy `m| ďc

M lnsM.

If α is such that ΦνpgtΛα,zq ď 1 for every t P N, then we get that pα, zq is pr, sq-approximable if and only if there exists infinitely many M for which there exists 0 ăt1 ă t2 ă . . . ă tr ďM satisfying pα, zq P

Şrj“1 AtjpMq.

But in general, for α and t ď M such that pα, zq P AtpMq, there may be multiplesolutions k such that 2t ă |k| ď 2t`1 for the same t. As in the case of Theorem 8.2 wehave to deal with this issue.

The following proposition shows that almost surely on pα, zq, multiple solutions donot occur. Its proof is based on Rogers identity for the second moment of the Siegeltransforms.

5Note that we do not ask in this affine setting that x1 ą 0 in the definition of Ea since the symmetriccontributions of ´k for every k P Zd that contributes to the Diophantine approximation countingproblem in the homogenous case of Theorem 8.2 do not appear in the inhomogeneous Diophantineapproximation problem of Theorem 8.5.


Proposition 8.14. For almost every pα, zq P Rd ˆ R, we have that for every M suffi-ciently large, for every t P r0,M s, it holds that ΦνpgtΛα,zq ď 1

Hence, Theorem 8.5 is equivalent to the following.

Theorem 8.15. If rs ď 1, then for almost every pα, zq P TdˆT, there exists infinitelymany M for which there exists 0 ă t1 ă t2 ă . . . ă tr ďM satisfying

α Prč

j“1

AtjpMq.

If rs ą 1, then for almost every pα, zq P Td ˆ T, there exists at most finitely manyM for which there exists 0 ă t1 ă t2 ă . . . ă tr ďM satisfying

α Prč

j“1

AtjpMq.

8.8. Modifying the initial distribution: inhomogeneous case. Since the horo-cycle directions of Λα,z, pα, zq P Td ˆ T account for all the strong unstable direction

of the diagonal flow gt acting on ĂM, we can transform the requirement of Proposition8.14 and Theorem 8.15 into a problem of multiple recurrence of the diagonal action gtwhen applied to a random lattice in ĂM.

We defineBtpMq :“ tL P ĂM : ΦνpgtLq ě 1u

Our goal becomes to prove the following.

Proposition 8.16. For µ-almost every L P ĂM, we have that for every M sufficientlylarge, for every t P r0,M s, it holds that ΦνpgtLq ď 1

Theorem 8.17. If rs ď 1, then for µ-almost every L P ĂM, there exists infinitely manyM for which there exists 0 ă t1 ă t2 ă . . . ă tr ďM satisfying

L Prč

j“1

BtjpMq.

If rs ą 1, then for µ-almost every L P ĂM, there exists at most finitely many M forwhich there exists 0 ă t1 ă t2 ă . . . ă tr ďM satisfying

L Prč

j“1

BtjpMq.

8.9. Proofs of Proposition 8.16 and Theorem 8.17. Again, the proofs of Propo-sition 8.16 and Theorem 8.17 are very similar to the proofs of their counterpart in thehomogeneous case, Proposition 8.10 and Theorem 8.11.

Similarly to the homogeneous case, we want to apply Corollary 3.7. For the system

pf,X, µq we take pg1, ĂM, µq, where µ is the Haar measure on ĂM. For the targets, wetake Ωρ “ tL : ΦρpLq ě 1u. Observe that from the invariance of the Haar measure bygt we have that µpΩt

ρq “ µpΩρq for any t.


The only difference in the proof of Proposition 8.16 and Theorem 8.17 compared tothat of Proposition 8.10 and Theorem 8.11, is in the application of Rogers identitiesto prove Proposition 8.16 as well as in the proof of (Mov) that is part of the proof ofTheorem 8.17.

We explain this difference now.In fact, Rogers identities are slightly simpler in the affine case, where there is no need

to pay a special attention to the multiples of a vector in the affine lattice. Recall (8.6).Rogers identities for affine lattices read ([110])

EpSpfqq “ż

Rd`1

fpuqdu

EpSpfq2q “ˆż

Rd`1

fpuqdu

˙2

`

ż

Rd`1

f 2puqdu.

(The idea behind the proof for the second moment identity is that the linear functionalson the space of continuous functions on Rd`1ˆRd`1 that are SLd`1pRq˙Rd`1 invariantcan be identified to invariant measures on Rd`1ˆRd`1 by the action of SLd`1pRq˙Rd`1.But the orbits of the latter action decomposes into pairs of independent vectors andpairs of equal vectors.)

Now for the proof Proposition 8.16, we have that

P´

ΦνpLq ą 1¯

ď E´

Φ2νpLq ´ ΦνpLq

¯

ď

ˆż

Rd`1

φνpuqdu

˙2

ď Cc2M´2 ln´2sM

and Proposition 8.16 then follows by a Borel Cantelli argument exactly as in the regularlattices case.

