Certain systems of three falling balls satisfy the Chernov-Sinai Ansatz
aa r X i v : . [ m a t h . D S ] S e p CERTAIN SYSTEMS OF THREE FALLING BALLS SATISFY THECHERNOV-SINAI ANSATZ
MICHAEL HOFBAUER-TSIFLAKOS
Abstract.
The system of falling balls is an autonomous Hamiltonian systemwith a smooth invariant measure and non-zero Lyapunov exponents almosteverywhere. For almost three decades now, the question of its ergodicity re-mains open. We contribute to the solution of the ergodicity conjecture forthree falling balls with a specific mass ratio in the following two points: First,we prove the ChernovSinai ansatz. Second, we prove that there is an abun-dance of sufficiently expanding points. It is of special interest that for theaforementioned specific mass ratio, the configuration space can be unfolded toa billiard table, where the proper alignment condition holds.
Contents
1. Introduction 12. Main results 43. The system of three falling balls 54. Lyapunov exponents 75. Ergodicity 105.1. Local Ergodicity 105.2. Abundance of sufficiently expanding points 146. Strict unboundedness - Part I 147. Particle falling in a wedge 187.1. Wide wedges 197.2. Projection 217.3. Proper alignment in wide wedges 228. Strict unboundedness - Part II 22Acknowledgements 25References 251.
Introduction
The system of falling balls was introduced by Wojtkowski [W90a, W90b]. Itdescribes the motion of N ≥ q , . . . , q N , momenta p , . . . , p N , and masses m , . . . , m N , moving up and down a vertical line and collid-ing elastically with each other. The bottom particle collides elastically with a rigidfloor placed at position q = 0. The standing assumptions are 0 ≤ q ≤ . . . ≤ q N and m ≥ . . . ≥ m N , m = m N . For convenience, we will refer to the point parti-cles as balls. The system is an autonomous Hamiltonian system, with Hamiltonian Mathematics Subject Classification.
Primary 37D50; Secondary 37J10. given by the sum of the kinetic and linear potential energy of each ball. It pos-sesses a smooth invariant measure with respect to the Hamiltonian flow and withrespect to a suitable Poincar´e map T , describing the movement of the balls fromone collision to the next. We denote the underlying Poincar´e section for this mapby M + and its invariant measure by µ .One aspect that makes the investigation of the dynamics cumbersome is thepresence of singularities. These are codimension one manifolds in phase space, onwhich the dynamics are not well-defined, in particular it has two different images.A point belongs to the singularity manifold, if its next collision is either betweenthree balls or two balls with the floor.Dynamicists first tried to answer the question whether the system of N , N ≥ N − N , N ≥
2, fallingballs, µ -a.e. point x ∈ M + has non-zero Lyapunov exponents [S96]. In [W98]Wojtkowski found an elegant way of proving the existence of non-zero Lyapunovexponents for a large class of falling balls systems. He first considers balls fallingnext to each other on a moving floor. By applying concrete stacking rules it ispossible to obtain a variety of falling ball systems, such as the original one intro-duced in [W90a] as a special case. The study of hyperbolicity is carried out byequivalently looking at the system of a particle falling in a wedge.The underlying motivation of this work is to contribute to the solution of thelong time open problem of ergodicity for three or more balls. For two balls, thesystem is already known to be ergodic [LW92, p. 70 -72], provided m > m . Sincethe system of three falling balls has non-zero Lyapunov exponents everywhere, thetheory of Katok-Strelcyn [KS86] yields, that the phase space partitions into at mostcountably many components on which the conditional smooth measure is ergodic.A reliable method to check the ergodicity of such systems is the local ergodictheorem [ChS87, KSSz90, LW92]. In the present work we will follow the localergodic theorem version of Liverani and Wojtkoswki [LW92]. For its application,the local ergodic theorem needs the following five conditions to hold, namely,(1) Chernov-Sinai ansatz,(2) Non-contraction property,(3) Continuity of Lagrangian subspaces,(4) Regularity of singularity sets,(5) Proper Alignment.The validity of these conditions guarantees the existence of an open neighbourhood,around a point with non-vanishing Lyapunov exponents, that lies (mod 0) in oneergodic component. To prove, that there is only one ergodic component needsthe validity of a transitivity argument. Namely, the set of points with a sufficientamount of expansion must have full measure and be arcwise connected. We willrefer to this property as the abundance of sufficiently expanding points. If the latteris true, one can build a chain of the aforementioned open neighbourhoods from any point with sufficient expansion to another. These neighbourhoods intersectpairwise on a subset of positive measure and, hence, there can only be one ergodiccomponent. For three or more balls only condition 3 is known [LW92] to be true.In their approach to ergodicity, Liverani and Wojtkowski introduced [LW92] theproperty of (strict) unboundedness for a sequence of derivatives ( d T n x T ) n ∈ N . Itroughly says, that the expansion (measured with respect to an indefinite quadraticform Q ) of any vector from the contracting cone field goes to infinity. In their termi-nology, it follows immediately that if ( d T n x T ) n ∈ N is strictly unbounded everywherethen the Chernov-Sinai ansatz holds. Additionally, the abundance of sufficientlyexpanding points follows as a simple corollary.The proof of the strict unboundedness property for every phase point is the maintask of this work (see Section 2 for more details). For this, we will partially usetechniques introduced in [W98], which allow us to study the system of falling ballsas a particle falling in a wedge. The results obtained from the latter analysis willbe used to slightly modify the approach to strict unboundedness in [LW92] for ourneeds.Another important issue, which we clarify in a separate subsection is the stateof the proper alignment condition (see Subsection 5.1.1). By some experts it hasbeen wrongly assumed not to hold. We will thoroughly explain that this conditioncan still be verified and is, thus, an open problem. Further, we will use the strictunboundedness property to analyze in Subsection 5.1.2, how the set of not properlyaligned points behaves under sufficiently large iterates. We point out that for a spe-cific mass ratio the configuration space of the falling balls systems can be unfoldedto a billiard table where the proper alignment condition holds (see Subsection 7.3).The latter was discovered by Wojtkowski [W16].On the same subject, Chernov formulated [Ch93], in the realm of semi-dispersingbilliards, a transversality condition, which can serve as a substitute for the properalignment condition. We will show, that in the framework of symplectic maps,Chernov’s transversality condition follows from the proper alignment condition (seeLemma 5.2).The paper is organized in the following way:In Section 2 we briefly summarize the main results of this paper, which are thestrict unboundedness for every orbit, the Chernov-Sinai ansatz and the abundanceof sufficiently expanding points. It will also be shown, that the latter two resultsfollow at once from the strict unboundedness property of every orbit.In Section 3 we introduce the system of three falling balls.In Section 4 we recall the standard method for studying Lyapunov exponents inHamiltonian systems [W91] and recall what has been done for the system of fallingballs so far.In Section 5 we explain the matter of ergodicity. It contains a detailed dis-cussion of the local ergodic theorem, the proper alignment condition, Chernov’stransversality condition and the abundance of sufficiently expanding points.In Section 6 we begin with the first part of the proof of the strict unboundednessproperty. This section is completely written in the language of Liverani and Wo-jtkowski [LW92] and explains how we use our new results in order to modify theirproof of the unboundedness property.In Section 7 we introduce the system of a particle falling in a three dimensionalwedge from [W98]. Its necessity stems from the fact, that for a special type of MICHAEL HOFBAUER-TSIFLAKOS wedges this system is equivalent to the system of falling balls with particular masses.In the last subsection we will explain that the proper alignment condition is validin these special wedges.In Section 8 we utilize the results of Section 6 and 7 to complete the proof of thestrict unboundedness property. 2.
