scholarly journals -openness of non-uniform hyperbolic diffeomorphisms with bounded -norm

2019 ◽  
Vol 40 (11) ◽  
pp. 3078-3104
Author(s):  
CHAO LIANG ◽  
KARINA MARIN ◽  
JIAGANG YANG
Keyword(s):  
Lyapunov Exponents ◽  
Dense Subset ◽  
Topological Properties ◽  
Hyperbolic Diffeomorphisms ◽  
Uniform Hyperbolicity ◽  
Partially Hyperbolic ◽  
The Stability ◽  
Bounded Norm ◽  
Volume Preserving

We study the $C^{1}$-topological properties of the subset of non-uniform hyperbolic diffeomorphisms in a certain class of $C^{2}$ partially hyperbolic symplectic systems which have bounded $C^{2}$ distance to the identity. In this set, we prove the stability of non-uniform hyperbolicity as a function of the diffeomorphism and the measure, and the existence of an open and dense subset of continuity points for the center Lyapunov exponents. These results are generalized to the volume-preserving context.

scholarly journals On the stable ergodicity of Berger–Carrasco’s example

2018 ◽  
Vol 40 (4) ◽  
pp. 1008-1056
Author(s):  
DAVI OBATA
Keyword(s):  
Lyapunov Exponents ◽  
Two Dimensional ◽  
Dominated Splitting ◽  
Standard Map ◽  
Hyperbolic Diffeomorphisms ◽  
Stable Ergodicity ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphism ◽  
Volume Preserving ◽  
Math Phys

We prove the stable ergodicity of an example of a volume-preserving, partially hyperbolic diffeomorphism introduced by Berger and Carrasco in [Berger and Carrasco. Non-uniformly hyperbolic diffeomorphisms derived from the standard map. Comm. Math. Phys.329 (2014), 239–262]. This example is robustly non-uniformly hyperbolic, with a two-dimensional center; almost every point has both positive and negative Lyapunov exponents along the center direction and does not admit a dominated splitting of the center direction. The main novelty of our proof is that we do not use accessibility.

scholarly journals Transitivity of conservative diffeomorphisms isotopic to Anosov on

2016 ◽  
Vol 38 (1) ◽  
pp. 1-9
Author(s):  
MARTIN ANDERSSON ◽  
SHAOBO GAN
Keyword(s):  
Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Volume Preserving

We prove transitivity for volume-preserving $C^{1+}$ diffeomorphisms on $\mathbb{T}^{3}$ which are isotopic to a linear Anosov automorphism along a path of weakly partially hyperbolic diffeomorphisms.

MINIMALITY OF STRONG STABLE AND UNSTABLE FOLIATIONS FOR PARTIALLY HYPERBOLIC DIFFEOMORPHISMS

2002 ◽  
Vol 1 (4) ◽  
pp. 513-541 ◽  
Author(s):  
Christian Bonatti ◽  
Lorenzo J. Díaz ◽  
Raúl Ures
Keyword(s):  
Dense Subset ◽  
Subject Classification ◽  
Compact Leaf ◽  
Unstable Foliation ◽  
Stable Foliation ◽  
Hyperbolic Diffeomorphisms ◽  
Primary 37D25 ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Homoclinic Tangencies

We give a topological criterion for the minimality of the strong unstable (or stable) foliation of robustly transitive partially hyperbolic diffeomorphisms.As a consequence we prove that, for $3$-manifolds, there is an open and dense subset of robustly transitive diffeomorphisms (far from homoclinic tangencies) such that either the strong stable or the strong unstable foliation is robustly minimal.We also give a topological condition (existence of a central periodic compact leaf) guaranteeing (for an open and dense subset) the simultaneous minimality of the two strong foliations.AMS 2000 Mathematics subject classification: Primary 37D25; 37C70; 37C20; 37C29

scholarly journals Center foliation: absolute continuity, disintegration and rigidity

2014 ◽  
Vol 36 (1) ◽  
pp. 256-275 ◽  
Author(s):  
RÉGIS VARÃO
Keyword(s):  
Absolute Continuity ◽  
Geodesic Flows ◽  
Absolutely Continuous ◽  
Rigidity Result ◽  
Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic ◽  
Smooth Conjugacy ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic Diffeomorphism ◽  
Volume Preserving

In this paper we address the issues of absolute continuity for the center foliation, as well as the disintegration on the non-absolute continuous case and rigidity of volume-preserving partially hyperbolic diffeomorphisms isotopic to a linear Anosov automorphism on $\mathbb{T}^{3}$. It is shown that the disintegration of volume on center leaves for these diffeomorphisms may be neither atomic nor Lebesgue, in contrast to the dichotomy (Lebesgue or atomic) obtained by Avila, Viana and Wilkinson [Absolute continuity, Lyapunov exponents and rigidity I: Geodesic flows. Preprint, 2012, arXiv:1110.2365v2] for perturbations of time-one of geodesic flow. In the case of atomic disintegration of volume on the center leaves of an Anosov diffeomorphism on $\mathbb{T}^{3}$, we show that it has to be one atom per leaf. Moreover, we show that not even a $C^{1}$ center foliation implies a rigidity result. However, for a volume-preserving partially hyperbolic diffeomorphism isotopic to a linear Anosov automorphism, assuming the center foliation is $C^{1}$ and transversely absolutely continuous with bounded Jacobians, we obtain smooth conjugacy to its linearization.

