On the work of Dolgopyat on partial and Nonuniform Hyperbolicity

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

This paper is a nontechnical survey and aims to illustrate Dolgo-pyat's profound contributions to smooth ergodic theory. I will discuss some of Dolgopyat's work on partial hyperbolicity and nonuniform hyperbolicity with emphasis on the interaction between the two-the class of dynamical systems with mixed hyperbolicity. On one hand, this includes uniformly partially hyperbolic diffeomorphisms with nonzero Lyapunov exponents in the center direction. The study of their ergodic properties has provided an alternative approach to the Pugh-Shub stable ergodicity theory for both conservative and dissipative systems. On the other hand, ideas of mixed hyper-bolicity have been used in constructing volume-preserving diffeomorphisms with nonzero Lyapunov exponents on any manifold.

Original languageEnglish (US)
Pages (from-to)227-241
Number of pages15
JournalJournal of Modern Dynamics
Volume4
Issue number2
DOIs
StatePublished - Apr 1 2010

Fingerprint

Partial Hyperbolicity
Nonuniform Hyperbolicity
Diffeomorphisms
Lyapunov Exponent
Dynamical systems
Conservative System
Ergodic Theory
Dissipative Systems
Hyperbolicity
Ergodicity
Dynamical system
Alternatives
Interaction

All Science Journal Classification (ASJC) codes

  • Analysis
  • Algebra and Number Theory
  • Applied Mathematics

Cite this

@article{278283262c1e40388c4f4472091c5735,
title = "On the work of Dolgopyat on partial and Nonuniform Hyperbolicity",
abstract = "This paper is a nontechnical survey and aims to illustrate Dolgo-pyat's profound contributions to smooth ergodic theory. I will discuss some of Dolgopyat's work on partial hyperbolicity and nonuniform hyperbolicity with emphasis on the interaction between the two-the class of dynamical systems with mixed hyperbolicity. On one hand, this includes uniformly partially hyperbolic diffeomorphisms with nonzero Lyapunov exponents in the center direction. The study of their ergodic properties has provided an alternative approach to the Pugh-Shub stable ergodicity theory for both conservative and dissipative systems. On the other hand, ideas of mixed hyper-bolicity have been used in constructing volume-preserving diffeomorphisms with nonzero Lyapunov exponents on any manifold.",
author = "Yakov Pesin",
year = "2010",
month = "4",
day = "1",
doi = "10.3934/jmd.2010.4.227",
language = "English (US)",
volume = "4",
pages = "227--241",
journal = "Journal of Modern Dynamics",
issn = "1930-5311",
publisher = "American Institute of Mathematical Sciences",
number = "2",

}

On the work of Dolgopyat on partial and Nonuniform Hyperbolicity. / Pesin, Yakov.

In: Journal of Modern Dynamics, Vol. 4, No. 2, 01.04.2010, p. 227-241.

Research output: Contribution to journalArticle

TY - JOUR

T1 - On the work of Dolgopyat on partial and Nonuniform Hyperbolicity

AU - Pesin, Yakov

PY - 2010/4/1

Y1 - 2010/4/1

N2 - This paper is a nontechnical survey and aims to illustrate Dolgo-pyat's profound contributions to smooth ergodic theory. I will discuss some of Dolgopyat's work on partial hyperbolicity and nonuniform hyperbolicity with emphasis on the interaction between the two-the class of dynamical systems with mixed hyperbolicity. On one hand, this includes uniformly partially hyperbolic diffeomorphisms with nonzero Lyapunov exponents in the center direction. The study of their ergodic properties has provided an alternative approach to the Pugh-Shub stable ergodicity theory for both conservative and dissipative systems. On the other hand, ideas of mixed hyper-bolicity have been used in constructing volume-preserving diffeomorphisms with nonzero Lyapunov exponents on any manifold.

AB - This paper is a nontechnical survey and aims to illustrate Dolgo-pyat's profound contributions to smooth ergodic theory. I will discuss some of Dolgopyat's work on partial hyperbolicity and nonuniform hyperbolicity with emphasis on the interaction between the two-the class of dynamical systems with mixed hyperbolicity. On one hand, this includes uniformly partially hyperbolic diffeomorphisms with nonzero Lyapunov exponents in the center direction. The study of their ergodic properties has provided an alternative approach to the Pugh-Shub stable ergodicity theory for both conservative and dissipative systems. On the other hand, ideas of mixed hyper-bolicity have been used in constructing volume-preserving diffeomorphisms with nonzero Lyapunov exponents on any manifold.

UR - http://www.scopus.com/inward/record.url?scp=77957093452&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77957093452&partnerID=8YFLogxK

U2 - 10.3934/jmd.2010.4.227

DO - 10.3934/jmd.2010.4.227

M3 - Article

AN - SCOPUS:77957093452

VL - 4

SP - 227

EP - 241

JO - Journal of Modern Dynamics

JF - Journal of Modern Dynamics

SN - 1930-5311

IS - 2

ER -