Hyperbolic measures and commuting maps in low dimension

Anatole Katok

doi:10.3934/dcds.1996.2.397

Hyperbolic measures and commuting maps in low dimension

Anatole Katok

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Abstract

We study invariant measures with non-vanishing Lyapunov characteristic exponents for commuting diffeomorphisms of compact manifolds. In particular we show that for k = 2, 3 no faithful ℤ^k real-analytic action on a k-dimensional manifold preserves a hyperbolic measure. In the smooth case similar statements hold for actions faithful on the support of the measure. Generalizations to higher dimension are proved under certain non-degeneracy conditions for the Lyapunov exponents.

Original language	English (US)
Pages (from-to)	397-411
Number of pages	15
Journal	Discrete and Continuous Dynamical Systems
Volume	2
Issue number	3
DOIs	https://doi.org/10.3934/dcds.1996.2.397
State	Published - 1996

All Science Journal Classification (ASJC) codes

Analysis
Discrete Mathematics and Combinatorics
Applied Mathematics

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10.3934/dcds.1996.2.397

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abstract = "We study invariant measures with non-vanishing Lyapunov characteristic exponents for commuting diffeomorphisms of compact manifolds. In particular we show that for k = 2, 3 no faithful ℤk real-analytic action on a k-dimensional manifold preserves a hyperbolic measure. In the smooth case similar statements hold for actions faithful on the support of the measure. Generalizations to higher dimension are proved under certain non-degeneracy conditions for the Lyapunov exponents.",

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N2 - We study invariant measures with non-vanishing Lyapunov characteristic exponents for commuting diffeomorphisms of compact manifolds. In particular we show that for k = 2, 3 no faithful ℤk real-analytic action on a k-dimensional manifold preserves a hyperbolic measure. In the smooth case similar statements hold for actions faithful on the support of the measure. Generalizations to higher dimension are proved under certain non-degeneracy conditions for the Lyapunov exponents.

AB - We study invariant measures with non-vanishing Lyapunov characteristic exponents for commuting diffeomorphisms of compact manifolds. In particular we show that for k = 2, 3 no faithful ℤk real-analytic action on a k-dimensional manifold preserves a hyperbolic measure. In the smooth case similar statements hold for actions faithful on the support of the measure. Generalizations to higher dimension are proved under certain non-degeneracy conditions for the Lyapunov exponents.

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U2 - 10.3934/dcds.1996.2.397

DO - 10.3934/dcds.1996.2.397

M3 - Article

AN - SCOPUS:0030501322

SN - 1078-0947

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EP - 411

JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

IS - 3

ER -

Hyperbolic measures and commuting maps in low dimension

Abstract

All Science Journal Classification (ASJC) codes

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