For the proof of (Mov) in the affine case we write for τ ě 1

Eτν “

px, yq P Rdˆ R | 2´τ |x| P r1, 2s, |x|d|y| P r0, νs

(

and for φτν the indicator function of Eτν , observe that

µpΩρ X f´τΩρq ď E

´

Φρ

´

Φρ ˝ gτ

¯¯

“

ż

M

ÿ

e2,e1PL

φρpe1qφτρpe2qdµpLq

“

ż

M

ÿ

e2‰e1PL

φρpe1qφτρpe2qdµpLq “

ˆż

Rd`1

φρpuqdu

˙2

ď CµpΩρq2

which is stronger than the required (Mov). l

8.10. Multiple recurrence for toral translations.

Proof of Theorem 4.4. Proof of part paq. We begin with several reductions. Let

z “ x´ y. Then dpx, y ` kαq “ dpz, kαq. Accordingly denoting dprqn pz, αq to be the r-th

smallest among tdpz, kαqun´1k“0 we need to show that for almost every pz, αq P pTdq2 we

have

(8.9) lim supnÑ8

| ln dp1qpz, αq| ´ 1d

lnn

ln lnn“

1

d,


(8.10) lim supnÑ8

| ln dprqpz, αq| ´ 1d

lnn

ln lnn“

1

2d, for r ě 2.

Next we claim that it suffices to prove (8.10) only for r “ 2. Indeed, since dprqn is non

decreasing in r, (8.10) with r “ 2 implies that for r ą 2

lim supnÑ8

| ln dprqpz, αq| ´ 1d

lnn

ln lnnď

1

2d.

To get the upper bound, suppose that dp2qn pz, αq ď ε. Then there are 0 ď k1 ă k2 ă n

such that kjα P Bpz, εq. Let k “ k2 ´ k1. Then

k2 ` pr ´ 2qα P Bpz, 1` 2pr ´ 2qεq.

Thus dprqpr´1qnpz, αq ď p2r´ 1qd

p2qn pz, αq. Taking lim sup we obtain that if (8.10) holds for

r “ 2 then it holds for arbitrary r. In summary, we only need to show (8.9) and

(8.11) lim supnÑ8

| ln dp2qpz, αq| ´ 1d

lnn

ln lnn“

1

2d.

The proofs of (8.9) and (8.11) is similar to but easier than the proof of Theorem 8.5so we only explain the changes. First, it is suffices to take lim sup for n of the form 2M

since for 2M´1 ď n ď 2M we have

dprq

2Mď dprqn ď d

prq

2M´1 .

Let νM “M´s for a suitable s and

(8.12) Eν “ te “ pe1, e2q P Rd

ˆ R : ||e1|| ď ν, e2 P r0, 1qu.

Then a direct inspection shows that

dprq

2Mě r ô Sp1EνM qpgMΛα,zZd`1

qq ě r

where gM is defined by (8.1) and Λα,z is defined by (8.5). As in the proof of Theorem

8.5 one can show that Sp1EνM qpgMΛα,zZd`1q ě r infinitely often for almost every pz, αq

iff Sp1EνM qpgM Lqq ě r infinitely often for almost every L P ĂM. Thus we need to show

that for almost every L P ĂM

(8.13) Sp1EνM qpgM Lq ě 1 infinitely often if s ă1

d,

(8.14) Sp1EνM qpgM Lq ě 1 finitely often if s ą1

d,

(8.15) Sp1EνM qpgM Lq ě 2 infinitely often if s ă1

2d,

(8.16) Sp1EνM qpgM Lq ě 2 finitely often if s ą1

2d.

To prove (8.13)-(8.16), we need the following fact.


Lemma 8.18. (a) PpSp1Eν q “ 1q “ cdνdp1` op1qq,

(b) c1ν2d ď PpSp1Eν q ě 2q ď c2ν2d.

Before we give the proof of the lemma, we see how it allows to obtain (8.13)–(8.16)and thus finish the proof of part paq of Theorem 4.4.

Indeed, Lemma 8.18 shows thatÿ

M

PpSp1EνM q “ 1q “ 8 iff s ă1

d,

ÿ

M

PpSp1EνM q ě 2q “ 8 iff s ă1

2d.

From there, (8.13)–(8.16) follow from the the classical Borel Cantelli Lemma, thatis, from the case r “ 1 in our Theorem 2.2. We note that in case r “ 1 Theorem 2.2 isa minor variation of standard dynamical Borel Cantelli Lemmas such as e.g., the BorelCantelli Lemma of [100].

It remains to verify the conditions of Theorem 2.2 with r “ 1. Thus we need to verifyconditions pM1q1, pM2q2 and pM3q1. This verification is very similar to the proof ofTheorem 8.5 so we leave it to the readers.

Proof of Lemma 8.18. Letting Φ “ Sp1Enuq we get by Rogers

EpΦq “ cdνd, EpΦ2

´ Φq “`

cdνd˘2.

It follows that

PpΦ ě 2q ďEpΦ2 ´ Φq

2ď Cν2d

proving the upper bound of part (b).In addition

E pΦ1Φě2q ď`

cdνd˘2

so that

(8.17) PpΦ “ 1q “ EpΦq ´ EpΦ1Φě2q “ cdνdÒ

`

ν2d˘

.

This proves part (a).To prove the lower bound in part (b) we need the following estimate. Let

E1 “ tpe1, e2q P Rd

ˆ R : |e1| ăν

10, e2 P r0.1, 0.2su,

E2 “ tpe1, e2q P Rd

ˆ R : |e1| ăν

10, |e2| ď 0.2u,

A “ tL PM : CardpLprime X E1pνqq “ CardpLprime X E2pνqq “ 1u.

Claim.We have

(8.18) PpAq “ cνd.

Assume the claim holds. For L P A, the fundamental domain of Rd`1L can bechosen to contain

E3pνq “ tpe1, e2q P Rd

ˆ R : |e1| ď 0.01ν, |e2| ď 0.01.u


We thus have

PppL` zq : CardppL` zq X Eνq ě 2qq ě PpAqPpCardppL` zq X Eνq ě 2|Aq

ě PpAqPpz P E3pνqq ě c1ν2d.

