Dynamics of some piecewise smooth Fermi-Ulam Models
DDYNAMICS OF SOME PIECEWISE SMOOTHFERMI-ULAM MODELS
JACOPO DE SIMOI AND DMITRY DOLGOPYAT
Abstract.
We find a normal form which describes the high energy dynamicsof a class of piecewise smooth Fermi-Ulam ping pong models; depending onthe value of a single real parameter, the dynamics can be either hyperbolicor elliptic. In the first case we prove that the set of orbits undergoing Fermiacceleration has zero measure but full Hausdorff dimension. We also show thatfor almost every orbit the energy eventually falls below a fixed threshold. Inthe second case we prove that, generically, we have stable periodic orbits forarbitrarily high energies, and that the set of Fermi accelerating orbits mayhave infinite measure. History and introduction
In this paper we study the dynamics of piecewise smooth Fermi-Ulam ping pongs;Fermi and Ulam introduced such systems as a simple mechanical toy model toexplain the occurrence of highly energetic particles coming from outer space anddetected on Earth (the so-called cosmic rays , see [18, 19]). The model describes themotion of a ball bouncing elastically between a wall that oscillates periodically and afixed wall, both of them having infinite mass. Fermi and Ulam performed numericalsimulations for the model and consequently conjectured (see [25]) the existence oforbits undergoing what is now called Fermi acceleration, i.e. orbits whose energygrows to infinity with time; we refer to such orbits as escaping orbits . Several yearslater, KAM theory allowed to prove that the conjecture is indeed false. Namely,provided that the wall motion is sufficiently smooth, there are no escaping orbitsbecause invariant curves prevent diffusion of orbits to high energy (see [20, 24, 23]).It was not many years (see [28]) before the existence of escaping orbits was proved insome examples of piecewise-smooth motions; it is worth noting that these exampleswere essentially the same that Fermi and Ulam were forced to investigate in theirnumerical simulations, due to the relatively limited computational power they coulduse . In this paper we study a more general class of piecewise smooth motions andwe investigate existence and abundance of escaping orbits in this setting.Our main result is that, for all possible wall motions having one discontinuity,there is a parameter ∆ which allows to describe the dynamics of the ping pong forlarge energies. Moreover, there exists a sharp transition so that for ∆ ∈ (0 ,
4) the
We thank Anthony Quas and Carlangelo Liverani for useful discussions. This work was par-tially supported by the European Advanced Grant Macroscopic Laws and Dynamical Systems(MALADY) (ERC AdG 246953) and by NSF. Both authors are pleased to thank the Fields Insti-tute in Toronto, Canada, for the excellent hospitality and working conditions provided in springsemester 2011. The processing power of the 1940’s state-of-the-art computers used by Fermi and Ulam isabout ten thousand times inferior than that of a low-end 2010 smartphone a r X i v : . [ m a t h . D S ] D ec JACOPO DE SIMOI AND DMITRY DOLGOPYAT system looks regular for large energies while for ∆ (cid:54)∈ [0 ,
4] the system is chaoticfor large energies (see Figure 2 in Section 2). Similar phenomena happen in awide class of piecewise smooth mechanical systems which for large energies can beviewed as small ( non-smooth ) perturbations of integrable systems such as, forexample, the impact oscillator [15]. However, in order to demonstrate the methodsand techniques in the simplest possible setting, we restrict our attention to theclassical Fermi-Ulam model. 2.
Results.
We consider the following one-dimensional system: a unit point mass moveshorizontally between two infinite mass walls, between collisions the motion is free,so that kinetic energy is conserved, collisions between the particle and the walls areelastic. The left wall moves periodically, while the right one is fixed. The distancebetween the two walls at time t is denoted by (cid:96) ( t ) which we assume to be strictlypositive, Lipshitz continuous and periodic of period 1. It is convenient to study theorbit only at the moments of collisions with the moving wall. Let t denote a time ofa collision of the ball with the moving wall, since (cid:96) is periodic we take t ∈ T = R / Z .Let v ∈ R be the velocity of the ball immediately after the collision. Introduce thenotation A = T × R ; the collision space is then given by M = { ( t, v ) ∈ A s.t. v > − ˙ (cid:96) ( t ) } . We can thus define the collision map f : M → M (1) f ( t n , v n ) = ( t n + δt ( t n , v n ) , v n − (cid:96) ( t n + δt ( t n , v n ))) = ( t n +1 , v n +1 )where, for large v , the function δt solves the functional equation(2) δt ( t, v ) = (cid:96) ( t ) + (cid:96) ( t + δt ( t, v )) v . It is a simple computation to check that the map f preserves the volume form ω = ( v + ˙ (cid:96) ( t ))d t ∧ d v . Throughout this work we assume ˙ (cid:96) to be piecewise smoothwith a jump discontinuity at t = 0 only. Define the singularity line S ⊂ M as S = { t = 0 } and R ∈ M as the infinite strip of width O (cid:0) v − (cid:1) bounded by S and f S ; introduce also ˜ R = f − R .As a first step to study the dynamics of the mapping f we describe the firstreturn map of f to the region R , which will be denoted by F : R → R . Ourmain result is a normal form for F for large values of v : introduce the notation (cid:96) = (cid:96) (0), ˙ (cid:96) ± = ˙ (cid:96) (0 ± ) and similarly for all derivatives. Define ∆ = J (cid:96) ( ˙ (cid:96) + − ˙ (cid:96) − )and ∆ = J (cid:96) (¨ (cid:96) + − ¨ (cid:96) − ) where J is given by J = (cid:90) (cid:96) − ( s )d s. (3)We introduce a useful shorthand notation. Let ψ ∈ C s ( A ⊂ A ); then we use thenotation ψ = O s (cid:0) v − k (cid:1) to indicate that v k ψ is bounded for sufficiently large v andthe same is true for all derivatives up to order s included. For our analysis it isimportant to ensure that all sub-leading terms vanish sufficiently fast for v → ∞ along with all partial derivatives up to the fifth order. large here means that the ball bounces off the fixed wall before the next collision with themoving wall IECEWISE SMOOTH FERMI-ULAM MODELS 3
Theorem 1.
There exist smooth coordinates ( τ, I ) on R , such that the first returnmap of f on R is given by F ( τ, I ) = ˆ F ( τ, I ) + F ( τ, I ) + r ( τ, I ) where ˆ F ( τ, I ) = (¯ τ , ¯ I ) with ¯ τ = τ − I mod 1 , ¯ I = I + ∆(¯ τ − / ,F is a correction of order O (cid:0) I − (cid:1) of the form F ( τ, I ) = I − (0 , ∆ ((¯ τ − / − / and r = O (cid:0) I − (cid:1) . Finally ω = d τ ∧ d I. Consequently, up to higher order terms, F coincides with ˆ F , where ˆ F is Z -periodic in appropriate “action-angle” variables and moreover d ˆ F = A is constant.Thus ˆ F covers a map (cid:101) F : T → T ; the map (cid:101) F is known in the literature as the“sawtooth map” or the “piecewise linear standard map” and it has been the subjectof a number of studies, see e.g. [2, 3, 4, 6, 5, 22, 27]. Notice that we haveTr( A ) = 2 − ∆ . Accordingly, d (cid:101) F is elliptic if ∆ ∈ (0 ,
4) and it is hyperbolic otherwise.
