# American Institute of Mathematical Sciences

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

## Hyperbolic measures and commuting maps in low dimension

 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.
Citation: Anatole Katok. Hyperbolic measures and commuting maps in low dimension. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 397-411. doi: 10.3934/dcds.1996.2.397
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