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Lectures on Smooth Ergodic Theory Christian Bonatti Contents 1 Elementary Facts from Ergodic Theory 2 1.1 Invariant Measure ................................ 2 1.2 Natural Extension ................................ 3 1.3 Regular Point and Ergodicity .......................... 4 1.4 Absolute Continuity of Measures ........................ 6 2 Expanding Maps on S 1 6 2.1 General Statement ................................ 7 2.2 A.C.I.P. for Expanding Maps .......................... 8 3 Non-Uniformly Expanding Maps on S 1 10 3.1 Regularity of the Inverse Map ......................... 11 3.2 Distortion Control at Hyperbolic Times .................... 11 3.3 Existence of Hyperbolic Times ......................... 12 3.4 Iteration of Lebesgue Measure ......................... 13 3.5 Erogodicity of the A.C.I.P . ........................... 14 4 Kan’s Example of SRB Measure 15 4.1 Stable Manifold and SRB Measure ....................... 16 4.2 Denseness of the Basins ............................. 17 4.3 Full Measure of the Basins ........................... 18 5 SRB Measures for Partially Hyperbolic Systems Whose Central Direction Is Mostly Contracting 19 5.1 Partial Hyperbolicity .............................. 19 5.2 Invariant Foliations ............................... 20 5.3 Distortion Control in the Unstable Direction .................. 21 5.4 Measures Absolutely Continuous in the Unstable Direction .......... 21 5.5 Systems Whose Central-Stable Direction Is Mostly Contracting ........ 22 1

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Page 1: Lectures on Smooth Ergodic Theorybonatti.perso.math.cnrs.fr/Beijing-2006-Ergodic.pdf1 Elementary Facts from Ergodic Theory In this section, we will introduce some basic concepts and

Lectures on Smooth Ergodic Theory

Christian Bonatti

Contents1 Elementary Facts from Ergodic Theory 2

1.1 Invariant Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Natural Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Regular Point and Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Absolute Continuity of Measures . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Expanding Maps on S 1 62.1 General Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 A.C.I.P. for Expanding Maps . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Non-Uniformly Expanding Maps on S 1 103.1 Regularity of the Inverse Map . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Distortion Control at Hyperbolic Times . . . . . . . . . . . . . . . . . . . . 113.3 Existence of Hyperbolic Times . . . . . . . . . . . . . . . . . . . . . . . . . 123.4 Iteration of Lebesgue Measure . . . . . . . . . . . . . . . . . . . . . . . . . 133.5 Erogodicity of the A.C.I.P. . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Kan’s Example of SRB Measure 154.1 Stable Manifold and SRB Measure . . . . . . . . . . . . . . . . . . . . . . . 164.2 Denseness of the Basins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.3 Full Measure of the Basins . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5 SRB Measures for Partially Hyperbolic Systems Whose Central Direction IsMostly Contracting 195.1 Partial Hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.2 Invariant Foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.3 Distortion Control in the Unstable Direction . . . . . . . . . . . . . . . . . . 215.4 Measures Absolutely Continuous in the Unstable Direction . . . . . . . . . . 215.5 Systems Whose Central-Stable Direction Is Mostly Contracting . . . . . . . . 22

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1 Elementary Facts from Ergodic TheoryIn this section, we will introduce some basic concepts and results in ergodic theory that willbe frequently cited in latter sections. Most of the results are stated without proof, since all ofthem can be found in any standard textbook on ergodic theory.

1.1 Invariant MeasureLet X be a compact metric space, B(X) be the Borel σ-algebra of X, and C(X) be the spaceof continuous functions on X with the usual maximal-module norm. P(X) denotes the spaceof Borel probability measures on X with weak-∗ topology, i.e. :

limn→∞

µn = µ in P(X) ⇐⇒ limn→∞

∫Xϕ dµn =

∫Xϕ dµ , ∀ϕ ∈ C(X) .

Proposition 1.1. P(X) is a compact metric space.

Proof. Let {ϕn}n∈N be a dense subset of C(X), and define:

d(µ, ν) =∞∑

n=1

12n‖ϕn‖

∣∣∣∣∣∫X

fn dµ −∫

Xfn dν

∣∣∣∣∣ , ∀µ, ν ∈ P(X) .

It is easy to verify that d is a metric on P(X) comparable to its weak-∗ topology. �

For any given continuous map f : X → Y , where both X and Y are compact metric spaces,there is an induced map f∗ : P(X)→ P(Y) , µ 7→ f∗µ, defined as:

f∗µ(E) = µ( f −1(E)), ∀E ∈ B(Y) .

It is clear that f∗ is continuous by definition.

Definition 1.1 (Invariant Measure). In the case that f : X , we say µ ∈ P(X) is f -invariant,or f preserves µ, if f∗µ = µ.

We denote the set{µ ∈ P(X)

∣∣∣ f∗µ = µ}

by P(X, f ).

Remark. Generally speaking, to define measure-preserving map, we only need that (X,B) isa measurable space and f : X is measurable. But in this lecture we are only interested inthe above case.

Proposition 1.2. If X is a compact space and f : X is continuous, then P(X, f ) , Ø.

Proof. Take an arbitrary µ ∈ P(X), and define µn =1n

n−1∑i=0

f i∗µ. Then {µn}n∈N ⊂ P(X). By

compactness of P(X), there exists a subsequence {µni } converging to some ν ∈ P(X). We onlyneed to show that f∗ν = ν. Notice that∫

Xϕ dµn =

1n

n−1∑i=0

∫Xϕ ◦ f i dµ , ∀ϕ ∈ C(X) .

So we have: ∣∣∣∣∣∫Xϕ d f∗µn −

∫Xϕ dµn

∣∣∣∣∣ = 1n

∣∣∣∣∣∫X

(ϕ ◦ f n − ϕ) dµ∣∣∣∣∣ ≤ 2

n‖ϕ‖ .

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Let n→ ∞, we get:∫Xϕ d f∗ν = lim

n→∞

∫Xϕ d f∗µn = lim

n→∞

∫Xϕ dµn =

∫Xϕ dν .

Since this is true for any ϕ, ν is f -invariant. �

1.2 Natural ExtensionLet (X, d) be a compact metric space. So is the product space XZ =

{(xi)i∈Z

∣∣∣ xi ∈ X}, whose

metric, still denoted by d, can be given as:

d((xi), (yi)

)=

∑i∈Z

12|i|

d(xi, yi) .

The left shift σ : XZ , (xi) 7→ (xi+1) is a homeomorphism preserving the product measure.There is a natural projection π : XZ → X, (xi) 7→ x0, which is continuous, surjective andmeasure preserving.

Definition 1.2 (Natural Extension). When f : X is continuous and surjective, we candefine a subspace X f of XZ, called the natural extension of (X, f ), as:

X f ={

(xi) ∈ XZ∣∣∣ xi+1 = f (xi)

}.

The following proposition holds automatically by definition.

Proposition 1.3. X f is a compact metric space, σf : X f is a homeomorphism, and thefollowing diagram commutes:

X fσ f

−−−−−−→ X f

π

y π

yX

f−−−−−−→ X

Here σf is the restriction of σ to X f . �

Theorem 1.4. Given µ ∈ P(X, f ), there exists a unique µ f ∈ P(X f , σf ), such that π∗(µ f ) = µ .