This gives the lower bound in part (b) of Lemma 8.18. To complete the proof,we nowgive the

Proof of the claim. We consider the cases d ą 1 and d “ 1 separately.In case d ą 1, denote Ψj “ Sp1Ejq. By Rogers

E pΨ1q “ 0.1cdνd, E

`

Ψ21 ´Ψ1

˘

“`

0.1cdνd˘2.

Thus arguing as in the proof of (8.17) we conclude that

(8.19) PpΨ1 “ 1q “ 0.1cdνdÒ

`

ν2d˘

.

Rogers identities also give

EpΨ1pΨ2 ´Ψ1qq “ O`

ν2d˘

.

Hence

(8.20) PpCardpLprime X E1q ě 1 and Card´

Lprime X´

E2zE1

¯¯

ě 1q “ O`

ν2d˘

.

Combining (8.19) and (8.20) we obtain (8.18) for d ą 1.In case d “ 1 we still have EpΨ1q “ cν ` Opν2q. On the other hand, for d “ 1 we

have CardpLprime X E2pνqq ď 1 since L is unimodular. ThusEpΨ1q “ PpΨ1 “ 1q “ PpΨ1 “ 1 and Ψ2 ´Ψ1 “ 0q “ cν. ˝

This completes the proof of Lemma 8.18 and thus of part paq of Theorem 4.4. ˝

Proof of part pbq. It is clear that for any r, if Er is not empty then it is equalto M . The fact that E1 “ M implies that Er “ M for all r is exactly similar to theimplication of (8.10) from (8.11), so we just focus on showing that E1 “ M . Adaptingthe beginning of the proof of part paq to the current homogeneous setting, we see thatwhat we want to prove boils down to showing that for almost every L PM

Sp1Eνn qpgnLq ě 1 infinitely often if s ă1

d,(8.21)

Sp1Eνn qpgnLq ě 1 finitely often if s ą1

d,(8.22)

where Eνn is as in (8.12), and S designates the Siegel transform as in (8.2). By Rogersidentity, Lemma 8.12(a), we have that E

`

Sp1Eνn qpgnLq˘

“ cn´sd, hence (8.21) and(8.22) follow by classical Borel Cantelli Lemma (see for example the Borel CantelliLemma of [100]) or by the case r “ 1 of our Theorem 2.2 .

This completes the proof of Theorem 4.4. ˝


8.11. Notes. A classical Khintchine–Groshev Theorem is given by (1.2)–(1.3). A lotof interest is devoted to extending this result to α lying in a submanifold of Rd (see e.g.[12, 17]). The applications of dynamics to Diophantine approximation is based on Danicorrespondence [38]. In particular, [99] discusses Khintchine–Groshev type results onmanifolds using dynamical tools. The use of Siegel transform as a convenient analytictool for applying Dani correspondence can be found in [110]. Surveys on applicationsof dynamics to metric Diophantine approximations include [15, 19, 45, 51, 52, 66, 96,101, 111]. Limit Theorems for Siegel transforms are discussed in [7, 10, 21, 46, 47]

9. Recurrence in configuration space.

9.1. The results. In this section we return to the study of compact manifolds, but wetreat targets which have more complicated geometry than the targets from Section 4.We will see that a richer geometry of targets leads to stronger results.

Let Q be a manifold of curvature a variable negative curvature and dimension d` 1.Denote SQ for the unitary tangent bundle over Q, π : SQÑ Q the canonical projection,φ the geodesic flow on SQ, and µ the Liouville measure on SQ.

Fix a small number ρ. Given a point a P Q and pq, vq P SQ, let tj be consecutivetimes where the function t Ñ dpa, πpφtpq, vqqq has a local minima such that dj :“

dpa, πpφtjpq, vqq ď ρ. Let dprqn pa, pq, vqq be the r-th minima among the numbers tdjutjďn.

Theorem 9.1. (a) For each a P Q and almost every pq, vq P SQ,

(9.1) lim supnÑ8

| ln dprqn pa, pq, vqq| ´ 1

dlnn

ln lnn“

1

rd;

For almost every pq, vq P SQ,

(9.2) lim supnÑ8

| ln dprqn pq, pq, vqq| ´ 1

dlnn

ln lnn“

1

rd.

Note that in contrast with Section 4 there are no exceptional points for hitting. Wealso obtain Poisson limit theorem. Denote Bpa, ρq “ tq P Q : dpa, qq ă ρu,

Bpa, ρq “ tpq, vq P SQ : dpa, qq ă ρ, v P SqQu,and

Apa, ρq “ YtPr0,εsφtBpa, ρq,

for fixed small ε.Let

(9.3) γ “1

εlimρÑ0

µpApa, ρqq

ρd.

We will show in §9.3 that this limit exists and does not depend on a P Q.

Theorem 9.2. For each a and almost every pq, vq if Na,ρ,τ pq, vq is the maximal r such

that dprq

τρ´dpa, pq, vqq ă ρ then

(9.4) limρÑ0

µ ppq, vq P SQ : Na,ρ,τ pq, vq “ rq “ e´τγpγτqr

r!


and

(9.5) limρÑ0

µ ppq, vq P SQ : Nq,ρ,τ pq, vq “ rq “ e´τγpγτqr

r!

