## Ergodicity of smooth systems with product measures

November 27, 2009 by jbuzzi

Federico RODRIGUEZ-HERTZ presented at the IHP conference for Katok’s 65th birthday several results pertaining to the ergodicity of systems with some product structure.

**Theorem (RHRHTU 2008).*** Let be a C^2 diffeomorphism of a compact manifold which preserves volume. Let be a hyperbolic periodic point. Define has a transverse point . Define similarly.*

*If and then these two sets are equal (up to a zero volume set) and the restriction of there, is ergodic with no zero exponent.*

**Corollary. ***Ergodicity is C^1-open and dense within C^2 partially hyperbolic diffeomorphisms with central dimension at most 2.*

**Theorem (M.A. Rodriguez-Hertz). ***A generic 3-dimensional volume-preserving either has all its Lyapunov exponents vanishing Lebesgue-almost everywhere, or has a dominated splitting separating positive and negative Lyapunov exponents and is ergodic with respect to the volume.*

*Remark. *The proof is expected to generalize to symplectomorphisms in arbitrary dimensions.

*Remark.* This is a stronger but less general result that that of AVILA and BOCHI according to which, in arbitrary dimension, the C^1-generic volume-preserving diffeomorphism either has a dominated splitting and a set of positive volume of non-uniformly hyperbolic points defining an ergodic component of the volume or there is, almost everywhere, at least one zero exponent.

Consider a bundle with fiber a smooth compact manifold and a bundle map acting smoothly on the fibers and preserving on the base a measure.

**Theorem. ***Assume that there are well-defined holonomy maps on the fibers: for . If the exponents along the fiber vanish, then these holonomy preserve the conditional measures. If the base map is hyperbolic and the base measure has a product structure then the conditional measures depend continuously on the base point.*

This extends a classical result of Ledrappier for matrix cocycles. The above has the following application:

**Theorem (RHRHTU).*** Let be partially hyperbolic with a central foliation into circles. If has the accessibility property then either is topologically conjugate to an isometric extension of an Anosov diffeomorphism (and has a unique maximal entropy measure which has a zero central exponent), or has at least 2 maximal entropy measures and finitely many of them and their Lyapunov exponents are bounded away from zero.*

**Question. **Is the set of diffeomorphism with finitely many ergodic maximal entropy measures C^r-dense?

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