Proof. Let An ={

(π−n)−1(B)∣∣∣ B ∈ B(X)

}, n ∈ N, where π−n : X f → X, (xi)i∈Z 7→ x−n is

the projection to the coordinate −n. Then {An} is a sequence of increasing sub-σ-algebras of

B(X f ), and their union A =∞⋃

n=0An is a set algebra, which generates B(X f ). To define µ f ,

we first notice that it is uniquely determined on A by the properties that (σf )∗(µ f ) = µ f , andπ∗(µ f ) = µ. This is because: for E ∈ A0, µ f (E) = µ(π(E)); for E ∈ An, µ f (E) = µ f (σf

−n(E)),where σf

−n(E) ∈ A0. Second, since B(X f ) is generated by A and clearly µ f is countablyadditive on A , there is a unique extension of µ f defined on B(X f ). So we complete theconstruction of µ f , which clearly satisfies the properties in the theorem. �

Remark. According to the construction of µ f in the proof above, it is easy to see that:

µ f (E) = limn→+∞

µ(π−n(E)), ∀E ∈ B(X f ).

Apparently the limit on the right hand always exists, since { µ(π−n(E)) }n∈N is a decreasingsequence.

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1.3 Regular Point and ErgodicityTheorem 1.5 (Birkhoff’s Ergodic Theorem). Suppose (X,B, µ) is a probability space, andf : X is measure-preserving. Then for any ϕ ∈ L1(µ), their exists some ϕ+ ∈ L1(µ), suchthat:

limn→∞

1n

n−1∑i=0

ϕ ◦ f i(x) = ϕ+(x) , µ-a.e. x ∈ X .

Clearly ϕ+ ◦ f = ϕ+, µ-a.e. . Moreover,∫

X ϕ dµ =∫

X ϕ+ dµ.

Corollary 1.6. If f is invertible, then ϕ− is similarly defined with f −1 instead of f in theabove theorem, and ϕ+ = ϕ−, µ-a.e. .

Proof. Suppose the conclusion is false.Without loss of generality, we can assume that E ={x ∈ X

∣∣∣ ϕ+(x) > ϕ−(x)}

has positive measure. Since E is an f -invariant set, we can applyBirkhoff’s ergodic theorem to both f |E and f −1|E . So we have:

∫E ϕ+ dµ =

∫E ϕ dµ =

∫E ϕ− dµ,

which contradicts to the definition of E. �

If X is a compact metric space and f is a homeomorphism of X, then we have the follow-ing definition.

Definition 1.3 (Regular Point). x ∈ X is called regular for f , if:

limn→∞

1n

n−1∑i=0

ϕ( f i(x)) = limn→∞

1n

n−1∑i=0

ϕ( f −i(x)) , ∀ϕ ∈ C(X) .

That is to say, there exists some µx ∈ P(X), such that:

limn→∞

1n

n−1∑i=0

δ f i(x) = limn→∞

1n

n−1∑i=0

δ f −i(x) = µx .

Moreover, if µx = µ, we say x is regular for µ.

Remark. If f is not invertible, we only reserve the forward-iteration part in the above defini-tion.

By defintion, the following is a direct corollary of Birkhoff’s ergodic theorem.

Corollary 1.7. ∀µ ∈ P(X, f ), µ({

x ∈ X∣∣∣ x is regular for f

})= 1. �

Definition 1.4 (Ergodic Measure). µ ∈ P(X, f ) is called ergodic, if ∀E ∈ B(X) with f −1(E) =E, µ(E) = 0 or 1.

We use the notation E(X, f ) to denote set of all the ergodic ones in P(X, f ).

Proposition 1.8. E(X, f ) , Ø.

Proposition 1.9. If µ ∈ P(X, f ), then the following statements are equivalent:

(1) µ is ergodic;

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(2) ∀E ∈ B(X) with µ(E) > 0, µ(∞⋃

n=0f −n(E)

)= 1;

(3) ∀E ∈ B(X) with f −1(E) ⊂ E, µ(E) = 0 or 1;

(4) ∀E ∈ B(X) with f −1(E) ⊃ E, µ(E) = 0 or 1;

(5) ∀ϕ ∈ L1(µ) with ϕ ◦ f = ϕ, µ-a.e., ϕ is constant µ-a.e.;

(6) µ({

x ∈ X∣∣∣ x is regular for µ

})= 1.

According to Birkhoff’s ergodic theorem and the proposition above, immediately we get:

Corollary 1.10. If µ ∈ E(X, f ), then ϕ+ =∫

X ϕ dµ, µ-a.e., ∀ϕ ∈ L1(µ). �

Corollary 1.11. In theorem 1.4, if µ ∈ E(X, f ), then µ f ∈ E(X f , σf ).

Proof. Let E ∈ B(X f ) with µ f (E) > 0 be such that σf−1(E) = E. We only need to show that

µ f (E) = 1. First, we notice that

σf−1(E) = E ⇒ π−n(E) = π0(E), ∀n ∈ N and f −1(π0(E)) ⊃ π0(E).

Then on the one hand, according to the remark following the proof of theorem 1.4, we haveµ f (E) = µ(π0(E)) > 0; on the other hand, µ is ergodic implies that µ(π0(E)) = 1. �

Exercise 1.1. We consider the following dynamical system on T 2 = S 1 × S 1:

f : T 2 (x, y) 7→ ( x + ϕ(x) mod Z , y + sin(2πx) mod Z ).

Here

ϕ : [0, 1]→ [0,+∞), ϕ(x) = exp(−

1x2(1 − x)2

), ∀x ∈ (0, 1) and ϕ(0) = ϕ(1) = 0.

Then we have:

• ∀(x, y) ∈ T 2, (x, y) is regular;

• x , 0⇒ µ(x,y) is the Lebesgue measure on {0} × S 1;

• all the ergodic measures are of the form δ(0,y), y ∈ S 1.

The proof is just some detailed calculation, so we skip it.

Theorem 1.12 (Ergodic Decomposition). For any µ ∈ P(X, f ) , there exists some Borel prob-ability measure θ on E(X, f ) , such that:∫

Xϕ dµ =

∫E(X, f )

(∫Xϕ dν

)dθ(ν), ∀ϕ ∈ C(X).

Sometimes we write µ =∫

E(X, f ) ν dθ(ν) for short.

Definition 1.5 (Basin of Measure). For µ ∈ P(X, f ), the set

B(µ) ={

x ∈ X∣∣∣ x is regular for µ

}is called basin of µ.

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Remark. If x ∈ B(µ) and limi→∞

d( f i(x), f i(y)) = 0, then y ∈ B(µ). Here∞ means both +∞ and

−∞, when f is invertible; only +∞, otherwise.

Exercise 1.2. Let f : M be a diffeomorphism on a compact manifold M, and X be anontrivial hyperbolic basic set, i.e. both of the stable and unstable manifolds of each periodicpoint are dense in X. Then the set

{x ∈ X

∣∣∣ x is not regular}

is residual in X.

Proof. Let γ1 and γ2 be two distinct periodic orbits in X. We can choose ϕ ∈ C(X), such thatϕ|γ1 = 0, and ϕ|γ2 = 1. Since both of the stable manifold of γ1 and of γ2 are dense in X, thesequence of sets

On =

x ∈ X∣∣∣ ∃n1 > n,

1n1

n1−1∑i=0

ϕ( f i(x)) <13

; ∃n2 > n,1n2

n2−1∑i=0

ϕ( f i(x)) >23

are all open and dense in X. So their intersection R =

∞⋂n=1

On is residual. By definition, it is

clear that ∀x ∈ R, x is not regular. �

There is a more general result as follows.

Theorem 1.13. For C1-generic diffeomorphism f : M ,{

x ∈ M∣∣∣ x is not regular

}is resid-

ual.

