Search results for " 37D25"
showing 3 items of 3 documents
A criterion for zero averages and full support of ergodic measures
2018
International audience; Consider a homeomorphism $f$ defined on a compact metric space $X$ and a continuous map $\phi\colon X \to \mathbb{R}$. We provide an abstract criterion, called control at any scale with a long sparse tail for a point $x\in X$ and the map $\phi$, which guarantees that any weak* limit measure $\mu$ of the Birkhoff average of Dirac measures $\frac1n\sum_0^{n-1}\delta(f^i(x))$ s such that $\mu$-almost every point $y$ has a dense orbit in $X$ and the Birkhoff average of $\phi$ along the orbit of $y$ is zero.As an illustration of the strength of this criterion, we prove that the diffeomorphisms with nonhyperbolic ergodic measures form a $C^1$-open and dense subset of the s…
Periodic measures and partially hyperbolic homoclinic classes
2019
In this paper, we give a precise meaning to the following fact, and we prove it: $C^1$-open and densely, all the non-hyperbolic ergodic measures generated by a robust cycle are approximated by periodic measures. We apply our technique to the global setting of partially hyperbolic diffeomorphisms with one dimensional center. When both strong stable and unstable foliations are minimal, we get that the closure of the set of ergodic measures is the union of two convex sets corresponding to the two possible $s$-indices; these two convex sets intersect along the closure of the set of non-hyperbolic ergodic measures. That is the case for robustly transitive perturbation of the time one map of a tr…
Entropy, Lyapunov exponents, and rigidity of group actions
2018
This text is an expanded series of lecture notes based on a 5-hour course given at the workshop entitled "Workshop for young researchers: Groups acting on manifolds" held in Teres\'opolis, Brazil in June 2016. The course introduced a number of classical tools in smooth ergodic theory -- particularly Lyapunov exponents and metric entropy -- as tools to study rigidity properties of group actions on manifolds. We do not present comprehensive treatment of group actions or general rigidity programs. Rather, we focus on two rigidity results in higher-rank dynamics: the measure rigidity theorem for affine Anosov abelian actions on tori due to A. Katok and R. Spatzier [Ergodic Theory Dynam. Systems…