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This question pertains to billiard dynamics and their invariant measures. Specifically, it concerns the oft-quoted 'fact' that billiard flow maps (built using specular reflection boundary conditions) admit the Liouville measure (also known as the restriction of the Lebesgue measure to the tangent bundle) as an invariant measure for all times.

To be precise, if the billiard table $\mathcal{P}\subset\mathbb{R}^{N}$ is a 'reasonably-nice' subset of Euclidean space (for instance, it admits the structure of a manifold with corners, an example of which would be the unit square if $N=2$), I'd like to prove that billiard flow maps $T^{t}:T\mathcal{P}\rightarrow T\mathcal{P}$ admit the property that $$T^{t}\#\mathscr{L}_{2N}\mathsf{L}T\mathcal{P}=\mathscr{L}_{2N}\mathsf{L}T\mathcal{P}$$ for all $t\in\mathbb{R}$, where:

  • $T\mathcal{P}$ is the tangent bundle of the table $\mathcal{P}$, considered as a subset of Euclidean space $\mathbb{R}^{2N}$;
  • $T^{t}$ is the Lebesgue almost everywhere-defined billiard map built using 'elastic reflection boundary conditions' on $\partial\mathcal{P}$, whose associated trajectories $t\mapsto T^{t}((x_{0}, v_{0}))$ are piecewise linear and continuous in the spatial variable, but lower semi-continuous in the velocity variable for any initial point $(x_{0}, v_{0})\in T\mathcal{P}$;
  • $\mathscr{L}_{2N}$ denotes the Lebesgue measure on $\mathbb{R}^{2N}$;
  • $\#$ denotes the pushforward operation, and $\mathsf{L}$ denotes the restriction measure operation, whence $\mathscr{L}_{2N}\mathsf{L}T\mathcal{P}$ is the restriction of the Lebesgue measure on $\mathbb{R}^{2N}$ to the set $T\mathcal{P}$.

One of the main issues in proving this statement is that the dynamics is not smooth, and one cannot appeal to the classical Liouville theorem of symplectic geometry to prove it.

Does anyone know of a reference in which the above statement is proved, at least for some class of tables $\mathcal{P}$? I have spent quite some time going through the literature, but I have so far come up empty handed. I was trained as a mathematical analyst, so I'm looking for a proof that someone in that community would consider as complete.

mark_wilkinson
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1 Answers1

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A good reference is Chaotic Billiards by Nikolai Chernov and Roberto Markarian. In chapter 2 of their book, they give a proof that the Liouville measure is an invariant measure for a class of billiards in domains (billiard tables) $\mathcal{D} \subset \mathbb{R}^2$, both for the flow and the billiard map. In particular they make the following 4 assumptions on the domains (these are cited, with some minor modifications to clarify things):

Assumption 1 (section 2.1, page 19): The boundary of the billiard domain $\partial \mathcal{D}$ is a union of smooth ($\mathcal{C}^{l}, l\geq 3$) compact curves: $$\partial \mathcal{D} = \Gamma = \Gamma_1\cup \dots \cup \Gamma_r.$$ More precisely, each curve $\Gamma_i$ is defined by a $\mathcal{C}^{l}$ map $f_i:\left[a_i,b_i\right] \rightarrow \mathbb{R}^{2}$, which is one-to-one on $\left[a_i, b_i\right)$ and has one-sided derivatives, up to order $l$, at the points $a_i$ and $b_i$.

Assumption 2 (section 2.1, page 20): The boundary components $\Gamma_i$ can intersect each other only at their endpoints; i.e. $$\Gamma_i \cap \Gamma_j \subset \partial\Gamma_i \cup \partial \Gamma_j, \text{ for } i\neq j.$$

Assumption 3 (section 2.1, page 20): On every $\Gamma_i$, the second derivative $f_i^{''}$ either never vanishes is is identically zero (thus, every wall $\Gamma_i$ is either a curve without inflection points or a line segment).

Assumption 4: Any billiard table $\mathcal{D}$ contains no cursps made by a focusing wall and a dispersing wall. (The reason for this is that this is the only type of cusp in which particles can get caught in a corner).

Now the invariance statement of the Liouville measure for the billiard flow under the above assumptions can be found in section 2.6 on page 28:

Theorem 2.21: The flow $\Phi^{t}$ preserves the volume form $dx\wedge dy \wedge d\omega $ on the phase space $\Omega = \mathcal{D}\times \mathcal{S}^1$, where $(x,y) \in \mathcal{D}$ represent the position in the billiard table and $\omega \in \mathcal{S}^{1}$ the angle with respect to the positive $x$-axis of the direction of the billiard particle. (Thus is preserves the Lebesgue measure $dxdyd\omega$ on $\Omega$.)

Even though this only covers the 2-dimensional case, I hope it helps somewhat.

Please let me know if I am violating any copyright restrictions (then I will remove the cited assumptions and the theorem). I typed out the assumptions and the theorem so that I could give some context to make it easier to understand and so that one would not have to read around it so much.

Nelus127
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