Main results
Denote by M + the phase space, which is partitioned (mod 0) into subsets M + i , i = 1 , ,
3, where each subset describes the moment right after collision of balls i − i . For i − q = 0. Themasses satisfy m > m > m and the special relation given in (7.9). Let T : M + (cid:9) be the Poincar´e map, describing the movement from one collision to thenext. After applying Wojtkowski’s convenient coordinate transformation ( q, p ) → ( h, v ) → ( ξ, η ) (see (4.1), (4.2)) we get a contracting cone field C ( x ) = { ( δξ, δη ) ∈ R × R : Q ( δξ, δη ) > , δξ = 0 , δη = 0 } ∪ { ~ } , where ( δξ, δη ) denote the coordinates in tangent space. The cone field is definedby the quadratic form Q ( δξ, δη ) = X i =1 δξ i δη i . Denote by C ( x ) the closure of the cone C ( x ) and let d x T n = d T n x T . . . d
T x
T d x T .The sequence or, more accurately, derivatives along the orbit( d T n x T ) n ∈ N = ( d x T, d
T x
T, d T x T, . . . ) , is called unbounded, iflim n → + ∞ Q ( d x T n v ) = + ∞ , ∀ v ∈ C ( x ) \ { ~ } , and strictly unbounded, iflim n → + ∞ Q ( d x T n v ) = + ∞ , ∀ v ∈ C ( x ) \ { ~ } . Main Theorem.
For every x ∈ M + , we have lim n → + ∞ Q ( d x T n ( δξ, δη )) = + ∞ , for all ( δξ, δη ) ∈ C ( x ) \ { ~ } . Due to Proposition 6.2 and Theorem 6.8 of [LW92], the Main Theorem alsoimplies the strict unboundedness for the orbit in negative time ( d T n x T ) n ∈ Z − (withrespect to the complementary cone field of C ( x )).The singularity manifold on which T resp. T − is not well-defined is given by S + resp. S − . Let µ S + resp. µ S − be the measures induced on the codimension onehypersurfaces S + resp. S − , from the smooth T -invariant measure µ .The validity of the Main Theorem immediately establishes the Chernov-Sinaiansatz, which is one of the conditions of the Local Ergodic Theorem. Chernov-Sinai ansatz.
For µ S − -a.e. x ∈ S − , we have lim n → + ∞ Q ( d x T n ( δξ, δη )) = + ∞ , for all ( δξ, δη ) ∈ C ( x ) \ { ~ } . The least expansion coefficient σ , for n ≥ x ∈ M + , is defined as σ ( d x T n ) = inf v ∈C ( x ) s Q ( d x T n v ) Q ( v ) . A point x ∈ M + , is called sufficiently expanding, if there exists n = n ( x ) ≥
1, suchthat σ ( d x T n ) > Abundance of sufficiently expanding points.
The set of sufficiently expandingpoints has full measure and is arcwise connected.
The abundance of sufficiently expanding points follows at once from the MainTheorem (see Theorem 4.4) and the proper alignment property (see Subsection7.3). The validity of the latter guarantees that the set of double singular collisionsis of codimension two.3.
The system of three falling balls
Let q i = q i ( t ) be the position, p i = p i ( t ) the momentum and v i = v i ( t ) thevelocity of the i -th ball. The balls are aligned on top of each other and are thereforeconfined to N ( q, p ) = { ( q, p ) ∈ R × R : 0 ≤ q ≤ q ≤ q } . The momenta and the velocities are related by p i = m i v i . We assume that themasses m i satisfy m > m > m . The movements of the balls are a result of alinear potential field and their kinetic energies. The total energy of the system isgiven by the Hamiltonian function H ( q, p ) = X i =1 p i m i + m i q i . The Hamiltonian equations are ˙ q i = p i m i , ˙ p i = − m i . (3.1)The dots indicate differentiation with respect to time t and the Hamiltonianvector field on the right hand side will be denoted as X H ( q, p ). The solutions tothese equations are q i ( t ) = − t t p i (0) m i + q i (0) ,p i ( t ) = − tm i + p i (0) , (3.2)which form parabolas in ( t, q i ( t )) ⊂ R × R + . It is clear from the choice of the linearpotential field, that the acceleration of each ball points downwards and, thus, theseparabolas cannot escape to infinity. Hence, for every initial condition ( q, p ) the MICHAEL HOFBAUER-TSIFLAKOS balls go through every collision in finite time and, thus, every collision happensinfinitely often. The energy manifold E c and its tangent space T E c are given by E c = { ( q, p ) ∈ R × R : H ( q, p ) = X i =1 p i m i + m i q i = c } , T ( q,p ) E c = { ( δq, δp ) ∈ R × R : d ( q,p ) H ( δq, δp ) = X i =1 p i δp i m i + m i δq i = 0 } . Including the restriction of the balls positions amounts to E c ∩ N ( q, p ).The Hamiltonian vector field (3.1) gives rise to the Hamiltonian flow φ : R × E c ∩ N ( q, p ) → E c ∩ N ( q, p ) , ( t, ( q, p )) φ ( t, ( q, p )) . For convenience, the image will also be written with the time variable as superscript,i.e. φ ( t, ( q, p )) = φ t ( q, p ).The standard symplectic form ω = P i =1 dq i ∧ dp i induces the symplectic volumeelement Ω = V i =1 dq i ∧ dp i . The volume element on the energy surface is obtainedby contracting Ω, by a vector u , where u is any vector satisfying dH ( u ) = 1.Denoting the contraction operator by ι , the volume element on the energy surfaceis given by ι ( u )Ω. Since the flow preserves the standard symplectic form, it preservesthe volume element and, hence, the Liouville measure ν on E c ∩ N ( q, p ) obtainedfrom it. We define the Poincar´e section, which describes the states right after acollision as M + = M +1 ∪ M +2 ∪ M +3 , with M +1 := { ( q, p ) ∈ E c ∩ N ( q, p ) : q = 0 , p /m ≥ } , M + i := { ( q, p ) ∈ E c ∩ N ( q, p ) : q i − = q i , p i − /m i − ≤ p i /m i } , i = 2 , . In the same way we define the set of states right before collision M − = M − ∪M − ∪ M − , by M − := { ( q, p ) ∈ E c ∩ N ( q, p ) : q = 0 , p /m < } , M − i := { ( q, p ) ∈ E c ∩ N ( q, p ) : q i − = q i , p i − /m i − > p i /m i } , i = 2 , . The ’+’ resp. ’-’ superscript refer to the states right after resp. before collision.The system of falling balls is considered as a hard ball system with fully elasticcollisions. During a collision of the balls i and i + 1 the momenta resp. velocitieschange according to p + i = γ i p − i + (1 + γ i ) p − i +1 ,p + i +1 = (1 − γ i ) p − i − γ i p − i +1 ,v + i = γ i v − i + (1 − γ i ) v − i +1 ,v + i +1 = (1 + γ i ) v − i − γ i v − i +1 , (3.3)where γ i = ( m i − m i +1 ) / ( m i + m i +1 ), i = 1 ,
2, and when the bottom particlecollides with the floor the sign of its momentum resp. velocity is simply reversed p +1 = − p − ,v +1 = − v − . (3.4) These collision laws are described by the linear collision mapΦ i − ,i : M − → M + , ( q, p − ) ( q, p + ) . We will write Φ if we do not want to refer to any specific collision. Let τ : M → R + be the first return time to M − . We define the Poincar´e map as T : M + → M + , ( q, p ) Φ ◦ φ τ ( q,p ) ( q, p ) .T is the collision map, that maps from one collision to the next. On M + , weobtain the volume element ι ( X H ) ι ( u )Ω, by contracting the volume element ι ( u )Ωon the energy surface with respect to the direction of the flow X H . This exteriorform defines a smooth measure µ on M + , which is T -invariant. Our dynamicalsystem can be stated as the triple ( M + , T, µ ). Matching the present state with thenext collision in the future (+) resp. the past (-), we obtain two (mod 0) partitionsof M + with elements M ± , = { x ∈ M +1 : T ± x ∈ M +1 } , M ± i,j = { x ∈ M + i : T ± x ∈ M + j } , i, j ∈ { , , } , j = i. It can be calculated, that µ ( M ± i,j ) >
