Hyperbolic measures and commuting maps in low dimension

Anatole Katok

doi:10.3934/dcds.1996.2.397

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1996, Volume 2, Issue 3: 397-411. Doi: 10.3934/dcds.1996.2.397

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Hyperbolic measures and commuting maps in low dimension

Anatole Katok^1,

1.
Department of Mathematics, Penn State University, University Park, State College, PA 16802

Received: April 30, 1996

Published: June 1996

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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 $\mathbb{Z}^k$ 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.

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Mathematics Subject Classification: 32Qxx.

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