Fundamental Theorem for smooth billiards in high dimensions

February 18, 2003

1 Problem

Prove the Fundamental Theorem of Semi-dispersing billiards for the case when scatterers are C_{3} smooth.

2 Motivation

The Fundamental Theorem (FTH) is the key tool in proving local ergodicity of semidispersing billiards and also many focusing billiards.

3 Discussion

The precise formulation of the theorem is long, so we postpone it after the discussion. To understand most of the discussion, a rough understanding of the statement is enough: Roughly speaking: The Fundamental Theorem states that through many points of the phase space, one can draw long stable and unstable manifolds.

3.1 History

The two-dimensional case had been settled in Sinai’s celebrated work, [S(1970)]. Seventeen years had elapsed until the multidimensional generalization given by Chernov and Sinai appeared, [S-Ch(1987)]. It offered a quite involved, but in essence very transparent formulation of the theorem and a delicate proof. A self-contained exposition of the original ideas with detailed conditions and arguments were provided in [K-S-Sz(1990)] where a slightly more general, the so called “transversal” version of the fundamental theorem was announced — mainly with its application to three-billiards in mind. Several other papers have appeared in the ’90s with nice expositions of the theorem, even for classes of dynamical systems more general than the original semi-dispersing billiard setting (eg. Hamiltonian systems with singularities in [L-W(1995)]).

3.2 Difficulty:

However, all of these papers assumed that for all powers of the dynamics the singularity set (the set of phase points where dynamics is not continuous) is a finite collection of one-codimensional smooth and compact submanifolds. This assumption was necessary to estimate the size of certain neighbourhoods of the singularity set on which unstable manifolds end. In [B-Ch-Sz-T(2000)] this was shown not to be satisfied in the high-dimensional case. In [B-Ch-Sz-T(2002)], a proof of the FTH was given with the smoothness assumption replaced by the so-called “Lipschitz decomposability” assumption. The assumption was shown to hold in the case when scatterers have algebraic boundaries. The case of more general scatterers remained open.

So the way to go seems to be to either check Lipscitz decomposability for the singularity set (of iterates of the dynamics) for high-dim. billiards with any C_{3} smooth scatterers, or find some other means to control the size of neighbourhoods of the singularity set.

4 (More) Precise formulation

of the version of the FTH in [B-Ch-Sz-T(2002)]

4.1 Prerequisites

Definition 4.1. Singularities. Consider the set of tangential reflections, i.e.

R := {(q, v) ∈ ∂M | ⟨v, n(q)⟩ = 0}.

It is easy to see that the map T is not continuous at the set T^{−1}R. As a consequence, the singularity set for a higher iterate T^{n} is

R^{(n)} = ∪_{i=1}^{n}R^{−i},

where in general R^{k} = T^{k}R.

Notation

For any n ∈ ℕ, Δ_{n} stands for the set of doubly singular phase points up to order n, i.e. x ∈ ∂M belongs to Δ_{n} whenever there are indices k_{1} ≠ k_{2}, |k_{i}| ≤ n such that both T^{k_{1}}x and T^{k_{2}}x are elements of R. We will need a notation for phase points with regular or with at most once singular trajectories, thus we consider the following set:

∂M^{∗} := ∂M \ ⋃_{n=1}^{∞} Δ_{n}

Definition 4.2. local orthogonal manifolds (or fronts) are defined in the flow phase space the following way.

Let x = (q, v) ∈ M \ ∂M and consider a C^{2}-smooth codimension 1 submanifold Σ′ ⊂ Q \ ∂Q such that q ∈ Σ′ and v = v(q) is the normal vector to Σ′ at q. Denote by Σ the normal section of the unit tangent bundle on Q restricted to Σ′. Σ is called a local orthogonal manifold or simply a front. A front is said to be (strictly) convex whenever its second fundamental form B_{Σ}(y) ≥ 0 (B_{Σ}(y) > 0) for every y ∈ Σ.