9.2. MultiLog Law.

Proof of Theorem 9.1. We shall check the conditions Corollary 3.7 for f “ φε, B thespace of Lipshitz functions, and the sets Apq, rq. Then (Prod) and (Gr) are clear. pEMq2is proven in [108] and pEMqr for r ą 2 follows from pEMq2 due to [42]. Note that Apa, ρqis a sublevel set of a Lipshitz function

hpq, vq “ mintPr0,εs

dpa, πφtpq, vqq

so (Appr) follows as in Remark 3.4. It remains to verify (Mov). That is, we need toprove the following Proposition.

Proposition 9.3. There exists η ą 0 and t0 ą 0 such that for any a P Q and ρsufficiently small, t

(9.6) µpApa, ρq X φtApa, ρqq ď µpApa, ρqq1`η,

for all t ą t0.

Denote Spq, tq “ φtSqQ and let Apq, εq “ď

tPr0,εs

Spq, tq. We shall show that for each

q P Q(9.7) mespApq, εq X φ´tApa, ρqq ď Cρη

which implies (9.6) by integration over q P Bpa, ρq.Denote γptq “ φtpq, vq The Jacobi field of γ are defined by the solution of the linear

equationJ2ptq `RpJptq, γ1ptqqγ1ptq “ 0,

where J 1 “ DdtJ and RpX, Y qZ denotes the curvature tensor, which is equivalent to

pJ iq2ptq `n´1ÿ

j“0

KijptqJ

jptq “ 0, i “ 1, . . . , n´ 1,

where the matrix Kij is symmetric and its spectrum lies between ´K2

1 and ´K22 when

Q has negative curvature.Note that Jp0q “ 0, J 1p0q ď C0 for some C0 ą 0, we have the following Proposition.

Proposition 9.4. There are constants C ą 0, t0 ą 0 such that J 1ptq ď CJptq fort ą t0.

Proof. Suppose J 1ptq “ V ptqJptq, Sptq “ xJptq, J 1ptqy and nptq “ J 1ptq2.

d

dtSptq “ J 1ptq2 ` xJptq, J2ptqy “ J 1ptq2 ` xJptq,´KptqJptqy

ě C1

`

J 1ptq2 ` Jptq2˘

ěC1

2Sptq


for some C1 ą 0. It follows that Sptq ą 0, t ą 0, and Sptq ą Spt0qeC12pt´t0q, t ą t0.

Besides,

d

dtnptq “ 2|xJ2ptq, J 1ptqy| “ 2|x´KptqJptq, J 1ptqy| ď C2

d

dtSptq,

for some C2 ą 0, which gives that nptqĆ ď C2Sptq and Sptq ě 2C2nptq, t ą t0 for some

t0 ą 0.Therefore,

J 1ptqJptq ě x Jptq, J 1ptqy ě2

C2

J 1ptq2,

J 1ptq ď CJptq

for some C ą 0. ˝

Let us now introduceApq, εq “ YtPr´ε,εsφtSqQ

By elementary geometry

µpApq, εq X φtApa, ρqq ď C

ρµpφtApq, εq XBpa, 2ρqq.

To estimate the measure of the last intersection we note that Σpt, q, εq :“ φtApq, εq isan embedded submanifold on SQ of dimension d. By Proposition 9.4, π : Σpt, q, εq Ñ Qis a local diffeomorphism and moreover ||dπ´1|| is uniformly bounded as a map TQÑ

TΣpt, q, εq. Since Σpt, q, εq approaches the weak unstable manifold of φ as tÑ 8, it hasuniformly bounded curvature. Hence, there exist uniform constants ε1, ε2 independentof t such that Σpt, q, εq can be cut into disjoint piece Σjptq satisfying that for each j,πΣjptq is contained in a ball of radius ε2 and contains a ball of radius ε1. Decreasing ε2

if necessary we obtain that the intersection πΣjptq X Bpa, ρq has only one componentand hence

µpπΣjptq XBpa, ρqq ď Cpε1qρd.

Since dπ´1 is bounded we also have

µpΣjptq X Bpa, ρqq ď Cpε1qρd.

By bounded distortion

µpφ´tpΣjptq X Bpa, ρqqq ď Cρdµpφ´tΣjptqq.

Summing over j we obtain (9.7) completing the proof of Theorem 9.1(a). ˝

In order to complete the proof of Theorem 9.1(b) it remains to verify the conditions

pApprq, pMovq and pSubq.Denote

Φpa, pq, vqq “ minsPr0,εs

d pa, πpφspq, vqqq γpaq1dc,

We regard Φ is a function of x “ pa, uq and y “ pq, vq on M ˆM even though Φ not

depend on u. For b ą 1 let φ˘px, yq be Lipschitz functions such that

1Bp0,ρ´ρbq ď φ´ ď 1Bp0,ρq ď φ` ď 1Bp0,ρ`ρbq.


Take A˘ρ “ φ˘pΦq, then

pρ´ ρbqd ď

ż

1Bp0,ρ´ρbqpΦpx, yqqdµpyq ď

ż

A´ρ px, yqdµpyq

ď

ż

A`ρ px, yqdµpyq ď

ż

1Bp0,ρ`ρbqpΦpx, yqqdµpyq ď pρ` ρbqd.

This proves pApprqpiíiiq.Denoting C0 “ sup

xPMγpxqγpyq, we get

ż

A`ρ px, yqdµpxq ď

ż

1Bp0,ρ`ρbqpΦpx, yqqdµpxq ď C0

ż

1Bp0,ρ`ρbqpΦpx, yqqdµpyq ď C0pρ`ρbqd,

which gives pApprqpivq.