0. The system of falling balls possessescodimension one singularity manifolds S +1 , = M +1 , ∩ M +1 , , S − , = M − , ∩ M − , , S +1 , = M +1 , ∩ M +1 , , S − , = M − , ∩ M − , , S +3 , = M +3 , ∩ M +3 , , S − , = M − , ∩ M − , . The states in S ± , face a triple collision next, while the states in S ± , , S ± , ex-perience a collision of the lower two balls with the floor next. The maps T resp. T − are not well-defined on the sets S +1 , , S +1 , , S +3 , resp. S − , , S − , , S − , , becausethey have two different images. This happens because the compositions Φ , ◦ Φ , and Φ , ◦ Φ , do not commute. When the trajectory hits a singularity, we willcontinue the system on both branches separately. In this way, the results obtainedin this work hold for every point.We abbreviate S ± = S ± , ∪ S ± , ∪ S ± , , S ± n = S ± ∪ T ∓ S ± ∪ . . . ∪ T ∓ ( n − S ± . Lyapunov exponents
We subject our system to two well-discussed coordinate transformations ( q, p ) → ( h, v ) → ( ξ, η ) introduced in [W90a]. The first one is given by h i = p i m i + m i q i , v i = p i m i , (4.1)while the second one is a linear coordinate transformation( ξ, η ) = ( A − h, A T v ) , (4.2) MICHAEL HOFBAUER-TSIFLAKOS where A is an invertible matrix depending only on the masses m i [W90a, p. 520].The energy manifold and its tangent space take the form E c = { ( ξ, η ) ∈ R × R : H ( ξ, η ) = ξ = c } , T E c = { ( δξ, δη ) ∈ R × R : dH ( δξ, δη ) = δξ = 0 } . The Hamiltonian vector field X H ( ξ, η ) = (0 , , , − , ,
0) becomes constant. Inthese coordinates, the derivative of the flow dφ t equals the identity map. Thus,only the derivatives of the collision maps d Φ i − ,i are relevant to the dynamics intangent space. In these coordinates the derivatives of the collision maps are givenby d Φ , = (cid:18) id B id (cid:19) , d Φ , = (cid:18) M U M T (cid:19) , d Φ , = (cid:18) M U M T (cid:19) . where B = β
00 0 0 , U = − α
00 0 0 , U = − α , id = , M = − γ , M = − γ − . The terms in the matrices are given by β = − m v − , α i = 2 m i m i +1 ( m i − m i +1 )( v − i − v − i +1 )( m i + m i +1 ) . (4.3)A Lagrangian subspace V is a linear space of maximal dimension on whichthe symplectic form vanishes. In general, every vector v ∈ R can be uniquelydecomposed by a pair of two given transversal Lagrangrian subspaces ( V , V ), i.e. v = v + v , v i ∈ V i , i = 1 ,
2. For a pair of transversal Lagrangian subspaces ( V , V )we can define a quadratic form Q by Q : R → R v Q ( v ) = ω ( v , v )The canonical pair of transversal Lagrangian subspaces in R is given by W = { ( δξ, δη ) ∈ R × R : δη = δη = δη = 0 } ,W = { ( δξ, δη ) ∈ R × R : δξ = δξ = δξ = 0 } . Restricting both to T E and excluding the direction of the flow gives L = { ( δξ, δη ) ∈ R × R : δξ = 0 , δη i = 0 , i = 1 , , } ,L = { ( δξ, δη ) ∈ R × R : δη = 0 , δξ i = 0 , i = 1 , , } . (4.4)For the pair ( L , L ), the quadratic form Q becomes the Euclidean inner product Q ( δξ, δη ) = h δξ, δη i . We see immediately that Q ( L i ) = 0. Also, Q is continuous and homogeneous ofdegree two. Using the quadratic form Q we can define the open cones C ( x ) = { ( δξ, δη ) ∈ L ⊕ L : Q ( δξ, δη ) > } ∪ { ~ } , C ′ ( x ) = { ( δξ, δη ) ∈ L ⊕ L : Q ( δξ, δη ) < } ∪ { ~ } . Denote by C ( x ) the closure of the cone C ( x ). Definition 4.1.
1. The cone field {C ( x ) , x ∈ M + } , is called invariant for x ∈ M + ,if d x T C ( x ) ⊆ C ( T x ) ,
2. The cone field {C ( x ) , x ∈ M + } , is called eventually strictly invariant for x ∈M + , if there exists a k ≥
1, such that d x T k C ( x ) ⊂ C ( T k x ) .
3. The monodromy map d x T is called Q -monotone for x ∈ M + , if Q ( d x T ( δξ, δη )) ≥ Q ( δξ, δη ) , for all ( δξ, δη ) ∈ L ⊕ L .4. The monodromy map d x T is called eventually strictly Q -monotone for x ∈ M + ,if there exists a k ≥
1, such that Q ( d x T k ( δξ, δη )) > Q ( δξ, δη ) , for all ( δξ, δη ) ∈ L ⊕ L \ { ~ } .Statement resp. is equivalent to statement resp. (see e.g. [LW92,Theorem 4.1]). The following lemma establishes eventual strict Q -monotonicity byusing only the evolution of the Lagrangian subspaces L and L (see e.g. [W90a,Lemma 2]). Lemma 4.2.
The monodromy map d x T is eventually strictly Q -monotone for x ∈M + , if there exists k ≥ , such that for all ( δξ, ∈ L and (0 , δη ) ∈ L , Q ( d x T k ( δξ, > and Q ( d x T k (0 , δη )) > . In order to get non-zero Lyapunov exponents Wojtkowski introduced [W90a, p.516] a criterion, which links eventual strict Q -monotonicity to nonuniform hyper-bolic behaviour Q-Criterion. If d x T is eventually strictly Q -monotone for µ -a.e. x ∈ M + , thenall Lyapunov exponents, except for two , are non-zero. For N , N ≥
2, balls, Wojtkowski proved [W90a], that d x T is Q -monotone forevery point in M + . Wojtkowski strengthened this statement in the case of threeballs with upward decreasing masses, by proving eventual strict Q -monotonicity forevery point in M + [W90a, Proposition 3]. Afterwards Sim´anyi proved [S96], that d x T is eventually strictly Q -monotone for µ -a.e. x ∈ M + and an arbitrary numberof balls.We close this subsection by formulating the (strict) unboundedness property andthe least expansion coefficient, which will be used to establish criteria for ergodicity.The least expansion coefficient σ , for n ≥ x ∈ M + , is defined as σ ( d x T n ) = inf v ∈C ( x ) s Q ( d x T n v ) Q ( v ) . (4.5) The exceptional directions with zero Lyapunov exponents are the direction of the flow andthe ones contained in the subset { v : dH ( v ) = 0 } . Even though Proposition 3 in [W90a] is stated for almost every point x ∈ M + , the readerwill discover, when carefully reading the proof, that it actually holds for every x ∈ M + . Definition 4.3.
1. The sequence ( d T n x T ) n ∈ N is called unbounded, iflim n → + ∞ Q ( d x T n v ) = + ∞ , ∀ v ∈ C ( x ) \ { ~ } .
2. The sequence ( d T n x T ) n ∈ N is called strictly unbounded, iflim n → + ∞ Q ( d x T n v ) = + ∞ , ∀ v ∈ C ( x ) \ { ~ } . The least expansion coefficient and the property of strict unboundedness relateto each other in the following way
Theorem 4.4 (Theorem 6.8, [LW92]) . The sequence ( d T n x T ) n ∈ N is strictly un-bounded if and only if lim n → + ∞ σ ( d x T n ) = + ∞ . Ergodicity
The theory of Katok-Strelcyn [KS86] implies, that since our system has non-zeroLyapunov exponents almost everywhere, we can partition the phase space M + intocountably many components on which the conditional smooth measure is ergodic.To prove that there is only one ergodic component the following two points needto be verified(1) Local Ergodicity.(2) Abundance of sufficiently expanding points.5.1. Local Ergodicity.
We start with the following
Definition 5.1.
