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Hyperbolic measures and commuting maps in low dimension

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  • 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.
    Mathematics Subject Classification: 32Qxx.

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