A conditional proof of the non-contraction property for N falling balls
aa r X i v : . [ m a t h . D S ] S e p A CONDITIONAL PROOF OF THE NON-CONTRACTIONPROPERTY FOR N FALLING BALLS
MICHAEL HOFBAUER-TSIFLAKOS
Abstract.
Wojtkowski’s system of N , N ≥
2, falling balls is a nonuniformlyhyperbolic smooth dynamical system with singularities. It is still an openquestion whether this system is ergodic. We contribute toward an affirmativeanswer, by proving the non-contraction property, conditioned by the assump-tion of strict unboundedness. For a certain mass ratio the configuration spacecan be unfolded to a billiard table where the daunting proper alignment con-dition is satisfied. We prove, that the aforementioned unfolded system withthree degrees of freedom is ergodic.
Contents
1. Introduction 12. Main results 43. The system of N falling balls 54. Lyapunov exponents 85. Ergodicity 125.1. Local Ergodicity 125.2. Abundance of sufficiently expanding points 136. Uniform lower bound of velocity differences 147. The non-contraction property 188. Ergodicity of a particle falling in a three dimensional wide wedge 23Acknowledgments 25References 251. Introduction
In [W90a, W90b], Maciej P. Wojtkowski introduced the system of N , N ≥ N point masses moving up and down avertical line, colliding with each other elastically and the lowest point mass collideswith a rigid floor placed at height zero. The system has N degrees of freedom,the positions q , . . . , q N and the momenta p , . . . , p N . The point masses are placedon top of each satisfying 0 ≤ q ≤ . . . ≤ q N . The overall standing assumptionon the masses is m > . . . > m N . Movement occurs due to kinetic energy anda linear potential field on a compact energy surface E c given by the Hamiltonian H ( q, p ) = P Ni =1 p i / m i + m i q i . The dynamics are further reduced to the Poincar´esection M containing the states right after a collision of two point masses or a Mathematics Subject Classification.
Primary 37D50; Secondary 37J10.
Key words and phrases.
Ergodic Theory, Hyperbolic dynamical systems with singularities. collision of the lowest point mass with the floor. Accordingly, the Poincar´e map T describes the dynamics from one collision to the next. It preserves the smoothmeasure µ , obtained from the symplectic volume form on R N × R N via symplecticreduction. Out of historic convenience we will refer to the falling point masses asfalling balls.An intrinsic obstacle, which makes the treatment of this system challenging, isthe presence of singular collisions. In physical terms, they occur in a triple collisionor when the two lower balls hit the floor simultaneously. The singular collisionsor singularities form codimension one submanifolds in phase space. The Poincar´emap is not well defined on the singularities because it has two images.The main question in Wojtkowski’s original paper [W90a] revolved around theexistence of non-zero Lyapunov exponents. Sim´anyi settled the general case byproving that an arbitrary number of falling balls have non-zero Lyapunov expo-nents almost everywhere [S96]. For a family of potential fields V ( q ), satisfying ∂V ( q ) /∂q > ∂ V ( q ) /∂q <
0, Wojtkowski proved the same result in [W90b].The latter family of potentials does not include the linear potential field.A new treatment, which ties in old and new ideas, can be found in Wojtkowski’slatest work on falling balls [W98]. He starts off with N , N ≥
2, horizontally alignedballs falling to a moving floor, establishes complete hyperbolicity, and then carriesthe result over to a variety of falling ball systems by applying stacking rules onthe balls. In the most extreme case he obtains his original system from [W90a].The billiard system of each falling ball system corresponds to a particle falling in aparticular wedge. The form of the wedge depends thereby on the masses and thephysical model.The main line of this work concerns the long time open conjecture whether three(or more) falling balls are ergodic. There are two results, confirming the ergodicityof two falling balls with mass configurations m > m : One for the linear potentialmentioned above [LW92] and one [Ch91] for the family of potentials consideredabove with the relaxed assumption ∂ V ( q ) /∂q ≤ < C ≤ ∂V ( q ) /∂q ≤ C < ∞ , 0 ≤ (cid:12)(cid:12) ∂ V ( q ) /∂q (cid:12)(cid:12) ≤ C < ∞ , for some constants C , C , C > From here, it is common to verify the Local ErgodicTheorem (LET) together with a transitivity argument to prove the existence of onlyone ergodic component of full measure and, thus, the ergodicity of the system. TheLET dates back to Sinai’s seminal proof of ergodicity for two discs moving uniformlyin the two dimensional torus [S70] and was later generalized in the framework ofsemi-dispersing billiards [ChS87, KSSz90, BChSzT02]. In order to prove ergodicitywe will use the LET, formulated for symplectic maps by Liverani and Wojtkowski[LW92].The LET claims, that one can find an open neighbourhood of a hyperbolic point p with sufficient expansion, which lies (mod 0) in one ergodic component, if thefollowing five conditions are satisfied:(C1) Regularity of singularity manifolds.(C2) Non-contraction property. An ergodic component is a set of positive measure in phase space on which the conditionalsmooth measure is ergodic. (C3) Continuity of transversal Lagrangian subspaces.(C4) Chernov-Sinai ansatz.(C5) Proper alignment.Assuming the validity of the LET and the abundance of sufficiently expandingpoints, ensures that the neighbourhoods of the LET can be connected to oneergodic component of full measure.The bulk of effort in this paper consists in giving a conditional proof of the non-contraction property (C2). We will use coordinate transformations (4.2), (4.3) forwhich the derivative of the flow equals the identity matrix. Hence, it is equivalentto verify the non-contraction property for the flow. The paramount advantage isthat it is easier for us to express results in finite times rather than arbitrarily manyderivative map compositions (cf. Section 7). There are two main ingredients for theproof of the non-contraction property: The first one requires that along every orbitand for every ball to ball collision there exists a subsequence of collision times, suchthat the pre-collisional velocity differences of the ball to ball collisions are uniformlybounded from below (cf. Theorem 6.1). The latter will imply that in every finiteinterval [0 , T ], T >
0, the number of ball to ball collisions is uniformly boundedfrom above by a constant which depends only on the length T and the energy c > Q along every orbit and every vectorof the closed expanding cone field (cf. Definition 4.4). For a constant E >
0, thestrict unboundedness property will help us to determine a time T = T ( E ) >
0, forwhich the Q -value of every vector from the expanding cone field has a uniform lowerbound E (cf. Lemma 7.1). This allows us to split the proof of the non-contractionproperty into two parts: First, we prove that the non-contraction property holdsfor every t ≤ T and, second, for every t > T . Note, that the uniform lower boundof the velocity differences and its implications is used for the first part only.Conditioning the validity of the non-contraction property to the validity of strictunboundedness has the advantage that we free ourselves from having to find aLyapunov semi-norm for this model, which is an inherently difficult task by itself.Additionally, as we discuss further below, the strict unboundedness property hasalready been verified for three falling balls with mass configuration given in (8.3)[HT20]. It is already known, that the continuity of Lagrangian subspaces (C3) is true foran arbitrary number of balls [W90a, W91]. For the special restriction of masses (8.3)the configuration space (3.1) can be unfolded to a wide wedge [W98, Definition 6.1].Wojtkowski’s insight [W16] allowed to verify condition (C5), by showing that, dueto the unfolding of the wedge, orbits hitting the unaligned triple collision singularitymanifold can be uniquely continued [HT20, Subsection 7.3]. Except for the missingtriple singularity manifold, this system is identical to the system of falling balls upto a Q -isometric coordinate transformation (8.1). In order to distinguish betweenthe two systems we follow Wojtkowski [W98] and call the former system a particlefalling in a wide wedge system. The abundance of sufficiently expanding points is equivalent to saying that the set of suffi-ciently expanding points has measure one and is arcwise connected (cf. Subsection 5.2). In fact, the bold reader may verify that the result of Theorem 6.1 can be implemented into[HT20], which will yield strict unboundedness for every mass configuration m > m > m . MICHAEL HOFBAUER-TSIFLAKOS
