On the measure of maximal entropy for finite horizon Sinai billiard maps. (English) Zbl 1444.37020
The Sinai billiard flow on the two-torus is a dispersing non-differentiable dynamical system where the singularities are due to the “nongrazing orbits,” that is, those that are not tangent to a scatterer.
In this paper the authors study the Sinai billiard map, which is the return map of the single point particle to the scatterers in the Sinai billiard flow.
The Sinai billiard map is discontinuous. It is the analogue of a periodic planar Lorentz gas model for the motion of a single dilute electron in a metal. The scatterers of the Sinai map correspond to the atoms of the metal and, in this paper, they are assumed to be strictly convex. The authors propose a definition for the topological entropy and they study measures of maximal entropy for this map.
They prove that this topological entropy is not smaller than the value given by the variational principle, and that it is equal to the definition given by R. Bowen [Trans. Am. Math. Soc. 184, 125–136 (1974; Zbl 0274.54030)] of topological entropy for continuous maps using spanning \(\epsilon\)-separated and \(\epsilon\)-spanning sets.
Under a mild condition of sparse recurrence to the singularities, the authors, using a transfer operator acting on a space of anisotropic distributions, construct an invariant probability measure of maximal entropy for the Sinai billiard map. They show that this measure has full support and is Bernoulli, that it is the unique measure of maximal entropy and that it is different from the smooth invariant measure except if all nongrazing periodic orbits have multiplier equal to the topological entropy. They also prove that the topological entropy of this map is equal to the Bowen-Pesin-Pitskel topological entropy of the restriction of the Sinai billiard map to a noncompact domain of continuity. Finally, they give a lower bound for the number of period orbits of order \(n\) of the Sinai billiard map. The lower bound is exponential in \(n\) multiplied by the topological entropy of the map.
In this paper the authors study the Sinai billiard map, which is the return map of the single point particle to the scatterers in the Sinai billiard flow.
The Sinai billiard map is discontinuous. It is the analogue of a periodic planar Lorentz gas model for the motion of a single dilute electron in a metal. The scatterers of the Sinai map correspond to the atoms of the metal and, in this paper, they are assumed to be strictly convex. The authors propose a definition for the topological entropy and they study measures of maximal entropy for this map.
They prove that this topological entropy is not smaller than the value given by the variational principle, and that it is equal to the definition given by R. Bowen [Trans. Am. Math. Soc. 184, 125–136 (1974; Zbl 0274.54030)] of topological entropy for continuous maps using spanning \(\epsilon\)-separated and \(\epsilon\)-spanning sets.
Under a mild condition of sparse recurrence to the singularities, the authors, using a transfer operator acting on a space of anisotropic distributions, construct an invariant probability measure of maximal entropy for the Sinai billiard map. They show that this measure has full support and is Bernoulli, that it is the unique measure of maximal entropy and that it is different from the smooth invariant measure except if all nongrazing periodic orbits have multiplier equal to the topological entropy. They also prove that the topological entropy of this map is equal to the Bowen-Pesin-Pitskel topological entropy of the restriction of the Sinai billiard map to a noncompact domain of continuity. Finally, they give a lower bound for the number of period orbits of order \(n\) of the Sinai billiard map. The lower bound is exponential in \(n\) multiplied by the topological entropy of the map.
Reviewer: Athanase Papadopoulos (Strasbourg)
MSC:
| 37C30 | Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc. |
| 37C55 | Periodic and quasi-periodic flows and diffeomorphisms |
| 37C25 | Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics |
| 37C83 | Dynamical systems with singularities (billiards, etc.) |
| 37E30 | Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces |
| 37B40 | Topological entropy |
| 37A25 | Ergodicity, mixing, rates of mixing |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 47B38 | Linear operators on function spaces (general) |
Keywords:
Sinai billiard map; topological entropy; topological entropy; maximal entropy measure; Bowen-Margulis measure; transfer operatorCitations:
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