NONUNIFORM MEASURE RIGIDITY

By B. Kalinin, A. Katok, F. R. Hertz. 2008

46 citations recorded by the source corpus.

We consider an ergodic invariant measure µ for a smooth actionof Z k , k � 2, on a (k+1)-dimensional manifold or for a locally free smooth action of R k , k � 2 on a (2k + 1)-dimensional manifold. We prove that if µ is hyperbolic with the Lyapunov hyperplanes in general position and if one element in Z k has positive entropy, then µ is absolutely continuous. The main ingredient is absolute continuity of conditional measures on Lyapunov foliations which holds for a more general class of smooth actions of higher rank abelian groups.

Subfields: Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bb8be0a583a435cdcc5ddc24b81367a3bc54454a

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In Mathematical Physics

1. Aaron W. Brown
2. F. R. Hertz
3. Artur Avila
4. Marcelo Viana

Citations in the corpus

• Measure rigidity for random dynamics on surfaces and related skew products (2015, methodology): “ surfaces and measures with non-zero exponents though we expect the results to hold in more generality. We rely heavily on the tools from the theory of nonuniformly hyperbolic diffeomorphisms used in [KKRH] and many ideas developedin [EM] includinga modified version(see [ EM, Section 16]) of the “exp”