Next, we verify pMovq. Denote

Ωρ “

"

ppa, uq, pq, vqq P SQˆ SQ : D s P r0, εs, d pa, πpφspq, vqqq ăcρ

γpaq

*

,

and

Ωtρ “

"

pq, vq P SQ : D s P r0, εs, d`

q, πpφs`tpq, vqq˘

ăcρ

γpqq

*

.

Take xi P Q, 1 ď i ď k such thatkď

i“1

Bpxi, ρq covers Q. By (9.6),

µpΩtρq ď

ÿ

i

"


q, πpφs`tpq, vqq˘

ăcρ

γpqq, q P Bi

*

ďÿ

i


xi, πpφs`tpq, vqq

˘

ă c0ρ, q P Bi

(

ďÿ

i

Cρdp1`ηq ď Cρη.

This verifies pMovq.It remains to verify pSubq. Denote M0 “ maxtPr´ε,εs Dφt,

Ωρ “

"

ppa, uq, pq, vqq P SQˆ SQ : D s P r´ε, εs, d pa, πpφspq, vqqq ăcρ

γpaq

*

,

and

Ωtρ “

"

pq, vq P SQ : D s P r´ε, εs, d`

q, πpφt`spq, vqq˘

ăcρ

γpqq

*

.

Note that when pq, vq P Ωt1ρ X Ωt2

ρ for t1 ă t2, we have some s1, s2 P r0, 1s such that

d`

q, πpφt1`s1pq, vqq˘

ďcρ

pγpqqq1d, d

`

q, πpφt2`s2pq, vqq˘

ďcρ

pγpqqq1d.


Hence

d`

πpφt1pq, vqq, πpφt2`s2´s1pq, vqq˘

ď M0d`

πpφt1`s1pq, vqq, πpφt2`s2pq, vqq˘

ď2M0cρ

pγpqqq1dď

c1ρ

pγpπpφt1pq, vqqqq1d.

It follows that Ωt1ρ X Ωt2

ρ P φ´t1Ωt2´t1

ρ , which complete the proof of Theorem 9.1(b) bya similar argument to Proposition 3.8 (ii).

9.3. Poisson regime.

Proof of Theorem 9.2. (9.4) follows from Theorem 2.8, since conditions pM1qr andpM2qr are satisfied for all r, due to the results of §9.2.

The proof of part (b) follows the same argument as the proof of Theorem 5.3 exceptthat now pM1qr is satisfied since the RHS of (5.3) takes form ρdλ because λ defined by(9.3) does not depend on a. It remains to verify that the limit in (9.3) indeed does notdepend on a base point.

If pq, vq P Bpa, ρq, denote

Lpq, vq “ L`pq, vq`L´pq, vq where L˘pq, vq “ suptt : φ˘spq, vq P Bpa, ρq for 0 ď s ď tu.

Then we have the following estimate

(9.8) µ´

Apa, ρq¯

“ ε

ˆż

Bpa,ρq

1

Lpq, vqdµ

˙

p1Òpρq

(see e.g. [33]). Note that µ is of the form dµpq, vq “ dλpqqdσpvqλpQq where λ is the Riemann

volume on Q and σ is normalized volume on the d dimensional sphere. If ρ is smallthen the integral in parenthesis equals to ρdγp1Òpρqq where

(9.9) γ “1

λpQq

ż

BˆSd

1

Lpx, vqdxdσpvq

where B is the unit ball in Rd`1 and Lp¨q is defined similarly Lp¨q with geodesics inQ replaced by geodesics in Rd`1. Specifically, an elementary plane geometry givesLpx, vq “

a

1´ r2min where rmin is the minimal distance between the line x` tv and the

origin. Thus rmin “ r sin θ where r is the distance from x to 0, θ is the angle betweenv and the segment from x to 0. This proves (9.3) with γ given by (9.9). ˝

9.4. Notes. In [113], Maucourant proved that for all p P Q and almost every pq, vq PSQ

(9.10) lim suptÑ`8

| ln d pp, π pφtpq, vqqq |

log t“

1

d.

[103] generalized Maucourant’s result to study a shrinking target problem for time hmap. The shrinking target problems for sets with complicated geometry is discussed in[62, 63, 65, 87, 88].

Concerning Poisson Limits we note that visits to sets with complicated geometrynaturally appears in Extreme Value Theory, see Section 10 for details. [135] provided


a general conditions for the number of visits to a small neighborhood of arbitrarysubmanifold to be asymptotically Poisson.

10. Extreme values.

10.1. From hitting balls to extreme values. Here we describe applications of ourresults to extreme value theory.

Let pf,M, µq be as in Definition 3.1. Recall the definitions of the sets Gr and Hpreceding the statement of Theorem 4.2, that satisfy by Theorems 4.2 and 4.3 : µpGrq “1 and, if the periodic points are dense, H contains a residual set.

Given a function φ and a point y P M let φprqn pyq be the r-th minimum among the

values tφpf jyqunj“0.

Theorem 10.1. (a) Suppose f is 2r-fold exponentially mixing. Then(i) There is a set G of full measure in M such that if φ is a function with a unique

non degenerate minimum at x P G then for almost every y PM

lim supnÑ8

| lnφprqn pyq ´ φpxq| ´ 2

dlnn

ln lnn“

2

rd.

(ii) If the periodic orbits of f are dense, then there is a dense Gδ set of H, we havesuch that if φ is a function with a unique non degenerate minimum at x P G then foralmost every y PM

lim supnÑ8


dlnn

ln lnn“

2

rd.