Since these systems relate to each other via a Q -isometric coordinate transforma-tion it suffices to verify the conditions of the LET and the abundance of sufficientlyexpanding points in only one of the systems.For the particular mass restrictions (8.3) we proved the strict unboundednessproperty, the Chernov-Sinai ansatz (C4) and the abundance of sufficiently expand-ing points [HT20]. Using [LW92, Lemma 7.7], it takes not much effort to check theregularity of singularity manifolds (C1) (cf. Section 8). Since the strict unbounded-ness assumption is valid, the new result of this work gives that the non-contractionproperty (C2) is valid as well. Therefore, we arrive at the conclusion that a particlefalling in a three dimensional wide wedge is ergodic.2. Main results
The phase space M is partitioned (mod 0) into subsets M i , i = 1 , . . . , N . M contains the states right after a collision with the floor and M i , i = 2 , . . . , N ,contains the states right after a collision of balls i − i . The Poincar´e map T : M (cid:9) describes the movement from one collision to the next. After applying Wojtkowski’sconvenient coordinate transformation ( q, p ) → ( h, v ) → ( ξ, η ) (4.2), (4.3), we obtainan expanding cone field {C ( x ) : x ∈ M} , explicitly given by C ( x ) = { ( δξ, δη ) ∈ R N × R N : Q ( δξ, δη ) > , δξ = 0 , δη = 0 } ∪ { ~ } , C ′ ( x ) = { ( δξ, δη ) ∈ R N × R N : Q ( δξ, δη ) < , δξ = 0 , δη = 0 } ∪ { ~ } . where ( δξ, δη ) = ( δξ , . . . , δξ N − , δη , . . . , δη N − ) denote the coordinates in tangentspace. The quadratic form Q is defined (cf. Definition 4.2) by a pair of constant,transversal Lagrangian subspaces (4.4) and the symplectic form ω . For this choiceof Lagrangian subspaces Q becomes the Euclidean inner product Q ( δξ, δη ) = h δξ, δη i = N − X i =1 δξ i δη i . Denote by C ( x ) the closure of the cone C ( x ), let d x T n = d T n x T . . . d
T x
T d x T and( d T n x T ) n ∈ N = ( d x T, d
T x
T, d T x T, . . . ). The sequence ( d T n x T ) n ∈ N is called un-bounded, if lim n → + ∞ Q ( d x T n v ) = + ∞ , ∀ v ∈ C ( x ) \ { ~ } , and strictly unbounded, iflim n → + ∞ Q ( d x T n v ) = + ∞ , ∀ v ∈ C ( x ) \ { ~ } . For the proof of the non-contraction property (cf. Theorem A below), we have toassume that ( d T n x T ) n ∈ N is strictly unbounded for every x ∈ M . Assumption (SU) . 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], Assumption (SU) also impliesthe strict unboundedness for the orbit in negative time ( d T n x T ) n ∈ Z − , i.e.lim n →−∞ Q ( d x T n v ) = −∞ , ∀ v ∈ 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 . We furtherabbreviate S ± n = S ± ∪ T ∓ S ± ∪ . . . ∪ T ∓ ( n − S ± . Under assumption (SU), we prove the non-contraction property which is one of thefive conditions (C1) - (C5) of the LET (cf. Section 5)
Theorem A (Non-contraction property) . Assume that assumption (SU) holds.Then, there exists ζ > , such that(1) 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 , (2) 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 . For three falling balls with the additional mass restriction (8.3), the configurationspace can be unfolded to a billiard table where the ominous proper alignmentcondition (C5) is satisfied [W16, HT20]. The reason is, that the unfolded systemmisses the unaligned triple collision singularity manifold, since every orbit passingthrough it can be smoothly continued. Except for the missing triple singularitymanifold, the system is identical to the system of falling balls up to a Q -isometriccoordinate transformation (8.1). In order to distinguish between the two systems wefollow Wojtkowski [W98] and call such a system a particle falling in a wide wedge.Assumption (SU) was proven for three falling balls with mass configurations (8.3)in [HT20]. Therefore, according to Theorem A, the non-contraction property holdsfor this system. Incorporating complementary previous results from [HT20] we willprove in Section 8 Theorem B (Ergodicity) . Consider the system of 3 falling balls with mass restric-tions (8.3). Then, the unfolded system of a particle falling in a three dimensionalwide wedge is ergodic. The system of N 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 = { ( q, p ) ∈ R N × R N : 0 ≤ q ≤ . . . ≤ q N } (3.1)where the subindex q in N q refers to the coordinates ( q, p ). The momenta and thevelocities are related by p i = m i v i . We assume that the masses m i decrease strictlyas we go upwards m > . . . > m N . The movement of the balls are the result of alinear potential field and their kinetic energies. The total energy of the system isgiven by the Hamiltonian function H ( q, p ) = N X i =1 p i m i + m i q i . MICHAEL HOFBAUER-TSIFLAKOS
The Hamiltonian equations are ˙ q i = p i m i , ˙ p i = − m i . (3.2)The dots indicate differentiation with respect to time t and the Hamiltonianvector field on the right hand side will be denoted as X H = X H ( q, p ). For someenergy value c >
0, the energy manifold E c and its tangent space T E c are given by E c = { ( q, p ) ∈ R N + × R N : H ( q, p ) = N X i =1 p i m i + m i q i = c } , T ( q,p ) E c = { ( δq, δp ) ∈ R N × R N : ∇ ( q,p ) H ( δq, δp ) = N X i =1 p i δp i m i + m i δq i = 0 } . (3.3)Including the restriction of the positions amounts to E c ∩ N q . The Hamiltonianvector field (3.2) gives rise to the Hamiltonian flow φ : R × ( E c ∩ N q ) → E c ∩ N q , ( 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 Ni =1 dq i ∧ dp i induces the symplectic volumeelement Ω = V Ni =1 ω . The volume element on the energy surface is obtained bycontracting Ω, by a vector u , where u is a vector satisfying dH ( u ) = 1. Denotingthe contraction operator by ι , the volume element on the energy surface is givenby ι ( u )Ω. Since the flow preserves the standard symplectic form, it preserves thevolume element and, hence, the Liouville measure ν on E c ∩ N q obtained from it.We define the Poincar´e section, which describes the states right after a collisionas M = M ∪ . . . ∪ M N , with M := { ( q, p ) ∈ E c ∩ N q : q = 0 , p /m ≥ } , M l := { ( q, p ) ∈ E c ∩ N q : q l − = q l , p l − /m l − ≤ p l /m l } , l = 2 , . . . , N. The set of states right before collision M b = M b ∪ . . . ∪ M bN , are defined by M b := { ( q, p ) ∈ E c ∩ N q : q = 0 , p /m < } , M bl := { ( q, p ) ∈ E c ∩ N q : q l − = q l , p l − /m l − > p l /m l } , l = 2 , . . . , N. The collision between balls i and i + 1 is fully elastic, i.e. the total momentum andthe kinetic energy are preserved. Therefore, the momenta resp. velocities changeaccording 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 , (3.4) where γ i = ( m i − m i +1 ) / ( m i + m i +1 ), i = 1 , . . . , N −
