Search results for "partial hyperbolicity"
showing 4 items of 4 documents
Chaotic dynamics and partial hyperbolicity
2017
The dynamics of hyperbolic systems is considered well understood from topological point of view as well as from stochastic point of view. S. Smale and R. Abraham gave an example showing that, in general, the hyperbolic systems are not dense among all differentiable systems. In 1970s, M. Brin and Y. Pesin proposed a new notion: partial hyperbolicity to release the notion of hyperbolicity. One aim of this thesis is to understand the dynamics of certain partially hyperbolic systems from stochastic point of view as well as from topological point of view. From stochastic point of view, we prove the following results: — There exists an open and dense subset U of robustly transitive nonhyperbolic …
Partially hyperbolic diffeomorphisms with a compact center foliation with finite holonomy
2011
The thesis classifies partially hyperbolic diffeomorphisms with a compact center foliation with finite holonomy. Under the further assumption of a one-dimensional unstable bundle we show the following: If the unstable bundle is oriented then the system fibers over a hyperbolic toral automorphism. We further establish that the system has a dense orbit of center leaves. During the proof we show a Shadowing Lemma and the dynamical coherence without restrictions of the dimensions.
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…