1.4 Absolute Continuity of MeasuresDefinition 1.6 (Absolute Continuity and Singularity). Let µ, ν be two measures on the samemeasurable space (X,B). We say ν is absolutely continuous with respect to µ, denoted byν << µ, if for any E measurable, µ(E) = 0⇒ ν(E) = 0. Moreover, µ, ν are called equivalent toeach other, if both µ << ν and ν << µ hold simultaneously. We say they are mutually singular,denoted by µ⊥ν, if there exists some E measurable, such that µ(E) = 0 and ν(Ec) = 0.

Theorem 1.14 (Radon-Nikodym). If µ, ν are two measures on (X,B) and ν << µ, then thereexists a positive measurable function ϕ on X, such that:

ν(E) =∫

Eϕ dµ , ∀E ∈ B.

Usually we writedνdµ= ϕ or ν = ϕµ for short.

Theorem 1.15 (Lebesgue Decomposition). If µ, ν ∈ P(X) , then there exist α ∈ [0, 1] andνa, νs ∈ P(X) , such that νa << µ , νs⊥µ , and µ = ανa + (1 − α)νs. This decomposition isuniquely determined by µ and ν.

2 Expanding Maps on S 1

In this and the next section , we only concentrate on the map f : S 1 without critical point,i.e. |D f | > 0 on S 1, where S 1 = R/Z is the unit cirlce. Without loss of generality, we canassume that D f > 0 for convenience. If f is of class C1 and D f > 1 on S 1, we say f is

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expanding. It is always helpful to introduce f : R , which is the natural lift of f , i.e. thefollowing diagram is commutative:

Rf

−−−−−−→ R

π

y π

yS 1 f−−−−−−→ S 1

Here π : R → S 1 x 7→ x mod Z is the natural universal covering. The trivial fact thatD f (x) = D f (π(x)) is frequently used.

Remark. Sometimes it is so subtle and even annoying to distinguish what we precisely discussabout is R or S 1, or the exact map is f or f , that we have to neglect it. We believe that thiswill cause no serious confusion.

2.1 General StatementTheorem 2.1 (Shub). Let f : S 1 be a covering of degree d, d > 1. Then there is acontinuous and increasing (i.e. orientation preserving) map h : S 1 of degree 1, such thatthe following diagram commutes:

S 1 f−−−−−−→ S 1

hy h

yS 1 x 7→d·x−−−−−−→ S 1

Furthermore, if f is expanding, then h is a homeomorphism.

Proof. d > 1⇒ f has a fixed point x0. Let h(x0) = 0. To construct h : S 1 , we identify thedomain with [x0, x0 + 1) and the range with [0, 1). Let

Xn = f −n(x0) ={

xi1i2···in

∣∣∣ 0 ≤ ik < d, k = 1, · · · , n},

where the permutation of {xi1i2···in } is such that f (xi1i2···in ) = xi2···in and the map

h|Xn : xi1i2···in 7→i1d+

i2d2 + · · · +

indn

is increasing. Then h is well defined on X =∞⋃

n=0Xn. Because h is increasing on X and h(X) is

dense in [0, 1), we can extend h continuously to [x0, x0 + 1) naturally, i.e.:

h(x) = infy∈X

x<y<x0+1

h(y) = supy∈X

x0<y<x

h(y), ∀x ∈ (x0, x0 + 1).

It is evident that the properties of h are satisfied automatically. If f is expanding, then X isdense in [x0, x0 + 1). Therefore h is injective, i.e. h is a homeomorphism. �

Definition 2.1 (Smale’s Solenoid). Let D2 ={

z ∈ C∣∣∣ |z| ≤ 1

}and f : S 1 is expanding.

We consider a map

F : S 1 × D2 (t, z) 7→ ( f (t), cz +e2πit

2),

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where c << inf{|t − s|

∣∣∣ f (t) = f (s) and t , s}, to ensure that F is a homeomorphism from

S 1 × D2 to its image. The F-invariant set Λ =∞⋂

n=1Fn(S 1 × D2) is called Smale’s Solenoid.

The following proposition is easy to prove.

Proposition 2.2. The dynamical system F : Λ topologically conjugates to the left shift ofthe natural extension of f : S 1 , denoted by σf : Π. The conjugation map is defined as:

h : Π→ Λ (xn)n∈Z 7→ the single point in∞⋂

n=1

Fn({x−n} × D2).

From this standpoint, it is easy to see that each connected component of Λ is homeomorphicto S 1.

So we can think of Smale’s solenoid as a geometrical realization of natural extension.

2.2 A.C.I.P. for Expanding MapsDefinition 2.2 (Absolutely Continuous Invariant Probability). Let f : M be a smoothdynamical system on some compact Riemannian manifold M. µ ∈ P(M, f ) is called anabsolutely continuous invariant probability, or a.c.i.p. for short, if µ << LebM .

Theorem 2.3. If f : S 1 is an expanding map of class C2, then there exists a unique a.c.i.p.ν ∈ P(S 1, f ) , such that it is equivalent to the Lebesgue measure on S 1 and ergodic .

Remark. If f is only of class C1+θ , 0 < θ ≤ 1, the above theorem still holds with no muchmodification of the proof. But the smoothness of f cannot be weaken to C1.

Theorem 2.4 (Bochi & Fayad). If f : S 1 is expanding, then C1-generically speaking,there exists no µ ∈ P(S 1, f ), such that µ << Leb, where Leb denotes the Lebesgue measureon S 1.

What we will do in the rest of this section is to prove theorem 2.3 .

Uniqueness is evident since any two distinct ergodic measures must be singular mutually.So we only need to prove existence, which will be divided into three steps.

(1) Distortion Control

f is expanding⇒ ∃λ > 1, such that D f > λ. f is C2 ⇒ log D f is C1 and ∃C > 0, suchthat

∣∣∣log D f∣∣∣ < C. Thus, ∀y0, z0 ∈ R, yi = f (y0), zi = f (z0), i = 0, 1, · · · , n, we have:

D f n(y0)D f n(z0)

=

n−1∏i=0

D f (yi)

n−1∏i=0

D f (zi)

⇒∣∣∣log D f n(y0) − log D f n(z0)

∣∣∣ ≤ n−1∑i=0

∣∣∣log D f (yi) − log D f (zi)∣∣∣

≤ C ·n−1∑i=0

d(yi, zi) ≤ C ·n∑

i=1

λ−i · d(yn, zn) ≤ C · d(yn, zn)

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Here C =C

λ − 1. So we get:

Lemma 2.5. log D f is a C-Lipschitz function. �

We will fix yn = y, zn = z in the following text, and let x0, y0 and n be variables.

(2) Iteration of Lebesgue Measure

Let µn = f n∗ (Leb), νn =

1n

n−1∑i=0

µi. Let ϕn =dµn

d Leb, ψn =

dνn

d Leb=

1n

n−1∑i=0

ϕi. We have the

following lemma.

Lemma 2.6. logϕn and logψn are C-Lipschitz functions.

Proof.

ϕn(y) =∑

f n(y′)=y

1D f n(y′)

.

Making use of the following elementary inequality,

n∑i=1

ai

n∑i=1

bi

≤ sup1≤i≤n

ai

bi, ∀ai, bi > 0, i = 1, 2, · · · , n

we have:

ϕn(y)ϕn(z)

=

∑f n(y′)=y

1D f n(y′)∑

f n(z′)=z

1D f n(z′)

≤ supy′,z′

D f n(z′)D f n(y′)

.