Example 2.1.
Consider the case where velocity is piecewise linear, that is(4) (cid:96)
A,B ( t ) = B + A (( t mod 1) − / . This is one of the cases which have been numerically investigated in [25]. Wecan choose the length unit so that B = 1 . In this case (cid:96) is positive for all t for A > − . Remarkably, we can obtain an explicit expression for ∆( A ), that is, for J ( A ). Namely J ( A ) = (cid:90) (1 + A ( s − / ) − d s = 2 | A | − / (cid:90) | A | / / (1 + sgn A · σ ) − d σ where σ = | A | / ( s − / J ( A ) = 2 A + 4 + (cid:40) ( | A | − / /
2) log | A | / −| A | / if − < A ≤ | A | − / arctan( | A | / /
2) if
A > . Recall that, by definition, ∆( A ) = − A (1 + A/ J ( A ). In particular, we find that∆( −
4) = 4 and lim A →− + ∆ (cid:48) ( A ) = lim A →− +
12 log 2 + | A | / − | A | / = + ∞ . The graph of dependence of ∆ on A is shown on Figure 1. It shows that thedynamics is hyperbolic for A ∈ ( − , − a ) where a ≈ − . A >
Theorem 2. If | Tr( A ) | > then (cid:101) F is ergodic, mixing and enjoys exponential decayof correlations for H¨older observables. JACOPO DE SIMOI AND DMITRY DOLGOPYAT − − − Figure 1.
Graph of ∆ as a function of A , where (cid:96) A,B is given by(4) with B = 1; the shaded area denotes the elliptic regime ∆ ∈ (0 , | Tr( A ) | < (cid:101) F is not ergodic. Namely, in this case, (cid:101) F is a piecewise isometry for the appropriate metric. Hence if p is a periodic pointof (cid:101) F , then a small ball around p is invariant by the dynamics. See Figure 2 for anexample of phase portrait of (cid:101) F in the two cases.Note that if p is periodic with period N for (cid:101) F , it need not be periodic for ˆ F .
Infact we have ˆ F N p = p + (0 , n )for some n ∈ Z . If n > p is a stable accelerating orbit ; if n < p is a stable decelerating orbit ; finally if n = 0 then p is periodic for ˆ F .
Consider for example the case N = 1: we can find a periodic orbit ( , >
2, we have a stable accelerating orbit (0 , + ) and stabledecelerating orbit (0 , − ) . To analyze periodic points we can use the duality between the accelerating anddecelerating periodic orbits. We have ˆ F = T ∆ ◦ G where G ( τ, I ) = ( τ − I mod 1 , I ) , T ∆ ( τ, I ) = ( τ, I + ∆( τ − / . (5)On the other hand ˆ F − (¯ τ , ¯ I ) = ( τ, I ) with I = ¯ I − ∆(¯ τ − / , τ = ¯ τ + I. Introducing σ = 1 − τ we rewrite the last equation as I = ¯ I + ∆( σ − / , σ = ¯ σ − I. In other words if J denotes the involution J ( τ, I ) = (1 − τ, I ) then( T ∆ ◦ G ) − = J ◦ G ◦ T ∆ ◦ J == J − ◦ G ◦ T ∆ ◦ J = ( T ∆ ◦ J ) − ◦ ( T ∆ ◦ G ) ◦ ( T ∆ ◦ J ) . The existence of periodic orbits for other small periods is summarized in table 1(here we use parameter θ such that Tr(d ˆ F ) = 2 cos θ, that is, d ˆ F is conjugated toa rotation by θ ). IECEWISE SMOOTH FERMI-ULAM MODELS 5
Figure 2.
On the top: phase portrait of a single orbit of the map (cid:101) F for ∆ = − .
3. On the bottom: phase portrait of selected orbitsof the map (cid:101) F for ∆ = 0 .
32. Notice the strong prevalence of ellipticbehavior; the “chaotic” region is given by forward and backwardimages of the singularity line.
JACOPO DE SIMOI AND DMITRY DOLGOPYAT N periodic accelerating/decelerating1 (0 , π ) ( π/ , π )2 (0 , π/
2) -3 (0 , π/
3) ( π/ , π/ , π/
4) (3 / π, π )5 (0 , π/
5) ( π/ , / π )6 (0 , π/ ∪ ( π/ , π ) (5 / π, π )7 (0 , π/
7) (3 / π, π/ , π/
8) (7 / π, π ) Table 1.
Numerically observed ranges of parameters for whichthere exist a periodic or accelerating/decelerating orbit of “pe-riod” N ; for N ≥ π Remark . We believe that stable escaping orbits should exist for arbitrarily smallpositive values of ∆, i.e. for each ∆ > < ∆ < ∆ such thatthe map ˆ F admits stable escaping orbits. However their “period” will necessarilyto grow to infinity as ∆ → + ; the smallest value of ∆ for which we were able tofind a stable escaping orbit is ∆ = 0 . escapingorbits E = { ( t , v ) : v n → ∞} . Theorem 3. If | Tr( A ) | < then(a) there exists a constant C so that for each ¯ v sufficiently large there exists aninitial condition ( t , v ) such that (6) C − ¯ v < v n < C ¯ v for all n ∈ Z . If additionally ∆ (cid:54) = 0 , then the same result holds for an adequately small ballaround the point ( t , v ) .(b) If (cid:101) F has a stable accelerating orbit then mes ( E ) = ∞ . In particular mes( E ) = ∞ if ∆ ∈ ( √ , . Theorem 4. If | Tr( A ) | > then(a) mes ( E ) = 0 .(b) there exists a constant C such that almost every orbit enters the region v < C. Moreover denote by T the first time velocity falls below C. If we fix the initialvelocity v (cid:29) and let the initial phase be random then T v converges to a stablerandom variable of index / , that is, there exists a constant ¯ D such that P ( T > ¯ Dv t ) → (cid:90) ∞ t e − / x √ πx d x as v → ∞ . The proof of second part of the last theorem relies on the following result whichis of independent interest.
Theorem 5.
Fix the initial velocity v (cid:29) and let the initial phase be randomthen Fix < a < < b. Consider the process defined by B v ( t ) = v ( v t ) v if v t is IECEWISE SMOOTH FERMI-ULAM MODELS 7 an integer with linear interpolation in between which is stopped when velocity goesabove bv or below av . Then, as v → ∞ , B v ( t ) converges to a Brownian Motionstarted from 1 and stopped when it reaches either a or b. Remark . Note that B v ( t ) is equal to v ( v t ) v only if v t is an integer. It seemsmore natural to use this formula for all values of t however this would lead to adifferent limit since as we shall see in the next section the ratio v ( n + ) /v ( n ) hasoscillations of order 1 while v ( n + 1) /v ( n ) − /v ( n ) . Theorem 5 makes Theorem 4 plausible since the time the Brownian Motionreaches a certain level has a stable distribution of index 1 / . However some work isneeded to deduce Theorem 4 from Theorem 5 since the proof of Theorem 5 relies ona perturbative argument near v = ∞ whereas Theorem 4 requires to handle smallvelocities as well since v ( T ) ≤ C. Theorem 4 shows that the set of escaping orbits has zero measure so it is naturalto ask about its Hausdorff dimension. The next result extends the work [13] wherea similar statement is proven for a smooth model of Fermi acceleration.
