Dynamics of Some Fermi Acceleration Models

Zoom Dynamical System Seminar

Dynamics of Some Fermi Acceleration Models

Jing ZhouPenn State University

Tuesday 22nd September, 2020

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Content

A Brief BackgroundThe original Fermi-Ulam modelElimination by KAM theoryAttempts of locating acceleration

A Rectangular Billiard with Moving SlitsA Rectangular Billiard with Moving SlitsTrapping regions and exponential accelerationThe Growth Lemma

Bouncing Ball in Gravity FieldBouncing ball in gravity fieldThe limit mapThe ergodic properties of the limit mapThe statistical properties of the limit map

Jing Zhou | Dynamics of Some Fermi Acceleration Models

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The Original Fermi-Ulam Model

Motivation: explain for the existence of high energy particles incosmic rays

§ Fermi (1949)– charged particles travel in moving magnetic fields

§ Ulam (1961)– a particle bounces elastically between two infinitely heavy walls

lptqlpt ` 1q “ lptqv “ v ´ 29l

Figure: The Fermi-Ulam Model


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3


t

lptq

1 2

Figure: Ulam’s Conjecture

Ulam’s ConjectureThere exist escaping orbits, i.e. E “ tvn Ñ8u.

Bounded orbits B “ tDM s.t. |vn| ă MuOscillatory orbits O “ tlim sup vn “ 8, lim inf vn ă 8u


3


t

lptq

1 2





3


t

lptq

1 2





4

Elimination by KAM Theory

Theorem (Pustyl’nikov 1983; Laederich, Levi 1991)If lptq P C5, then E “ H.

t

v

1

Figure: Invariant Curves by KAM Theory

- Zharnitsky (2000): KAM stability in quasi-periodic motions withDiophantine frequencies- Kunze, Ortega (2018): mpEq “ 0 with rationally independentfrequencies


4

Elimination by KAM Theory

Theorem (Pustyl’nikov 1983; Laederich, Levi 1991)If lptq P C5, then E “ H.

t

v

1

Figure: Invariant Curves by KAM Theory

- Zharnitsky (2000): KAM stability in quasi-periodic motions withDiophantine frequencies- Kunze, Ortega (2018): mpEq “ 0 with rationally independentfrequencies


5

Attempts of Locating Acceleration

§ Nonsmooth wall motions§ Introduce background potential§ Increase dimension


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Attempts of Locating Acceleration

§ Nonsmooth wall motions§ Introduce background potential <- bouncing ball in gravity§ Increase dimension <- rectangular billiard with slits


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Attempts of Locating AccelerationBreak smoothness

t

lptq

1 2

§ Zharnitsky (1998)– linearly escaping orbit in a

piecewise linear model

t

lptq

1 2

§ Dolgopyat, de Simoi (2012)– Elliptic: mpEq “ 8,

mpBq “ 8– Hyperbolic: mpEq “ 0,

HDpEq “ 2


7

Attempts of Locating AccelerationBreak smoothness

t

lptq

1 2

§ Zharnitsky (1998)– linearly escaping orbit in a

piecewise linear model

t

lptq

1 2

§ Dolgopyat, de Simoi (2012)– Elliptic: mpEq “ 8,

mpBq “ 8– Hyperbolic: mpEq “ 0,

HDpEq “ 2


8

Attempts of Locating AccelerationIntroduce background potential

A single periodically oscillating wall + potential Upxq “ xα

§ Pustyl’nikov 1977 (bouncing ball in gravity)– α “ 1, D open set of (periodic analytic) lptq s.t. mpEq “ 8

§ Z. 2019 (bouncing ball in gravity)– α “ 1, lptq nonsmooth + hyperbolicity, then mpEq “ 0, but E and B

coexist at arbitrarily high energy levels§ Dolgopyat 2008

– α ą 1, α ‰ 2, lptq smooth, then E “ H– α ă 1{3, lptq sinusoid, then mpEq “ 0

§ de Simoi 2009– α ă 1, lptq sinusoid, then HDpEq “ 2

§ Ortega 2002– α “ 2 + resonance, conditions for existence of E


8





coexist at arbitrarily high energy levels

§ Dolgopyat 2008– α ą 1, α ‰ 2, lptq smooth, then E “ H– α ă 1{3, lptq sinusoid, then mpEq “ 0




