# Markov approximations and statistical properties of billiards

## TL;DR

This paper surveys the development of Markov partitions and their variants, such as sieves and towers, which are crucial for understanding the statistical properties of hyperbolic billiards with singularities.

## Contribution

It provides a comprehensive overview of the evolution and various constructions of Markov partitions for hyperbolic systems with singularities.

## Key findings

- Markov partitions are essential for analyzing hyperbolic billiards.
- Various flexible and simpler Markov partition variants have been developed.
- These methods facilitate the study of statistical properties in complex dynamical systems.

## Abstract

Markov partitions designed by Sinai(1968) and Bowen(1970) proved to be an efficient tool for descibing statistical properties of uniformly hyperbolic systems. For hyperbolic systems with singularities, in particular, for hyperbolic billiards the construction of a Markov partition by Bunimovich and Sinai(1980) was a delicate and hard task. Therefore later more and more flexible and simple variants of Markov partitions appeared: Markov sieves (Bunimovich-Chernov-Sinai, 1990), Markov towers (Young, 1998), standard pairs (Dolgopyat). This remarkable evolution of Sinai's original idea is surveyed in this paper.