In the above and following expressions, y′ and z′, as subscipts under the sup symbol, arerequired to be in the same connected components of f −n ([y, z]). Then:∣∣∣logϕn(y) − logϕn(z)

∣∣∣ ≤ supy′,z′

∣∣∣logϕn(y′) − logϕn(z′)∣∣∣ ≤ Cd(y, z)

∣∣∣logψn(y) − logψn(z)∣∣∣ =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣log

1n

n−1∑i=0

ϕn(y)

1n

n−1∑i=0

ϕn(z)

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣≤

∣∣∣∣∣∣log(

sup0≤i<n

ϕi(y)ϕi(z)

)∣∣∣∣∣∣ ≤ Cd(y, z)

So logϕn and logψn are C-Lipschitz functions. �∫S 1 ψn dLeb =1 ⇒ ∃y ∈ S 1, ψn(y) =1 ⇒

∣∣∣logψn

∣∣∣ ≤ C, i.e. ψn(z) ∈ [e−C , eC], ∀z ∈S 1. Then by Arzela-Ascoli theorem, ∃ni ↗ +∞ such that ψni → ψ uniformly as i → ∞.Apparently logψ is also a C-Lipschitz function. So lim

n→∞νni = ν, where ν = ψLeb is equivalent

to Leb.

(3) Ergodicity of ν

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To prove ν is ergodic, only to show that:

ν

S 1 \

∞⋃n=0

f n(E)

= 0, ∀ν(E) > 0.

Since Leb is equivalent to ν, we can take Leb instead of ν. For any Leb(E) > 0, let x ∈ E bea Lebesgue density point of E. Then ∀ε > 0, ∃δ0 > 0, such that:

Leb([x − δ, x + δ] \ E)2δ

< ε, ∀0 < δ < δ0.

We can choose δ small enough and n accordingly, such that f n|[x−δ,x+δ] is injective, andLeb ( f n([x − δ, x + δ])) > 1 − ε. Then we have:

Leb ( f n([x − δ, x + δ])) =

∫[x−δ,x+δ]

D f n dLeb

Leb ( f n([x − δ, x + δ] \ E)) =

∫[x−δ,x+δ]\E

D f n dLeb

Since log D f n is C-Lipschitz, we have:

Leb ( f n([x − δ, x + δ] \ E))Leb ( f n([x − δ, x + δ]))

≤ e2C f n([x − δ, x + δ] \ E)2δ

< e2Cε.

Hence Leb( f n(E)) > (1 − ε)(1 − e2Cε). For ε is arbitrary, we complete the proof of theorem2.3 .

Remark. As a direct application of theorem 2.3, we consider the Smale’s solenoid. Accordingto corollary 1.11, the invariant measure ν f on the solenoid induced by ν is also ergodic.Moreover, we can prove that almost every conditional measure of ν on its correspondingconnected component, which is homeomorphic to S 1, is absolutely continuous.

3 Non-Uniformly Expanding Maps on S 1

In this section, we consider C1+θ map f : S 1 , D f > 0. For every x ∈ S 1, we can define the

so called lower Lyapunov exponent λ−(x) = lim infn→+∞

1n

log D f n(x).

Definition 3.1. f is called non-uniformly expanding, if ∃X ⊂ S 1 with Leb(X) > 0, such thatλ−(x) > 0,∀x ∈ X.

The main result in this section is the following theorem.

Theorem 3.1. If f satisfies the conditions above, then there is an unique ergodic a.c.i.p. onS 1.

The proof will be composed of the following four subsections.

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3.1 Regularity of the Inverse MapLet f : R be the natural lift of f mentioned before, then f is a C1+θ-diffeomorphism of R,and f (x + 1) = f (x) + d, where d is the topological degree of f . Let g = f −1, which is also aC1+θ-diffeomorphism of R.

Lemma 3.2. ∃C > 0, such that for every x, y ∈ R with |x − y| < 1, we have:∣∣∣log D f (x) − log D f (y)∣∣∣ ≤ C · d(x, y)θ∣∣∣log Dg(x) − log Dg(y)∣∣∣ ≤ C · d(x, y)θ

Proof.

f ∈ C1+θ and(log x

)′=

1x

⇒∣∣∣log D f (x) − log D f (y)

∣∣∣ ≤ supz∈R

1D f (z)

·∣∣∣D f (x) − D f (y)

∣∣∣ ≤ C1 · d(x, y)θ

g ∈ C1 and Dg(x) =1

D f (g(x))⇒

∣∣∣log Dg(x) − log Dg(y)∣∣∣ ≤ ∣∣∣log D f (g(x)) − log D f (g(y))

∣∣∣ ≤ C1 · d (g(x), g(y))θ ≤ C2 · d(x, y)θ

We can take C = max{C1,C2}. �

3.2 Distortion Control at Hyperbolic Times

Lemma 3.3. Given σ > 1, 0 < ε ≤14

logσ, ∃δ > 0, such that: if x ∈ R and t ∈ N satisfy∣∣∣Dgi(x)∣∣∣ ≤ σ−i, i = 1, 2, · · · , t, then:∣∣∣log Dgt(y) − log Dgt(z)

∣∣∣ < Cσd(y, z)θ < ε, ∀y, z ∈ [x − δ, x + δ], i = 1, 2, · · · , t .

Proof. Let σ =∞∑j=0σ−

j θ2 . Take δ ∈ (0,

12

), such that Cσ(2δ)θ < ε. When i = 0, the inequality

holds automatically. Assume for j = 1, 2, · · · , i − 1, we have proved that∣∣∣log Dg j(y) − log Dg j(z)∣∣∣ < ε, ∀y, z ∈ [x − δ, x + δ].

Then for j = 1, 2, · · · , i − 1:

Dg j(x) ≤ σ− j

⇒ Dg j(w) < exp(− j logσ + ε

)< σ−

j2 , ∀w ∈ [x − δ, x + δ]

⇒ d(g j(y), g j(z)

)≤ sup

w∈[y,z]Dg j(w) · d(y, z) < σ−

j2 d(y, z)

⇒∣∣∣log Dgi(y) − log Dgi(z)

∣∣∣ ≤ i−1∑j=0

∣∣∣∣log Dg(g j(y)

)− log Dg

(g j(z)

)∣∣∣∣≤ C · d

(g j(y), g j(z)

)θ< C ·

i−1∑j=0

σ−j θ2 · d(y, z)θ < Cσ(2δ)θ < ε .

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Definition 3.2. t ∈ N is called a σ-hyperbolic time for x ∈ S 1, if: D f t− j(

f j(x))≥ σt− j, j =

0, 1, · · · , t − 1.

Corollary 3.4. If t is a σ-hyperbolic time for x, and ϕt =d f t∗(d Leb)d Leb

, then:∣∣∣logϕt(y) − logϕt(z)∣∣∣ < Cσd(y, z)θ < ε, ∀y, z ∈ [ f t(x) − δ, f t(x) + δ].

Proof. Noticing that ϕn(y) =∑

f n(y′)=y

1D f n(y′)

and applying lemma 3.3, to get the conclusion

we just need to do the same as in lemma 2.6. �

3.3 Existence of Hyperbolic Times

Definition 3.3. Given A > 0, ai ≤ A, i ∈ N and c > 0, we say i is a c-top time, if:i∑

k= j+1ak ≥

c(i − j), j = 0, 1 · · · , i − 1.

Lemma 3.5 (Pliss). Given c < b < A and i0 ∈ N , suppose thati0∑

k=1ak ≥ b i0 . Then:

#{

1 ≤ i ≤ i0∣∣∣ i is a c-top time

}≥

b − cA − c

· i0.

