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On Some Aspects of the Theory of Anosov Systems

With a Survey by Richard Sharp: Periodic Orbits of Hyperbolic Flows

  • Grigoriy A. Margulis

Part of the Springer Monographs in Mathematics book series (SMM)

Table of contents

  1. Front Matter
    Pages i-vii
  2. Richard Sharp
    Pages 73-138
  3. Back Matter
    Pages 139-142

About this book

Introduction

 In this book the seminal 1970 Moscow thesis of Grigoriy A. Margulis is published for the first time. Entitled "On Some Aspects of the Theory of Anosov Systems", it uses ergodic theoretic techniques to study the distribution of periodic orbits of Anosov flows. The thesis introduces the "Margulis measure" and uses it to obtain a precise asymptotic formula for counting periodic orbits. This has an immediate application to counting closed geodesics on negatively curved manifolds. The thesis also contains asymptotic formulas for the number of lattice points on universal coverings  of compact manifolds of negative curvature.

The thesis is complemented by a survey by Richard Sharp, discussing more recent developments in the theory of periodic orbits for hyperbolic flows, including the results obtained in the light of Dolgopyat's breakthroughs on bounding transfer operators and rates of mixing.

Keywords

Anosov Flows Hyperbolic Flows Lebesgue measures Periodic Orbits Riemannian Geometry dynamical systems ergodic theory

Authors and affiliations

  • Grigoriy A. Margulis
    • 1
  1. 1.Department of MathematicsYale UniversityNew HavenUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-662-09070-1
  • Copyright Information Springer-Verlag Berlin Heidelberg 2004
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-07264-2
  • Online ISBN 978-3-662-09070-1
  • Series Print ISSN 1439-7382
  • Buy this book on publisher's site
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