(b) If f is an expanding map of T and µ is a non-conformal Gibbs measure of di-mension d then there is a set Gµ with µpGq “ 1 and such that if φ is a function with aunique non degenerate minimum at x P G then for µ–almost every y PM

lim supnÑ8


dlnn

a

plnnq ln ln lnn“ 2c.

(c) Part (a) remains valid for the geodesics flow on a compact pd ` 1q dimensionalmanifold Q and functions φ : Q Ñ R which have unique non-degenerate minimum atsome point on Q.

(d) For toral translations we have that for almost all α and almost all y we have

lim supnÑ8


dlnn

ln lnn“

#

2d

r “ 11d

r ě 2.

Proof. Since x is non degenerate we have that for

(10.1)d2py, xq

Kď φpyq ´ φpxq ď Kd2

py, xq

so (a) holds for x P Gr and (b) holds for x P H. Hence, the part (a) follows fromTheorems 4.2 and 4.3, part (b) follows from Theorem 6.1, part (c) follows from Theorem4.4, and part (d) follows from Theorem 9.1. ˝


Theorem 10.2. Under the assumptions of Theorem 10.1(a) or Theorem 10.1(d) thereis a set of points x of full measure such that if φ has a non-degenerate minimum at xthen the process

φp1qn pyq ´ φpxq

ρ2,φp2qn pyq ´ φpxq

ρ2, . . . ,

φprqn pyq ´ φpxq

ρ2, . . .

with n “ τρ´d converges as ρ Ñ 0 to the Poisson process on R` with intensity

γpφqτ 2tpd2q´1

d.

Proof. (10.1) does not provide enough information to deduce the result from (5.1) of

Theorem 5.1 however it is not difficult to verify directly conditions ĆpM1qr and pM2qrfrom §2.5 for targets

(10.2) Ωn,i“ ty : φpyq ´ φpxq P

«

r´jρ2,r`jρ2

ff

using the results of Section 3. Since (Mov) for targets (10.2) follows from (Mov) forballs, only (Appr) needs to be checked but it follows immediately from Remark 3.4. ˝

Next, we consider functions of the form

(10.3) ψpyq “c

dspx, yq` rψpyq, where c ă 0 and rψ P Lip.

Theorem 10.3. Let f be 2r-fold exponentially mixing then(a) there is a set G or full measure such that if ψ satisfies (10.3) with x P G then foralmost all y

lim supnÑ8

ln |ψprqn pyq| ´ s

dlnn

ln lnn“

s

rd.

(b) there is a Gδ set H such that if ψ satisfies (10.3) with x P H then for almost all y

lim supnÑ8

ln |ψprqn pyq| ´ s

dlnn

ln lnn“s

d.

(c) If x P G then

ρsψp1qn pyq

c,ρsψ

p2qn pyq

c, . . . ,

ρsψprqn pyq

c, . . . where n “ τρ´d

converges as ρÑ 0 to the Poisson process on R` with intensity

1dstpdsq`1

.

The proofs of the above results is similar to the proofs of Theorem 10.1 and 10.2 sowe will leave them to the readers.

The next result is an immediate consequence of Theorems 10.2 and 10.3(c).


Corollary 10.4. (a) (Frechet Law for smooth functions) If f is r´ fold expo-nentially mixing, φ is a smooth function with non-degenerate minimum at some x P Gthen there is σ “ σpxq such that for each t ą 0

limnÑ8

µpy : φp1qn pyq ą n´2dtq “ e´σzd2

.

(a) (Weibull Law for unbounded functions) If f is r´ fold exponentiallymixing, φ is given by (10.3) with x P G then there is σ “ σpxq such that for each t ą 0

limnÑ8

µpy : φp1qn pyq ą ń´sdtq “ e´σz

´ds

.

10.2. Notes. A classical Fisher–Tippett–Gnedenko theorem says that for independentidentically distributed random variables the only possible limit distributions of nor-malized extremes are the Gumbel distribution the Frechet distribution, or the Weibulldistribution. Corollaries 7.3 and 10.4(a) and (b) provide typical examples where one canencounter each of these three types. We refer to [105] for the proof of Fisher–Tippett–Gnedenko theorem as well as for extensions of this theorem to weakly dependent randomvariables. The weak dependence conditions used in the book have a similar sprit to ourconditions (M1) and (M2). More discussions about relations of extreme value theory toPoisson limit theorems in the context of dynamical systems can be found in [56]. Thebook [109] discusses extreme value theory for dynamical systems and lists various appli-cations. One application of extreme value theory, is that for non-integrable functions,such as described in Theorem 10.3 above, the growth of ergodic sums are dominatedby extreme values, see [1, 25, 39, 89, 90, 114] and references wherein.

Appendix A. Multiple exponentially mixing

A.1. Basic properties. Let f be a smooth map of a compact manifold M preservinga smooth measure µ. In the dynamical system literature f is called r fold exponentiallymixing if there are constant s, C and θ ă 1 such that for any Cs functions A1, . . . Arfor any r tuple k1 ă k2 ă ¨ ¨ ¨ ă kr

(A.1)

ˇ

ˇ

ˇ

ˇ

ˇ

ż rź

j“1

`

Aj ˝ fkj˘

dµ´rź

j“1

ż

Ajdµ

ˇ

ˇ

ˇ

ˇ

ˇ

ď Cθlrź

j“1

AjCs ,

where m “ minjpkj ´ kj´1q with kj “ 0.

