Abstract
We study the size of the set of ergodic directions for the directional billiard flows on the infinite band $\R\times [0,h]$ with periodically placed linear barriers of length $0<\lambda<h$. We prove that the set of ergodic directions is always uncountable. Moreover, if $\lambda/h\in(0,1)$ is rational the Hausdorff dimension of the set of ergodic directions is greater than 1/2. In both cases (rational and irrational) we construct explicitly some sets of ergodic directions.
Highlights
In this paper we consider the following infinite periodic billiard, whose ergodic properties have been the object of recent investigation
A billiard trajectory is the trajectory of a point-mass which moves freely inside the table on segments of straight lines and undergoes elastic collisions when it hits the boundary of the table
The billiard flowt∈R is defined on the subset of the phase space T (h, a, λ) × S1 that consists of the points (x, θ ) ∈ T (h, a, λ) × S1 such that if x belongs to the boundary of T (h, a, λ) θ is an inward direction
Summary
In this paper we consider the following infinite periodic billiard, whose ergodic properties have been the object of recent investigation (see e. g. [1,3,9]). If λ/ h is rational or belongs to a set ⊂ (0, 1) of full Lebesgue measure for almost every direction θ the billiard flow (φtθ )t∈R on T (h, a, λ) is not ergodic. We prove that the set of ergodic directions is uncountable when λ/ h is irrational (see Theorem 4.1) and for rational λ/ h we prove that its Hausdorff dimension is greater than 1/2 (see Theorem 5.1). 5 with the approach introduced recently by Hooper in [7] one might be able to describe all invariant ergodic Radon measures for (φtθ )t∈R whenever λ/ h is rational and θ belongs to a set of positive Hausdorff dimension
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