A compact subset X ⊂ M + , is called regular if(1) X = S ni =1 I i , where I i are compact submanifolds, with I i = int I i ,(2) dim I i = 3,(3) I i ∩ I j ⊂ ∂I i ∪ ∂I j , i = j ,(4) ∂I i = S mj =1 H i,j , where dim H i,j = 2 and H i,j is compact.Local ergodicity amounts to showing that around a point with least expansioncoefficient larger than three, it is possible to find an open neighbourhood, which lies(mod 0) in one ergodic component. To claim this, one needs to check the followingfive conditions Condition 1 (Regularity of singularity sets) . The singularity sets S + n and S − n areboth regular sets for every n ≥ . Condition 2 (Non-contraction property) . There exists ζ > , such that for every n ≥ , x ∈ M + \ S + n , and ( δξ, δη ) ∈ C ( x ) , we have k d x T n ( δξ, δη ) k ≥ ζ k ( δξ, δη ) k . Condition 3 (Chernov-Sinai Ansatz) . For µ S − -a.e. x ∈ S − , we have lim n → + ∞ Q ( d x T n ( δξ, δη )) = + ∞ , for all ( δξ, δη ) ∈ C ( x ) . Condition 4 (Continuity of Lagrangian subspaces) . The ordered pair of transversalLagrangian subspaces ( L ( x ) , L ( x )) varies continuously in int M + . Condition 5 (Proper Alignment) . There exists N ≥ , such that for every x ∈ S + resp. S − , we have d x T − N v + x resp. d x T N v − x belong to C ′ ( T − N x ) resp. C ( T N x ) ,where v + x resp. v − x are the characteristic lines of T x S + resp. T x S − . At the moment, for three or more falling balls, only Condition 4 has been verified.This is in fact easy to see, because the canonical pair of transversal Lagrangiansubspaces (4.4) does not depend on the base point x and is therefore constant in M + . Note, that Conditions 2 and 3 also have to hold in negative time. Local Ergodic Theorem.
If Conditions 1 - 5 are satisfied, then for any x ∈ M + and n ≥ , such that σ ( d x T n ) > , there exists an open ergodic neighbourhood U ( x ) , that lies (mod 0) in one ergodic component. Chernov postulated in [Ch93] a weaker condition to Condition 5. Denote by W u ( x ) resp. W s ( x ) the unstable resp. stable manifolds at point x . Condition 6 (Transversality) . For µ S ± -a.e. x , the stable subspace W s ( x ) resp.unstable subspace W u ( x ) is transversal to S − resp. S + . Lemma 5.2.
The proper alignment condition implies the transversality condition.Proof.
Assume that at point x ∈ S − , the singularity manifold and the stable man-ifold W s ( x ) are not transversal but still properly aligned, i.e. T W s ( x ) ⊂ T S − and v − x ∩ T W s ( x ) = ∅ . Since transversality is not satisfied and v − x is the characteristicline, we have ω ( v − x , v ) = 0, for all v ∈ T W s ( x ). This means, that v − x ∈ ( T W s ( x )) ⊥ ω ,where ( T W s ( x )) ⊥ ω is the ω -orthogonal complement of T W s ( x ). But T W s ( x ) is aLagrangian subspace and, thus, ( T W s ( x )) ⊥ ω = T W s ( x ). Hence, v − x ∈ T W s ( x ), acontradiction. (cid:3) Even though the proper alignment implies transversality, it is presently unclearwhether it is enough for the local ergodic theorem (in the Liverani-Wojtkowskiframework) to hold by considering the validity of the proper alignment conditiononly on a set of full measure with respect to the measure µ S ± .5.1.1. The current state of proper alignment.
There has been a substantial miscon-ception whether the system of falling balls is properly aligned or not. In brief, thecorrect answer to this question is that on some part of the singularity manifoldthe system is properly aligned and on the complementary part we simply do notknow. The latter affects only the singularity manifolds S ± , , since every point on S ± , and S ± , is properly aligned. The original formulation of the proper alignmentcondition in [LW92] is more restrictive than the one stated above. Namely, it de-mands the characteristic line v − x resp. v + x to lie in C ( x ) resp. C ′ ( x ) for every pointof the singularity manifolds. Below of the original proper alignment condition itsays ([LW92, p. 37])It will be clear from the way in which the proper alignment ofsingularity sets is used in Section 12 that it is sufficient to assumethat there is N such that T N S − and T − N S + are properly aligned. The characteristic line v ± x is a vector of T x S ± that has the property of annihilating everyother vector w ∈ T x S ± with respect to the symplectic form ω , i.e. ω ( v ± x , w ) = 0, ∀ w ∈ T x S ± . Al-ternatively stated, it is the ω -orthogonal complement of T x S ± . Note, that in symplectic geometrythe ω -orthogonal complement of a codimension one subspace is one dimensional. In Section 12 of [LW92] the authors remind the reader, that, in their constructiveargument, the size of the neighbourhood U ( x ), appearing in the Local ErgodicTheorem, was chosen small enough, such that U ( x ) ∩S − N = ∅ . Due to the regularityof singularity manifolds (see Condition 1), for every M > N , there exists a finite p = p ( M ) >
0, such that S Mi = N T i S − = S pk =1 I k , where I k are compact submanifolds(see Definition 5.1). In the proof of Proposition 12.2, Liverani and Wojtkowskimake use of the fact, that every point x ∈ I k is properly aligned (see [LW92, p.185]). Hence, the relaxed version of the proper alignment condition (see Condition5) is justified.The authors continue (see [LW92, p. 37]) with the following assertionWe will show, in section 14, that for the system of falling balls eventhis weaker property (see Condition 5) fails.The content of the last quotation is wrong. We will now illustrate what Liverani andWojtkowski really did in section 14: The argument is carried out for the singularitymanifold S − , . The characteristic line at point x ∈ S − , is given by v − x = { ( δq, δp ) ∈ T x S − , : δq = δq = δq = 0 , X i =1 δp i = 0 , X i =1 p i δp i m i = 0 , p m ≤ p m ≤ p m } . The restrictions of the momenta follow from S − , ⊂ M +2 ∩ M +3 . We will look at theset of momenta in a little bit more detail: Without loss of generality let t < t , x = x ( t ) ∈ S − , and T x = x ( t ) ∈ M +1 . Since p +1 ( t ) /m ≤ p +2 ( t ) /m ≤ p +3 ( t ) /m ,applying the equations of motion (3.2) yields p − ( t ) /m ≤ p − ( t ) /m ≤ p − ( t ) /m .Due to x ( t ) ∈ M +1 , we have p − ( t ) /m <
0. Incorporating the latter, we (mod 0)partition the set of eligible momenta at time t into the subsets Mom ( q ( t ) , p − ( t )) = n p − ( t ) m < ≤ p − ( t ) m ≤ p − ( t ) m o , Mom ( q ( t ) , p − ( t )) = n p − ( t ) m < p − ( t ) m ≤ ≤ p − ( t ) m o , Mom ( q ( t ) , p − ( t )) = n p − ( t ) m < p − ( t ) m ≤ p − ( t ) m ≤ o . Using again the equations of motion, we obtain in time t Mom ( q ( t ) , p + ( t )) = n p +1 ( t ) m < t − t ≤ p +2 ( t ) m ≤ p +3 ( t ) m o , Mom ( q ( t ) , p + ( t )) = n p +1 ( t ) m < p +2 ( t ) m ≤ t − t ≤ p +3 ( t ) m o , Mom ( q ( t ) , p + ( t )) = n p +1 ( t ) m < p +2 ( t ) m ≤ p +3 ( t ) m ≤ t − t o . Observe that all the momenta can only be simultaneously negative on the set
Mom ( q ( t ) , p + ( t )).The quadratic form Q of the contracting cone field in coordinates ( q, p ) equals Q ( δq, δp ) = X i =1 δq i δp i + p i ( δp i ) m i . Inserting v − x into Q results in Q ( v − x ) = X i =1 p i ( δp i ) m i . (5.1)Ths singularity manifold S − , at point x is properly aligned if and only if Q ( v − x ) ≥ Mom i ( q ( t ) , p + ( t )) contains a subset on which S − , is not properly aligned, i.e. Q ( v − x ) <
0. Hence, depending on the point x ∈ S − , ,(5.1) can obtain non-negative and negative values on every set Mom i ( q ( t ) , p + ( t )).Additionally note, that the image of the characteristic line is the characteristicline of the image, i.e. d x T n v − x = v − T n x . (5.2)Combining this with the fact, that d x T is Q -monotone for every point x ∈ M + (see Definition 4.1.3) we obtain, that once a point is properly aligned, it remainsproperly aligned.We summarize, that on some parts of S − , the system of falling balls is properlyaligned and on the complement we do not know, since an iterate of the characteristicline could very well be mapped into the contracting cone field. This is exactly whatLiverani and Wojtkowski prove in section 14. More importantly, they do not examine whether any iterate of v − x gets mapped into the contracting cone field ornot. This is currently not known.5.1.2. Iterates of the characteristic line.