Random entropy expansiveness for diffeomorphisms with dominated splittings

2019 ◽  
Vol 20 (02) ◽  
pp. 2050014
Author(s):  
Zeya Mi
Keyword(s):  
Dynamical Systems ◽  
Lyapunov Exponents ◽  
Random Dynamical Systems ◽  
Local Entropy ◽  
Dominated Splitting ◽  
One Dimensional ◽  
Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic ◽  
Bowen Balls ◽  
Partially Hyperbolic Diffeomorphisms

We study the local entropy of typical infinite Bowen balls in random dynamical systems, and show the random entropy expansiveness for [Formula: see text] partially hyperbolic diffeomorphisms with multi one-dimensional centers. Moreover, we consider [Formula: see text] diffeomorphism [Formula: see text] with dominated splitting [Formula: see text] such that [Formula: see text] for every [Formula: see text], and all the Lyapunov exponents are non-negative along [Formula: see text] and non-positive along [Formula: see text], we prove the asymptotically random entropy expansiveness for [Formula: see text].

A volume preserving flow with essential coexistence of zero and non-zero Lyapunov exponents

2012 ◽  
Vol 33 (6) ◽  
pp. 1748-1785 ◽  
Author(s):  
JIANYU CHEN ◽  
HUYI HU ◽  
YAKOV PESIN
Keyword(s):  
Lyapunov Exponents ◽  
Smooth Manifold ◽  
Dense Subset ◽  
Three Dimensional ◽  
Frequency Vector ◽  
Almost Everywhere ◽  
Invariant Submanifolds ◽  
Hyperbolic Behavior ◽  
Volume Preserving ◽  
Full Volume

AbstractWe demonstrate essential coexistence of hyperbolic and non-hyperbolic behavior in the continuous-time case by constructing a smooth volume preserving flow on a five-dimensional compact smooth manifold that has non-zero Lyapunov exponents almost everywhere on an open and dense subset of positive but not full volume and is ergodic on this subset while having zero Lyapunov exponents on its complement. The latter is a union of three-dimensional invariant submanifolds, and on each of these submanifolds the flow is linear with Diophantine frequency vector.

Stably ergodic approximation: two examples

2000 ◽  
Vol 20 (3) ◽  
pp. 875-893 ◽  
Author(s):  
MICHAEL SHUB ◽  
AMIE WILKINSON
Keyword(s):  
Compact Manifold ◽  
Standard Map ◽  
Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Volume Preserving

It has been conjectured that the stably ergodic diffeomorphisms are open and dense in the space of volume-preserving, partially hyperbolic diffeomorphisms of a compact manifold. In this paper we deal with two recalcitrant examples; the standard map cross Anosov and the ergodic automorphisms of the 4-torus. In both cases we show that they may be approximated by stably ergodic diffeomorphisms which have the stable accessibility property.

scholarly journals On stable transitivity of finitely generated groups of volume-preserving diffeomorphisms

2017 ◽  
Vol 39 (2) ◽  
pp. 554-576
Author(s):  
ZHIYUAN ZHANG
Keyword(s):  
Riemannian Manifold ◽  
Compact Riemannian Manifold ◽  
Dense Subset ◽  
Duke Math ◽  
Large Integer ◽  
Finitely Generated Groups ◽  
Partially Hyperbolic ◽  
New Criterion ◽  
Partially Hyperbolic Diffeomorphism ◽  
Volume Preserving

In this paper, we provide a new criterion for the stable transitivity of volume-preserving finite generated groups on any compact Riemannian manifold. As one of our applications, we generalize a result of Dolgopyat and Krikorian [On simultaneous linearization of diffeomorphisms of the sphere. Duke Math. J. 136 (2007), 475–505] and obtain stable transitivity for random rotations on the sphere in any dimension. As another application, we show that for $\infty \geq r\geq 2$, for any $C^{r}$ volume-preserving partially hyperbolic diffeomorphism $g$ on any compact Riemannian manifold $M$ having sufficiently Hölder stable or unstable distribution, for any sufficiently large integer $K$ and for any $(f_{i})_{i=1}^{K}$ in a $C^{1}$ open $C^{r}$ dense subset of $\text{Diff}^{r}(M,m)^{K}$, the group generated by $g,f_{1},\ldots ,f_{K}$ acts transitively.

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