Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A family of natural equilibrium measures for Sinai billiard flows

Jérôme Carrand

Abstract

The Sinai billiard flow on the two-torus, i.e., the periodic Lorentz gas, is a continuous flow, but it is not everywhere differentiable. Assuming finite horizon, we relate the equilibrium states of the flow with those of the Sinai billiard map $T$ -- which is a discontinuous map. We propose a definition for the topological pressure $P_*(T,g)$ associated to a potential $g$. We prove that for any piecewise Hölder potential $g$ satisfying a mild assumption, $P_*(T,g)$ is equal to the definitions of Bowen using spanning or separating sets. We give sufficient conditions under which a potential gives rise to equilibrium states for the Sinai billiard map. We prove that in this case the equilibrium state $μ_g$ is unique, Bernoulli, adapted and gives positive measure to all nonempty open sets. For this, we make use of a well chosen transfer operator acting on anisotropic Banach spaces, and construct the measure by pairing its maximal eigenvectors. Last, we prove that the flow invariant probability measure $\bar μ_g$, obtained by taking the product of $μ_g$ with the Lebesgue measure along orbits, is Bernoulli and flow adapted. We give examples of billiard tables for which there exists an open set of potentials satisfying those sufficient conditions.

A family of natural equilibrium measures for Sinai billiard flows

Abstract

The Sinai billiard flow on the two-torus, i.e., the periodic Lorentz gas, is a continuous flow, but it is not everywhere differentiable. Assuming finite horizon, we relate the equilibrium states of the flow with those of the Sinai billiard map -- which is a discontinuous map. We propose a definition for the topological pressure associated to a potential . We prove that for any piecewise Hölder potential satisfying a mild assumption, is equal to the definitions of Bowen using spanning or separating sets. We give sufficient conditions under which a potential gives rise to equilibrium states for the Sinai billiard map. We prove that in this case the equilibrium state is unique, Bernoulli, adapted and gives positive measure to all nonempty open sets. For this, we make use of a well chosen transfer operator acting on anisotropic Banach spaces, and construct the measure by pairing its maximal eigenvectors. Last, we prove that the flow invariant probability measure , obtained by taking the product of with the Lebesgue measure along orbits, is Bernoulli and flow adapted. We give examples of billiard tables for which there exists an open set of potentials satisfying those sufficient conditions.
Paper Structure (29 sections, 42 theorems, 254 equations, 2 figures)

This paper contains 29 sections, 42 theorems, 254 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Billiards and equilibrium states
  3. Statement of main results -- Organization of the paper
  4. Topological Pressure, Variational Principle and Abramov Formula
  5. Easy Direction of the Variational Principle for the Pressure
  6. Abramov Formula and Choice of the Potential $g$
  7. Growth Lemma and Fragmentation Lemmas
  8. Growth Lemma
  9. Fragmentation Lemmas
  10. Exact Exponential Growth of Thermodynamic Sums -- Cantor Rectangles
  11. Estimates on norms of the potential
  12. The Banach Spaces ${\mathcal{B}}$ and ${\mathcal{B}}_w$ and the Transfer Operators $\mathcal{L}_g$
  13. Motivation and heuristics
  14. Definition of the Banach spaces and embeddings into distribution
  15. The transfer operators
  16. ...and 14 more sections

Key Result

Theorem 1.2

If $g$ is a bounded, piecewise HölderSee precise definition in the beginning of Section sect:pressure. potential such that $P_*(T,g) - \sup g > s_0 \log 2$ and $\log \Lambda > \sup g - \inf g$, then there exists a probability measure $\mu_g$ such that If the assumption $\log \Lambda > \sup g - \inf g$ is weakened into the condition SSP.1 (Definition def:SSP1), then item (i) still holds. If the as

Figures (2)

  • Figure 1: (a) The Sinai billiard on a triangular lattice studied in Baras95 with angle $\pi/3$, scatterer of radius 1, and distance $d$ between the centers of adjacent scatterers. (b) The Sinai billiard on a square lattice with scatterers of radius $R < R'$ studied in Garrido97. The boundary of a single cell is indicated by dashed lines in both tables. (Figure taken from BD2020MME.)
  • Figure 2: Two examples of two periodic trajectories.

Theorems & Definitions (96)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof : Proof of Lemma \ref{['lemma:sup_less_poly_inf']}
  • proof : Proof of Theorem \ref{['thm:pressure']}
  • proof : Proof of Proposition \ref{['prop:pressure_bowen_definitions']}.
  • Lemma 2.4
  • ...and 86 more