Theorem 6. If Tr( A ) > then HD ( E ) = 2 . In other words, even though the set of escaping points is small from the measuretheoretical point of view, it is large from the point of view of dimension.3.
The first return map If (cid:96) were a smooth function, KAM theory would allow to conjugate the dynam-ics of f for most initial conditions with large values of v to the dynamics of thecompletely integrable map g : T × R + → T × R + g : ( ϑ, J ) (cid:55)→ ( ϑ + J − , J ) . Consider the vertical line S (cid:48) ⊂ T × R + given by S (cid:48) = { ϑ = 0 } ; let moreover R (cid:48) bethe infinite strip of width O (cid:0) J − (cid:1) bounded between S (cid:48) and gS (cid:48) i.e. R (cid:48) = { ≤ ϑ < J − } . As a preparatory step we study the first return map of g to the region R (cid:48) . Proposition 3.1.
Let τ = Jϑ and consider coordinates ( τ, J ) on R (cid:48) . Then thefirst return map of g to the region R (cid:48) is given by the map G defined in (5) .Proof. Let k = (cid:98) J (cid:99) and J = k + ˆ J . We claim that G ( τ, J ) = (cid:40) g k ( τ, J ) if τ ≤ − ˆ Jg k +1 ( τ, J ) otherwise.In fact, we can check by simple inspection that, denoting g k ( ϑ, J ) = ( ϑ k , J ) wehave ϑ k = ϑ + kk + ˆ J = ϑ + 1 − ˆ JJ , ϑ k +1 = ϑ + 1 + 1 − ˆ JJ which implies our claim. (cid:3) In our systems, (cid:96) is only piecewise smooth, consequently we expect to be able todefine action-angle coordinates outside ˜ R only. JACOPO DE SIMOI AND DMITRY DOLGOPYAT
Lemma 3.2 (Approximate reference coordinates) . There exists a smooth coordi-nates change h : ( t, v ) (cid:55)→ ( ϑ, J ) such that if ( t, v ) (cid:54)∈ ˜ R , h conjugates the collisionmap f to the reference map g up to high order terms (7) g − h ◦ f ◦ h − = ( r ϑ , r J ) with r ϑ = O (cid:0) v − (cid:1) , r J = O (cid:0) v − (cid:1) .Proof. Recall the definition of J given by (3) and introduce the notation ( t (cid:48) , v (cid:48) ) = f ( t, v ). Define the two functions (see e.g. [28] for a motivation of the formuladefining ϑ ) ϑ ( t ) = J − (cid:90) t (cid:96) − ( s )d s, I ( t, v ) = J (cid:34)(cid:90) t (cid:48) t (cid:96) − ( s )d s (cid:35) − . (8)It is immediate to observe that ϑ ( t (cid:48) ) = ϑ ( t ) + I − ( t, v ). Since the expressiondefining I is implicit, we find it convenient to use a suitable approximation in ourcomputations. Define J : A \ S → R as(9) 2 J − J ( · , v ) = v(cid:96) + (cid:96) ˙ (cid:96) + 13 (cid:96) ¨ (cid:96)v . We claim that h : ( t, v ) (cid:55)→ ( ϑ ( t ) , J ( t, v )) is the required change of coordinates. Thefirst step is to obtain an approximate solution of (2). Since (cid:96) is Lipshitz continuous,we can find the solution by iteration. Let δt (0) ≡ n > δt ( n ) ( t, v ) = (cid:96) ( t ) + (cid:96) ( t + δt ( n − ( t, v )) v . Then (cid:107) δt ( n ) − δt ( n − (cid:107) = O ( v − n ) and thus δt ( n ) → δt uniformly. Consequently, ifwe express the solution as(10) δt = ∞ (cid:88) n =1 δt n with δt n ( t, v ) = a n ( t ) v n , we can then find the functions a n by the previous argument. In particular, outside˜ R we obtain that δt ( · , v ) = 2 (cid:96)v , δt ( · , v ) = 2 (cid:96)v ˙ (cid:96), δt ( · , v ) = 2 (cid:96)v ( ˙ (cid:96) + (cid:96) ¨ (cid:96) ) . (11)Assume now that ( t, v ) (cid:54)∈ ˜ R . By expanding (8) in Taylor series and using equations(11) it is immediate to check that(12) J = I + O (cid:0) v − (cid:1) . Recall that r ϑ ( t, v ) = ϑ ( t (cid:48) ) − ϑ ( t ) − J ( t, v ) − , r J ( t, v ) = J ( t (cid:48) , v (cid:48) ) − J ( t, v ) . Thus estimate (12) immediately yields r ϑ = O (cid:0) v − (cid:1) . The proof of Lemma 3.2 isthus complete once we prove that J ( t (cid:48) , v (cid:48) ) − J ( t, v ) = O (cid:0) v − (cid:1) . We begin by introducing a convenient notation. Fix ( t, v ). Recall that r J = J ◦ f − J ;denote J = J ( t, v ) , J (cid:48) = J ( t (cid:48) , v (cid:48) ) , (cid:96) = (cid:96) ( t ) , (cid:96) (cid:48) = (cid:96) ( t (cid:48) ) and likewise for all derivatives.Notice that Jv − is a polynomial in v − with coefficients given by smooth functions IECEWISE SMOOTH FERMI-ULAM MODELS 9 of t . Using (10) we can express δt in similar form, thus, by expanding in Taylorseries the smooth function (cid:96) and its derivatives we can write r J ( t, v ) = b ( t ) + b ( t ) v + b ( t ) v + r ∗ J ( t, v )where r ∗ J = O (cid:0) v − (cid:1) . It amounts to a simple but tedious computation to showthat our choice of J implies b ≡ b ≡ b ≡
0. Here we will only sketchthe main steps of the computation. First, we obtain an expression for δvδv = v (cid:48) − v = − (cid:96) (cid:48) = δv + δv + δv + O (cid:0) v − (cid:1) where δv = − (cid:96) δv = − (cid:96) ¨ (cid:96)v δv = − (cid:96) ˙ (cid:96) ¨ (cid:96) + (cid:96) ... (cid:96)v . Next, we expand r J in Taylor series and collect terms of order v − or higher in thefunction r ∗ J r J = ∂ t J ( δt + δt + δt ) + ∂ v J ( δv + δv + δv )++ 12 ∂ tt J (cid:0) δt + 2 δt δt (cid:1) + ∂ tv J ( δt δv + δt δv + δt δv )++ 16 ∂ ttt Jδt + 12 ∂ ttv Jδt δv + r ∗ J . Using the explicit form (9) it is then simple to obtain b = ˙ (cid:96)vδt + (cid:96)δv b = ˙ (cid:96)v δt + ( ˙ (cid:96) + (cid:96) ¨ (cid:96) ) vδt + 12 ¨ (cid:96)v δt + (cid:96)vδv + ˙ (cid:96)vδt δv . and finally b = ˙ (cid:96)v δt + ( ˙ (cid:96) + (cid:96) ¨ (cid:96) ) v δt + 13 (2 (cid:96) ˙ (cid:96) ¨ (cid:96) + (cid:96) ... (cid:96) ) vδt ++ ¨ (cid:96)v δt δt + 12 (3 ˙ (cid:96) ¨ (cid:96) + (cid:96) ... (cid:96) ) v δt + 16 ... (cid:96) v δt ++ (cid:96)v δv − (cid:96) ¨ (cid:96)δv + ˙ (cid:96)v δt δv + ˙ (cid:96)v δt δv + 12 ¨ (cid:96)v δt δv . Now it is possible to conclude by substituting b j into the formulae obtained previ-ously. (cid:3) Proof of Theorem 1.
