scholarly journals $C^r$-density of (non-uniform) hyperbolicity in partially hyperbolic symplectic diffeomorphisms

2016 ◽  
Vol 91 (2) ◽  
pp. 357-396 ◽  
Author(s):  
Karina Marin
Keyword(s):  
Symplectic Diffeomorphisms ◽  
Uniform Hyperbolicity ◽  
Partially Hyperbolic

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 C1-generic symplectic diffeomorphisms: partial hyperbolicity and zero centre Lyapunov exponents

2009 ◽  
Vol 9 (1) ◽  
pp. 49-93 ◽  
Author(s):  
Jairo Bochi
Keyword(s):  
Random Walks ◽  
Lyapunov Exponents ◽  
Probabilistic Method ◽  
International Congress ◽  
Partial Hyperbolicity ◽  
Symplectic Diffeomorphisms ◽  
Symplectic Diffeomorphism ◽  
Partially Hyperbolic

AbstractWe prove that if f is a C1-generic symplectic diffeomorphism then the Oseledets splitting along almost every orbit is either trivial or partially hyperbolic. In addition, if f is not Anosov then all the exponents in the centre bundle vanish. This establishes in full a result announced by Mañé at the International Congress of Mathematicians in 1983. The main technical novelty is a probabilistic method for the construction of perturbations, using random walks.

scholarly journals NON-UNIFORM HYPERBOLICITY FOR INFINITE DIMENSIONAL COCYCLES

2013 ◽  
Vol 13 (03) ◽  
pp. 1250026 ◽  
Author(s):  
MÁRIO BESSA ◽  
MARIA CARVALHO
Keyword(s):  
Hilbert Space ◽  
Probability Measure ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space ◽  
The Family ◽  
Uniform Hyperbolicity ◽  
Partially Hyperbolic

Let [Formula: see text] be an infinite dimensional Hilbert space, X a compact Hausdorff space and f : X → X a homeomorphism which preserves a Borel ergodic probability measure which is positive on non-empty open sets. We prove that non-uniformly Anosov cocycles are C0-dense in the family of partially hyperbolic cocycles with non-trivial unstable bundles.

scholarly journals A link between topological entropy and Lyapunov exponents

2017 ◽  
Vol 39 (3) ◽  
pp. 620-637
Author(s):  
THIAGO CATALAN
Keyword(s):  
Lyapunov Exponents ◽  
Topological Entropy ◽  
Lower Semicontinuous ◽  
Periodic Points ◽  
Partial Hyperbolicity ◽  
Symplectic Diffeomorphisms ◽  
Symplectic Diffeomorphism ◽  
Partially Hyperbolic ◽  
Generic Set ◽  
Hyperbolic Sets

We show that a $C^{1}$-generic non-partially hyperbolic symplectic diffeomorphism $f$ has topological entropy equal to the supremum of the sum of the positive Lyapunov exponents of its hyperbolic periodic points. Moreover, we also prove that $f$ has topological entropy approximated by the topological entropy of $f$ restricted to basic hyperbolic sets. In particular, the topological entropy map is lower semicontinuous in a $C^{1}$-generic set of symplectic diffeomorphisms far from partial hyperbolicity.

The Topology of Symplectic Partially Hyperbolic Automorphisms of the 4-Torus

2020 ◽  
Vol 108 (3-4) ◽  
pp. 462-464
Author(s):  
L. M. Lerman ◽  
K. N. Trifonov
Keyword(s):  
Partially Hyperbolic

scholarly journals Diffeomorphism cocycles over partially hyperbolic systems

2020 ◽  
pp. 1-24
Author(s):  
VICTORIA SADOVSKAYA
Keyword(s):  
Hyperbolic System ◽  
Compact Manifold ◽  
Hyperbolic Systems ◽  
Riemannian Metrics ◽  
Hölder Continuous ◽  
Group Of Diffeomorphisms ◽  
Small Loss ◽  
Partially Hyperbolic ◽  
Loss Of Regularity

Abstract We consider Hölder continuous cocycles over an accessible partially hyperbolic system with values in the group of diffeomorphisms of a compact manifold $\mathcal {M}$ . We obtain several results for this setting. If a cocycle is bounded in $C^{1+\gamma }$ , we show that it has a continuous invariant family of $\gamma $ -Hölder Riemannian metrics on $\mathcal {M}$ . We establish continuity of a measurable conjugacy between two cocycles assuming bunching or existence of holonomies for both and pre-compactness in $C^0$ for one of them. We give conditions for existence of a continuous conjugacy between two cocycles in terms of their cycle weights. We also study the relation between the conjugacy and holonomies of the cocycles. Our results give arbitrarily small loss of regularity of the conjugacy along the fiber compared to that of the holonomies and of the cocycle.

scholarly journals Dynamical incoherence for a large class of partially hyperbolic diffeomorphisms

2020 ◽  
pp. 1-17
Author(s):  
THOMAS BARTHELMÉ ◽  
SERGIO R. FENLEY ◽  
STEVEN FRANKEL ◽  
RAFAEL POTRIE
Keyword(s):  
Large Class ◽  
Universal Cover ◽  
Isotopy Class ◽  
Seifert Manifold ◽  
Hyperbolic Diffeomorphisms ◽  
Surface Homeomorphisms ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic Diffeomorphism

Abstract We show that if a partially hyperbolic diffeomorphism of a Seifert manifold induces a map in the base which has a pseudo-Anosov component then it cannot be dynamically coherent. This extends [C. Bonatti, A. Gogolev, A. Hammerlindl and R. Potrie. Anomalous partially hyperbolic diffeomorphisms III: Abundance and incoherence. Geom. Topol., to appear] to the whole isotopy class. We relate the techniques to the study of certain partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds performed in [T. Barthelmé, S. Fenley, S. Frankel and R. Potrie. Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part I: The dynamically coherent case. Preprint, 2019, arXiv:1908.06227; Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part II: Branching foliations. Preprint, 2020, arXiv: 2008.04871]. The appendix reviews some consequences of the Nielsen–Thurston classification of surface homeomorphisms for the dynamics of lifts of such maps to the universal cover.

scholarly journals Kupka–Smale diffeomorphisms at the boundary of uniform hyperbolicity: a model

Nonlinearity ◽  
2017 ◽  
Vol 30 (10) ◽  
pp. 3895-3931
Author(s):  
Renaud Leplaideur ◽  
Isabel Lugão Rios
Keyword(s):  
Uniform Hyperbolicity
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