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## Ergodicity of smooth systems with product measures

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 $f$ be a C^2 diffeomorphism of a compact manifold which preserves volume. Let $p$ be a hyperbolic periodic point. Define $\Lambda^u(p):=\{x\text{ backward-Lyapunov regular}: W^u(x)\cap W^s(p)$ has a transverse point $\}$. Define $\Lambda^s(p)$ similarly.

If $vol(\Lambda^s(p))>0$ and $vol(\lambda^u(p))>0$ then these two sets are equal (up to a zero volume set) and the restriction of $f$ 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: $h^s_{xy}:F_x\to F_y$ for $y\in W^s(x)$. 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 $f:S^1\times\mathbb T^2\to S^1\times\mathbb T^2$  be partially hyperbolic with a central foliation into circles. If $f$ has the accessibility property then either $f$ is topologically conjugate to an isometric extension of an Anosov diffeomorphism (and $f$ has a unique maximal entropy measure which has a zero central exponent), or $f$ 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?