8










8










8










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Attempts of Locating AccelerationIncrease dimension

§ Gelfreich, Turaev 2008– D exponentially escaping orbits for billiards with breathing boundary

§ Akinshin, Loskutov, Ryabov 2000– Lorentz gas with perturbed scatterer boundaries– average velocity grows linearly for stochastic perturbation and

quadratically for periodic perturbation


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§ Gelfreich, Turaev 2008– D exponentially escaping orbits for billiards with breathing boundary

§ Akinshin, Loskutov, Ryabov 2000– Lorentz gas with perturbed scatterer boundaries– average velocity grows linearly for stochastic perturbation and

quadratically for periodic perturbation


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§ Rom-Kedar, Turaev, Shah 2010– Ideal approximation: Ppup/downq 9 |openings|– E

`

log vv

˘

matches the numerics in non-resonant cases– exp rate is significantly higher in resonant cases

Figure: Ideal Probabilistic Approximation

-> Rectangular billiard with moving slits


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§ Rom-Kedar, Turaev, Shah 2010– Ideal approximation: Ppup/downq 9 |openings|– E

`

log vv

˘

matches the numerics in non-resonant cases– exp rate is significantly higher in resonant cases

Figure: Ideal Probabilistic Approximation

-> Rectangular billiard with moving slits


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A Rectangular Billiard with Moving Slits

hL,hR P C2 have period 2, vH “ 1 ùñ 1:1 resonance

Parameters: λ (length of left platform), x0 (starting horizontal position)

0

1

λ 1

hLptqhRptq

Figure: A Square Billiard with Moving Platforms

t = time of collision at the platforms, v = vertical velocity right after

Collision dynamics G : pt , vq Ñ pt , vq


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A Rectangular Billiard with Moving SlitsTrapping regions

0

1

λ 1

t˚1 t˚2

Figure: The Trapping Lower Chamber:t˚1 “ λ´ x0, t˚2 “ 2´ λ´ x0

-> Exponential acceleration!


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0

1

λ 1

t˚1 t˚2


-> Exponential acceleration!


12


0

1

λ 1

t˚1 t˚2


-> Exponential acceleration!Jing Zhou | Dynamics of Some Fermi Acceleration Models

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A Rectangular Billiard with Moving SlitsTrapping regions and exponential acceleration

hL

hRhL

hR

t˚1 t˚2

Theorem (Exponential acceleration)Assume that either

p1q hLpt˚1 q ă hRpt˚1 q and hLpt˚2 q ą hRpt˚2 q

or p2q hLpt˚1 q ą hRpt˚1 q and hLpt˚2 q ă hRpt˚2 q.

Then there exists V˚ " 1 s.t. almost every orbit with |v0| ą V˚

eventually gains energy exponentially in time.


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Theorem (Waiting time estimate)Assume λ and x0 are such that the relative positions of two slitschange at two critical jumps and that |Tr | ą 2. Then there areK , ζ ą 0 such that for any ε ą 0, there exists V0 “ V0pεq and T “ T pεqsuch that for each V ě V0 the complement of set

"

pt0, v0q : |v0| P rV ,V ` 1s : @t ě T |vptq| ě|vpT q|

Keζt

*

has measure less than ε.

T “ T pεq “ 2pkl ` 1q s.t.

k ąlogp0.25ε{Lq

log Dand

kl ` 1

Λl{2u

ă0.25ε

C˚ ` L2

where L,D,C˚ are constants and Λu ą 1 is the rate of expansion inunstable direction.