It is simple to check by inspection that ( τ, I ) with τ = Iϑ aresmooth coordinates on R for sufficiently large I . Let ( t, v ) ∈ R , (¯ t, ¯ v ) = F ( t, v )and (˜ t, ˜ v ) = f − (¯ t, ¯ v ). We use the convenient shorthand notation J = J ( t, v ),˜ J = J (˜ t, ˜ v ) and ¯ J = J (¯ t, ¯ v ) and similarly for ϑ , ˜ ϑ and ¯ ϑ . By iteration of Lemma3.2 we obtain J (˜ t, ˜ v ) − J ( t, v ) = O (cid:0) v − (cid:1) . We then claim that¯ J − ˜ J = 12 J ( ˙ (cid:96) + − ˙ (cid:96) − ) (cid:104) ¯ t ¯ v (1 − ˙ (cid:96) + ¯ v ) − (cid:96) (cid:105) +(13) + 14 J ¨ (cid:96) + − ¨ (cid:96) − ¯ v (cid:20) (¯ t ¯ v − (cid:96) ) − (cid:96) (cid:21) + O (cid:0) ¯ v − (cid:1) . In fact notice that¨ (cid:96) (¯ t ) = ¨ (cid:96) + + O (cid:0) ˜ v − (cid:1) ¨ (cid:96) (˜ t ) = ¨ (cid:96) − + O (cid:0) ˜ v − (cid:1) ˙ (cid:96) (¯ t ) = ˙ (cid:96) + + ¨ (cid:96) + ¯ t + O (cid:0) ˜ v − (cid:1) ˙ (cid:96) (˜ t ) = ˙ (cid:96) − + ¨ (cid:96) − ˜ t + O (cid:0) ˜ v − (cid:1) (cid:96) (¯ t ) = (cid:96) + ˙ (cid:96) + ¯ t + 12 ¨ (cid:96) + ¯ t + O (cid:0) ˜ v − (cid:1) (cid:96) (˜ t ) = (cid:96) + ˙ (cid:96) − ˜ t + 12 ¨ (cid:96) − ˜ t + O (cid:0) ˜ v − (cid:1) and moreover˜ t = ¯ t − (cid:96) ˜ v + ˙ (cid:96) + + ˙ (cid:96) − ˜ v ¯ t − (cid:96) ˙ (cid:96) − ˜ v + O (cid:0) v − (cid:1) ¯ v = ˜ v − (cid:96) (¯ t ) . (14)By the definition of J we thus obtain¯ J − ˜ J = 12 J (cid:104) ( (cid:96) (¯ t ) − (cid:96) (˜ t ))˜ v + − ( (cid:96) (¯ t ) ˙ (cid:96) (¯ t )) + (cid:96) (˜ t ) ˙ (cid:96) (˜ t )++ 13 (cid:96) (¯ t )¨ (cid:96) (¯ t ) − (cid:96) (˜ t )¨ (cid:96) (˜ t )˜ v (cid:105) + O (cid:0) ˜ v − (cid:1) from which (13) follows by a straightforward computation. Notice that by definitionwe have 2 J − ¯ J = ¯ v(cid:96) + ˙ (cid:96) + ¯ v ¯ t + (cid:96) ˙ (cid:96) + + O (cid:0) ¯ v − (cid:1) J − ˜ J = ˜ v(cid:96) + ˙ (cid:96) − ˜ v ˜ t + (cid:96) ˙ (cid:96) − + O (cid:0) ˜ v − (cid:1) . Next, by definition of ϑ ¯ t = J (cid:96) ¯ ϑ (1 + J (cid:96) ˙ (cid:96) + ¯ ϑ ) + O (cid:0) v − (cid:1) from which we obtain (cid:16) ¯ t ¯ v (1 + ˙ (cid:96) + / ¯ v ) − (cid:96) (cid:17) = 2 (cid:96) (¯ τ − /
2) + O (cid:0) v − (cid:1) . Therefore we can rewrite ¯ J as follows¯ J = J + ∆(¯ τ − /
2) + ∆ J ((¯ τ − / − /
12) + O (cid:0) J − (cid:1) . Using estimate (12) we thus conclude that I (¯ t, ¯ v ) − I ( t, v ) = I (¯ t, ¯ v ) + O (cid:0) I − (cid:1) . We now prove that(15) ¯ τ = τ − J mod 1 . By Lemma 3.2 and definition we have˜ J ˜ ϑ = τ − J − J ¯ ϑ = ¯ τ . On the other hand, using the definition of ϑ and the approximate expressions for J given above, we obtain˜ J ˜ ϑ = ˜ v + ˙ (cid:96) − (cid:96) ˜ t + O (cid:0) J − (cid:1) ¯ J ¯ ϑ = ¯ v + ˙ (cid:96) + (cid:96) ¯ t + O (cid:0) J − (cid:1) . From the last equation we obtain, using (14), that¯ J ¯ ϑ = ˜ J ˜ ϑ + 1 + O (cid:0) J − (cid:1) . Now (15) follows from (12). (cid:3)
IECEWISE SMOOTH FERMI-ULAM MODELS 11 Hyperbolic case. Properties of the limiting map
The proof of ergodicity of the map (cid:101) F has been first given in [26]. Strongerstatistical properties claimed by Theorem 2 follow from the following general result.Let G be a piecewise linear hyperbolic automorphism of T and denote by S + and S − the discontinuity curves of G − and G , respectively; let S = S − ∪ S + . For anypositive n ∈ N let S n = G n − S + and S − n = G − ( n − S − ; assume for convenience S = ∅ ; let S ( n ) = (cid:83) nk = − n S k . Proposition 4.1 (Chernov, [10]) . Assume:(a) S i ∩ S j is a finite set of isolated points if i (cid:54) = j ;(b) S is everywhere transversal to the invariant stable and unstable directions;(c) for every m ≥ , the number of components of S ( n ) meeting at a single point isbounded by Km for some constant K ;then G is ergodic, mixing and enjoys exponential decay of correlations for H¨olderobservables.Proof of Theorem 2. If Tr( A ) > (cid:101) F is a piecewise linear hyperbolic automor-phism of T ; we recall the explicit formula: (cid:101) F : ( τ, I ) (cid:55)→ ( τ − I mod 1 , I + ∆(( τ − I mod 1) − / . Thus it is easy to check that S − is given by the diagonal circle τ = I and S + isgiven by the vertical circle τ = 0. It is then a simple linear algebra computation toprove that the stable and unstable slopes are given by the solution of the quadraticequation h − ∆ h + ∆ = 0; thus we immediately obtain item (b) in the hypothesesof Proposition 4.1. Since d (cid:101) F is constant at any point where it can be defined, the n -th image of any line segment is a finite disjoint union of line segments parallel toeach other; hence each point p ∈ T can meet at most two of such segments, whichproves item (c). Finally, unless the initial line segment is aligned to an invariantdirection (stable or unstable), the slopes of line segments belonging to images atdifferent times are different, which proves item (a) and concludes the proof. (cid:3) Elliptic case. Growth of energy.