Proof. Let ak = ak − c ≤ A − c,i0∑

k=1ak ≥ (b − c)i0. i is a c-top time ⇔

i∑k= j+1

ak ≥ 0, j =

0, 1 · · · , i − 1. Let s j =j∑

k=1ak, so i is a c-top time⇔ si ≥ s j, j = 1, 2, · · · , i − 1. Therefore if

j < i are two adjacent c-top times, then s j ≥ si−1 and hence s j + ai ≥ si. So we get:

#{

1 ≤ i ≤ i0∣∣∣ i is a c-top time

}× (A − c) ≥

∑1≤i≤i0i is a

c-top time

ai ≥

i0∑i=1

ai ≥ (b − c)i0.

Since Leb(X) > 0 and ∀x ∈ X, λ−(x) > 0, then ∃Y ⊂ X, N ∈ N and λ > 0, such that

β = Leb(Y) > 0 and ∀x ∈ Y , n > N,1n

log D f n(x) > λ. If we chooseσ so small that logσ < b,

take A = maxx∈S 1

log D f (x), b = λ, c = logσ, and denote α =b − cA − c

, then we get:

Corollary 3.6. ∀x ∈ Y, n > N, we have at least α · n σ-hyperbolic times in {1, 2, · · · , n}. �

The parameters σ, λ, etc will be fixed in the following text.

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3.4 Iteration of Lebesgue MeasureIn the following text we will denote measures on R by symbols with superscript ‘˜’ andthe corresponding ones on S 1 by the same symbols without ‘˜’. For all n ≥ 0, we denote

µn = f n∗

(Leb

∣∣∣[0,1)

), µn = π∗µn = f n

∗ (Leb) and νn =1n

n−1∑i=0

µi. From now on, we fix δ given

in lemma 3.3 such that1δ∈ N and fix ε =

14

logσ. We denote by ηi the restriction of µi

on the union of intervals of the form [kδ, (k + 1)δ], k ∈ Z such that ∃x ∈ [0, 1) ∩ π−1(Y),

f n(x) ∈ [kδ, (k + 1)δ] and i is a σ-hyperbolic time for x. Let ηi = π∗ηi and γi =1i

i−1∑j=0η j.

Notice that ηi, ηiand γi are generally not probability measures.It is clear that the support of ηiand γi are union of intervals of the form [kδ, (k + 1)δ], and

ηi, γi << Leb. Suppose ηi = φi Leb and γi = ψi Leb, then the following lemma holds:

Lemma 3.7. log φi and logψi are (Cσ, θ)-Holder continuous on each interval [kδ, (k + 1)δ]contained in the support of ηi and γi respectively. Furthermore, ∀x, y ∈ [kδ, (k + 1)δ] , wehave: ∣∣∣log φi(x) − log φi(y)

∣∣∣ < ε and∣∣∣logψi(x) − logψi(y)

∣∣∣ < εProof. Let ηi = φi Leb. Corollary 3.4 tells us that log φi is (Cσ, θ)-Holder continuous on eachinterval [kδ, (k + 1)δ] ⊂ supp ηi. Suppose [kδ, (k + 1)δ] ⊂ supp ηi.

π−1([kδ, (k + 1)δ]) =⋃n∈Z

[kδ + n, (k + 1)δ + n]⇒ φi

∣∣∣[kδ,(k+1)δ] =

∑j

φi

∣∣∣[kδ+n j,(k+1)δ+n j]

.

Here n j’s are such that [kδ + n j, (k + 1)δ + n j] ⊂ supp ηi. Following what we did in lemma2.6, we can get the conclusion. �

Lemma 3.8. ∀n ∈ N, γn(S 1) > (1 −1n

)αβ.

Proof. By definition of γi and ηi, we have:

γn(S 1) =1n

n−1∑i=0

ηi(S 1) ≥1n

n−1∑i=0

Leb({

x ∈ Y∣∣∣ i is a σ- hyperbolic time for x

})=

1n

∫Y

#{

0 ≤ i < n∣∣∣ i is a σ-hyperbolic time for x

}dLeb(x)

≥1n

∫Yα(n − 1) dLeb = (1 −

1n

)αβ .

The last inequality is due to corollary 3.6. �

By compactness, ∃ni ↗ +∞, such that limi→∞

νni = ν and limi→∞

γni = γ. It is clear that

f∗(ν) = ν, f∗(γ) = (γ) and γ << Leb, γ(S 1) = limi→∞

γni (S1) ≥ αβ . Moreover, γ ≤ ν, i.e. :∫

S 1ϕ dγ ≤

∫S 1ϕ dν, ∀ϕ : S 1 → [0,+∞).

Consider the Lebesgue decomposition: ν = νa + νs, where νa << Leb and νs ⊥ Leb.γ ≤ ν, γ << Leb ⇒ γ ≤ νa, hence νa > 0. On the other hand, ν = f∗ν = f∗νa + f∗νs. Notice

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that D f > 0 ⇒ f∗(Leb) << Leb. Therefore, f∗νa << f∗(Leb) << Leb. Next we show thatf∗νs ⊥ Leb. This is because νs ⊥ Leb ⇒ ∃E ⊂ S 1, such that Leb(E) = νs(Ec) = 0, andtherefore:

f∗(νa)(E) = 0⇒ f∗νs(E) = f∗ν(E) = ν(E) = νs(E) = νs(S 1) = f∗νs(S 1)⇒ f∗νs(Ec) = 0.

So by the uniqueness of Lebesgue decomposition, we conclude that f∗νs = νs. Finally we

obtain an a.c.i.p. ν0 =1

νa(S 1)νa,.

3.5 Erogodicity of the A.C.I.P.

Lemma 3.9. Leb(∞⋃

n=0f n(Y)

)= 1.

Proof. Let x ∈ Y be a Lebesgue density point of Y . Then

limε→0+

Leb ([x − ε, x + ε] \ Y)2ε

= 0.

Let ni ↗ +∞ be a sequence of σ-hyperbolic times for x, and denote f ni (x) by xi. For each i,there exists δi > 0, such that f ni ([x − δi, x + δi]) = [xi − δ, xi + δ]. ni’s are σ-hyperbolic times⇒ lim

i→∞δi = 0. We will follow what we did at the end of the proof of theorem 2.3 .

First, by lemma 3.3:D f ni (y)D f ni (z)

≤ e2ε ,∀y, z ∈ [x − δi, x + δi].

Second:Leb ([xi − δ, xi + δ] \ f ni (Y)) =

∫[x−δi,x+δi]\Y

D f ni dLeb

2δ = Leb ([xi − δ, xi + δ]) =∫

[x−δi,x+δi]D f ni dLeb

Then we get:

Leb ([xi − δ, xi + δ] \ f ni (Y))2δ

≤ e2ε Leb ([x − δi, x + δi] \ Y)2δi

→ 0, i→ ∞.

Let x∞ be some limit point of { xi }. Without loss of generality we can assume that limi→∞

xi = x∞.

Denote [x∞ − δ, x∞ + δ] \∞⋃

n=0f n(Y) by E. Then we have:

Leb(E) ≤ limi→∞

Leb ([xi − δ, xi + δ] \ f ni (Y)) = 0

By the definition of Y , λ−(y) > 0, a.e. y ∈ [x∞ − δ, x∞ + δ]. So in R, the length of the intervalf k([x∞ − δ, x∞ + δ]) → ∞ as k → ∞, which means that f k([x∞ − δ, x∞ + δ]) = S 1 for ksufficiently large. We choose such a k, finally:

D f > 0 & Leb(E) = 0⇒ Leb( f k(E)) = 0⇒ Leb(∞⋃

n=0f n(Y)

)= 1

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Corollary 3.10. Leb({

x ∈ S 1∣∣∣ λ−(x) > 0

})= 1

Proof. The definition of Y tells us that x ∈∞⋃

n=0f n(Y)⇒ λ−(x) > 0. �

Corollary 3.11. ∀X1 ⊂ S 1, Leb(X1) > 0⇒ Leb(∞⋃

n=0X1

)= 1.