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Bouncing Ball in Gravity Field

f pt ` 1q “ f ptq, f P C3p0,1q:f ptq ą 0 for all t or :f ă ´g for all t

g

Figure: Bouncing Pingpong in Gravity Field

t “ time collision, v “ velocity immediately after collisionCollision map F : pt , vq Ñ pt , vq


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Bouncing Ball in Gravity FieldThe limit map

For large velocities,

F :

#

t1 “ t0 ` 2v0g Ò

´

1v0

¯

v1 “ v0 ` 29f pt1q

Ignoring the error, we obtain F8 on the cylinder R{Zˆ R`

F8 :

#

t1 “ t0 ` 2v0g

v1 “ v0 ` 29f pt1q

Projecting on the torus R{Zˆ R{gZ,

F8 :

#

t1 “ t0 ` 2v0g

v1 “ v0 ` 29f pt1q

where t “ t (mod 1), v “ v (mod gq


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F :

#

t1 “ t0 ` 2v0g Ò

´

1v0

¯

v1 “ v0 ` 29f pt1q


F8 :

#

t1 “ t0 ` 2v0g

v1 “ v0 ` 29f pt1q


F8 :

#

t1 “ t0 ` 2v0g

v1 “ v0 ` 29f pt1q



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F :

#

t1 “ t0 ` 2v0g Ò

´

1v0

¯

v1 “ v0 ` 29f pt1q


F8 :

#

t1 “ t0 ` 2v0g

v1 “ v0 ` 29f pt1q


F8 :

#

t1 “ t0 ` 2v0g

v1 “ v0 ` 29f pt1q



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Bouncing Ball in Gravity FieldThe ergodic properties of the limit map

Theorem (Ergodicity)

F8 is ergodic.

Theorem (Recurrence)The set E of escaping orbits of F has zero measure. Consequently, Fis recurrent, i.e. almost every orbit comes arbitrarily close to its initialpoint.


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Bouncing Ball in Gravity FieldThe statistical properties of the limit map

Theorem (Exp decay of correlations)F8 enjoys exponential decay of correlations for dynamically Höldercontinuous observables: D b ą 0 such that for any pair of dynamicallyHölder continuous observables ϕ, φ, D Cϕ,φ such that

ˇ

ˇ

ˇ

ˇ

ż

Tpϕ ˝ F n

8qφd µ´ż

Tϕd µ

ż

Tφd µ

ˇ

ˇ

ˇ

ˇ

ď Cϕ,φe´bn, n P N.


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Theorem (CLT)F8 satisfies central limit theorem for dynamically Hölder observables,i.e.

1?

n

n´1ÿ

i“0

ϕ ˝ F i8

distá N p0, σ2

ϕq

where

σ2ϕ :“

8ÿ

n“´8

ż

Tϕ ¨ pϕ ˝ F n

8qd µ ă 8.

Theorem (Bounded and escaping orbits)F possesses escaping and bounded orbits with arbitrarily highenergy.


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A function Φ is global if it is bounded, uniformly continuous and has afinite average Φ in the following sense: for any ε there exists N solarge that for any rectangle V “ r0,1q ˆ ra,bs with b ´ a ą N we have

ˇ

ˇ

ˇ

ˇ

1µpV q

ż

VΦdµ´ Φ

ˇ

ˇ

ˇ

ˇ

ď ε.

We denote by GU the space of all such global functions.

Theorem (Global global mixing)F is global global mixing with respect to GU , i.e. for any Φ1,Φ2 P GU ,the following holds

limnÑ8

lim supµpVqÑ8

1µpV q

ż

VΦ1 ¨ pΦ2 ˝ F nqdµ “

limnÑ8

lim infµpVqÑ8

1µpV q

ż

VΦ1 ¨ pΦ2 ˝ F nqdµ “ Φ1Φ2.


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When you are not about to jump...

Lemma (Adiabatic Coordinate)For pt , vq R Ri Y Ri pi “ 1,2q and v ! ´1, there exists an adiabaticcoordinate pζ, Jq “ ΨLpt , vq P R{2Zˆ R` such that

ζn`1 “ ζn `2JnÒ

ˆ

1J4

n

˙

, Jn`1 “ Jn Òˆ

1J3

n

˙

.