Definition 4.3. Let us consider a nonsingular finite trajectory segment for the flow: S^{[a,b]}x, where a < 0 < b and a, b, 0 are not moments of collision. N_{0}(S^{[a,b]}x), the neutral subspace at time 0 for the segment S^{[a,b]}x is defined as follows:

N_{0}(S^{[a,b]}x) := { w ∈ ℝ^{d} : ∃(δ > 0)s.t.∀α ∈ (−δ, δ)
v(S^{a}(q(x) + αw, v(x))) = v(S^{a}x)&
v(S^{b}(q(x) + αw, v(x))) = v(S^{b}x)}.

Observe that v(x) ∈ N_{0}(S^{[a,b]}x) is always true, the neutral subspace is at least 1 dimensional. Neutral subspaces at time moments different from 0 are defined by N_{t}(S^{[a,b]}x) := N_{0}(S^{[a−t,b−t]}(S^{t}x)), thus they are naturally isomorphic to the one at 0.

Definition 4.4. The non-singular trajectory segment S^{[a,b]}x is sufficient if for some (and in that case for any) t ∈ [a, b] : dim(N_{t}(S^{[a,b]}x)) = 1. A point x ∈ M^{0} is said to be sufficient if its entire trajectory S^{(−∞,∞)}x contains a finite sufficient segment. Singular points are treated by the help of trajectory branches (see [K-S-Sz(1990)]): a point x ∈ M^{1} (this precisely means that the entire trajectory contains one singular reflection) is sufficient if both of its trajectory branches are sufficient.

All these concepts have their natural counterparts for the billiard map phase space ∂M. For example, a smooth piece Σ ∈ ∂M of the image of a local orthogonal manifold in M is referred to as a front as well.

To precisely formulate the property that we propose instead of smoothness of the singularities, we need the following two definitions:

Definition 4.5. A function f : D ⊂ ℝ^{m−1} → ℝ will be called locally Lipschitz (with Lipschitz contant L), if for any x ∈ D there exists a neighbourhood U ⊂ ℝ^{m−1} of x such that the restricted function f|_{D∩U} is Lipschitz (with Lipschitz constant L).

In all applications D will be open. Notice that in this case, f typically cannot even be extended to D̄ in a continuous way.

Definition 4.6. H ⊂ ℝ^{m} will be called a (one-codimensional) open locally Lipschitz graph (with Lipschitz constant L), if we can choose an appropriate Cartesian coordinate system so that H becomes the graph of a locally Lipschitz function:

H = {(x, f(x) | x ∈ D}

with some D ⊂ ℝ^{m−1} and f : D → ℝ locally Lipschitz (with constant L).

We will be mainly interested in the case when the domain D is an open set in ℝ^{m−1}, then

– H will be called an open locally Lipschitz graph (even though it is not an open set in ℝ^{m}),

– and we will denote by ∂H the boundary of H as of a surface: ∂H = H̄ \ H.

Now we are able to define the regularity property that should replace the smoothness of singularities. This property, called ‘Lipschitz decomposability’ will be defined for subsets of ℝ^{m} here. For Lipschitz decomposability of subsets of a Riemannian manifold, see Remark 4.8.

Definition 4.7. Consider H ⊂ ℝ^{m}, and L ∈ ℝ. H will be called ‘Lipschitz decomposable’ (one-codimensional) subset with constant L if it can be decomposed into a finite number of open locally Lipschitz graphs and a small remainder set in the following way: There exist H^{∗} and H_{1}, ..., H_{K} such that:

• H ⊂ ⋃_{i=1}^{K} H̄_{i} ⋃ H^{∗},

• H_{i} ∩ H_{j} = ∅ for any i ≠ j,

• every H_{i} is a one-codimensional open locally Lipschitz graph (with constant L),

• L^{m} ((( ⋃_{i=1}^{K} ∂H_{i}) ∪ H^{∗})^{[δ]}) = o(δ).

