scholarly journals Cohomology of fiber-bunched twisted cocycles over hyperbolic systems

2020 ◽  
Vol 63 (3) ◽  
pp. 844-860
Author(s):  
Lucas Backes
Keyword(s):  
Lie Group ◽  
Hyperbolic Systems ◽  
Hölder Continuous ◽  
Periodic Data ◽  
Twisted Cocycles

AbstractA twisted cocycle taking values on a Lie group G is a cocycle that is twisted by an automorphism of G in each step. In the case where G = GL(d, ℝ), we prove that if two Hölder continuous twisted cocycles satisfying the so-called fiber-bunching condition have the same periodic data then they are cohomologous.

scholarly journals Cohomology of fiber bunched cocycles over hyperbolic systems

2014 ◽  
Vol 35 (8) ◽  
pp. 2669-2688 ◽  
Author(s):  
VICTORIA SADOVSKAYA
Keyword(s):  
Vector Bundle ◽  
Small Perturbation ◽  
Closed Subgroup ◽  
Hyperbolic Systems ◽  
Anosov Diffeomorphism ◽  
Continuous Fiber ◽  
Hölder Continuous ◽  
Periodic Data ◽  
And Shifts ◽  
Hyperbolic Sets

We consider Hölder continuous fiber bunched $\text{GL}(d,\mathbb{R})$-valued cocycles over an Anosov diffeomorphism. We show that two such cocycles are Hölder continuously cohomologous if they have equal periodic data, and prove a result for cocycles with conjugate periodic data. We obtain a corollary for cohomology between any constant cocycle and its small perturbation. The fiber bunching condition means that non-conformality of the cocycle is dominated by the expansion and contraction in the base. We show that this condition can be established based on the periodic data. Some important examples of cocycles come from the differential of a diffeomorphism and its restrictions to invariant sub-bundles. We discuss an application of our results to the question of whether an Anosov diffeomorphism is smoothly conjugate to a $C^{1}$-small perturbation. We also establish Hölder continuity of a measurable conjugacy between a fiber bunched cocycle and a uniformly quasiconformal one. Our main results also hold for cocycles with values in a closed subgroup of $\text{GL}(d,\mathbb{R})$, for cocycles over hyperbolic sets and shifts of finite type, and for linear cocycles on a non-trivial vector bundle.

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 Statistical properties of the maximal entropy measure for partially hyperbolic attractors

2016 ◽  
Vol 37 (4) ◽  
pp. 1060-1101 ◽  
Author(s):  
ARMANDO CASTRO ◽  
TEÓFILO NASCIMENTO
Keyword(s):  
Probability Measure ◽  
Hyperbolic Systems ◽  
Maximal Entropy ◽  
Entropy Measure ◽  
Hölder Continuous ◽  
Hyperbolic Diffeomorphisms ◽  
The Central Limit Theorem ◽  
Exponential Decay Of Correlations ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphisms

We show the existence and uniqueness of the maximal entropy probability measure for partially hyperbolic diffeomorphisms which are semiconjugate to non-uniformly expanding maps. Using the theory of projective metrics on cones, we then prove exponential decay of correlations for Hölder continuous observables and the central limit theorem for the maximal entropy probability measure. Moreover, for systems derived from a solenoid, we also prove the statistical stability for the maximal entropy probability measure. Finally, we use such techniques to obtain similar results in a context containing partially hyperbolic systems derived from Anosov.

scholarly journals Equilibrium stability for non-uniformly hyperbolic systems

2018 ◽  
Vol 39 (10) ◽  
pp. 2619-2642 ◽  
Author(s):  
JOSÉ F. ALVES ◽  
VANESSA RAMOS ◽  
JAQUELINE SIQUEIRA
Keyword(s):  
Hyperbolic Systems ◽  
Equilibrium States ◽  
Topological Pressure ◽  
Equilibrium Stability ◽  
Hölder Continuous ◽  
Skew Products ◽  
Uniformly Hyperbolic Systems ◽  
Wide Family

We prove that for a wide family of non-uniformly hyperbolic maps and hyperbolic potentials we have equilibrium stability, i.e. the equilibrium states depend continuously on the dynamics and the potential. For this we deduce that the topological pressure is continuous as a function of the dynamics and the potential. We also prove the existence of finitely many ergodic equilibrium states for non-uniformly hyperbolic skew products and hyperbolic Hölder continuous potentials. Finally, we show that these equilibrium states vary continuously in the $\text{weak}^{\ast }$ topology within such systems.