1. When the bottom particlecollides with the floor the sign of its momentum resp. velocity is reversed p +1 = − p − ,v +1 = − v − . (3.5)This is derived from (3.4), by setting the floor velocity v zero and letting thefloor mass m go to infinity. As a result, the floor collision does not preserve thetotal momentum.These collision laws are described by the linear, symplectic, involutory collisionmap Φ i − ,i : M b → M , ( q, p − ) ( q, p + ) . We will write Φ if we do not want to refer to any specific collision. Let τ : E c ∩ N q → R + , (3.6)be the first return time to M b . 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 the state from right after one collision to the next.On M , we obtain the volume element ι ( X H ) ι ( u )Ω, by contracting the volumeelement ι ( u )Ω on the energy surface with respect to the direction of the flow X H .This exterior form defines a smooth measure µ on M , which is T -invariant.Matching the present state with the next collision in the future resp. the past,we obtain two (mod 0) partitions of MM = M +1 , ∪ N [ i =1 N [ j =1 j = i M + i,j = M − , ∪ N [ i =1 N [ j =1 j = i M − i,j , where M +1 , = { x ∈ M : T x ∈ M } , M + i,j = { x ∈ M i : T x ∈ M j } , ≤ i, j ≤ N, j = i, M − , = { x ∈ M : T − x ∈ M } , M − i,j = { x ∈ M i : T − x ∈ M j } , ≤ i, j ≤ N, j = i. For some instances, it is useful to define the subset M m, +1 , := M +1 , ∩ T − M +1 , ∩ . . . ∩ T − m M +1 , ⊂ M +1 , , m ≥ , which contains the states returning ( m + 1)-times to the floor.Each partition element M ± i,j has a boundary ∂ M ± i,j and the intersection of twoelements of the same partition is strictly contained in the intersection of theirboundaries, i.e. M ± i,j ∩ M ± k,l ⊂ ∂ M ± i,j ∩ ∂ M ± k,l , ( i, j ) = ( k, l ) . MICHAEL HOFBAUER-TSIFLAKOS
The boundary of each partition consists of a regular part R ± and a singular part S ± , where we set ∂ M ± = R ± ∪ S ± . The singular part comprises the followingcodimension one submanifolds S + j,i = M + j,i ∩ M + j,i +1 , S − i,j = M − i,j ∩ M − i +1 ,j ,i = 2 , . . . , N − , j = 1 , . . . , N, j = i, i + 1 , S + k, = M + k, ∩ M + k, , S − ,k = M − ,k ∩ M − ,k ,k = 1 , . . . , N, k = 2 . These sets are called singularity manifolds. The states in S ± j,i face a triple collisionnext, while the states in S + k, , S − ,k experience a collision of the lower two balls withthe floor next. The maps T resp. T − have two different images and are thereforenot well-defined on the sets S + j,i , S + k, resp. S − i,j , S − ,k , because the compositionsΦ i − ,i ◦ Φ i,i +1 and Φ , ◦ Φ , do not commute. In this case, we follow the convention,that the orbit branches into two suborbits and we continue the system on eachbranch separately. We abbreviate, for n ≥ S ± = N − [ i =2 N [ j =1 j = i,i +1 S ± i,j ∪ N [ k =1 k =2 S + k, ∪ N [ k =1 k =2 S − ,k , S ± n = S ± ∪ T ∓ S ± ∪ . . . ∪ T ∓ ( n − S ± . In the upcoming sections we also want to refer to the singularity manifolds for theflow. We define them, informally, as S ± t , for every t ∈ R + .Similarly to S ± , the T ± -image of all points in R ± consists of two simultaneouscollisions. The key difference to singular points is that the derivatives of the involvedcollision maps commute. This follows from the fact, that the two pairs of collisionsdo not share a common ball. Hence, for regular points our orbit does not split intotwo suborbits and can therefore be continued uniquely. Since the collision mapsof the simultaneous collisions for points in R ± commute and T is well-defined on S − \ S + , the regularity properties of the flow and the collision map imply that, for n ≥ T n : M \ S + n → M \ S − n (3.7)is a symplectomorphism, i.e. T extends diffeomorphically to R + .4. Lyapunov exponents
The study of Lyapunov exponents was carried out using a method developed byWojtkowski in the string of papers [W85, W88, W91, LW92, W00]. This methodhas been successfully implemented to derive that an arbitrary number of fallingballs has non-zero Lyapunov exponents almost everywhere [S96]. The basic toolsof the Lyapunov exponent machinery were further advanced and are inevitable inthe study of ergodicity of Hamiltonian systems [LW92]. We are therefore goingto formulate the fundamentals of this method and how it applies to the system offalling balls.The standard symplectic form ω = P Ni =1 dq i ∧ dp i is given by ω ( v , v ) = h v , , v , i − h v , , v , i , where v i = ( v i, , v i, ) ∈ R N × R N , i = 1 ,
2. A Lagrangian subspace V is a subspaceof dimension N which is the ω -orthogonal complement to itself, i.e. the symplectic form is zero for every input from V [LM87, Definition 6.4]. It is equivalently thesubspace of maximal dimension on which ω vanishes. Note, that for two transversalLagrangian subspaces V , V , every vector v ∈ R N × R N has a unique decomposition v = v + v , v ∈ V , v ∈ V . Definition 4.1.
For two transversal Lagrangian subspaces V , V we define thecone between V and V by C V ,V = { v ∈ R N × R N : ω ( v , v ) > , v = v + v , v i ∈ V i , i = 1 , } ∪ { ~ } . Definition 4.2.
The quadratic form Q V ,V , or Q V ,V -form, associated to a pairof transversal Lagrangian subspaces V , V is given by Q V ,V : R N × R N → R ,v ω ( v , v ) , where v = v + v , v i ∈ V i , i = 1 , Q V ,V -form is indefinite with signature ( N, N ) on R N × R N . With the definitions above, the quadratic form can be used to define thecone C V ,V = { v ∈ R N × R N : Q V ,V ( v ) > } ∪ { ~ } . The complementary cone of C V ,V is given by C ′ V ,V = { v ∈ R N × R N : Q V ,V ( v ) < } ∪ { ~ } . The arguably simplest expression of Q V ,V can be obtained by associating it to thestandard Lagrangian subspaces given by L = R N × { ~ } , L = { ~ } × R N . (4.1)For this choice of transversal Lagrangian subspaces we will abbreviate Q = Q L ,L and C = C L ,L . Further, for v = v + v , the Q -form reads Q ( v ) = h v , v i . In [W90a], Wojtkowski introduced two coordinate transformations, i = 1 , . . . , N , h i = p i m i + m i q i , v i = p i m i , (4.2)and ( ξ , ξ , . . . , ξ N − ) T = A − ( h , h , . . . , h N ) T ( η , η , . . . , η N − ) T = A T ( v , v , . . . , v N ) T , (4.3)where A is an invertible matrix depending only on the masses [W90a, p. 520].In order to keep calculations concise and lucid, we will work exclusively in ( ξ, η )-coordinates.The energy manifold, its tangent space and the Hamiltonian vector field takethe form E c = { ( ξ, η ) ∈ R N − × R N − : H ( ξ, η ) = ξ = c } , T E c = { ( δξ, δη ) ∈ R N − × R N − : ∇ ( ξ,η ) H ( δξ, δη ) = δξ = 0 } ,X H ( ξ, η ) = (0 , . . . , , − , , . . . , . Intersecting the standard Lagrangian subspaces (4.1) in ( δξ, δη )-coordinates withthe tangent space of the energy manifold and quotienting out the flow direction gives L = { ( δξ, δη ) ∈ R N × R N : δξ = 0 , δη i = 0 , i = 0 , . . . , N − } ≃ R N − , L = { ( δξ, δη ) ∈ R N × R N : δη = 0 , δξ i = 0 , i = 0 , . . . , N − } ≃ R N − . (4.4)Thus, the Q -form given by L , L reduces to R N − × R N − and now amounts to Q ( δξ, δη ) = h δξ, δη i = N − X i =1 δξ i δη i , (4.5)with no further restrictions, when inserting a vector from L ⊕ L .In these coordinates, the derivative of the flow dφ t equals the identity map. Thus,only the derivatives of the collision maps d Φ i,i +1 are relevant to the dynamics intangent space. Since δξ = 0, δη = 0 we can reduce the derivatives of the collisionmaps to (2 N − × N − d Φ , = (cid:18) id N − B id N − (cid:19) , d Φ i,i +1 = (cid:18) D i F i D Ti (cid:19) , i = 1 , . . . , N − , (4.6)where B = ( b m,n ) N − m,n =1 , F i = ( f m,n ) N − m,n =1 have the structure of the zero matrix,except for the entries b , = β , f i,i = − α i and D i = ( d m,n ) Nm,n =1 has the structureof the identity matrix, except for the following entries in the i -th row d i,i − = 1 − γ i , d i,i = − , d i,i +1 = 1 + γ i . The terms α , . . . , α N and β in the matrices are non-negative and 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 ) . (4.7)Observe, that the strict inequality m > . . . > m N of the mass configurationsimplies, that α i >
0, since v − i − v − i +1 > Q , we define the open cone C and the complementarycone C ′ associated to the Lagrangian subspaces L , L by C = { ( δξ, δη ) ∈ L ⊕ L : Q ( δξ, δη ) = h δξ, δη i > } ∪ { ~ } , C ′ = { ( δξ, δη ) ∈ L ⊕ L : Q ( δξ, δη ) = h δξ, δη i < } ∪ { ~ } . The cone field {C ( x ) : x ∈ M} is constant and therefore continuous in M . Denoteby C the closure of the cone C . Definition 4.3.