Proof. From corollary 3.10 we know that λ−(x) > 0, a.e. x ∈ X1 . Then all the conclu-sions we have got about X still hold when it is replaced by X1. So lemma 3.9 tells us that

Leb(∞⋃

n=0f n(X1)

)= 1. �

Noticing that ν0 << Leb, we can conclude that ν0 is ergodic from the corollary above atonce. So we complete the proof of theorem 3.1 .

4 Kan’s Example of SRB MeasureIn this section we consider the dynamical system

f : S 1 × [0, 1] (r, s) 7→ (3r, fr(s)),

where f is of class C1+θ. If we regard π : S 1 × [0, 1] → S 1 as a bundle with fiber [0, 1],where π is the bundle projection, then f is a fiber-preserving map. Moreover, we need severaladditional assumptions below:

• 0 < D fr < 3 ∀r ∈ S 1;

• fr(0) = 0, fr(1) = 1 ∀r ∈ S 1;

• λ0 =∫

S 1 log D fr(0) dr < 0, λ1 =∫

S 1 log D fr(1) dr < 0;

• f0 and f 12

has no fixed point on (0, 1);

• f0 − id < 0, f 12− id > 0 on (0, 1);

• D f0(0) < 1, D f 12(1) < 1.

From these assumptions we know that (0, 0) and (12, 1) are hyperbolic fixed points of f . Let

us denote by µi the Lebesgue measure on S 1 × {i}, i = 0, 1. Notice that µi is ergodic and wedenote by B(µi) the basin of µi. Recall that B(µ0) ∩ B(µ1) = Ø.

The main result in this section is:

Theorem 4.1 (Kan). Under the given assumptions, we have:

Leb(S 1 × [0, 1] \ (B(µ0) ∪ B(µ1))

)= 0 and B(µi) = S 1 × [0, 1], i = 0, 1.

More precisely, we will actually prove that for every open set U ⊂ S 1 × [0, 1], one has:Leb (U ∩ B(µi)) > 0, i = 0, 1.

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Definition 4.1 (Sinai-Ruelle-Bowen Measure). Let f : M be a smooth dynamical systemon some compact Riemannian manifold M. µ ∈ P(M, f ) is called an Sinai-Ruelle-Bowenmeasure, or simply SRB measure, if Leb (B(µ)) > 0.

Remark. According to proposition 1.9 and definition, it is easy to see that an ergodic a.c.i.p.in P(M, f ) must be its unique SRB measure. So the a.c.i.p.’s we got in the sections before areexamples of SRB measure.

So µ0 and µ1 are the unique SRB measures for f , and they are “intermingled”.We will take all the three subsections to prove this theorem. Due to the similarity between

µ0 and µ1, we will only state and prove the results on ν0 and omit the corresponding ones onµ1 in subsections 4.1 and 4.2 .

4.1 Stable Manifold and SRB MeasureFor every x = (r, s) ∈ S 1 × [0, 1], we consider the so called stable manifold

W s(x) ={

y ∈ S 1 × [0, 1]∣∣∣ lim

n→+∞d( f n(y), f n(x)) = 0

}and local stable manifold

W s0(x) =

{y ∈ {r} × [0, 1]

∣∣∣ y ∈ W s(x)}.

W s0(x) is an subinterval of {r} × [0, 1], since f ({r} × [s1, s2]) ⊂ {3r} × [ fr(s1), fr(s2)]. We will

always denote by l the length of subinterval of a fiber. Because

limn→+∞

d( f n(y), f n(x)) = 0 ⇐⇒ π( f n(y)) = π( f n(x)), for some n ∈ N,

we have W s(x) =⋃

n∈Nf −n

(f n

(W s

0(x)))

. It is clear that for each x ∈ S 1 × {0} regular for µ0,

W s(x) ⊂ B(µ0).

Proposition 4.2. ∃δ > 0 and Xδ ⊂ S 1 × {0} with µ0(Xδ) > 0, such that l(W s

0(x))≥ δ.

Proof. Let us denote by Dc f n the nth derivative of f in the “central” direction, i.e. Dc f n =∂ f n

∂s. Then:

f n(r, s) = (3nr, f3n−1r ◦ · · · f3r ◦ fr(s))⇒ Dc f n(r, s) =n−1∏i=0

D f3ir(si).

Here s0 = s, si+1 = f3ir(si), i = 0, 1, · · · , n − 1.

Lemma 4.3. Given σ > 1, ∃δ > 0 such that: if x = (r, s) satisfies Dc f n(x) < σ−n, ∀n > 0,then {r} × [0, δ] ⊂ W s

0(x).

Proof. It is more or less a copy of lemma 3.3, with the same parameters ε, δ, σ,C etc. So weneed not to show all the details. First notice that log D fr is a (C, θ)-Holder continuous functionas we showed in lemma 3.2. Followed by the proof in lemma 3.3, we know that ∀y ∈ {r} ×

[0, δ],∣∣∣log Dc f n(y) − log Dc f n(x)

∣∣∣ < 14

logσ. Immediately we get that l ( f n({r} × [0, δ])) <

σ−n2 δ, which implies the conclusion. �

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For every x = (r, s) ∈ S 1 × [0, 1], we introduce the upper central Lyapunov exponent:

λc+(x) = lim sup

n→+∞

1n

log Dc f n(x).

Lemma 4.4. Suppose x ∈ S 1 × {0} and σ > 1 are such that λc+(x) < −logσ < 0. Then there

is i ∈ N such that Dc f n( f i(x)) < σ−n, ∀n ∈ N.

Proof. Since λc+(x) < −logσ, we know that i = sup

{j ∈ N

∣∣∣ Dc f j(x) ≥ σ− j}

is finite. Thelemma holds for this i , because:

Dc f i(x) · Dc f n(

f i(x))≤ σ−(n+i) = Dc f n+i(x)⇒ Dc f n( f i(x)) < σ−n , ∀n ∈ N.

According to Birkhoff’s ergodic theorem:

λc+(x) =

∫S 1

log D fr(0) dr = λ0 < 0, µ0-a.e. x ∈ S 1 × {0}.

Then fixing σ = e−λ02 , lemma 4.3 and 4.4 imply that for almost every x ∈ S 1 × {0}, there

is i ∈ N such that l(W s

0( f i(x)))≥ δ. That is to say, if we define

Xδ ={

x ∈ S 1 × {0}∣∣∣ l

(W s

0( f i(x)))≥ δ

},

then µ0

(∞⋃

i=0f −i(Xδ)

)= 1. The ergodicity of µ0 implies µ0(Xδ) > 0, so we complete the proof

of proposition 4.2. �

Since λc+(x) > 0 for µ0-a.e. x ∈ S 1 × {0}, in fact in the the proof of proposition 4.2 we

have proved that:

Corollary 4.5. l(W s

0(x))> 0, µ0-a.e. x ∈ S 1 × {0}. �

According to the above corollary, Fubini’s Theorem tells us that Leb (B(µ0)) > 0, i.e. µ0is a SRB measure.

4.2 Denseness of the BasinsTo obtain that Leb (U ∩ B(µ0)) > 0, for any open set U ⊂ S 1 × [0, 1], we only need to prove:

Proposition 4.6. LebS 1×{s} ([r − ε, r + ε] × {s} ∩ B(µ0)) > 0, ∀ε > 0, (r, s) ∈ S 1 × (0, 1) .