Proof of Theorem 3.
In order to prove item (a) we will prove that for each ¯ v suf-ficiently large there exists a stable periodic point ( t ∗ , v ∗ ) ∈ R whose orbit satisfiescondition (6); stability of the fixed point then implies that (6) holds for any initialcondition ( t , v ) in a small ball around ( t ∗ , v ∗ ). We already noticed that the point(ˆ τ , ˆ J ) is a stable fixed point of the map ˆ F if ˆ τ = 1 / J ∈ Z ; in order to proveexistence of a stable fixed point of the first return map F we would need to provethat the fixed point of ˆ F satisfies the non-degenerate twist condition. However,since ˆ F is piecewise linear, we actually need to consider the first return map F as a O (cid:0) J − (cid:1) perturbation of the map ¯ F = ˆ F + F and check that ¯ F satisfies the twistcondition. Since the perturbation term is small up to derivatives of sufficiently highorder, we can conclude.Fix once and for all ˆ J ∈ N such that | J (0 , ¯ v ) − ˆ J | <
2; let ˆ λ = exp( i ˆ θ ) be themultiplier at the fixed point (ˆ τ , ˆ J ) of ˆ F . Since ∆ ∈ (0 ,
4) we have ˆ λ (cid:54) = 1; then ¯ F willhave a fixed point close to (ˆ τ , ˆ J ) that we denote by (¯ τ , ¯ J ); introduce the parameter ε = ∆ /I ; by inspection is is easy to see ¯ J = ˆ J and ¯ τ = ˆ τ + ε/ (12∆) + O (cid:0) ε (cid:1) . Introduce coordinates ( σ, J ) in a neighborhood of the fixed point (¯ τ , ¯ J ) such that( τ, J ) = (¯ τ + σ, ˆ J + J ). The expression for d ¯ F in these new coordinates is:d F ( σ, J ) = (cid:18) − κ + εσ − κ − εσ (cid:19) where κ = ∆ + ε / (6∆); denote by ¯ λ = exp( i ¯ θ ) the multiplier of the map ¯ F at(¯ τ , ¯ J ): it is immediate to check that(16) cos ¯ θ = cos ˆ θ − ε / (12 / ∆) . In order to check the twist condition we perform a complex change of variables( σ, J ) (cid:55)→ ( z, ¯ z ) such that the map can be expressed as follows¯ F : z (cid:55)→ z + λz + A z + A z ¯ z + A ¯ z . Then (see e.g. [12]) we need to ensure that:Υ = 3 | A | ¯ λ + 1¯ λ − | A | ¯ λ + 1¯ λ − (cid:54) = 0;Notice that from the fact that ¯ F is symplectic we obtain that | A | = | A | ; thusthere are two possibilities for the twist condition to fail; either A = A = 0 or ¯ λ solves the equation 3 ¯ λ + 1¯ λ − λ + 1¯ λ − . It is easy to check that the above condition is given by either ¯ θ = 0 or cos ¯ θ = − / ε , according to (16),possibly by choosing a different ˆ J . Therefore, we just need to check that A (cid:54) = 0:from elementary linear algebra we find that: z = σ + (1 − λ ) J z = σ + (1 − ¯ λ ) J ;changing variables we obtain: A = − λ − λ − ¯ λ ) ε A = − − λ − ¯ λλ − ¯ λ ε A = ¯ λ − λ − ¯ λ ) ε Which implies | A | (cid:54) = 0 and concludes the proof of item (a).The proof of item (b) is analogous to the proof of the corresponding resultobtained in [15], Section 3, and will therefore be omitted. (cid:3) Hyperbolic case. Measure of accelerating orbits.
Note that part (a) of Theorem 4 follows from part (b), however since the proof ofpart (b) is rather involved we give a direct proof in this section. We expect Lemma6.1 below to be useful for a wide range of mechanical systems. In particular,Theorem 4(a) is a direct consequence of Theorem 2 and Lemma 6.1.Let X be a Borel space and Y be a subset of X × N containing { ( x, m ) : m ≥ ¯ m } for some ¯ m. Let Φ : Y → Y be the mapΦ( x, m ) = ( φ ( x, m ) , m + γ ( x, m )) . Assume that Φ is asymptotically periodic in the following sense. Denote T k ( x, m ) =( x, m + k ) and consider Ψ k = T − k Φ T k . Assume that there exist a map ψ : X → X IECEWISE SMOOTH FERMI-ULAM MODELS 13 preserving a probability measure µ, and a function γ : X → Z such that for each M for each function h supported on X × [0 , M ] and each l we have(17) || h ◦ Ψ lk − h ◦ Ψ l || L ( (cid:101) µ ) → x, m ) = ( ψ ( x ) , m + γ ( x ))and (cid:101) µ is a product of µ and a counting measure on Z . Denote ( x n , m n ) = Φ n ( x , m )and let E = { ( x , m ) : m n → + ∞} . Lemma 6.1.
Assume that(i) ψ is ergodic with respect to µ ;(ii) (cid:82) X γ ( x )d µ ( x ) = 0; (iii) Φ preserves a measure (cid:101) ν with bounded density with respect to (cid:101) µ ; (iv) || γ ( x, m ) || L ∞ ≤ K. Then (cid:101) ν ( E ) = 0 . Proof.