1. The cone C is called invariant at x ∈ M , if d x T C ⊆ C ,
2. The cone C is called strictly invariant at x ∈ M , if d x T C ⊆ C ,
3. The cone C is called eventually strictly invariant at x ∈ M , if there exists apositive integer k = k ( x ) ≥
1, such that d x T k C ⊆ C ,
4. The map d x T is called Q -monotone , if Q ( d x T v ) ≥ Q ( v ) , for all v ∈ L ⊕ L .5. The map d x T is called strictly Q -monotone , if Q ( d x T v ) > Q ( v ) , for all v ∈ L ⊕ L \ { ~ } .6. The map d x T is called eventually strictly Q -monotone , if there exists a positiveinteger k = k ( x ) ≥
1, such that Q ( d x T k v ) > Q ( v ) , for all v ∈ L ⊕ L \ { ~ } .In the definition above, statements 1, 2, 3 are equivalent to statements 4, 5,6 [LW92, Section 4]. In order to obtain non-zero Lyapunov exponents we repeatWojtkowski’s criterion [W85], which links eventual strict Q -monotonicity to non-zero Lyapunov exponents Q-Criterion (Theorem 5.1, [W85]) . If d x T is eventually strictly Q -monotone for µ -a.e. x, then all Lyapunov exponents are non-zero almost everywhere. The derivative d x T is Q -monotone for every x ∈ M and any number of fallingballs [W90a]. Sim´anyi established that N , N ≥
2, falling balls have non-zeroLyapunov exponents for µ -a.e. x ∈ M , by verifying the Q -criterion [S96].Observe, that the coordinate transformation (4.3) is Q -isometric, i.e. Q ( δξ, δη ) = Q ( A − δh, A T δv ) = Q ( δh, δv ) , which represents a change of basis inside of both Lagrangian subspaces. Therefore,it does not make a difference in terms of the Q -form’s value whether we operate in( δh, δv ) or ( δξ, δη )-coordinates.We close this subsection by formulating the (strict) unboundedness propertyand the least expansion coefficients, which will be used to establish criteria forergodicity.The least expansion coefficients σ , σ C ′ , for n ≥
1, are defined as σ ( d x T n ) = inf = v ∈C ( x ) s Q ( d x T n v ) Q ( v ) , σ C ′ ( d x T − n ) = inf = v ∈C ′ ( x ) s Q ( d x T − n v ) Q ( v ) . Definition 4.4.
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 4.1 (Theorem 6.8, [LW92]) . The following assertions are equivalent:(1) The sequence ( d T n x T ) n ∈ N is strictly unbounded.(2) lim n →∞ σ ( d x T n ) = ∞ . Remark . The strict unboundedness property can also be stated in negativetime, i.e. lim n →−∞ Q ( d x T n v ) = −∞ , ∀ v ∈ C ′ ( x ) \ { ~ } . Following the proof of [LW92, Theorem 6.8], Theorem 4.1 also extends to this case,i.e. (1) The sequence ( d T n x T ) n ∈ Z − is strictly unbounded.(2) lim n →∞ σ C ′ ( d x T − n ) = ∞ .5. Ergodicity
Due to the theory of Katok-Strelcyn [KS86] we know that our phase space decom-poses into at most countably many components on which the conditional smoothmeasure is ergodic. The strategy to prove ergodicity involves two steps:(1) Proving local ergodicity (or the Local Ergodic Theorem), which impliesthat every ergodic component is a (mod 0) open set.(2) Proving that the set of sufficiently expanding points (Definition 5.1) isarcwise connected and of full measure, which implies that any two (mod0) open ergodic components can be connected with each other, such thattheir intersection is of positive µ -measure.The validity of both points above proves the existence of only one ergodic compo-nent of full measure.5.1. Local Ergodicity.
We use the Local Ergodic Theorem (LET) of [LW92] andbegin with the definition of a sufficiently expanding point.
Definition 5.1.
A point p ∈ M is called sufficient (or sufficiently expanding ) ifthere exists a neighbourhood U = U ( p ) and an integer N = N ( p ) > U ∩ S − N = ∅ and σ ( d y T N ) >
3, for all y ∈ T − N U , or(4) U ∩ S + N = ∅ and σ C ′ ( d y T − N ) >
3, for all y ∈ T N U .Note, that in the sufficiency definition the requirements U ∩ S − N = ∅ in (3) and U ∩ S + N = ∅ in (4) additionally demand, that the orbit meets no singular manifoldin the first N ( p ) − Local Ergodic Theorem.
Let p ∈ M be a sufficient point and let U = U ( p ) bethe neighbourhood from Definition 5.1. Suppose conditions (C1) - (C5) below aresatisfied.(C1) ( Regularity of singularity manif olds ) :
The sets S + n and S − n , n ≥ , areregular subsets. (C2) ( N on − contraction property ) : There exists ζ > , such that(a) 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 , For the definition of a regular subset refer to [LW92, Definition 7.1] (b) 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 . (C3) ( Continuity of Lagrangian subspaces ) :
The ordered pair of transversalLagrangian subspaces ( L ( x ) , L ( x )) varies continuously in int M .(C4) ( Chernov − Sinai ansatz ) :
For µ S ∓ -a.e. x ∈ S ∓ , lim n →±∞ Q ( d x T n v ) = ±∞ , for every v ∈ C ( x ) \ { } , if x ∈ S − and for every v ∈ C ′ ( x ) \ { } , if x ∈ S + .(C5) ( P roper alignment ) :
There exists M ≥ , such that for every x ∈ S + resp. S − , we have d x T − M v + x resp. d x T M v − x belong to C ′ ( T − M x ) resp. C ( T M x ) ,where v + x resp. v − x are the characteristic lines of T x S + resp. T x S − .Then, the open neighbourhood U ( p ) is contained (mod 0) in one ergodic component. Abundance of sufficiently expanding points.
The notion of a sufficientlyexpanding point was given in Definition 5.1. Once local ergodicity is established wededuce that every ergodic component is (mod 0) open. One possibility to obtain asingle ergodic component is
Theorem 5.1 (Abundance of sufficiently expanding points) . The set of sufficientlyexpanding points has full measure and is arcwise connected.
The abundance of sufficiently expanding points can be proven at once by requir-ing the strict unboundedness assumption (SU), the proper alignment property (C5)and the explicit construction of the neighbourhood lying in one ergodic componentfrom the LET in the beginning of Section 8 in [LW92].
Proof of Theorem 5.1.