Proof. Due to corollary 4.5 and Dc f n > 0, we only need to prove the following lemma. �

Lemma 4.7. ∃n ∈ N, such that f n ([r − ε, r + ε] × {s}) meets each interval {t} × [0, δ].

Proof. We fix some n ∈ N such that 3nε > 1, then the length of the projection to S 1 of thecurve f n([r − ε, r + ε] × {s}) is large than 2. This implies that the curve intersect transversallywith W s

0((0, 0)) = {0} × [0, 1) . Since (0, 0) is a hyperbolic fixed point, the inclination lemmasays that there is a subsequence of { f n([r − ε, r + ε] × {s}) }n∈N converging to S 1 × {0} in thesense of C1-topology, which leads to the conclusion. �

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4.3 Full Measure of the BasinsWe remains to prove that Leb(Y) = 0, where Y = S 1 × [0, 1] \ (B(µ0) ∪ B(µ1)). In fact we canprove a stronger proposition which implies it:

Proposition 4.8. ∀s ∈ (0, 1), LebS 1×{s} Y = 0 .

First we need the following lemma and corollary. Notice that there is a condition D fr < 3unused till now.

Lemma 4.9. ∃τ > 0 such that: ∀x = (r, s) ∈ S 1 × [0, 1] , ∀v ∈ Tx

(S 1 × [0, 1]

), v =

vr ∂

∂r+ vs ∂

∂sand w = D f (v) = wr ∂

∂r+ ws ∂

∂s, we have |vs| ≤ τ|vr | ⇒ |ws| ≤ τ|wr | .

Proof. It only needs some calculation. First we have:

D f (v) = vrD f(∂

∂r

)+ vsD f

(∂

∂s

)= 3vr ∂

∂r+

(vr ∂ fr∂r+ vsDc f

)∂

∂s= wr ∂

∂r+ ws ∂

∂s.

Let α, β be universal positive constants such that 0 < Dc f < α and∣∣∣∣∣∂ fr∂r

∣∣∣∣∣ < β, and in particular

we can choose α < 3. Taking τ =β

3 − α> 0, then we have:

|vs| ≤ τ|vr | ⇒ |ws| = |vr ∂ fr∂r+ vsDc f | ≤ β|vr | + α|vs| ≤ (τα + β)|vr | = τ|wr | .

This lemma implies the following corollary immediately.

Corollary 4.10. If the graph of a curve γ : [0, 1] → S 1 × [0, 1] is τ-Lipschitz when it isregarded as a function from some segment in S 1 to [0, 1] , so is f (γ). �

Proof of proposition 4.8. We use reduction to absurdity and suppose that ∃s ∈ (0, 1), suchthat Leb

(S 1 × {s} ∩ Y

)> 0. Then ∃r ∈ S 1, such that (r, s) is a Lebesgue density point of

S 1 × {s} ∩ Y . We denote γn = [r −13n , r +

13n ]× {s} endowed with probability ηn =

3n

2·Lebγn .

We will consider its nth iteration Γn = f n(γn), the corresponding probability νn = f n∗ (ηn), and

the length of whose projection curve to S 1 is 2.Firstly, noticing that f −n (Y) = Y , then the Lebesgue density point theorem implies that

limn→∞

ηn (γn ∩ Y) = 1. As we did in the last section, the uniform distortion control for all

f n∣∣∣γn

tells us that, roughly speaking, νn on Γn are uniformly “equivalent” to ηn, and thereforelimn→∞

νn (Γn ∩ Y) = 1. Here the details are omitted to save trouble.Secondly, by corollary 4.10, Γn is the graph of some τ-Lipschitz function ϕn. So by

Aezela-Ascoli theorem, there is an subsequence of {ϕn}n∈N converging to a τ-Lipschitz func-tion ϕ uniformly, which is the graph of some curve Γ with projective length 2. If Γ ⊂ S 1×{0},then ∃n large enough, such that Γn ⊂ S 1 × [0, δ], where δ is given in proposition 4.2. Sodue to proposition 4.2, since ϕn is τ-Lipschitz, we can get that νn (Γn ∩ B(µ0)) > µ0 (Xδ),which makes a contradiction. If Γ ⊂ S 1 × {1}, the argument is all the same. Otherwise,Γ * S 1 × {0, 1}. In this case Γ intersects transversally with W s

0((0, 0)), so as in lemma 4.7,the inclination lemma tells us that there exist a subsegment Γ of Γ and m ∈ N, such that

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f m(Γ) ⊂ S 1 × [0,δ

2], and the length of whose projective curve is 1. Instead of Γ, an argu-

ment on f m(Γ) similar to the above one leads to a contradiction again, so we complete theproof. �

Up to now we have finished the whole proof of theorem 4.1 .

Exercise 4.1. In theorem 4.1, to obtain the same conclusion, the smoothness of f can bereduced to C1.

Clue to the proof. Review the whole proof of theorem 4.1, we only need to prove that lemma4.3 still holds when f is only C1. This can be achieved if we make use of absolute continuityof log Dc f instead of Holder continuity. �

Exercise 4.2. f : S 1 is of class C1+θ or even C1 with D f > 0, and µ is an ergodic invariantmeasure for f such that

∫S 1 log D f dµ < 0. Then µ is a Dirac measure supported on some

sink, i.e. an attracting periodic orbit.

5 SRB Measures for Partially Hyperbolic Systems WhoseCentral Direction Is Mostly Contracting

5.1 Partial HyperbolicityLet (M, g) be a compact Riemannian manifold and f : M is a diffeomorphism. ‖ · ‖ denotesthe norm on M induced by g. A splitting of the tangent bundle, denoted by TM = E ⊕ F, iscalled D f -invariant, or invariant shortly, if D f (Ex) = E f (x) and D f (Fx) = F f (x), ∀x ∈ M.

Let us denote by D f∣∣∣Ex

the restriction of D f to Ex for each x ∈ M, and it is similar toF.

Definition 5.1. An compact invariant subset K of M is called partially hyperbolic, if thetangent bundle restricted to K has a D f -invariant splitting TK M = E ⊕ F, such that ∃C > 1and 0 < λ < 1, for each x ∈ K and n ∈ N, we have:

∥∥∥∥D f n∣∣∣Ex

∥∥∥∥ · ∥∥∥∥D f −n∣∣∣F f n (x)

∥∥∥∥ ≤ Cλn, i.e. E is dominated by F;

• either∥∥∥∥D f n

∣∣∣Ex

∥∥∥∥ ≤ Cλn, when E is uniformly contracting,

or∥∥∥∥D f −n

∣∣∣Fx

∥∥∥∥ ≤ Cλn, when F is uniformly expanding.

In the first case we denote E = E s and F = Ecu; in the second one E = Ecs and F = Eu. Heres = stable, c = center, and u = unstable.

Remark. In the above definition, the splitting of TM is usually not unique. For instance, whenTK M = E s ⊕ Ec ⊕ Eu.

Proposition 5.1. In the above definition we can always choose another norm | · | on M equiv-alent to ‖ · ‖ , such that C = 1. Such a norm is called adapted to the partially hyperbolicstructure.

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Proof. Taking any τ ∈ (λ, 1) and N ∈ N such that N ≥log τ − log λ

log C, we define:

|v| =N−1∑i=0

τ−i∥∥∥D f i(v)

∥∥∥ , v ∈ E and |v| =N−1∑i=0

τi∥∥∥D f −i(v)

∥∥∥ , v ∈ F .