By [1] we know that conditions (i) and (ii) imply that Ψ : X × Z → X × Z is conservative. That is for each subset ¯ Y of finite (cid:101) µ measure the Poincare mapˆΨ : ¯ Y → ¯ Y is defined almost everywhere. Let ¯ Y = X × [0 , K + 1] where K isthe constant from condition (iv). By Rohlin Lemma applied to ˆΨ for each ε thereexists a set Ω ε ⊂ ¯ Y and a number L ε such that (cid:101) µ (Ω ε ) < ε and (cid:101) µ (( x, m ) : ˆΨ l ( x, m ) (cid:54)∈ Ω ε for 0 ≤ l ≤ L ε ) < ε. In view of (17) and condition (iii) there exists a constant C (independent of ε ) anda number k ( ε ) such that for k ≥ k ( ε ) we have (cid:101) ν ( ¯Ω k,ε ) < Cε where¯Ω k,ε = { ( x, m ) ∈ T k ¯ Y : Φ l (cid:54)∈ T k Ω ε for l ∈ N } . Let Ω = (cid:91) n ∈ N (cid:16) T k (1 /n ) Ω /n (cid:91) ¯Ω k (1 /n ) , (1 /n ) (cid:17) . Note that (cid:101) ν (Ω) < ∞ . On the other hand if ( x, m ) ∈ E then, due to condition(iv), its orbit Orb( x, m ) visits T k ¯ Y for all k except for finitely many k . HenceOrb( x, m ) ∩ (Ω ∩ E ) (cid:54) = ∅ . Accordingly it suffices to show that (cid:101) ν (Ω ∩ E ) = 0 . However, by the foregoing discussion, the first return map ˆΦ : (Ω ∩ E ) → (Ω ∩ E )is defined almost everywhere and by Poincare recurrence theorem for almost all( x, m ) ∈ Ω ∩ E we have Orb( x, m ) ∩ X × { m } (cid:54) = ∅ . Thus for almost all points inΩ ∩ E we have ( x, m ) (cid:54)∈ E. Therefore ν (Ω ∩ E ) = 0 as claimed. (cid:3) Hyperbolic case. Time of deceleration.
Plan of the proof.
Here we prove Theorem 4(b). The argument of thissection has many similarities with the arguments in [8, 14, 16] so we just indicatethe key steps.The proof relies on the notion of standard pair . A standard pair is a pair (cid:96) = ( γ, ρ )there γ is a curve such that | γ | < | γ | denotes the length of γ, γ (cid:48) belongsto an unstable cone, | γ (cid:48)(cid:48) | ≤ K , and ρ is a probability density on γ satisfying || ln ρ || C ( γ ) ≤ K . We let | (cid:96) | denote the length of γ. We denote by E (cid:96) the expectationwith respect to the standard pair E (cid:96) ( A ) = (cid:90) γ A ( x ) ρ ( x ) dx and by P (cid:96) the associated probability measure, that is, P (cid:96) (Ω) = E (cid:96) (1 Ω ) . An easy computation shows that if I is sufficiently large on γ then the standardpairs are invariant by dynamics, that is E (cid:96) ( A ◦ F n ) = (cid:88) j c j E (cid:96) j ( A )where (cid:80) j c j = 1 and (cid:96) j are standard pairs. We need to know that most of γ j inthis decomposition are long. To this end let r n ( x ) be the distance from x n to theboundary of the component γ j containing x n . Lemma 7.1 (Growth lemma) . (a) There exist constants C > and θ < such that P (cid:96) ( r n ( x ) < ε ) ≤ Cε + P (cid:96) ( r ( x ) < εθ n ) . (b) There exists a constant ε such that if n > K | ln | γ || then P (cid:96) ( r n ( x ) < ε for n = n , . . . , n + k ) ≤ Cθ k . The Growth Lemma is the key element of proving exponential mixing for (cid:101) F (see[11, 9]) and the argument used to prove the Growth Lemma for (cid:101) F shows that thisproperty is also valid for small perturbations of (cid:101) F .
Given a point x let T a be the first time I n < a if I > a and be the first time I n > a if I < a. Let T a,b be the first time then either I n < a or I n > b. The proof of part (b) of Theorem 4 depends on two propositions. The first oneis an extended version of Theorem 5. It will allow to handle large velocities. Thesecond one gives an a priori bounded needed to handle small velocities.Fix 0 < a < < b. Denote(18) D = ∞ (cid:88) n = −∞ (cid:90) (cid:90) T A ( x ) A ( F n x ) dx where F = G ◦ T ∆ and G and T ∆ are defined by (5). Proposition 7.2.
Let x be distributed according to a standard pair (cid:96) such that I ∼ I on (cid:96) and | (cid:96) | > I − . Then(a) The process B I ( t ) = I min( I t,T aI ,bI ) I converges to the Brownian Motion with zero mean and variance D t which isstarted from 1 and is stopped when it reaches either a or b ; (b) There exists δ > such that | E (cid:96) ( I T aI ,bI ) − I | ≤ CI − δ ; (c) There exists θ < such that P (cid:96) ( T aI ,bI > kI ) ≤ max( θ k , I − ); (d) Let T ∗ a,b = min( T aI ,bI , I ) . Then P (cid:96) ( r T ∗ a,b ( x ) < ε ) ≤ CI ε. Proposition 7.3.
Given ε > there exists K ( ε ) > such that if | (cid:96) | > I − then P (cid:96) ( T C > K ( ε ) I ) ≤ ε. Note that Proposition 7.2 implies that P (cid:96) ( T δI ≥ tI ) → P ( T Bδ ≥ t ) IECEWISE SMOOTH FERMI-ULAM MODELS 15 where T Bδ denotes the first time the Brownian Motion from Proposition 7.2 reaches δ. Indeed | P (cid:96) ( T δI ≥ tI ) − P (cid:96) ( T δI ≥ tI and T δI ≤ T AI ) | ≤ P (cid:96) ( T δI ≥ T AI ) . By Proposition 7.2 the RHS can be made as small as we wish by taking A large.On the other hand by Proposition 7.2 P (cid:96) ( T δI ≥ tI and T δI ≤ T AI ) → P (cid:96) ( T Bδ ≥ t and T Bδ ≤ T BA )and the last expression can be made as close to P ( T Bδ ≥ t ) as we wish by taking A large.Next note that it is enough to prove Theorems 4(b) and 5 with v replaced by I. Indeed, in view of (12), (9) and (3), we have(19) I ≈ J (cid:96) (0)2 v which shows that v can be replaced by I in Theorem 5. Also (19) allows us tosqueeze the first time v goes below C between the time I goes below C and thetime I goes below C and, in view of Proposition 7.3, the times to go below C and C satisfy the same estimates.We are now ready to derive part (b) of Theorem 4 from Propositions 7.2 and7.3.We have P (cid:96) ( T C ≤ tI ) ≤ P (cid:96) ( T δI ≤ tI ) → P ( T Bδ ≤ t )and the last expression can be made as close to P ( T B ≤ t ) as we wish by taking δ small.Conversely P (cid:96) ( T C ≥ tI ) ≤ P (cid:96) ( T δI ≥ ( t − K ( ε ) δ ) I ) + P (cid:96) ( T C ( x T δI ) ≥ K ( ε ) δ I )where K ( ε ) is given by Proposition 7.3. The first term can be made as close to P ( T B ≥ t ) as we wish by taking small δ. To estimate the second term note that P (cid:96) ( T C ( x T δI ) ≥ K ( ε ) δI )= P ( T C ( x T δI ) ≥ K ( ε ) δ I and r T δI ( x ) < ( δI ) − )+ P ( T C ( x T δI ) ≥ K ( ε ) δ I and r T δI ( x ) ≥ ( δI ) − ) = I + II.