Recall that a point x ∈ M is sufficient if there exists apositive integer N = N ( x ) >
0, such that either (3) or (4) from Definition 5.1 aresatisfied. Due to the strict unboundedness (SU), Theorem 4.1 and Remark 4.1, σ ( d x T n ) and σ C ′ ( d x T − n ) diverge to infinity for every x ∈ M . Therefore, everyorbit which experiences at most one singular collision satisfies either(5) σ ( d x T N ( x ) ) > T k U ∩ S + = ∅ , 0 ≤ k ≤ N ( x ), or(6) σ C ′ ( d x T − N ( x ) ) > T − k U ∩ S − = ∅ , 0 ≤ k ≤ N ( x ).It follows that the only non-sufficient orbits lie in a subset of double singular colli-sions. Due to the proper alignment property (C5), S + and S − are transversal forevery point, thus, the points of double singular collisions form a set of (at least)codimension two. Hence, there is an arcwise connected set of measure one suchthat the least expansion coefficient is larger than three. For the last part, the prooffollows the beginning of Section 8 in [LW92]:Without loss of generality assume that σ C ′ ( d x T N ) >
3. We can choose a smallenough neighbourhood U around the point x such that T N : T − N U → U is a dif-feomorphism. This implies that
U ∩ S − N = ∅ and T − N U ∩ S + N = ∅ . Further, thefunctional y σ ( d y T N ) is continuous on U and by making U smaller, if necessary,we obtain σ ( d y T N ) >
3, for every y ∈ T − N U . (cid:3) 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. Uniform lower bound of velocity differences
The investigation regarding a uniform lower bound of velocity differences v − i − v − i +1 , for any i ∈ { , . . . , N − } , is of main interest for the non-contraction property.Denote by ( i, i + 1) the collision between ball i and ball i + 1, i.e. when q i = q i +1 .Let x = x ( t ) ∈ M i +1 , i ∈ { , . . . N − } . The velocity difference v − i ( t ) − v − i +1 ( t )is non-negative and due to the collision laws (3.4), changes sign after the collision,i.e. 0 ≤ v − i ( t ) − v − i +1 ( t ) = − ( v + i ( t ) − v + i +1 ( t )) . (6.1)The Hamiltonian equations imply, that during free flight this quantity remainspreserved (3.2). Using (3.2), (3.4), we see that the term v − i ( t ) − v − i +1 ( t ) is onlyaffected by a ( i − , i ) resp. ( i + 1 , i + 2) collision when being expanded backwards,i.e. 0 ≤ v − i ( t ) − v − i +1 ( t ) = (1 + γ i − )( v − i − ( t c ) − v − i ( t c ))+ ( v − i ( t c ) − v − i +1 ( t c )) , resp.0 ≤ v − i ( t ) − v − i +1 ( t ) = (1 − γ i +1 )( v − i +1 ( t c ) − v − i +2 ( t c ))+ ( v − i ( t c ) − v − i +1 ( t c )) , (6.2)where t c < t is the collision time of the ( i − , i ) resp. ( i + 1 , i + 2) collision. Sincewe stopped our expansion right before a ( i − , i ) resp. ( i + 1 , i + 2) collision, wehave v − i − ( t c ) − v − i ( t c ) ≥ , v − i +1 ( t c ) − v − i +2 ( t c ) ≥ . (6.3)Formula (6.2) can be generalized in the following way. Let t < t be collision timesof two successful ( i, i + 1) collisions and m, n ∈ N . Assume that in between thosetwo ( i, i + 1) collisions we have m ( i − , i ) collisions and n ( i + 1 , i + 2) collisions,with collision times r , . . . , r m and u , . . . u n . Expanding only the ( i, i + 1) velocitydifference backwards without changing the appearing ( i − , i ) and ( i + 1 , i + 2)velocity differences, we obtain for i ≥ ≤ v − i ( t ) − v − i +1 ( t )= (1 + γ i − ) P mj =1 (cid:0) v − i − ( r j ) − v − i ( r j ) (cid:1) + (1 − γ i +1 ) P nl =1 (cid:0) v − i +1 ( u l ) − v − i +2 ( u l ) (cid:1) + v + i ( t ) − v + i +1 ( t ) . (6.4)In between two (1 ,
2) collisions, we assume to have one floor collision, m full returnsto the floor of the lowest ball and n (2 ,
3) collisions, again with collision times r , . . . , r m and u , . . . u n . Expanding (1 ,
2) at t backwards yields0 ≤ v − ( t ) − v − ( t )= 2 P mj =1 jv +1 ( r j )+ (1 − γ ) P nl =1 (cid:0) v − ( u l ) − v − ( u l ) (cid:1) + 2 q ( v +1 ( t )) + 2 q ( t ) + v +1 ( t ) − v +2 ( t ) . (6.5) If there is at least one floor collision between two (1 ,
2) collisions, then the squareroot term 2 q ( v +1 ( t )) + 2 q ( t ) appears in (6.5). The latter is part of the timethe lowest ball needs to fall to the floor after a (1 ,
2) collision. Remember, that v +1 ( r j ) ≥ , ∀ j ∈ { , . . . , m } , (6.6)since this is the velocity of the first ball right after taking off from the floor.At the heart of this work lies the following Theorem 6.1.
There exists a constant
C > , such that for every x ∈ M and every i ∈ { , . . . , N − } , there exists a divergent sequence of collision times ( t n ) n ∈ N =( t n ( x, i )) n ∈ N : v − i ( t n ) − v − i +1 ( t n ) ≥ C .Outline of the proof : We start describing a certain collision pattern. Since everycollision happens infinitely often, this collision pattern can be found (non-uniquely)infinitely often in every orbit. The time interval of this pattern in the proof belowis given by [ t − (1 , , t (1 , ]. At t − (1 , , t (1 , we have a (1 ,
2) collision and somewherein between is at least one (0 ,
1) collision.We start a proof by contradiction assuming that every pre-collisional velocitydifference of (1 ,
2) collisions in [ t − (1 , , t (1 , ] is arbitrarily small. Using the aboveformulas (6.2) - (6.5) this will amount to having every ball arbitrarily close to thefloor with velocities being arbitrarily close to each other at time t − (1 , . Since thereis at least one (0 ,
1) collision between the two (1 ,
2) collisions at t − (1 , , t (1 , thesquare root term in (6.5) exists. This implies that at t − (1 , all the balls will havearbitrarily small velocities, which results in a contradiction since the energy of thesystem would be arbitrarily small. Hence, the velocity difference v − − v − of at leastone (1 ,
2) collision in [ t − (1 , , t (1 , ] is bounded from below.Repeatedly using the above formulas, we obtain lower bounds for at least onevelocity difference v − i − v − i +1 , for every i ∈ { , . . . , N − } . Since this collisionpattern appears infinitely often along every orbit we can extend these considerationsobtaining the result from Theorem 6.1. Proof.
Pick an arbitrary (1 ,
2) collision and mark the time as t − (1 , . Then, pickthe next (2 ,
3) collision in the future and mark the time as t − (2 , . Continuingthis procedure for the next (3 , , . . . ( N − , N ) collisions, gives us collision times t − (3 , , . . . , t − ( N − ,N ) . After that we pick the first (0 ,
1) collision and mark itscollision time with t . We now reverse the order of collisions after t and markthe future collision times of the first consecutively appearing ( N − , N ) , . . . , (1 , t ( N − ,N ) , . . . , t (1 , . Note, that in the intervals [ t − ( i,i +1) , t − ( i +1 ,i +2) ], i ∈ { , . . . , N − } , exactly one ( i + 1 , i + 2) collision occurs, while in the interval[ t , t ( N − ,N ) ] resp. [ t ( i,i +1) , t ( i − ,i ) ], i ∈ { , . . . , N − } , exactly one ( N − , N ) resp.( i − , i ) collision occurs, but there is no restriction on other collisions happening.The collision times of each ( i, i + 1) collision, including floor collisions, inducea partition P i of the time interval [ t − (1 , , t (1 , ]: For every i ∈ { , . . . N − } ,there exists a positive integer n = n ( i ) ≥
2, such that the collision times of allthe ( i, i + 1) collisions in the interval [ t − ( i,i +1) , t ( i,i +1) ] are given by s i, , . . . , s i,n , The exact time the lowest ball needs to fall to the floor is v +1 ( t ) + q ( v +1 ( t )) + 2 q ( t ). Note, that we can have a floor collision between two (1 ,
2) collision without a full return ofthe lowest ball to the floor, i.e. the square root term is present in (6.5) but m = 0 in the first sum. with s i, := t − ( i,i +1) and s i,n := t ( i,i +1) . For i = 1, n = n (1) ≥
0, and by default s , = t − (1 , , s ,n +1 = t (1 , . For i = 0, s , , . . . , s ,n , n = n (0) ≥
1, are simply thecollision times of the lowest ball with the floor in the open interval ( t − (1 , , t (1 , ).We augment the collision time sequences by a first element s i, := t − (1 , and alast element s i,n +1 := t (1 , , which yields the partitions P i = S nk =0 [ s i,k , s i,k +1 ], forevery i ∈ { , . . . , N − } .If at time s i,k of our partition, we face a singular collision, we might have torepartition. The details of this procedure are described in the last four paragraphsof the proof.Assume that for every ε > k ∈ { , . . . , n + 1 } , where n = n (1), wehave v − ( s ,k ) − v − ( s ,k ) < ε, (6.7)that is, the velocity differences right before every (1 ,
2) collision in [ t − (1 , , t (1 , ]are arbitrarily small.In order to apply (6.5) we need to quantify how many (2 ,