In fact, ζ “ ζptq “2

M˚

ż t

0

dsmpsq2

mod 2,

J “ Jpt , vq “M˚

2

ˆ

mv `m 9m `m2 :m3v

˙

,

where m “ ´h and M˚ “şt0

dsmpsq2 .


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Now when you jump...

Lemma (Normal form)Suppose the lower chamber is trapping. Then the Poincaré mapP12

LL : R´1 Ñ R´2 is given by

pρ2, J2q “ G12LLpρ1,J1q ` H12

LL pρ1,J1q ÒpJ ´21 q

where

G12LLpρ1,J1q “

ˆ

´m´

2

m`2tM˚J1pζ

˚2 ´ ζ

˚1 q ´ ρ1u2 ` 1`

m´2

m`2,

m`2

m´2J1 `Υ2pρ2 ´ 1q

˙

andH12

LL pρ1,J1q “`

0,Υ12pρ2 ´ 1q2{J1 `Υ22{J1˘

.


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t˚1 t˚2

hLpt˚1 qhRpt˚1 q

hLpt˚2 q

hRpt˚2 q

Figure: The Trapping Lower Chamber and Exponential Acceleration

Jt˚1ÝÑ

hRpt˚1 qhLpt˚1 q

J Òp1qt˚2ÝÑ

hLpt˚2 qhRpt˚2 q

hRpt˚1 qhLpt˚1 q

J Òp1q


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A Rectangular Billiard with Moving SlitsHyperbolicity and waiting time

τ

I

Op 1I2 q

Figure: Hyperbolicity and Waiting Time in Upper Chamber


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A Rectangular Billiard with Moving SlitsThe Growth Lemma

Suppose γ is an unstable line in a box. For x P γ,

rnpxq “ distpxn, Bγnq

where γn the nearest boundary of the component of Gnγ containingxn.

Lemma (Growth Lemma)There exists a constant C˚ s.t. for any small ε ą 0 and any n P N

mesγtx P γ : rnpxq ă εu ď C˚ε


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Complexity controlknpδq “ max number of the pieces that an unstable line of length lessthan δ can be cut into after n iterates.Define kn “ limδÑ0 knpδq.

Figure: Complexity control

Then kn ď 8n and thus Dδ0 so small that knpδq ď 16n for any δ ă δ0.Choose n0 so that 32n0

Λn0uă 1, where Λu is the expansion rate of G. We

can assume n0 “ 1 by replacing G with Gn0 .


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Proof of Growth Lemma.Cut a long unstable line into pieces shorter than δ0.rn`1pxq is less than ε if xn`1 either passes a real or artificialsingularity. The former is controlled by 2k1pδ0qmesγtrn ă

εΛuu while

the latter by 2εδ0

. Therefore

mesγtrn`1 ă εu ď 32mesγ

"

rn ăε

Λu

*

`2εδ0.

Thus by induction

mesγtrn ă εu ď

ˆ

32Λu

˙n

mesw tr0 ă εu `2εδ0

ˆ

1`32Λu` ¨ ¨ ¨ `

ˆ

32Λu

˙n˙


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A Rectangular Billiard with Moving SlitsProof of waiting time estimate

Good: γ has more than two pieces that remains in the upper chamberin the next period. Otherwise, bad.

Proof of waiting time estimate.Fix k , l " 1 and take N “ kl ` 1. Suppose x remains in the upperchamber for N periods. If x lands on good lines more than k times,then mesγtx lands on good lines more than k timesu ă Dk L.If x lands on good lines less than k times, it must land on bad linesconsecutively l times. Suppose xn, ¨ ¨ ¨ , xn`l are bad. Bn,L collectspoints with |γn| ě Λ

´l{2u and Bn,S collects points with |γn| ă Λ

´l{2u .

By Growth Lemma, |Bn,S| ď C˚Λ´l{2u . On the other hand,

|Bn,L| ď|γn`l |

Λlu|γn|

ÿ

|γn|ěΛ´l{2u

|GńU γn| ď

L

Λl{2u

|γ| ďL2

Λl{2u

.