The Main Theorem allows us to comparethe set of iterated singular points, which are not properly aligned, to not properlyaligned points of the iterated singularity manifold. For this, an immediate con-sequence of the Main Theorem is, that the monodromy matrix d x T is eventuallystrictly Q -monotone for every point (see e.g. (6.1b) in Theorem 6.1), i.e. for every x ∈ M + , there exists k = k ( x ) ≥ Q ( d x T k v ) > Q ( v ), for all v ∈ L ⊕ L . Define,for n ≥
1, the sets A ( n, S − , ) = { x ∈ S − , : Q ( v − x ) < , Q ( d x T n v ) > Q ( v ) , ∀ v ∈ L ⊕ L } , [ n ≥ A ( n, S − , ) = A ( S − , ) . The sets A ( n, S − , ) consist of all points in S − , , which are not properly alignedand have an eventually strictly Q -monotone monodromy matrix after n steps. Weremark, that the sets A ( n, S − , ) are empty for small values of n . Once A ( n, S − , ) = ∅ , the Q -monotonicity of d x T for every point implies that A ( n, S − , ) ⊆ A ( n +1 , S − , ). Due to the eventually strict Q -monotonicity of d x T , we have Q ( d T n x T − n v − T n x ) < Q ( v − T n x ) , ∀ T n x ∈ T n A ( n, S − , ) . Using the last statement together with (5.2), we obtain T − n A ( T n S − , ) ⊂ A ( n, S − , ) ⊂ A ( S − , ) . However, the size of T − n A ( T n S − , ) and whether there exists a fixed N ≥
1, suchthat A ( T N S − , ) = ∅ , remains unknown. Abundance of sufficiently expanding points.
Liverani and Wojtkowskirequire the point in the local ergodic theorem to have least expansion coefficientlarger than three. However, after their formulation of the local ergodic theoremthey point out (see [LW92, p. 39]) that there is no loss in generality in actuallydemanding that the least expansion coefficient is only larger than one. The reasonfor this is due to the fact, that the set of points with non-zero Lyapunov exponentshas full measure (see [S96], [W98]). We quoteLet us note that the conditions of the last theorem are satisfied foralmost all points p ∈ M . Indeed, let M n,ǫ = { p ∈ M| σ ( D p T n ) > ǫ } . Since almost all points are strictly monotone, then + ∞ [ n =1 [ ǫ> M n,ε has full measure. By the Poincar´e Recurrence Theorem and thesupermultiplicativity of the coefficient σ we conclude that + ∞ [ n =1 M n, has also full measure. Definition 5.3.
Under the assumption µ ( { x ∈ M + : ∃ n = n ( x ) ≥ , σ ( d x T n ) > } ) = 1, a point x ∈ M + is called sufficiently expanding, if there exists an n ≥ σ ( d x T n ) > Theorem 5.4 (Abundance of sufficiently expanding points) . The set of sufficientlyexpanding points has full measure and is arcwise connected.
More precisely, this implies, that one can connect any two sufficiently expandingpoints by a curve, which lies completely in the set of sufficiently expanding points.Consequently the points on the curve can be chosen in such a way, that the openneighbourhoods, from the local ergodic theorem, intersect pairwise on a set ofpositive measure. Hence, there can only be one ergodic component. For a moredetailed proof see e.g. [ChM06, p. 151 - 152].6.
Strict unboundedness - Part I
In this section we will begin with the proof of the strict unboundedness of thesequence ( d T n x T ) n ∈ N , for every x ∈ M + . Due to [LW92, Theorem 6.8] we have thefollowing equivalence Theorem 6.1.
For every x ∈ M + , the sequence ( d T n x T ) n ∈ N is strictly unboundedif and only ifFor every x ∈ M + , the sequence ( d T n x T ) n ∈ N is unbounded. (6.1a) For every x ∈ M + , there exist k , k ∈ N , such that Q ( d x T k ( δξ, > and Q ( d x T k (0 , δη )) > , for all ( δξ, ∈ L , (0 , δη ) ∈ L . We will prove the strict unboundedness by equivalently proving properties (6.1a)and (6.1b). Let k · k denote the Euclidean norm.The most important ingredient for (6.1a) is the following
Theorem 6.2.
There exists a positive constant Λ > , such that for all x ∈ M + ,there exists a sequence of strictly increasing positive integers ( n k ) k ∈ N = ( n k ( x )) k ∈ N and for all (0 , δη ) ∈ L : Q ( d T n k − x T n k − − n k − (0 , δη )) > Λ k (0 , δη ) k . (6.2)In fact, we will prove, that dT n k − − n k − either equals d Φ , d Φ , , d Φ , d Φ , , d Φ , d Φ , d Φ , or d Φ , d Φ , d Φ , . δξ n , δη n ) = dT ( δξ n − , δη n − ),with ( δξ , δη ) = ( δξ, δη ) and q n = Q ( δξ n , δη n ). From [W90a] we know, that d x T is Q -monotone for every x ∈ M + , therefore, q n +1 ≥ q n . Hence, in order to provelim n → + ∞ q n = + ∞ , it is enough to prove this divergence along a subsequence( q n k − ) k ∈ N . We define this subsequence by setting q n k − = Q ( d T n k − x T n k − − n k − ( δξ n k − , δη n k − )) . (6.3)We will postpone the proof of Theorem 6.2 and property (6.1b) to section 8, asthey will both follow from our analysis of a particle moving inside a wedge (seesection 7). Here we will show how Theorem 6.2 is utilized to prove the unbound-edness property (6.1a). In fact, (6.1a) will be obtained by using the estimate fromTheorem 6.2 in a modified version of the unboundedness proof in [LW92, p. 159 -160]. Beforehand we need to take some preparatory steps. Proposition 6.1.
For every x ∈ M + , we have q n k +1 > q n k + Λ k (0 , δη n k ) k . (6.4) Proof.
Without loss of generality let d T n k x T n k +1 − n k be the product of d Φ , d Φ , .Using (6.2), we estimate q n k +1 = Q ( d T n k x T n k +1 − n k ( δξ n k , δη n k ))= Q (cid:16)(cid:18) M M M U + U M T M T M T (cid:19) (cid:18) δξ n k δη n k (cid:19)(cid:17) = h M M δξ n k + ( M U + U M T ) δη n k , M T M T δη n k i = h M M δξ n k , M T M T δη n k i + Q (cid:16)(cid:18) M M M U + U M T M T M T (cid:19) (cid:18) δη n k (cid:19)(cid:17) > h δξ n k , δη n k i + Λ k (0 , δη n k ) k = q n k + Λ k (0 , δη n k ) k . (cid:3) Proposition 6.2.
Let ( a n k ) k ∈ N be a sequence of positive numbers and C a positiveconstant. If + ∞ X i =0 a n i = + ∞ then + ∞ X k =0 a n k C + P ki =0 a n i = + ∞ . The results remain valid if we allow multiple collisions with the floor, i.e. d Φ k , for every k ≥ Proof.
For 1 ≤ j ≤ l , we have l X k = j a n k C + P ki =0 a n i > P lk = j a n k C + P j − i =0 a n i + P li = j a n i → , as l → + ∞ . The tail of the series does not tend to zero, hence the series diverges. (cid:3)
Consider the subsequence ( q n k − ) k ∈ N introduced in (6.3). Since Q + ∞ k =1 q n k − /q n k − =+ ∞ implies lim n → + ∞ q n k − = + ∞ , we will estimate + ∞ Y k =1 q n k − q n k − ≥ + ∞ Y k =1 r k , and further prove, that P + ∞ k =1 r k = + ∞ , which yields the unboundedness.Before we start with the proof of property (6.1a) we need to recall and calculatesome preliminary necessities:(1) From the definition of the monodromy maps, we immediately obtain d Φ , ( δξ n − , δη n − ) = (cid:18) δξ n − Bδξ n − + δη n − (cid:19) = (cid:18) δξ n δη n (cid:19) ,d Φ i,i +1 ( δξ n − , δη n − ) = (cid:18) M i δξ n − + U i δη n − M Ti δη n − (cid:19) = (cid:18) δξ n δη n (cid:19) , i = 1 , . (6.5)(2) Cheng and Wojtkowski introduced in [ChW91] the norm k δξ k CW = X i =1 ( δξ i +1 − δξ i ) m i . The maps M i are invariant with respect to this norm, i.e. k M i δξ k CW = k δξ k CW . (6.6)(3) The equivalence of norms gives us constants D , D >
0, such that D k δξ k max ≤ k δξ k CW ≤ D k δξ k max , (6.7) where k · k max denotes the maximum norm.(4) Using the definitions of the Hamiltonian and the terms α i (4.3), we calculatemax { α , α } ≤ √ cm m √ m , (6.8) where c > i, i + 1), i = 0 , ,
2, stand for a collision of ball i with ball i + 1, i.e.when q i = q i +1 . When i = 0 the system experiences a collision with thefloor. Proof of property (6.1a).