Next I ≤ P ( r T δI ( x ) < ( δI ) − )) = O ( I − )by Proposition 7.2(d) and II = P ( r T δI ( x ) ≥ I − )) P (cid:96) ( T C ( x T δI ) ≥ K ( ε ) δ I | r T δI ≥ ( δI ) − )) ≤ P (cid:96) ( T C ( x T δI ) ≥ K ( ε ) δ I | r T δI ≥ ( δI ) − )) ≤ ε where the last inequality follows by definition of K ( ε ) . This completes the derivation of Theorem 4(b) from Propositions 7.2 and 7.3. Itremains to establish the propositions. Proposition 7.2 is proven in section 7.2 andProposition 7.3 is proven in section 7.3.
Central Limit Theorem.
Let F † I ( I, τ ) = ˆ F ( I, τ ) + [ I ] − (0 , ∆ ((¯ τ − / − / . Note that F † I approximates F up to error O ( I − ) . Next consider a mapping of the T given by ¯ F N ( I, τ ) = (cid:101) F ( I, τ ) + N − (0 , ∆ ((¯ τ − / − / . Then F † I locally covers (cid:101) F [ I ] ; also ¯ F N preserves the measure d I d τ. The proof ofTheorem 2 shows that ¯ F N is exponentially mixing. In particular, if | (cid:96) | ≥ ε then E (cid:96) ( A ◦ ¯ F nN ) = (cid:90) (cid:90) T A d I d τ + O ( θ n ) . We use this property to establish the following estimate
Lemma 7.4 (Averaging Lemma) . Suppose that | (cid:96) | > I − . Let n = K ln I where K is sufficiently large. Let A be a piecewise smooth periodic function.(a) E (cid:96) ( A ◦ F n ) = (cid:82)(cid:82) T A d I d τ + O ( I − δ ); (b) There is L > such that E (cid:96) ( A ( F n x ) A ( F n + k x )) = (cid:90) (cid:90) T A ( x ) A ( (cid:101) F k x )d I d τ + O ( I − β L k ) . The proof of this lemma is similar to the proof of Proposition 3.3 in [7]. Theproof of part (a) proceeds in two steps. First, if | (cid:96) | > ε then we use the shadowingargument to show that(20) E (cid:96) ( A ◦ F n ) = E (cid:96) ( A ◦ ¯ F n [ I ] ) + O ( I − δ )and then use exponential mixing of ¯ F [ I ] . In the general case we find a function n ( x ) < K ln I such that E (cid:96) ( A ( F n ( x ) x )) = (cid:88) j c j E (cid:96) j ( A )and (cid:80) | (cid:96) j |≤ ε c j ≤ I − and then apply (20) to all long components (cid:96) j . To prove part (b) we first use the foregoing argument to show that E (cid:96) ( A ( F n x ) A ( F n + k x )) = (cid:90) (cid:90) T A ( x ) A ( F k x )d I d τ + O ( I − δ L k )(the factor L k accounts for the exponential growth of the Lipshitz norm of A ( A ◦ F k )) and then use the shadowing argument again to show that (cid:90) (cid:90) T A ( x ) A ( F k x )d I d τ = (cid:90) (cid:90) T A ( x ) A ( (cid:101) F k x )d I d τ + O ( I − β ) . It is shown in [8], Appendix A that Lemmas 7.1 and 7.4 imply parts (a) and (b) ofProposition 7.2. We note that the error bound O ( I − (2 − δ ) ) (cid:28) I − is needed to compute the drift of the limiting process; to compute its variance it isenough that F = ˆ F + O ( I − ) and that ˆ F covers (cid:101) F which satisfies the CLT in thesense that I n √ n ⇒ Normal(0 , D ) IECEWISE SMOOTH FERMI-ULAM MODELS 17 where the diffusion coefficient D is given by the Green-Kubo formula (18). (Infact (18) is the Green-Kubo formula for F = G (cid:101) F G − but F and (cid:101) F clearly have thesame transport coefficients.)Next, part (a) of Proposition 7.2 implies part (c) with k = 1 , that is, there is θ < E (cid:96) ( T aI ,bI < I ) ≤ θ. For k > F ( k − I γ which have not escaped by the time ( k − I . Finally P (cid:96) ( T ∗ a,b ≤ ε ) ≤ I (cid:88) m = K ln I P (cid:96) ( r m ( x ) < ε )so part (d) follows from part (a) of Lemma 7.1.7.3. A priori bounds for the return time.
Let σ be the first time when | I σ − m | ≤ ∆ . For j ≥ σ j inductively as follows. Assume that σ j − wasalready defined so that | I σ j − − m j − | ≤ ∆ . Let ˆ σ j be the first time after σ j − wheneither | I σ − m j +1 | ≤ ∆ or | I σ − m j − | ≤ ∆ . Let σ j = min(ˆ σ j , σ j − + 2 m j − ) . If either ˆ σ j ≥ σ j − + 2 m j − or r σ j < − m j or 2 m j < ¯ I then we stop otherwisewe continue and proceed to define σ j +1 . If we stop we let j ∗ = j be the stoppingmoment. If we stop for the first or the second reason we say that we have anemergency stop, otherwise we have a normal stop. By the discussion at the endof section 7.1 the lower cutoff in Proposition 7.3 is not important so to prove theproposition it is enough to control the first time when I n is close to 2 ¯ m with 2 ¯ m < ¯ I. In other words we need to control σ j ∗ , especially if it is a normal stop. Also since σ is unlikely to be large by part (c) of Proposition 7.2 (in fact, part (a) would alsobe sufficient for our purposes) we need to control σ j ∗ − σ . Let F j be the σ -algebra generated by ( m , σ ) , · · · , ( m j , σ j ) . Proposition 7.2implies that P (cid:96) ( m j +1 = m j + 1 | F j ) = 13 + o (1) , ¯ I → ∞ P (cid:96) ( m j +1 = m j − | F j ) = 23 + o (1) , ¯ I → ∞ P (cid:96) ( σ j +1 is an emergency stop | F j ) = O (cid:0) − m j (cid:1) . Let ξ j be a random walk with ξ = m and P ( ξ j +1 = ξ j + 1) = 0 . , P ( ξ j +1 = ξ j −
1) = 0 . j be iid random variables independent of ξ s such that P (Λ j = k ) = (cid:40) Kθ k if k ≥ k k is sufficiently large and K = − θθ k . Let ¯Λ j = min(2 ξ j Λ j , ξ j ) . Proposition7.2 allows us to construct a coupling such that for j ≤ j ∗ m j ≤ ξ j , σ j ≤ σ + j − (cid:88) m =0 ¯Λ m . Now a standard computation with random walks shows that Proposition 7.3 is validfor the random walk itself. Consequently, given ε , there exists K ( ε ) such that P (cid:96) (cid:18) σ j ∗ − σ ≥ m K ( ε )2 (cid:19) ≤ ε . Unfortunately j ∗ need not to be a normal stop, it can be an emergency stop aswell. To deal with this problem let p k = P (cid:96) ( j ∗ is an emergency stop and m j ∗ = k ) . Denote Ω kl = { j ∗ is an emergency stop, m j ∗ = k and j ∗ is the l -th visit to k } ,V kl = { k is visited at least l times } . Then p k ≤ ∞ (cid:88) l =1 P (cid:96) (Ω kl ) = ∞ (cid:88) l =1 P (cid:96) ( V kl ) P (cid:96) (Ω kl | V kl ) . By Proposition 7.2(d) P (cid:96) (Ω kl | V kl ) ≤ C − k while the existence of the coupling with the random walk discussed above impliesthat P (cid:96) ( V kl ) ≤ θ l . Therefore(22) p k ≤ C − k . Accordingly, by choosing ¯ I large enough we can make the probability of an emer-gency stop less than 0 . . However we can not decrease that probability below ε/ I is fixed, so more work is needed.First, we note that P (cid:96) ( m j ∗ > m /
2) = O ( I − ) so it can be neglected. Secondly,an argument similar to one leading to (22) shows that P (cid:96) ( r σ j ∗ < I − ) = O ( I − ) . Next, if r σ j ∗ > I − , m j ∗ < m / j ∗ is an emergency stop let ¯ σ be the firsttime after σ j ∗ such that r ¯ σ ( x ) ≥ ε . By the Growth Lemma P (cid:96) (¯ σ − σ j ∗ > K ln I ) ≤ I − if K is large enough.If ¯ σ − σ j ∗ < K ln I then we can repeat the procedure described above with x replaced by x ¯ σ . If the second stop is a normal one we are done, otherwise we trythe third time and so on. We have P (cid:96) (First k stops are emergency stops) ≤ (0 . k which can be made less than ε/
10 if k is large enough. Next we have P (cid:96) ( k ∗ < k, T ¯ I ≤ K ( ε ) I ) ≤ ε K ( ε )2 I with probability greater than 1 − ε andall other tries take time O ( I ) since with overwhelming probability we start thosetries below level O ( √ I ) . This concludes the proof of Proposition 7.3.
