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July  1996, 2(3): 397-411. doi: 10.3934/dcds.1996.2.397

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  May 1996 Published  May 1996

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.
Keywords: compact manifolds, commuting diffeomorphisms, hyperbolic measure..
Mathematics Subject Classification: 32Qx.
Citation: Anatole Katok. Hyperbolic measures and commuting maps in low dimension. Discrete and Continuous Dynamical Systems, 1996, 2 (3) : 397-411. doi: 10.3934/dcds.1996.2.397
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