Livšic theorem for matrix cocycles over nonuniformly hyperbolic systems

2019 ◽  
Vol 19 (02) ◽  
pp. 1950010 ◽  
Author(s):  
Rui Zou ◽  
Yongluo Cao
Keyword(s):  
Periodic Point ◽  
Measurable Function ◽  
Hyperbolic Systems ◽  
Type Theorem ◽  
Hölder Continuous ◽  
Nonuniformly Hyperbolic

We prove a nonuniformly hyperbolic version of the Livšic-type theorem, with cocycles taking values in [Formula: see text]. To be more precise, let [Formula: see text] Diff[Formula: see text] preserving an ergodic hyperbolic measure [Formula: see text], and [Formula: see text] be Hölder continuous satisfying [Formula: see text] for each periodic point [Formula: see text], then there exists a measurable function [Formula: see text] satisfying [Formula: see text] for [Formula: see text]-almost every [Formula: see text].

scholarly journals Cohomology of dominated diffeomorphism-valued cocycles over hyperbolic systems

2015 ◽  
Vol 36 (6) ◽  
pp. 1703-1722 ◽  
Author(s):  
LUCAS H. BACKES ◽  
ALEJANDRO KOCSARD
Keyword(s):  
Compact Manifold ◽  
Hyperbolic Systems ◽  
Rigidity Theorem ◽  
Diffeomorphism Groups ◽  
Periodic Data

We prove a rigidity theorem for dominated Hölder cocycles with values on diffeomorphism groups of a compact manifold over hyperbolic homeomorphisms. More precisely, we show that if two such cocycles have equal periodic data, then they are cohomologous.

scholarly journals On equilibrium states for partially hyperbolic horseshoes

2016 ◽  
Vol 38 (1) ◽  
pp. 301-335 ◽  
Author(s):  
I. RIOS ◽  
J. SIQUEIRA
Keyword(s):  
Existence And Uniqueness ◽  
Hyperbolic Systems ◽  
Three Dimensional ◽  
Small Variation ◽  
Equilibrium States ◽  
Dimensional System ◽  
Hölder Continuous ◽  
The Family ◽  
Partially Hyperbolic ◽  
Uniqueness Of Equilibrium

We prove the existence and uniqueness of equilibrium states for a family of partially hyperbolic systems, with respect to Hölder continuous potentials with small variation. The family comes from the projection, on the center-unstable direction, of a family of partially hyperbolic horseshoes introduced by Díaz et al [Destroying horseshoes via heterodimensional cycles: generating bifurcations inside homoclinic classes. Ergod. Th. & Dynam. Sys.29 (2009), 433–474]. For the original three-dimensional system we consider potentials with small variation, constant on local stable manifolds, obtaining existence and uniqueness of equilibrium states.

scholarly journals Simplicity of Lyapunov spectrum for linear cocycles over non-uniformly hyperbolic systems

2019 ◽  
Vol 40 (11) ◽  
pp. 2947-2969 ◽  
Author(s):  
LUCAS BACKES ◽  
MAURICIO POLETTI ◽  
PAULO VARANDAS ◽  
YURI LIMA
Keyword(s):  
Gibbs Measure ◽  
Hyperbolic System ◽  
Hyperbolic Systems ◽  
Affirmative Answer ◽  
Lyapunov Spectrum ◽  
Continuous Linear ◽  
Hölder Continuous ◽  
Generic Fiber ◽  
Uniformly Hyperbolic Systems

We prove that generic fiber-bunched and Hölder continuous linear cocycles over a non-uniformly hyperbolic system endowed with a $u$-Gibbs measure have simple Lyapunov spectrum. This gives an affirmative answer to a conjecture proposed by Viana in the context of fiber-bunched linear cocycles.

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