In this paper we need to consider a larger class of functions, namely we need thatthere are constants s, C and θ ă 1 such that for any B P CspM r`1q we have(A.2)ˇ

ˇ

ˇ

ˇ

ż

Bpx0, fk1x0, ¨ ¨ ¨ , f

krx0qdµpx0q ´

ż

Bpx0, ¨ ¨ ¨ , xrqdµpx0q ¨ ¨ ¨ dµpxrq

ˇ

ˇ

ˇ

ˇ

ď CsθmBCs .

In this section we show equivalence of (A.1) and (A.2). We use the following fact.

Remark A.1. If (A.1) holds for some s then it holds for all s (with different θ). Thesame applies for (A.2).


Indeed suppose that (A.2) for some Cs functions. Pick some α ă s. We claim thatit also holds for Cα functions. Indeed pick a small ε and approximate a Cα function Bwith BCα “ 1 by a Cs function B, so that

B ´ BC0 ď e´εαm, BCs ď Keεsm.

Thenż

Bpx0, fk1x0, ¨ ¨ ¨ , f

krx0qdµpx0q “

ż

Bpx0, fk1x0, ¨ ¨ ¨ , f

krx0qdµpx0q ˘ e´εαm

“

ż

Bpx0, x1, . . . xrqdµpx0qdµpx1q . . . dµpxrq Ò`

e´εαm˘

Ò pθmeεsmq

“

ż

Bpx0, x1, . . . xrqdµpx0qdµpx1q . . . dµpxrq Ò`

e´εαm˘

Ò pθmeεsmq .

and the second error term is exponentially small if ε is small enough. The argumentfor (A.1) is identical.

We now ready to show that (A.1) implies (A.2).

Theorem A.2. Suppose that (A.1) holds and s is sufficiently large. Then (A.2) holds.

Proof of Theorem A.2. Since B P CspM r`1q it also belongs to HspM r`1q. Hence we candecompose

B “ÿ

λ

bλφλ

where φλ are eigenfunctions of Laplacian on M r`1 with eigenvalues λ2 and φλL2 “ 1.The eigenfunctions φλ are of the form

φλpx0, x1, . . . , xrq “r`1ź

j“0

ψjpxjq

where ∆Mψj “ ζ2jψj and λ2 “

ř

j ζ2j . Recall that by Sobolev Embedding Theorem for

compact manifolds HspMq Ă Cs´ d2´1´εpMq. Since ψjHs “ ζsj we have

ψjC1 ď Cuζuj ď Cuλ

u if u ą 1`d

2.

It follows from (A.1) that if φ ı 1 thenˇ

ˇ

ˇ

ˇ

ż

φλpx, fk1x, . . . , fkrxqdµpxq

ˇ

ˇ

ˇ

ˇ

ď Cλupr`1qθm.

Thereforeˇ

ˇ

ˇ

ˇ

ż

Bpx, fk1x, . . . , fkrxqdµpxq ´

ż

Bpx0, ¨ ¨ ¨ , xrqdµpx0q ¨ ¨ ¨ dµpxrq

ˇ

ˇ

ˇ

ˇ

ď Cθmÿ

λ

bλλupr`1q

ď Cθm||B||Hupr`1qpMr`1q.

This proves the result if s ą`

1` d2

˘

pr ` 1q. ˝


A.2. Mixing for Gibbs measures.

Proof of Proposition 6.2. Denote by LippTq the set of Lipschitz functions on T.The proof consists of three steps.Step 1. By the same argument as in [132, Proposition 3.8], we have that for ψ1 P

LippT, aq, ψ2 P L1pµq,

(A.3)

ˇ

ˇ

ˇ

ˇ

ż

ψ1pψ2 ˝ fnqdµ´

ż

ψ1dµ

ż

ψ2dµ

ˇ

ˇ

ˇ

ˇ

ď Cψ1Lipψ2L1 θn, n ě 0.

Step 2. We proceed to show inductively that for each r ą 0 and ψi P LippTq fori “ 1, . . . , r

(A.4)

ˇ

ˇ

ˇ

ˇ

ˇ

ż

˜

rź

i“1

ψipfkixq

¸

dµpxq ´rź

i“1

ż

ψidµ

ˇ

ˇ

ˇ

ˇ

ˇ

ď Cθmrź

i“1

||ψi||Lip,

where m “ min1ďiďr´1pki`1 ´ kiq, k0 “ 0.

By invariance of µ we may assume that k1 “ 0. Applying (A.3) with ψ1 “ ψ1,

ψ2 “

rź

j“2

ψjpfkj´k2xq we get

ˇ

ˇ

ˇ

ˇ

ˇ

ż

˜

rź

i“1

ψipfkixq

¸

dµpxq ´

ˆż

ψ1pxqdµpxq

˙

«

ż

˜

rź

i“2

ψipfkixq

¸

dµpxq

ffˇ

ˇ

ˇ

ˇ

ˇ

ď Cθmψ1Lip

›

›

›

›

›

˜

rź

i“2

ψipfkixq

¸›

›

›

›

›

L1

ď Cθmrź

i“1

||ψi||Lip.

Applying inductive estimate toż

˜

rź

i“2

ψipfkixq

¸

dµpxq

we obtain (A.4).Step 3. Applying the same argument as in proof of Theorem A.2 we get pEMqr. ˝

A.3. Examples of exponentially mixing systems. There are many results aboutdouble (=twofold) exponential mixing. Many examples of those systems are partiallyhyperbolic. In particular, they expand an invariant foliation W s by unstable manifolds.The next result allows to promote double mixing to r fold mixing.