3) collisions and floorreturns of the lowest ball are in between two successful (1 ,
2) collisions. We intro-duce, for i ∈ { , . . . , N − } , j ∈ { , . . . , N − } , k ∈ { , . . . , n } , where n = n ( j ),the functional c i : P j → N [ s j,k , s j,k +1 ] c i ([ s j,k , s j,k +1 ]) =: c i,j,k . The term c i,j,k counts how many ( i, i +1) collisions appear in the interval [ s j,k , s j,k +1 ]of the partition P j , i.e. in between two successful ( j, j + 1) collisions happening attime s j,k and s j,k +1 . Applying this notation, we expand the velocity differences in(6.7) backwards and according to (6.5) obtain for every k ∈ { , . . . , n (1) + 1 } ≤ v − ( s ,k ) − v − ( s ,k )= 2 P c , ,k j =1 jv +1 ( s ,g ( j ) )+ (1 − γ ) P c , ,k l =1 (cid:0) v − ( s ,g ( l ) ) − v − ( s ,g ( l ) ) (cid:1) + 2 q ( v +1 ( s ,k − )) + 2 q ( s ,k − ) + v +1 ( s ,k − ) − v +2 ( s ,k − ) , (6.8)where the functions g ( j ) ∈ { , . . . , n (0) } and g ( l ) ∈ { , . . . , n (2) } enumerate thecollision times subindices. Using (6.8) together with (6.3), (6.6) and our assumption(6.7), implies for every ε > v − ( s ,k ) − v − ( s ,k ) < ε, ∀ k ∈ { , . . . , n (2) } , (6.9a) v +1 ( s ,k ) < ε, ∀ k ∈ { , . . . , n (0) } , (6.9b) v +1 ( s ,k − ) < ε, q ( s ,k − ) < ε, ∀ k ∈ { , . . . , n (1) + 1 } . (6.9c)We repeat step (6.8), by expanding the remaining velocity differences v − i ( s i,k ) − v − i +1 ( s i,k ), for all i ∈ { , . . . , N − } , k ∈ { , . . . , n ( i ) } backwards. Using (6.3), (6.4),(6.9a), this leads to v − i ( s i,k ) − v − i +1 ( s i,k ) < ε, (6.10) for all ε > i ∈ { , . . . , N − } , k ∈ { , . . . , n ( i ) } . Every pre-collisional velocitydifference v − i − v − i +1 occurring in [ t − (1 , , s i, ) resp. ( s i,n , t (1 , ] can be expanded for-ward resp. backward and by using (6.10) will be arbitrarily small as well. Therefore,(6.7) implies that every ball to ball pre-collisional velocity difference in [ t − (1 , , t (1 , ]is arbitrarily small.If the next ball to ball collision is ( i, i + 1), i ∈ { , . . . , N − } , the collision timeis given by q i +1 − q i v i − v i +1 . If the denominator v i − v i +1 is arbitrarily small, q i +1 , q i , has to be arbitrarilysmall as well, otherwise the collision time would be arbitrarily large, which wouldresult in arbitrarily large velocities and contradict the finite energy assumption.Since every velocity difference in [ t − (1 , , t (1 , ] is arbitrarily small, at time t − (1 , ,all the balls are lying arbitrarily close to the floor with velocities being arbitrarilyequal. Due to our construction, there is at least one (0 ,
1) collision in [ t − (1 , , t (1 , ].Hence, the square root term in (6.8) is present, which further implies (6.9c). Thus,at time t − (1 , , every ball lies arbitrarily close to the floor with arbitrarily smallvelocity. In this way, H ( q ( t ) , p ( t )) < ε , for every ε >
0, which means that ourorbit would break through the constant energy surface. Since this is impossible,we obtain a contradiction to our beginning assumption (6.7), hence, there exists aconstant C > k ∈ { , . . . , n (1) + 1 } , such that v − ( s ,k ) − v − ( s ,k ) ≥ C . (6.11)In order to obtain the existence of a constant C > i, i + 1)collision, such that v − i − v − i +1 ≥ C , for all i ∈ { , . . . N − } , we first pick theprevious resp. next (2 ,
3) collision before resp. after the (1 ,
2) collision in (6.11).Let the past resp. future (2 ,
3) collision happen at t p resp. t f . Using (6.4) weexpand v − ( t f ) − v − ( t f ) backwards and obtain0 < v − ( t f ) − v − ( t f )= (1 + γ ) P mj =1 (cid:0) v − ( r j ) − v − ( r j ) (cid:1) + (1 − γ ) P nl =1 (cid:0) v − ( u l ) − v − ( u l ) (cid:1) + v +2 ( t p ) − v +3 ( t p ) , where r , . . . , r m resp. u , . . . , u n are the collision times of the (1 ,
2) resp. (3 , ,
3) collisions occurring at times t p , t f . Note, thatthe reason we denoted these collision times as r j resp. u l (and not s ,j resp. s ,l ) isbecause one of the (2 ,
3) collisions may lie outside of [ t − (1 , , t (1 , ]. This dependson the position of the (1 ,
2) collision at time s ,k from (6.11).Assuming that both (2 ,
3) velocity differences in the past and future are arbitrar-ily small yields a contradiction since v − ( s ,k ) − v − ( s ,k ) ≥ C . Hence, there existsa constant C >
0, such that either v − ( t f ) − v − ( t f ) ≥ C or v − ( t p ) − v − ( t p ) ≥ C .Successfully continuing this procedure we find positive constants C , . . . , C N − > i, i + 1) collision, for all i ∈ { , . . . , N − } , such that v − i − v − i +1 ≥ min { C , . . . , C N − } . (6.12) It follows from the way we obtained (6.12), that the collision times of all ( i, i + 1)collisions satisfying (6.12) do not necessarily belong to [ t − (1 , , t (1 , ].The above steps can be repeated, thus, creating infinitely many compact intervalswith a sequence of constant positive lower bounds for at least one v − − v − percompact interval. This holds along every orbit. Those lower bounds have a globalminimum, i.e. min x ∈M min n ∈ N v − ( t n ( x, − v − ( t n ( x, C >
0, for every pre-collisional velocity difference.In the event of a singular collision between balls i − i , i + 1, i ≥
1, whichhappens at s i − ,k = s i,k , for some k ∈ { , . . . , n − } , our orbit branches intotwo suborbits and the procedure above works for both branches, because there arefurther ( i − , i ), ( i + 1 , i ) collisions flanking the singular collisions in the past andthe future.If the singularity occurs at the last possible collision time s i − , = s i, or s i − ,n = s i,n , we have to repartition the collision times for one of the suborbits. We onlyoutline s i − , = s i, since s i − ,n = s i,n works in a similar way: If, for the firstsuborbit, the collision order is ( i − , i ) → ( i, i + 1), nothing changes. If, ( i, i + 1) → ( i − , i ), then we do not consider the ( i, i + 1) collision but rather set s i − , = s i, to be the collision time of ( i − , i ). Then, continue as described in the beginning ofthe proof by picking the next collisions ( i, i + 1) , . . . , ( N − , N ) with corresponding(and possibly new) collision times s i, , . . . , s N − , .Note, that if i = 1 in the last paragraph we face no problem with either collisionorder (0 , → (1 , , → (0 , s , = t − (1 , with the (1 ,
2) collision (and in exactly the same manner, we associate s ,n +1 = t (1 , either (1 ,
2) in the future).The same procedure as in the last three paragraphs is initiated if the orbitexperiences a singularity involving more than three balls. (cid:3)
Remark . We want to bring to the readers attention, that it may be possible(depending on the dynamics), for some x ∈ M , i ∈ { , . . . , N − } , to have divergingcollision time subsequences ( u n ) n ∈ N , which satisfy, for instancelim n →∞ v − i ( u n ( x, i )) − v − i +1 ( u n ( x, i )) = 0 . The importance is, that such a behaviour may only happen along a collision timesubsequence, since there must be enough space left for ( t n ) n ∈ N from Theorem 6.1to exist. 7. The non-contraction property
We begin this section by pointing out, that it is sufficient for the non-contractionproperty to hold if we only prove it for every v ∈ C ( x ) ∩ ∂B k·k (0 , ∂B k·k (0 , k · k , in tangent space.Since the flow derivative between collisions is equal to the identity matrix, it isequivalent to formulate the non-contraction property in terms of the flow, i.e. ∃ ζ > , ∀ t > , ∀ x ∈ M \ S + t , ∀ v ∈ C ( x ) ∩ ∂B k·k (0 ,
1) : k d x φ t v k ≥ ζ. We know [HT19, Remark 10.3], that arbitrarily many (0 ,
1) collisions can occurin finite time. This is why we prefer to formulate the non-contraction property interms of the flow, because we rather deal with finite times than arbitrarily manyderivative map compositions.Assume now that the strict unboundedness property (SU) holds for every point.We fix E > τ + E : M → R + ,x τ + E ( x ) , where τ + E ( x ) = min { t > Q ( d x φ t v ) > E , ∀ v ∈ C ( x ) } . (7.1)The assumption of strict unboundedness (SU) together with the compactness of M and C ( x ) ∩ ∂B k·k (0 ,
1) will help us to assert that τ + E is uniformly bounded fromabove, i.e. ∃ T > , ∀ x ∈ M : τ + E ( x ) ≤ T . (7.2)This information is then utilized to split up the proof of the non-contraction prop-erty into two parts: First, we prove the non-contraction property for every collisionof every feasible orbit in the fixed time interval [0 , T ] and, second, for every t > T .We begin with the proof of the uniform upper bound for τ + E . Lemma 7.1.
