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Markov Approximations and Statistical Properties of Billiards

  • Domokos Szász
Chapter
Part of the The Abel Prize book series (AP)

Abstract

Markov partitions designed by Sinai (Funct Anal Appl 2:245–253, 1968) and Bowen (Am J Math 92:725–747, 1970) proved to be an efficient tool for describing 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 (Commun Math Phys 78:247–280, 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, Russ Math Surv 45(3):105–152, 1990), Markov towers (Young, Ann Math (2) 147(3):585–650, 1998), standard pairs (Dolgopyat). This remarkable evolution of Sinai’s original idea is surveyed in this paper.

Notes

Acknowledgements

The author is highly indebted to Yasha Pesin for his careful reading of the manuscript and many useful remarks and, moreover, to Péter Bálint and Thomas Gilbert for kindly helping him out with the figures and to Ferenc Wettl for technical help with them. This research was supported by Hungarian National Foundation for Scientific Research grant No. K 104745 and OMAA-92öu6 project.

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Authors and Affiliations

  • Domokos Szász
    • 1
  1. 1.Budapest University of Technology and EconomicsBudapestHungary

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