The proof is based on the scheme given in [LW92, p. 159- 160].We first give an estimate for k δξ n k − k CW in between points T n k − x and T n k − x . Without loss of generality we set d T n k − x T n k − − n k − to be the product of d Φ , d Φ , for every k ∈ N . We estimate k δξ n k − k CW = k M M δξ n k − + ( M U + U M T ) δη n k − k CW ≤ k δξ n k − k CW + k ( M U + U M T ) δη n k − k CW ≤ k δξ n k − k CW + D k (cid:18) α (1 + γ ) α + (1 − γ ) α α (cid:19) δη n k − k max ≤ k δξ n k − k CW + D { α , α }k δη n k − k max ≤ k δξ n k − k CW + D √ cm m √ m k δη n k − k max (6.9)We abbreviate the constant factor in the last inequality by K = D √ cm m √ m . In between points T n k x and T n k − x we have one of the following situations:Either a floor collision occurs, in which k δξ n k k CW = k δξ n k − k CW or a ball to ballcollision occurs, in which k δξ n k k CW ≤ k δξ n k − k CW + k U κ ( n k − δη n k − k CW (see(6.5)). Thereby, κ : N → { , } , depends on the point and describes whether wehave a (1,2) or (2,3) collision. Combining this with (6.9) we obtain k δξ n k k CW ≤ k δξ n k CW + k X i =1 X j ∈ I i k U κ ( n i − j ) δη n i − j k CW + K k X i =1 k δη n i − k CW , (6.10)where | I i | are the number of ball to ball collisions happening between points T n i x and T n i − x . If | I i | = 0, we set k U κ ( n i ) δη n i k CW = 0.The Cauchy-Schwarz inequality gives us q n k = h δξ n k , δη n k i ≤ k δξ n k kk δη n k k , which yields k δη n k k ≥ q n k k δξ n k k . (6.11)From Proposition 6.1 and the Cauchy-Schwarz inequality, we get q n k +1 > q n k + Λ k δη n k k ≥ q n k + Λ k δη n k k max q n k k δξ n k k max ≥ q n k (cid:16) D k δη n k k max k δξ n k k CW (cid:17) . Utilizing the above, we estimate q n k +1 q n k ≥ D k δη n k k max k δξ n k CW + P ki =1 P j ∈ I i k U κ ( n i − j ) δη n i − j k CW + K P ki =1 k δη n i − k CW . Let r k = Λ D k δη n k k max k δξ n k CW + P ki =1 P j ∈ I i k U κ ( n i − j ) δη n i − j k CW + K P ki =1 k δη n i − k CW . Without loss of generality assume that the sum P + ∞ i =1 P j ∈ I i k U κ ( n i − j ) δη n i − j k CW is finite. The only thing left to show is that P + ∞ k =1 r k = + ∞ . In view of Proposition6.2, it will follow once we prove, that P + ∞ i =0 k δη n i k max = + ∞ . Assume on thecontrary, that this is not true. Then, by (6.9), the sequence ( k δξ n k − k CW ) k ∈ N isbounded from above. This and (6.11) imply, that ( k δη n k k max ) k ∈ N is bounded awayfrom zero, which contradicts our assumption. This yields the unboundedness. (cid:3) Particle falling in a wedge
Wojtkowski analyzed in [W98] the hyperbolicity of a particle moving along para-bolic trajectories in a variety of wedges. The particle is subject to constant acceler-ation and collides with the walls of the wedge. We adopt his notation and call sucha system particle falling in a wedge- or, abbreviated, PW system. Heuristicallyspeaking, for special wedges, namely simple ones, the PW system is equivalent toa falling balls system (or FB system) with particular masses. After introducingthe basic setup in three dimensions we are going to recall and expand some of theresults in [W98] in order to prove Theorem 6.2 and property (6.1b) in section 8.Let E be the three dimensional Euclidean space. For three linearly independentvectors { e , e , e } we define the wedge W ( e , e , e ) ⊂ E by W ( e , e , e ) = { e ∈ E : e = λ e + λ e + λ e , λ i ≥ , i = 1 , , } . The set of vectors { e , e , e } are called the generators of the wedge. We denote by S ( e , . . . , e i ), 1 ≤ i ≤
3, the linear subspace spanned by the linearly independentvectors { e , . . . , e i } . A three dimensional wedge is called simple, if the generatorscan be ordered in such a way that the orthogonal projection of e resp. e onto S ( e , e ) resp. S ( e ) is a positive multiple of e resp. e . The simplicity of a wedgecan be verified with the following Proposition 7.1 (Proposition 2.3, [W98]) . Let { e , e , e } be a set of linearlyindependent unit vectors. The wedge W ( e , e , e ) is simple if and only if h e i , e i +1 i > , i = 1 , , (7.1a) h e , e i = h e , e ih e , e i . (7.1b)The angles α i = ∢ ( e i , e i +1 ), i = 1 ,
2, completely determine the geometry of thewedge. In a simple wedge the angles satisfy 0 < α i < π and if { e , e , e } are unitvectors we have cos α i = h e i , e i +1 i . (7.2)We also give another geometric characterization of the wedge by introducing asecond pair of angles β , β . Thereby, β i is the angle between subspaces S ( e i , e i +2 )and S ( e i +1 , e i +2 ), where for i = 2, we set β = α . If the wedge is simple, theysatisfy 0 < β i < π . The relation between β and α , α is given bytan β = tan α sin α . (7.3) If the sum is infinite, then we can apply the argument in [LW92, p. 159 - 160] directly. Thekey point is, that we do not have control over this sum, so we assume the worst case, namely, itsfiniteness. Consider the FB system from Section 2. Its Hamiltonian is given by H ( q, p ) = h Kp, p i + h c , q i , K = diag ( m , m , m ), c = ( m , m , m ). Thereby, K is thediagonal matrix with diagonal entries m , m , m . The unit vectors e = 1 √ , e = 1 √ , e = span the configuration space W q ( e , e , e ) = { ( q , q , q ) ∈ R : 0 ≤ q ≤ q ≤ q } . It carries the natural Riemannian metric given by the kinetic energy h K · , ·i . Wesubject the system to the coordinate transformation x i = √ m i q i , w i = p i √ m i , (7.4)and obtain the Hamiltonian H ( x, w ) = h w, w i + h c , x i , c = ( √ m , √ m , √ m ).The natural Riemannian metric in these coordinates is the standard Euclidean innerproduct. The new generators of length one are h = 1 √ M √ m √ m √ m , h = 1 √ M √ m √ m , h = , (7.5)where M i = m i + · · · + m , i = 1 ,
2. The configuration space changes to W x ( h , h , h ) = { ( x , x , x ) ∈ R : 0 ≤ x √ m ≤ x √ m ≤ x √ m } . (7.6)With respect to the Euclidean inner product we have h h i , h j i = p M j √ M i , ≤ i < j ≤ , which immediately yields properties (7.1a), (7.1b) from Proposition 7.1, provingthat W x ( h , h , h ) is a simple wedge. Further, using properties (7.2) and (7.3) weget a direct link between the angles characterizing the wedge and the masses of theFB system cos α i = M i +1 M i , sin α i = m i M i , tan β i = m i m i +1 . (7.7)Notice, that the direction of the acceleration vector is along the first generator.We arrived at the important conclusion, that a PW system in a simple wedgewith acceleration vector along the first generator is equivalent to a FB system withappropriate masses.7.1. Wide wedges.Definition 7.1.