The function τ + E is uniformly bounded from above.Proof. The assertion of strict unboundedness (SU) is equivalent to ∀ K ≥ , ∀ x ∈ M , ∀ v ∈ C ( x ) , ∃ s = s ( K, x, v ) : Q ( d x φ t v ) > K, ∀ t ≥ s . Since the Q -form is homogeneous (of degree two), the previous statement does notlose its general validity if we only assume it for v ∈ C ( x ) ∩ ∂B k·k (0 , ∃ T > , ∀ x ∈ M , ∀ v ∈ C ( x ) ∩ ∂B k·k (0 ,
1) : s ( E , x, v ) ≤ T . Assume on the contrary, that ∀ T > , ∃ x = x ( T ) ∈ M , ∃ v = v ( T ) ∈ C ( x ( T )) ∩ ∂B k·k (0 ,
1) : s ( E , x ( T ) , v ( T )) > T . Due to compactness, the limits lim T →∞ x ( T ) = x ∗ resp. lim T →∞ v ( T ) = v ∗ lie in M resp. C ( x ∗ ) ∩ ∂B k·k (0 , s in the strict unboundednessstatement, our assumption implies ∃ x ∗ ∈ M , ∃ v ∗ ∈ C ( x ∗ ) ∩ ∂B k·k (0 ,
1) : lim t →∞ Q ( d x ∗ φ t v ∗ ) ≤ E , which clearly yields a contradiction to the strict unboundedness property. (cid:3) We introduce the norm k ( δξ, δη ) k HT := k ( δξ, δη ) k + k δη k CW , where k δη k CW = N − X i =1 ( δη i +1 − δη i ) m i , is a norm on R N − introduced by Cheng and Wojtkowski in [ChW91, (11)]. It isinvariant with respect to the submatrices D i , D Ti of the ball to ball collision mapderivatives given in (4.6). The norm k · k refers to the Euclidean norm.We start with the first part of the proof by investigating how the fixed length ofa vector changes, when it is subjected to floor or ball to ball collisions. Lemma 7.2.
There exists a constant E > , such that for all i ∈ { , . . . , N } , x ∈ M + i, , ( δξ, δη ) ∈ C ( x ) ∩ ∂B k·k HT (0 , and n ≥ , we have k d x Φ n , ( δξ, δη ) k HT ≥ E . Proof.
Using the definition of the floor derivative d Φ , (4.6), we estimate k d x Φ n , ( δξ, δη ) k HT ≥ k ( δξ, nBδξ + δη ) k ≥ max {k δξ k , k nBδξ + δη k } . We will be proving the following statement: There exists a constant E >
0, suchthat for all i ∈ { , . . . , N } , x ∈ M + i, , ( δξ, δη ) ∈ C ( x ) ∩ ∂B k·k HT (0 ,
1) and n ≥
1, wehave k δξ k ≥ E ∨ k nBδξ + δη k ≥ E . (7.3)Assume on the contrary that the previous statement does not hold, i.e. for every E >
0, there exists an i ∈ { , . . . , N } , x ∈ M + i, , ( δξ, δη ) ∈ C ( x ) ∩ ∂B k·k HT (0 , n ≥
1, such that k δξ k < E ∧ k nBδξ + δη k < E . (7.4)For E sufficiently small, conditions (7.4) imply | δξ | , | δξ | , . . . , | δξ N − | < E , | nβδξ + δη | , | δη | , . . . , | δη N − | < E . (7.5)Since ( δξ, δη ) ∈ C ( x ) ∩ ∂B k·k HT (0 , δη com-ponent of the vector ( δξ, δη ) is concentrated on the first entry δη , i.e. there existsa constant E = E ( E ) >
0, such that | δη | ≥ E . (7.6)If δξ , δη > , δξ , δη < δξ = 0 , then | nβδξ + δη | = nβ | δξ | + | δη | ≥ | δη | ≥ E , which contradicts (7.5). Assume therefore that δξ δη < C ( x ) ∩ ∂B k·k HT (0 , P N − i =1 δξ i δη i ≥ > δξ δη ≥ − ( δξ δη + . . . + δξ N − δη N − ) . (7.7)Due to (7.5), (7.6) the second inequality in (7.7) is violated, because the right handside is of quadratic order O ( E ), while the δξ δη term is of linear order O ( E ).Hence, for sufficiently small E , this implies ( δξ, δη ) / ∈ C ( x ) ∩ ∂B k·k HT (0 , (cid:3) A uniform lower bound for multiple ball to ball collisions can only be establishedfor a fixed number of ball to ball collisions. Let v max > Lemma 7.3.
For a fixed n ≥ , let d x T n be a product of n ball to ball collisionderivatives. Then, there exists a constant E > , such that for all m ≥ , x ∈M \ ( S Ni =2 M + i, ∪ M m, +1 , ) , ( δξ, δη ) ∈ C ( x ) ∩ ∂B k·k HT (0 , , n ≤ n , we have k d x T n ( δξ, δη ) k HT ≥ E . Proof.
Using the ball to ball collision map derivatives (4.6), a first estimate gives, k d x T n ( δξ, δη ) k HT ≥ max {k D n δξ + U n δη k , k D Tn δη k CW } . If E > k δη k CW ≥ E , then the invariance with respect to D Tn of the norm k · k CW immediately yields, that the vector is bounded from below.Assume therefore that k δη k CW < E , for a value E , which will be chosensufficiently small. Since ( δξ, δη ) ∈ C ( x ) ∩ ∂B k·k HT (0 , E = E ( E ) >
0, such that k δξ k CW ≥ E . It is clear, that if E decreases, E increases.The matrix product U n is recursively defined by U = F i , U n = D i n U n − + F i n D Tn − , for some i , . . . , i n ∈ { , . . . , N } depending on x and D n = D i n . . . D i . Repeatedlyusing the triangle inequality and the D i -invariance of the k·k CW -norm, we estimate k U n k CW ≤ k D i n k CW k U n − k CW + k F i n k CW k D Tn − k CW = k U n − k CW + k F i n k CW ≤ k F i k CW + . . . + k F i n k CW . Remembering the definition of F i k (4.6) and α k (4.7), we obtain the upper bound k U n k CW < n v max max i ∈{ ,...,N − } m i m i +1 ( m i − m i +1 )( m i + m i +1 ) . We abbreviate E := max i ∈{ ,...,N − } m i m i +1 ( m i − m i +1 )( m i + m i +1 ) , and estimate k D n δξ + U n δη k ≥ k δξ k − k U n kk δη k ≥ c k δξ k CW − c k U n k CW k δη k CW ≥ c E − c n v max E E ≥ c E − c n v max E E , where c , c > E sufficiently small and obtain alower bound E > (cid:3) To conclude the first step of the non-contraction property it remains to prove
Lemma 7.4.
Let
T > . The number of ball to ball collisions in [0 , T ] is boundedfrom above by a constant, which depends only on the length of the interval and theenergy of the system. Proof.