IECEWISE SMOOTH FERMI-ULAM MODELS 19 Hyperbolic case. Dimension of accelerating orbits.
Proof of Theorem 6.
Foliate the phase space by line segments parallel to the un-stable direction of the limiting map ˆ
F .
It suffices to show that, given s < I such that if Γ is a leaf of our foliation and I ≥ ¯ I on γ then HD(Γ ∩ E ) > s. By Theorem 2 the limiting map satisfies CLT. That is, for any unstable curve γ , if the initial conditions are distributed uniformly on γ then ˆ I n − I √ n converges toa normal distribution with zero mean and some variance D (here we are using thenotation ˆ F n ( τ , I ) = (ˆ τ n , ˆ I n )). In particular there exists a constant κ > n , we have(23) P (cid:96) ( ˆ I n − I > κ √ n ) > (cid:96) denotes the standard pair ( γ, Const) . Moreover, given δ < ¯ δ , we can find n so that (23) holds uniformly for all curves of length between δ and ¯ δ. Let ˆ r n ( x )denote the distance from ˆ F n x to the boundary of the component of ˆ F n Γ contain-ing ˆ F n x. By the Growth Lemma (Lemma 7.1) if δ is sufficiently small than forsufficiently large n we have P (cid:96) (ˆ r n ( x ) < δ ) < γ is longer than δ. By Theorem 1 we can take ¯ I so large that if I > ¯ I on γ then P (cid:96) ( I n − I > κ √ n and r n ( x ) > δ ) ≥ F ( I , τ ) = ( I n , τ n ) and r n ( x ), as before, denotes the distance from F n x tothe boundary of the component of F n Γ containing F n x. Note that any curve oflength greater than 3 δ can be decomposed as a disjoint union of curves with lengthsbetween δ and 2 δ. Hence F n γ ⊃ (cid:83) j γ j where on each γ j the action grew up by atleast κn and the total measure of (cid:83) j F − n γ j is at least mes( γ ) / . Next, supposethat δ ≤ | γ | ≤ δ. Then we have | F − n γ j | > λ + ε ) n | γ | and the number of curvesis at least ( λ − ε ) n where λ is the expansion coefficient of ˆ F and ε can be madeas small as we wish by taking ¯ I large.Continuing this procedure inductively we construct a Cantor set inside Γ suchthat each interval has at least ( λ − ε ) n children and ratios of the lengths ofchildren to the length of the parent are at least λ + ε ) n . It follows that the resultingCantor set has dimension at leastln (cid:0) ( λ − ε ) n (cid:1) ln (2( λ + ε ) n ) . This number can be made as close to 1 as we wish by taking n large and thentaking ¯ I large to make ε as small as needed. (cid:3) Remark . The Cantor set above is constructed by taking as children the sub-interval where energy grows by κ √ n . However, the same estimate remains validif we take sometimes children with increasing energy and sometimes children withdecreasing energy as long as I n always stays above ¯ I. For example we can requirethat the energy grows until it reaches 2 ¯ I then decays until it falls below I thengrows above 3 ¯ I then decays below I then grows above 4 ¯ I etc. Then the argumentpresented above shows that the set of oscillatory orbits has full Hausdorff dimension. We expect that this set has also positive measure but the proof of this fact seemsout reach at the moment. 9.
Conclusions.
In this paper we considered piecewise smooth Fermi-Ulam ping pong systems.Near infinity this system can be represented as a small perturbation of the identitymap. Small smooth perturbations of the identity were studied in the context of inner[21] and outer ([17]) billiards. In this case, after a suitable change of coordinates,the problem can be reduced to the study of small perturbations of the map τ n +1 = τ n + ω ( J n ) , J n +1 = J n . This map is integrable so the above mentioned problems fall in the context ofsmall smooth perturbations of integrable systems (i.e. KAM theory). In the caseof piecewise smooth perturbations the normal form also exists: it is a piecewiselinear map of a torus. However in contrast with the smooth case the dynamicsof the limiting map is much more complicated and, in fact, it is not completelyunderstood, especially then the linear part is not hyperbolic. In this paper wedescribed for a simple model example:(i) how to obtain the limiting map and(ii) how the properties of the limiting map can be translated to results about thediffusion for the actual systems.However, there are plenty of open question on both stages of this procedure. Forexample, for piecewise smooth Fermi-Ulam ping pongs it is unknown if there is apositive measure set of oscillatory orbits such thatlim inf v n < ∞ , lim sup v n = ∞ in fact no such orbit is known for ∆ ∈ (0 , . This demonstrates that more effortis needed in order to develop a general theory of piecewise smooth near integrablesystems.
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Figure 3.
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Jacopo De Simoi, Dipartimento di Matematica, II Universit`a di Roma (Tor Vergata),Via della Ricerca Scientifica, 00133 Roma, Italy.
E-mail address : [email protected] Dmitry Dolgopyat, Department of Mathematics, University of Maryland, 4417 Math-ematics Bldg, College Park, MD 20742, USA
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