A three dimensional wedge with generators { g , g , g } is wide ifthe angle of the generators exceeds π/
2, i.e. h g i , g j i < , ≤ i < j ≤ W x ( h , h , h ) (7.6). We will unfold W x ( h , h , h ) to a wide wedge by continuously reflecting it in the faces, which areequipped with the first generator, i.e. W ( h , h ) and W ( h , h ). It is not hard to see, that this procedure creates a wide wedge if and only if the angle between thesubspaces S ( h , h ) and S ( h , h ) is exactly π/
3. This translates to the condition12 = cos π h n S ( h ,h ) , , n S ( h ,h ) , i , (7.8)where n S ( h ,h ) , resp. n S ( h ,h ) , are the unit normal vectors of the subindexedsubspace. Using (7.5) in (7.8) we obtain for the appropriate masses of the corre-sponding FB system 2 √ m √ m = √ m + m √ m + m . (7.9)In this way we obtain new generators { g , g , g } and the wedge W x ( g , g , g ),which consists exactly of six simple wedges. With the help of (7.5) and elementarylinear algebra it follows rather easily that the wedge W x ( g , g , g ) is wide.The two dimensional inner faces of the simple wedges possessing the first gener-ator h correspond to a collision of two balls in the associated FB system. Whenthe particle hits one of the inner faces we allow the particle to pass through theface to the adjacent wedge.A collision of the particle with one of the faces of the wide wedge correspondsto a collision with the floor in the associated FB system. In this case, we do notallow the particle to pass through the face, but instead reflect the velocity vectoracross the face by using w +1 = − w − .Since the trajectory is parabolic, a natural question to ask is, whether or notgrazing collisions can occur. For our purposes we will confine ourselves to the simplewedge W x ( h , h , h ). The definition of a grazing collision is as follows Definition 7.2.
A collision of the trajectory x ( t ), at time t , with one of the facesof the simple wedge W x ( h , h , h ) is grazing, if the velocity vector ˙ x ( t ) lies in theface of collision.The next result gives equivalent conditions of a grazing collision with one of thefaces possessing the first generator. Proposition 7.2.
Let t < t be consecutive collision times of the trajectory inthe simple wedge W x ( h , h , h ) and assume that x ( t ) ∈ W x ( h , h ) or x ( t ) ∈ W x ( h , h ) . The following statements are equivalent:1. A collision with the face W x ( h , h ) resp. W x ( h , h ) , at time t , is grazing.2. The differences w +1 ( t ) √ m − w +2 ( t ) √ m resp. w +2 ( t ) √ m − w +3 ( t ) √ m are equal to zero .
3. The trajectory segment { x ( t ) : t ∈ [ t , t ] } is confined to W x ( h , h ) resp. W x ( h , h ) . Proof. ⇒ x ( t ) ∈ W x ( h , h ) or x ( t ) ∈ W x ( h , h ).Further, let the particle experience a grazing collision with the face W x ( h , h ) attime t . In a grazing collision the velocity w − ( t ) = −√ m ( t − t ) + w +1 ( t ) −√ m ( t − t ) + w +2 ( t ) −√ m ( t − t ) + w +3 ( t ) Otherwise the unfolded simple wedges would overlap. is parallel to the face W x ( h , h ) = { ( x , x , x ) ∈ W x ( h , h , h ) : x √ m = x √ m } . This is equivalent to w +1 ( t ) √ m = w +2 ( t ) √ m . The argument for a grazing collision with the face W x ( h , h ) is exactly the same.2 ⇒ x ( t ) ∈ W x ( h , h ) or x ( t ) ∈ W x ( h , h )and let the particle collide with the face W x ( h , h ) at time t . From the Hamil-tonian equations, we calculate the first collision time t − t = x ( t ) / √ m − x ( t ) / √ m w +1 ( t ) / √ m − w +2 ( t ) / √ m . Since the energy is fixed, t − t < ∞ . It follows, that if w +1 ( t ) / √ m − w +2 ( t ) / √ m →
0, then x ( t ) / √ m − x ( t ) / √ m → x ( t ) / √ m = x ( t ) / √ m , which implies thatthe trajectory moves inside the face W x ( h , h ).The argument for w +2 ( t ) / √ m − w +3 ( t ) / √ m = 0 is exactly the same.3 ⇒ (cid:3) Projection.
The Hamiltonian equations imply that the flow is an invertedparabola. Let [ t , t c ] be the time from one collision to the next. We define theplanar subspace P x ([ t ,t c ]) = S ( ˙ x ( t ) , ˙ x ( t )) , ˙ x ( t ) = ˙ x ( t ) , t ≤ t < t ≤ t c . (7.10)The movement of the parabolic trajectory is confined to the planar subspace, i.e. { x ( t ) : t ∈ [ t , t c ] } ⊂ P x ([ t ,t c ]) . The acceleration vector a = ¨ x ( t ) is always element of P x ([ t ,t c ]) : Set n P ( t ) = ˙ x ( t ) × ¨ x ( t ) , k ˙ x ( t ) k = k ¨ x ( t ) k = 1 , ∀ t ∈ [ t , t c ] . The vector n P ( t ) has unit length and since the trajectory moves inside a planarsubspace, n P ( t ) is constant for all choices t ∈ [ t , t c ]. Thus, ˙ n P ( t ) = 0. Observe,that h n x ( t ) , ˙ x ( t ) i = 0 , ∀ t ∈ [ t , t c ] , (7.11)where n x ( t ) is a normal vector to ˙ x ( t ) at point x ( t ). Differentiating (7.11) gives h n x ( t ) , ¨ x ( t ) i = −h ˙ n x ( t ) , ˙ x ( t ) i . Substituting ¨ x ( t ) with a and n x ( t ) with n P ( t ) gives h a, n P ( t ) i = −h ˙ n P ( t ) , ˙ x ( t ) i = 0 . We will use this fact to project the configuration space W x ( g , g , g ) along thefirst generator h to the plane spanned by the normal vectors n S ( h ,h ) , n S ( h ,h ) of the subspaces S ( h , h ), S ( h , h ). The projected configuration space becomesan equilateral triangle. Its algebraic form is given by △ : √ m x + √ m x + √ m x = d, d > , (7.12) where d determines its displacement from the origin. Since the acceleration vectorlies in the plane spanned by two velocity vectors of the flow, the parabola projectedto △ becomes a straight line (see Figure 1). h h h g g g h Figure 1: The projected parabolamoving inside the projectedconfiguration space △ . (2 , ,
3) (2 , ,
2) (1 , , Proper alignment in wide wedges.
The idea to unfold the simple wedge W x (7.6) into a wide wedge stems from Wojtkowski [W16]. It is evident, that thetriple collision states in the configuration space, which are represented by the firstgenerator h , disappear in the wide wedge. More precise, each trajectory, whichpasses through the spot where h was, has a smooth continuation. Since the triplecollision singularity manifold is the only obstacle in proving the proper alignmentcondition, the system of a particle falling in the wide wedge, obtained for the specialmass configuration (7.9), satisfies the proper alignment condition. However, in thesimple wedge W x , once a trajectory hits the corner h it is impossible to continueit uniquely, since it has two images after the singular collision. The latter holds forany possible mass configuration. Therefore, the validity of the proper alignmentcondition cannot be immediately deduced from the dynamics of the wide wedge. Itremains unknown at the moment (see Subsection 5.1.1 for more details).8. Strict unboundedness - Part II
Consider a PW system in the simple wedge W x ( h , h , h ) (7.6) and mass restric-tions given by (7.9). Due to the results of the last section we reflect the simple wedgein its faces possessing the first generator to obtain a wide wedge W x ( g , g , g ).For the strict unboundedness, it remains to prove Theorem 6.2 and property(6.1b) from Section 6. The latter was already proven as part of the Main Theorem6.6 in [W98, p. 327 - 331]. In essence, Wojtkowski proved, that every orbit willeventually hit every face of the wide wedge. Subsequently, this yields all necessarycollisions for eventually mapping the Lagrangian subspaces L and L inside theinterior of the contracting cone field .To prove Theorem 6.2 we first establish how many different collisions, involvingall the balls, are possible in between two consecutive collisions of the lowest ball One can directly calculate, that all ( δξ, ∈ L get mapped into C ( x ) after at most threereturns to the floor and all (0 , δη ) ∈ L as soon as the trajectory experiences the first two ball toball collisions. with the floor. Using the projection to △ (see (7.12)), we encounter the followingfour different possibilities (see Figure 2)I. (0 , −→ (1 , −→ (2 , −→ (0 , , −→ (1 , −→ (2 , −→ (1 , −→ (0 , , −→ (2 , −→ (1 , −→ (0 , , −→ (2 , −→ (1 , −→ (2 , −→ (0 , Proof of Theorem 6.2.