Choose k , l such that Dk L ă 0.5ε and pkl ` 1qC˚`L2

Λl{2u

ă 0.5ε.


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Bouncing Ball in Gravity FieldProof of ergodicity

Suppose pX , ωq is a compact symplectic manifold and T : X ý is asymplectic map preserving ω.

LW1 The phase space X is a finite disjoint union of compact subsetsof a linear symplectic space R2 with dense and connectedinterior and regular boundaries.

LW2 For every n ě 1 the singularity sets S`n and Sń of T n and Tń

respectively are regular.LW3 Almost every point p P X possesses strictly monotone cones

Cppq and its complementary C1ppq.LW4 The singularity sets S` and S´ are properly aligned, i.e. the

tangent line of S´ at any p P S´ is contained strictly in the coneCppq and those of S` at any p P S` is contained strictly in C1ppq.

LW5 Noncontraction: There is a constant a P p0,1s such that for everyn ě 1 and for every p P XzS`n , ||dpT nv || ě a||v || @v P Cppq.

LW6 Sinai-Chernov Ansatz: For almost every p P S´ with respect tothe measure µS , its least coefficient of expansion satisfieslimnÑ8 σpdpT nq “ 8.


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Bouncing Ball in Gravity FieldProof of ergodicity

Theorem (Liverani, Wojtkowski 1995)Suppose pX ,T q satisfies Conditions LW1-6. For any n ě 1 and forany p P XzS`n such that σpdpT nq ą 3 there is a neighborhood of pwhich is contained in one ergodic component of T .

Proof of ergodicity.Local ergodicity: find invariant cones.Global ergodicity: no nontrivial ergodic component.


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Bouncing Ball in Gravity FieldProof of recurrence

Decompose v into integer part and fractional part, i.e.

v “ v `mg where v P r0,gq

Then decompose the limit map F8 on the cylinder into its projectionF8 on the torus and a map γ on integers Z, i.e.

pt1, v1,m1q “ F8pt0, v0,m0q “ pF8pt0, v0q,m0 ` γpt0, v0qq.

Lemma (de Simoi, Dolgopyat 2012)

Suppose that F8 is ergodic with respect to the measure µ “ dtdv onthe torus. If the energy change of F8 has zero average, i.e.ş

T γpt0, v0qd µ “ 0, then the escaping set of F has zero measure.

Proof of recurrence.Null set of escaping orbits: by Lemma above.Recurrence: argue by contradiction + Poincare recurrence.


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Bouncing Ball in Gravity FieldProof of exp decay of correlations and CLT

Let T : M Ñ M be a C2 diffeomorphism of a two dimensionalRiemannian manifold M with singularities S.

CZ1 Uniform hyperbolicity of T . There exist two continuous families ofunstable cones Cu

x and stable cones Csx in the tangent spaces

TxM for all x P M, and D a constant Λ ą 1 such thatCZ1.1 DT pCu

x q Ă CuTx , and DT pCs

x q Ą CsTx whenever DT exists;

CZ1.2 ||Dx Tv|| ě ˜||v|| @v P Cux , and ||Dx T´1v|| ě ˜||v|| @v P Cs

x;CZ1.3 The angle between Cu

x and Csx is uniformly bounded away from zero.

CZ2 Singularities S˘ of T and T´1. The singularities S˘ have thefollowing properties:

CZ2.1 T : MzS` Ñ MzS´ is a C2 diffeomorphism;CZ2.2 S0 Y S` is a finite or countable union of smooth compact curves in

M;CZ2.3 Curves in S0 are transverse to the stable and unstable cones.

Every smooth curve in S` (S´) is a stable (unstable) curve. Everycurve in S` terminates either inside another curve of S` or on S0;

CZ2.4 D β P p0, 1q and c ą 0 such that for any x P MzS`,||Dx T || ď cdpx ,S`q´β .


