Ergodic directions for billiards in a strip with periodically located obstacles

TL;DR

This paper investigates the set of ergodic directions in a billiard system with periodic obstacles, proving uncountability and providing lower bounds on Hausdorff dimension, with explicit constructions for these directions.

Contribution

It establishes the uncountability of ergodic directions and provides explicit constructions and dimension bounds for these directions in a periodic billiard system.

Findings

The set of ergodic directions is uncountable.

Hausdorff dimension of ergodic directions exceeds 1/2 when obstacle length ratio is rational.

Explicit sets of ergodic directions are constructed.

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.