The set H^{∗} is included in the decomposition for technical reasons: we want to allow for sets H having parts of strictly higher codimension. This occurs generically if H is an algebraic subvariety of ℝ^{n}. Nevertheless we would like to note that such higher codimensional parts are not present in the singularities of semi-dispersing billiards.

Remark 4.8. Lipschitz decomposability in Riemannian manifolds.

Throughout the paper – and in particular in conjecture 4.9 below – subsets of a compact Riemannian manifold M are considered. For H ⊂ M Lipschitz decomposition is understood in terms of coordinate charts.

To be more precise, let us fix some convention related to the atlas {U_{t}, ψ_{t}}_{t=1}^{T} for M first. It is important that M is compact thus we may consider a finite atlas. We say that the atlas is bi-Lipschitz if all charts ψ_{t} : U_{t} → ℝ^{m} are bi-Lipschitz maps, i.e. both ψ_{t} and (ψ_{t})^{−1} are Lipschitz with some constant K > 1. All atlases considered are assumed to be bi-Lipschitz with a fixed constant. This ensures that Euclidean distance on ℝ^{m} is comparable to Riemannian metric on the manifold. Note that bi-Lipschitzness – with Lipschitz constant arbitrarily close to one – can always be obtained by choosing the coordinate patches sufficiently small.

As to the problem of Lipschitz decomposition, we will say that H ⊂ M is Lipschitz decomposable whenever a finite bi-Lipschitz atlas can be chosen, such that for all charts ψ_{t}(H ∩ U_{t}) is Lipschitz decomposable as a subset of ℝ^{m}, in the sense of Definition 4.7. ^{1}

The precise property that we expect the singularities of semi-dispersing billiards to have is formulated in the form of a conjecture:

Conjecture 4.9. For any semi-dispersing billiard with a finite horizon there exists an L ∈ ℝ such that for any integer N the set ⋃_{|n|≤N} R^{n} of singularities of order up to N is ‘Lipschitz decomposable’ with constant L.

It is worth noting that by introducing “transparent walls” (cf. [S-Ch(1987)]) any semi-dispersing billiard can be reduced to one with a finite horizon.

The statement of this conjecture will appear word by word among the conditions of the fundamental theorem for semi-dispersing billiards.

4.2 Formulation of the theorem

Before its formulation its is important to point out the conditions under which the modified proof of the theorem works. We use the notations just introduced.

Condition 4.10. (Chenov-Sinai Ansatz, Condition 3.1 from [K-S-Sz(1990)]). For almost every point x ∈ R we have x ∈ ∂M^{∗} and, moreover, the positive semi-trajectory of the point x is sufficient.

Condition 4.11. There exists an L ∈ ℝ such that for every N ∈ ℕ the singularity set ⋃_{|n|≤N} R^{n} is ‘Lipschitz decomposable’ with constant L (cf. Conjecture 4.9).

To formulate the fundamental theorem, we introduce the notion of regular coverings. Note that m = 2d − 2 is the dimension of the (Poincaré) phase space ∂M. The next definition will not be absolutely precise, for we omit some technical details for the sake of easier understanding. For a precise formulation please see Definition 3.4 in [K-S-Sz(1990)].

Definition 4.12. Let us assume that for a point x ∈ ∂M^{∗} and its neighborhood U(x) a smooth foliation U(x) = ∪_{α∈B^{d−1}}Γ_{α} is given. The foliae Γ_{α} are d − 1-dimensional manifolds uniformly transversal to all possible local stable manifolds (B^{d−1} is the standard d − 1-dimensional open ball).

The parametrized family of finite coverings

G^{δ} = {G_{i}^{δ} | i = 1, ..., I(δ) } 0 < δ < δ_{0}

is a family of regular coverings iff:

1The delicate question how sensitive this notion of Lipschitz-decomposition is to the choice of the atlas needs further investigation.