We know from [G78, G81, V79], that the number of ball to ball collisions in[0 , T ], without any floor interaction, is bounded from above. Assume therefore thatwe have arbitrarily many ball to ball collisions and floor interactions in [0 , T ]. Let i k ∈ { , . . . , N − } , k ∈ N , and ( i k , i k + 1) be the aforementioned diverging collisionsequence in decreasing order, i.e. ( i k , i k + 1) happens prior to ( i k − , i k − + 1).Additionally, we let each ( i k , i k + 1) happen at time t ( i k ,i k +1) . Due to the energyrestriction it is clear that v max > | v + i ( t ( i ,i +1) ) | . Using the formulas of the velocitytime evolution (3.2) and elastic collisions (3.4), (3.5), we expand v + i ( t ( i ,i +1) ) n -times backward and obtain v max > | v + i ( t ( i ,i +1) ) | = |− ( t ( i ,i +1) − t ( i n ,i n +1) ) + v + i n ( t ( i n ,i n +1) ) + c ( n ) X k =1 γ i k ( v − i k − v − i k +1 ) | . (7.8)The positive integer function c ( n ) counts how many ball to ball collisions happenedwithin the n -backward iterations. Since we also consider floor collisions c ( n ) < n ,for n large enough. Observe that each ball to ball collision adds a velocity differenceterm to the positive sum in (7.8). Hence, since we assume to have arbitrarily manyball to ball collisions, lim n →∞ c ( n ) = ∞ . Letting n go to infinity in (7.8), wefirst obtain that lim n →∞ − ( t ( i ,i +1) − t ( i n ,i n +1) ) is bounded since [0 , T ] is. Second,due to Theorem 6.1, the sequence ( γ i k ( v − i k − v − i k +1 )) k ∈ N does not converge to zeroand, thus, lim n →∞ P c ( n ) k =1 γ i k ( v − i k − v − i k +1 ) = ∞ , which results in the contradiction v max > ∞ .We want to supplement the details for the reader, that the sum will be largeenough for (7.8) to be violated after a uniform number of summations. The proofof this fact is similar to the proof of Lemma 7.1. Abbreviate a c ( n ) ( x ) := c ( n ) X k =1 γ i k ( v − i k ( t ( i k ,i k +1) ) − v − i k +1 ( t ( i k ,i k +1) )) . The divergence of a c ( n ) ( x ) is equivalent to ∀ K > , ∀ x ∈ M , ∃ M = M ( K, x ) ∈ N : a c ( n ) ( x ) > K, ∀ n ≥ M ( K, x ) . (7.9)Due to the energy c > K = K ( c ) > a c ( n ) for whichinequality (7.8) is violated. Observe that K ( c ) additionally depends on the termof opposite sign − ( t ( i ,i +1) − t ( i n ,i n +1) ) and, thus, on the length of [0 , T ], hence,we have the depence K = K ( c, T ).We want to prove ∀ c > , ∀ T > , ∃ M = M ( c, T ) ∈ N , ∀ x ∈ M : M ( K , x ) < M ( c, T ) . Assume on the contrary, that this does not hold, i.e. ∃ c > , ∃ T > , ∀ M ∈ N , ∃ x = x ( M ) : M ( K , x ( M )) ≥ M . Since M is compact, the limit lim M →∞ x ( M ) = x ∗ lies in M . For this x ∗ , weobtain from (7.9) lim n →∞ a c ( n ) ( x ∗ ) ≤ K , which contradicts the divergence of a c ( n ) ( x ∗ ). Therefore, the number of ball to ballcollisions in [0 , T ] are bounded by a constant M ( c, T ), which depends only on thelength T of [0 , T ] and the energy c > (cid:3) Combining the last three lemmas yields
Corollary 7.1.
Let
T > and c > be the energy of the system. Then, thereexists a constant ζ = ζ ( T, c ) > , such that the non-contraction property holds forevery finite time interval [0 , T ] , i.e. ∃ ζ > , ∀ t ≤ T, ∀ x ∈ M \ S + t , ∀ v ∈ C ( x ) ∩ ∂B k·k HT (0 ,
1) : k d x φ t v k ≥ ζ . The corollary applies directly to the interval [0 , T ], where T is the uniform upperbound of τ + E (7.2). We will now conclude the proof of the non-contraction propertyby proving it for all t > T .Due to h δξ − δη, δξ − δη i ≥
0, the Euclidean norm k · k and the Q -form can berelated via k ( δξ, δη ) k ≥ √ p Q ( δξ, δη ) . Using the Q -monotonicity of the derivative and (7.1), (7.2), we obtain k d x φ t ( δξ, δη ) k HT ≥ k d x φ t ( δξ, δη ) k ≥ √ p Q ( d x φ t ( δξ, δη )) ≥ √ p Q ( d x φ T ( δξ, δη )) ≥ p E , ∀ t > T . This immediately proves the non-contraction property
Corollary 7.2.
The non-contraction property formulated in terms of the norm k · k HT holds with constant ζ = min { ζ , √ E } . Ergodicity of a particle falling in a three dimensional wide wedge
In [W98], Wojtkowski investigated the hyperbolicity of dynamical systems, whichdescribe the motion of a particle subjected to constant acceleration in a variety ofwedges. We start by recapitulating the necessary prerequisites from [W98, HT20]to prove our results. For a thorough introduction to the subject we recommendreading [W98].The unrestricted configuration space N q (3.1) of N falling balls has the form ofa wedge. Abbreviating q = ( q , . . . , q N ), we can alternatively formulate it as W ( b , . . . , b N ) = n q ∈ R N : q = N X i =1 d i b i , d i ∈ R N + , i ∈ { , . . . , N } o , where the set of linearly independent vectors { b , . . . , b N } , called generators , aregiven by b i = ( b i,k ) Nk =1 with b i, = . . . = b i,i − = 0, b i,i = . . . = b i,N = 1. Observe,that for every mass configuration ( m , . . . , m N ), the system of falling balls hasthe same unrestricted configuration space. As before, we obtain the dynamics byintersecting the wedge with the energy surface E c (3.3).We introduce the Q -isometric coordinate transformation x i = √ m i q i , v i = p i √ m i . (8.1)The newly obtained unit generators { e , . . . , e N } in these coordinates become √ M i e i =(0 , . . . , , √ m i , . . . , √ m N ), where M i = m i + . . . + m N . We observe that, up to a scalar multiple, every mass configuration defines a different wedge W ( e , . . . , e N ),since the angles between the generators now depend on the masses. The inner prod-uct given by the kinetic energy in these coordinates is the standard scalar product h· , ·i in R N . Since h e i , e j i = p M j / √ M i , it is easy to verify that h e i , e i +1 i > , ∀ i ∈ { , . . . , N − } , h e i , e j i = Q j − l = i h e l , e l +1 i , ∀ i, j ∈ { , . . . , N } , i = j. (8.2)Wedges satisfying (8.2) are called simple [W98, Proposition 2.3].In the three dimensional case, the hitting of the face W ( e , e ), resp. W ( e , e )corresponds in the physical model to a (1 ,
2) resp. (2 ,
3) collision. The triplecollision states are given by the intersection of the former faces, which amounts tothe first generator, i.e. W ( e , e ) ∩ W ( e , e ) = e .It was shown in [HT20] that for the mass restriction2 √ m m = √ m + m + m , (8.3)the configuration wedge can be unfolded, by continuously reflecting it in the faces W ( e , e ) resp. W ( e , e ), into a wide wedge [W98, Definition 6.1]. This widewedge consists exactly of six simple wedges [HT20, Figure 1]. This idea is due toWojtkowski and can be generalized to N dimensions [W16].The triple collision states in the configuration space, which are represented bythe first generator e , disappear in the wide wedge. More precisely, each trajectorywhich passes through the spot where e was has a smooth continuation. Sincethe triple collision singularity manifold is the only obstacle to proper alignment[LW92, HT20], the system of a particle falling in the wide wedge, obtained for thespecial mass configuration (8.3), satisfies the proper alignment condition (C5).For the Chernov-Sinai ansatz (C4) to be valid, we need that the orbit for µ S ± -a.e. x ∈ S ± emerging from the singularity manifold is strictly unbounded. Thiscertainly holds since strict unboundedness is established for every orbit in M [HT20,Main Theorem].As was proven earlier in this work, the non-contraction property (C2) followsdirectly from the validity of the strict unboundedness for every point.The Lagrangian subspaces L , L (4.4) of the eventually strictly invariant conefield C are both constant in M and therefore continuous. This verifies condition(C3).For the regularity of singularity manifolds (C1), we employ [LW92, Lemma 7.7].The aforementioned lemma states that if T : M\S + → M\S − is a diffeomorphism,the proper alignment condition (C5) holds and d x T is Q -monotone for every x ∈ M ,then the regularity of singularity subsets follows. The last two conditions havealready been affirmatively addressed. For the first one, observe that we outlinedat the end of Section 3, that T is a symplectomorphism up to and including theregular boundary R + (3.7).Since the proper alignment condition (C5) holds and every point is strictly un-bounded [HT20, Main Theorem] it follows from Subsection 5.2 that the set ofsufficient points has measure one and is arcwise connected. Thus, the model of aparticle falling in a three dimensional wedge is ergodic. Acknowledgments.
I want to thank Nandor J. Sim´anyi and Maciej P. Wojtkowskifor their consultation and hospitality. Additionally my gratitude is expressed to-wards Marius N. Nkashama and UAB for providing the academic support, duringthe completion of this work in the summer of 2020.
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Michael Hofbauer-Tsiflakos
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