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CZ3 Regularity of smooth unstable curves. We assume there exists aT -invariant class of unstable curves W such that

CZ3.1 Bounded curvature. The curvature of W is uniformly bounded fromabove;

CZ3.2 Distortion control. D γ P p0, 1q and C ą 1 such that for any regularunstable curve W and any x , y P W

ˇ

ˇ logJW pxq ´ logJW pyqˇ

ˇ ď Cdpx , yqγ ,

where JW pxq “ |Dx T |W | denotes the Jacobian of T at x P W ;CZ3.3 Absolute continuity of the holonomy map. Let W1,W2 be two

regular unstable curves that are close to each other. We denote

W 1i “ tx P Wi : W s

pxq XW3í ‰ Hu, i “ 1, 2.

The holonomy map h : W 11 Ñ W 1

2 is defined by sliding along thestable manifold. We assume that h‹µW 11

! µW 12and that for some

constant C and ϑ ă 1ˇ

ˇ logJ hpxq ´ logJ hpyqˇ

ˇ ď Cϑs`px,yq, x , y P W 11

where J h is the Jacobian of h;Jing Zhou | Dynamics of Some Fermi Acceleration Models

34


CZ4 SRB measure. µ is an SRB measure, i.e. the induced measureµW u on any unstable manifold W u is absolutely continuous withrespect to LebW u . We also assume that µ and is mixing.

CZ5 One-step expansion. Let ξn denote the partition of M intoconnected components of MzS`n . Denote as Vα the connectedcomponent of TW with index α P M{ξ1 and Wα “ T´1Vα.D q P p0,1s such that

lim infδÑ0

supW :|W |ăδ

ÿ

αPM{ξ1

ˆ

|W |

|Vα|

˙q|Wα|

|W |ă 1,

where the supremum is taken over all unstable curves W .


35


Theorem (Chernov 1999; Chernov, Zhang 2009)Under the Conditions CZ1-5, the system pT ,Mq above enjoysexponential decay of correlations and central limit theorem fordynamically Hölder continuous observables.

Proof of exp decay, CLT and GGM.Exp decay, CLT: by Theorem above.GGM: F8 admits a Young tower of exp tail by Chen, Wang, Zhang(2019). F8 well approximates F at infinity, hence by Theorem 2.4, 2.9in Dolgopyat-Nandori (2018), F is global gloabl mixing.


36

Bouncing Ball in Gravity FieldExistence of bounded and oscillatory orbits

Lemma (Dolgopyat, Nandori 2016)There exists δ2 " 1 such that on any unstable curve W with |W | ą δ2we have the following central limit theorem for dynamically Hölderobservables, i.e.

1?

n

n´1ÿ

i“0

ϕ ˝ F i8

distá N p0, σ2

ϕq

where ϕ is dynamically Hölder with zero averageş

T ϕd µ “ 0.

Lemma (Growth Lemma)Suppose W is an unstable curve with length |W | ă δ0. Then for anyε ą 0,

mesW trnpxq ă εu ď pϑ1ΛqnmesW

!

r0pxq ăε

Λn

)

` Cε|W |

where ϑ1 ă 1 and Λ is the expansion rate of F8.Jing Zhou | Dynamics of Some Fermi Acceleration Models

37

Bouncing Ball in Gravity FieldExistence of bounded and escaping orbits

Proof for existence of bounded and escaping orbits.By local CLT, Dn0,A such that for an unstable curve W longer than δ2

PW`

v8n0ą v0 ` A

?n0˘

ą13.

By Growth Lemma, if δ2 is small, then for large n0

PW prn0 ă 4δ2q ă110.

Recall that F8 well approximates F , thus we can choose v˚ " 1 solarge that if v0 ą v˚ everywhere on W , then

PW pvn0 ą v0 ` A?

n0, rn0 ą 4δ2q ą14.

=> At least one component W1 Ă F n08W contains a segment W1

longer than δ2 and vn0 ą v0 ` A?

n0 holds everywhere on W1.Jing Zhou | Dynamics of Some Fermi Acceleration Models

Thank you!

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Dynamics of Some Fermi Acceleration Models