Anatole Katok (Penn State)

Measure rigidity beyond uniform hyperbolicity and positive entropy

Abstract. In the last decade, and especially in the last few years a great progress has been achieved in the classification of invariant ergodic measures of algebraic Anosov and to a lesser extent partially hyperbolic actions of higher rank abelian groups, i.e. $\mathbb{Z}^k$ and $\mathbb{R}^k$ for $k\ge 2$ . At the same time progress in the differentiable rigidity for hyperbolic actions of such groups indicates that extension of the measure rigidity theory in this direction may turn out to be essentially vacuous and may not be worth pursuing. On the other hand, there are actions of higher rank abelian groups with hyperbolic invariant measures, i.e. measures with non-vanishing Lyapunov exponents. Topological structure of such actions may be rather complicated but some rigidity appears for hyperbolic invariant measures with positive entropy. An essential modification of the methods of the measure rigidity theory is needed to tackle this case. I will discuss the results for measures for Cartan type (actions of $\mathbb{Z}^{n-1}$ on $n$ -dimensional manifolds) with Lyapunov exponents in general position which I have obtained several years ago but whose detailed proofs are still not published. I will outline open problems and possible approaches to more general situations.

Another notoriously resistant problem concerns zero entropy measures even for such simple systems as the Furstenberg $\times2,\,\,\times 3$ action on the circle. I offer some speculations on how partial progress in this direction might be achieved.