Theorem A.3. ([42, Theorem 2]) Suppose that for each subset D in a single unstableleave of bounded geometry6 and any Holder probability density ρ on D we have

ˇ

ˇ

ˇ

ˇ

ż

D

Apfnxqρpxqdx´ µpAq

ˇ

ˇ

ˇ

ˇ

ď CθnACsρCα

then f is r fold exponentially mixing for all r P Z.6We refer the reader to [42] for precise requirements on D since those requirements are not essentialfor the present discussion.


Examples of maps satisfying the conditions of Theorem A.3 include expanding maps,volume preserving Anosov diffeomorphsims [22, 117], time one maps of contact Anosovflows [108], mostly contracting systems [26, 41], partially hyperbolic translations onhomogeneous spaces [98], and partially hyperbolic automorphisms of nilmanifolds [67].

We also not the following fact

Theorem A.4. A product of exponentially mixing maps is exponentially mixing.

The proof of this theorem is very similar to the proof of Theorem A.2 so we leaveit to the reader. We also note that instead of direct products one can also considercertain skew products (so called generalized T, T´1 transformations) provided that theskewing function has positive drift. We refer the reader to [43] for more details.

Another source of exponential mixing is spectral gap for transfer operators (cf. §A.2as well as [117, 132]). This allows to handle non-uniformly hyperbolic systems admittingYoung tower with exponential tails [136] as well as piecewise expanding maps [132].

We note that the maps described in the last paragraph do not fit in the framework ofthe present paper due to either lack of smoothness or lack of smooth invariant measure.It is interesting to extend the result of the paper to cover those systems as well as someslower mixing system and this is a promising direction for a future work.

Appendix B. Fluctuations of local dimension. Proof of Lemma 6.6.

We start with the proof of pbq. Denote µn “ µpBnpx, εqq. Since P pgq “ 0 we have

(B.1) lnµn “n´1ÿ

j“0

gpf jxq Òp1q.

Denote

rn “ suprą0tr | Bpx, rq Ă Bnpx, εqu, rn “ inf

rą0tr | Bpx, rq Ą Bnpx, εqu.

By bounded distortion property, there exists C0 ą 0 such that if dpfny, fnxq ă ε then

pC0 exp εαq´1ă|Dfnpyq|

|Dfnpxq|ă C0 exp εα,

Thus

e´řn´1j“0 ϕpf

jxqpC0 exp εαq´1 dpx, yq ď dpfnx, fnyq ď e´

řn´1j“0 ϕpf

jxqpC0 exp εαq dpx, yq.

Henceε0C

´10 e´

řn´1j“0 ϕpf

jxq´εαď rn ď rn ď ε0C

´10 e´

řn´1j“0 ϕpf

jxq`εα .

It follows that

(B.2) ln rn “n´1ÿ

j“0

ϕpf jxq Òp1q, ln rn “n´1ÿ

j“0

ϕpf jxq Òp1q.

Accordingly lnµn´d ln rn “ lnµn´d ln rn “řn´1j“0 ψpf

jxq Òp1q. By Law of Iterated

Logarithm [83],

lim supnÑ8

řn´1j“0 ψpf

jxq?

2n ln lnn“ σ, lim inf

nÑ8

řn´1j“0 ψpf

jxq?

2n ln lnn“ ´σ.


Since Bpx, rnq Ă Bnpx, εq Ă Bpx, rnq

lim supnÑ8

| lnµBpx, rnq| ´ d| ln rn|?

2n ln lnnď σ ď lim sup

nÑ8

| lnµBpx, rnq| ´ d| ln rn|?

2n ln lnn.

Using (B.2) again we conclude that there exists k such that for every sufficiently smallδ there exists npδq such that rn`k ď δ ď rn. Then

σ ď lim supδÑ0

| lnµBpx, rnq| ´ d| ln rn|a

2npδq ln lnnpδqď lim sup

δÑ0

| lnµBpx, δq| ´ d| ln δ|a

2npδq ln lnnpδq

ď lim supδÑ0

| lnµBpx, rnq| ´ d| ln rn|a

2npδq ln lnnpδqď σ.

It follows that all inequalities above are in fact equalities. In particular,

lim supδÑ0


2npδq ln lnnpδq“ σ.

On the other hand by (B.2) and the ergodic theorem we see that limnÑ8

| ln rn|

n“ λ.

Therefore limnÑ8

| ln rn|pln ln | ln rn|q

n ln lnn“ λ. Since rnC ď δ ď rn we have

limδÑ0

d

npδq ln lnnpδq

| ln δ|pln ln | ln δ|q“

1?λ.

Multiplying the last two displays we obtain

lim supδÑ0


2| ln δ|p| ln ln | ln δ|q“

σ?λ.

Likewise

lim infδÑ0


2| ln δ|p| ln ln | ln δ|q“ ´

σ?λ.

This proves part (b).To prove part (a) suppose that σ2 “ 0. Since we also have that

ş

ψdµ “ 0 [117,Proposition 4.12] shows that ψ is a coboundary, that is, there exists a Holder function

η such that ψpxq “ ηpxq´ηpfxq. Thusn´1ÿ

k“0

ψpfkxq “ ηpxq´ηpfnxq is uniformly bounded

with respect to both n and x. Recalling the definition of ψ we see that in this case

n´1ÿ

k“0

gpfkxq “

«

dn´1ÿ

k“0

´fupfkxq

ff

Òp1q.

Now (B.1) and (B.2) show that µ is conformal. l


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