It is easy to verify that |D f (v)| ≤ τ|v|, v ∈ E and |D f −1(v)| ≤ τ|v|, v ∈ F. So we can define:

|v|2 = |vE |2 + |vF |

2, ∀v = vE + vF ∈ E ⊕ F .

Clearly this new norm is equivalent to the original one. �

According to the proposition above, we will always assume the norm ‖ · ‖ is adapted.

Proposition 5.2. The partially hyperbolic structure on K is robust under perturbation, i.e.there exists an open neighborhood U of K, such that for any C1-diffeomorphism g : U →g(U) ⊂ M sufficiently close to f in C1-topology, if K ⊂ U is an compact invariant set for g,then K is also partially hyperbolic.

Clue to proof. Fixing α > 0, for each x ∈ K, we introduce a cone at x :

CFα (x) =

{v = vE + vF ∈ E ⊕ F

∣∣∣ ‖vF ‖ ≥ α‖vE ‖}.

Clearly it is forward invariant, i.e. D f(CFα (x)

)⊂ CF

α ( f (x)). It is easy to see that:

Eu(x) =∞⋂

n=0

D f nf −n(x)

(CFα ( f −n(x))

).

We can prove that if g is a small C1-perturbation of f , then the right hand of the above equalityalso represents the unstable subbundle for g, when f is replaced by g in the expression. Theanalogous discussion on Ecs is just the same. �

5.2 Invariant FoliationsFrom now on we assume that f is of class C2 and there is a partially hyperbolic splittingTK M = Ecs ⊕ Eu over K. Moreover, K is an topological attractor, i.e. there is an open

neighborhood U of K such that f (U) ⊂ U and K =∞⋂

n=0f n(U). Then the following theorem

holds:

Theorem 5.3 (Brin & Pesin). K admits a foliation structure whose leaves are C2-embeddedsubmanifolds tangent to Eu, and the curvatures of the leaves are uniformly bounded.

Remark. Locally K is of the form C × D u, where C ⊂ Dcs is compact, Dcs is a disk tangentto Ecs and Du is similar.

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5.3 Distortion Control in the Unstable DirectionWe introduce a new function Ju f n, which is defined as Ju f n(x) =

∣∣∣∣det D f n∣∣∣Eu(x)

∣∣∣∣ .Lemma 5.4. There is τ > 0 such that for each n ∈ N, log Ju f −n is τ-Lipschitz as a functionon any certain unstable leaf L of the foliation of K.

Proof. Let us denote by du the distance on each leaf. Because Eu is uniformly expanding,du( f −n(x), f −n(y)) < λndu(x, y), ∀x, y ∈ L close enough. Compared with lemma 2.5, Ju f −n

here is the analogue of f there. So we can follow the proof before. �

Corollary 5.5. As functions on L × L, we have:

limn→∞

(log Ju f −n(x) − log Ju f −n(y)

)= logψ(x, y), ∀x, y ∈ L .

Moreover, logψ is also τ-Lipschitz . �

5.4 Measures Absolutely Continuous in the Unstable DirectionLet µ be a probability on M with supp µ ⊂ K, and C×D u is a local chart of K. Then there is apositive measure η on C, which is the projection of µ

∣∣∣C×D u to C, and a measure decomposition:

µ∣∣∣C×D u =

∫Cµξ dη(ξ) .

Here ξ denotes the local unstable leaf and µξ is the corresponding conditional measure withrespect to µ.

Definition 5.2. Under the above conditions, µ is called absolutely continuous in the unstabledirection, if for η-a.e. ξ, µξ << Lebξ .

Definition 5.3 (u-Gibbs State). Let µ be absolutely continuous in the unstable direction andC × D u be a local chart, we say µ is a u-Gibbs state, if η-a.e. ξ ∈ C, µξ = ϕξ Lebξ andϕξ(x)ϕξ(y)

= ψ(x, y), ∀x, y ∈ ξ.

Lemma 5.6. Every invariant probability absolutely continuous in the unstable direction is au-Gibbs state.

Lemma 5.7. For any D ⊂ U being a C2-embedded disk transversal to Ecs and of dimension

dimEu, let µ0 be the normalized Lebesgue measure on D, and µn = f∗µ0 , νn =1n

n−1∑i=0

µi . Then

every limit point ν of {νn}n∈N is an invariant probability absolutely continuous in the unstabledirection, and hence a u-Gibbs state.

The main difficulty here is the following lemma.

Lemma 5.8. The curvatures of the disks f n(D) are uniformly bounded.

Lemma 5.9. Let ν be a u-Gibbs state. Then in the sense of ergodic decomposition theorem,almost all its ergodic components are u-Gibbs states.

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Proof. Let Rν ={

x ∈ K∣∣∣ x is regular for ν

}. Then ν(Rν) = 1, and LebLx (Lx \ Rν) = 0, for

ν-a.e. x ∈ K, since ν is a u-Gibbs state. Here Lx denotes the unstable leaf containing x, andsuch an Lx is regular, i.e. :

limn→∞

1n

n−1∑i=0

δ f i(y) = limn→∞

1n

n−1∑i=0

δ f i(x) , ∀y ∈ Rν ∩ Lx.

Since the above limit is an ergodic measure, the support of almost every ergodic componentof ν is a union of entire leaves, which implies the conclusion. �

In fact we have the following much stronger result.

Theorem 5.10 (Bonatti & Viana). For Leb-a.e. x ∈ U, any limit point of{

1n

n−1∑i=0

δ f i(x)

}n∈N

is

a u-Gibbs state.

5.5 Systems Whose Central-Stable Direction Is Mostly ContractingBy mostly contracting, we mean that there is a nonuniform contraction in the central-stabledirection.

Theorem 5.11 (Bonatti & Viana). Assume that for every unstable leaf L there is a subset Xwith LebL(X) > 0, such that ∀x ∈ X, its upper central-stable Lyapunov exponent

λcs+ (x) = lim sup

n→+∞

1n

log∥∥∥∥D f n

∣∣∣Ecs(x)

∥∥∥∥ < 0 .

Then we have:

(1) there are only finitely many ergodic u-Gibbs states µ1, · · · , µn;

(2) each µi is an SRB measure;

(3) Leb(K \

n⋃i=1B(µi)

)= 0.

Clue to proof. ∀x ∈ L with λcs+ (x) < 0, there is a C2-embedded disk Wcs

0 (x), the so called“Pesin stable manifold”, tangent to Ecs and y ∈ Wcs

0 (x) ⇒ limn→+∞

d( f n(x), f n(y)) = 0. As aresult, for each unstable leaf L, there are δ > 0 and X ⊂ L with LebL(X) > 0, such that ∀x ∈ X,the radius of Wcs

0 (x) is large than δ. According to a theorem of Pugh & Shub, when f is C2,the foliation of local central-stable manifolds is absolutely continuous.

Therefore, if µ is an ergodic u-Gibbs state, then we can take a regular leaf L for µ and acorresponding X ⊂ L as above. So we have:

B(µ) ⊃⋃

x∈X∩B(µ)

Wcs0 (x),

and the latter is of positive Lebesgue measure, i.e. µ is an SRB measure.If there are infinitely many ergodic u-Gibbs states, then we can pick a convergent sequence

{µn}n∈N of them, such that limn→∞

µn = µ. Then µ is also a u-Gibbs state, and so is one of itsergodic components ν. Let L ⊂ supp ν ⊂ supp µ be an unstable leaf. Then when n is large

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enough, L ⊂ supp µn. It contradicts to the assumption that µn’s are distinct, since they are allergodic.

To prove Leb(K \

n⋃i=1B(µi)

)= 0, we can follow what we did in subsection 4.3, here we

do not show the details again. �

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