Since every collision in the FB system happens infinitelyoften we distinguish between two sets of orbits O and O :(1) O consists of all the orbits where at least one of the cases I-IV abovehappens infinitely often.(2) O consists of all the orbits where each case I-IV happens at most finitelyoften. O can be considered as a special case, where in between two consecutive collisionsof the lowest ball with the floor occurs at most one ball to ball collision. Webegin with O . Due to symmetry it is enough to consider only the first two cases(8.1). Without loss of generality we start at time t on the face W x ( h , g ). Incase I, the order of faces crossed by the trajectory is W x ( h , g ), W x ( h , h ) beforethe particle hits the last face W x ( h , g ). In case II, the trajectory crosses faces W x ( h , g ), W x ( h , h ), W x ( h , g ) before it reaches the last face W x ( h , g ). Wecompactly display the latter information as Case I . W x ( g , h ) −→ W x ( h , g ) −→ W x ( h , h ) −→ W x ( h , g ) , Case II . W x ( g , h ) −→ W x ( h , g ) −→ W x ( h , h ) −→ W x ( h , g ) −→ W x ( h , g ) . Case I.
Let t < t < t , be the collision times with the faces W x ( h , g ), W x ( h , h ) and W x ( h , g ). When the particle crosses the face W x ( h , g ) resp. W x ( h , h ), we have w − ( t ) √ m − w − ( t ) √ m > w − ( t ) √ m − w − ( t ) √ m > . (8.2)The velocity differences are invariant in between collision, i.e. w − ( t ) √ m − w − ( t ) √ m = w +1 ( t ) √ m − w +2 ( t ) √ m ,w − ( t ) √ m − w − ( t ) √ m = w +2 ( t ) √ m − w +3 ( t ) √ m , (8.3)Due to Proposition 7.2, the quantities (8.2) are arbitrarily close to zero if and onlyif the collisions with the respective faces are arbitrarily close to grazing ones. Thefirst collision with the face W x ( h , g ) is almost grazing if and only if the planarsubspace P x ([ t ,t ]) (see (7.10)) is almost perpendicular to the face W x ( g , g ), i.e. x ( t ) ∈ W x ( g , g ). But this contradicts the fact of the trajectory reaching the lastface W x ( h , g ). Therefore, there exists ψ >
0, such that for all x ( t ) ∈ W x ( h , g ): ∢ ( P x ([ t ,t ]) , W x ( h , g )) > ψ . (8.4)The second collision with the face W x ( h , h ) is almost grazing if and only if P x ([ t ,t c ]) is almost perpendicular to the face W x ( g , g ), i.e. x ( t ) ∈ W x ( h , g ). But this contradicts x ( t ) ∈ W x ( h , g ). Therefore, there exists ψ >
0, such thatfor all x ( t ) ∈ W x ( h , g ): ∢ ( P x ([ t ,t ]) , W x ( h , h )) > ψ . (8.5)Using the projection along the first generator (see (7.12) and Figure 1) we conclude,that ψ = ψ = π/ Case II.
Let t < t < t < t be the collision times of the particle with thefaces W x ( h , g ), W x ( h , h ), W x ( h , g ) and W x ( h , g ). It is sufficient to provethat either w − ( t ) √ m − w − ( t ) √ m and w − ( t ) √ m − w − ( t ) √ m (8.6)or w − ( t ) √ m − w − ( t ) √ m and w − ( t ) √ m − w − ( t ) √ m (8.7)are uniformly bounded away from zero.In order to reach the last face W x ( h , g ), the quantity w − ( t ) / √ m − w − ( t ) / √ m is always uniformly bounded away from zero. Otherwise, due to Proposition 7.2, P x ([ t ,t ]) would be perpendicular to the face W x ( g , g ) and, thus, never reach thelast face W x ( h , g ).Due to Proposition 7.2, w − ( t ) / √ m − w − ( t ) / √ m is arbitrarily close to zeroif and only if the planar subspace P x ([ t ,t ]) is almost perpendicular to the face W x ( g , g ). But this implies that w − ( t ) / √ m − w − ( t ) / √ m is uniformly boundedaway from zero.If w − ( t ) / √ m − w − ( t ) / √ m is arbitrarily close to zero, then by the samereasoning as above, w − ( t ) / √ m − w − ( t ) / √ m is uniformly bounded away fromzero. Thus, in case II., either (8.6) or (8.7) are always uniformly bounded awayfrom zero.It is clear, due to the coordinate transformation (7.4), that w i / √ m i − w i +1 / √ m i +1 is uniformly bounded from below if and only if v i − v i +1 is uniformly bounded frombelow.Consider the FB system in x = ( ξ, η ) coordinates. Along every orbit ( T n x ) n ∈ N we have obtained two subsequences ( T n k x ) k ∈ N and ( T n k +1 x ) k ∈ N , where we set( T n k x ) k ∈ N to be the phase points before- and ( T n k +1 x ) k ∈ N right after, two con-secutive collisions with velocity differences bounded away from zero. This means,that the derivative map dT n k − − n k − equals either d Φ , d Φ , or d Φ , d Φ , . Bothof the latter maps are upper triangular matrices of the form (cid:18) X X X T (cid:19) .X depends only on the masses, while X = X ( α , α ) depends on the massesand the velocity differences v i − v i +1 in α , α (see (4.3)). Each pair of consecutivecollisions with velocity differences bounded away from zero belongs to one of thecases I-IV (8.1). Each of these velocity differences has a uniform lower bound. Setthe minimum of these lower bounds to be Θ >
0. Observe, that Q ( d T n k − x T n k − − n k − (0 , δη )) = h X k (0 , δη ) k δη, X T k (0 , δη ) k δη ik (0 , δη ) k . Let X (Θ) be the matrix in which the velocity differences in X ( α , α ) are replacedby Θ. Since X ( X ( α , α ) − X (Θ)) is positive semi-definite, we have h X ( α , α ) 1 k (0 , δη ) k δη, X T k (0 , δη ) k δη i > h X (Θ) 1 k (0 , δη ) k δη, X T k (0 , δη ) k δη i . Denote by ∂B k·k ( ~ ,
1) the boundary of the unit ball in tangent space with respect tothe indicated norm. The functional f ( u ) = h X (Θ) u, X T u i is positive, independentof x and continuous on the compact space ∂B k·k (0 , > Q ( d T n k − x T n k − − n k − (0 , δη )) > Λ k (0 , δη ) k . (8.8)The special case O reduces to the analysis of the reappearing collision sequence(1 , −→ (0 , −→ (2 , −→ (0 , −→ (1 , . Using the Hamiltonian flow, the collision laws and the collision times, it can bequickly calculated that both velocity differences of the (1 ,
2) collisions can not be-come arbitrarily small, otherwise this state would leave the constant energy surface.Hence, one of them has a uniform lower bound. The same idea can be applied toobtain a uniform lower velocity difference bound of the (2 ,
3) collision, i.e. we ob-serve a loss of energy when sufficiently reducing the value of the velocity differencesof (2 ,
3) above and its successive (2 ,
3) collision. For orbits in O , dT n k − − n k − either takes the form d Φ , d Φ , d Φ , or d Φ , d Φ , d Φ , . Considering that d Φ , and d Φ , commute, the invariance of L under d Φ , and the Q -monotonicity, weobtain the same estimate (8.8) with a different Λ >
0. Setting Λ = min { Λ , Λ } finishes the proof of Theorem 6.2 and therefore also Theorem 6.1. (cid:3) As it was outlined in Section 2, the strict unboundedness for every orbit sub-sequently implies the Chernov-Sinai ansatz and the abundance of sufficiently ex-panding points.
Acknowledgements.
I wish to cordially thank Maciej P. Wojtkowski for his helpin outlining the difficulties of the problem and his hospitality during my visit toOpole in October 2017. Further, my gratitude is expressed to my advisor HenkBruin and P´eter B´alint, Nandor Sim´anyi, Domokos Sz´asz and Imre P´eter T´othfor many helpful discussions. The author was supported by a Marietta-Blau andMarshall Plan Scholarship.
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