1. each G_{i}^{δ} is an open parallelepiped of dimension 2d − 2;

2. the d − 1-dimensional faces of G_{i}^{δ} are all cubes with edge-length δ, moreover, they may belong to two different categories: the s-faces are ’parallel with leaves of the stable foliation’ while the Γ-faces are ’parallel’ with the leaves of the foliation Γ;^{2}

3. For any point, the maximal number of parallelepipeds covering it is 2^{2d−2};

4. if G_{i}^{δ} ∩ G_{j}^{δ} ≠ ∅, then

μ_{1}(G_{i}^{δ} ∩ G_{j}^{δ}) ≥ c_{1}δ^{2d−2}

with c_{1} independent of δ.

Some further convention: Given any G_{i}^{δ} its s-jacket, ∂^{s}(G_{i}^{δ}) is the union of those (2d − 3)-dimensional faces of G_{i}^{δ} which contain at least one s-face of it. The Γ-jacket, ∂^{Γ}(G_{i}^{δ}) is defined similarly. Clearly, ∂(G_{i}^{δ}) = ∂^{s}(G_{i}^{δ}) ∪ ∂^{Γ}(G_{i}^{δ}). We say that a stable manifold γ^{s}(y) intersects G_{i}^{δ} correctly if:

∂(G_{i}^{δ} ∩ γ^{s}(y)) ⊂ ∂^{Γ}(G_{i}^{δ}).

Theorem 4.13. (The Fundamental Theorem) We assume that:

- conditions 4.10 and 4.11 are satisfied;

- a sufficient phase point x ∈ ∂M^{∗} is given;

- a smooth transversal foliation Γ in a neighborhood U_{0} of x is fixed;

- a constant 0 < ϵ_{1} < 1 is chosen.

Then there is a sufficiently small neighborhood U_{ϵ_{1}}(x) such that for any U(x) ⊂ U_{ϵ_{1}}(x) and for any family of regular coverings, the covering G^{δ} can be divided into two disjoint subsets, G_{g}^{δ} and G_{b}^{δ} (called ‘good’ and ‘bad’), in such a way that: (I) For any G_{i}^{δ} ∈ G_{g}^{δ} and any s-face E^{s} of it, the set:

{y ∈ G_{i}^{δ} | ρ(y, E^{s}) < ϵ_{1}δ and γ^{s}(y) intersects correctly}

has positive relative μ_{1}-measure in G_{i}^{δ}. (II)

μ_{1} ( ⋃_{G_{i}^{δ}∈G_{b}^{δ}} G_{i}^{δ}) = o(δ).

2More precisely if we consider the center of each parallelepiped w_{i}^{δ} ∈ G_{i}^{δ}, the s- and Γ- faces are parallel with the tangent planes T_{w_{i}^{δ}}γ^{s}(w_{i}^{δ}) and T_{w_{i}^{δ}}Γ(w_{i}^{δ}), respectively.

References

[B-Ch-Sz-T(2000)] P. Bálint, N. Chernov, D. Szász, P. Tóth; Geometry of Multi-dimen-sional Dispersing Billiards, to appear in Astérisque (2000)

[B-Ch-Sz-T(2002)] P. Bálint, N. Chernov, D. Szász, P. Tóth; Multidimensional Semi-Dispersing Billiards: Singularities and the Fundamental Theorem, Annales Henri Poincaré 3 (2002) 451–482

[K-S-Sz(1990)] A. Krámli, N. Simányi, D. Szász; A “Transversal” Fundamental Theorem for Semi-Dispersing Billiards, Comm. Math. Phys. 129 (1990) 535–560

[L-W(1995)] C. Liverani, M. Wojtkowski; Ergodicity in Hamiltonian Systems, Dynamics Reported 4 (New series) (1995) 130–202

[S(1970)] Ya.G. Sinai; Dynamical Systems with Elastic Reflections, Russian Mathematical Surveys, (2) 25 (1970) 137–189

[S-Ch(1987)] Ya. G. Sinai, N. Chernov; Ergodic Properties of Certain Systems of 2–D Discs and 3–D Balls, Russain Mathematical Surveys, (3) 42 (1987) 181–201