Related papers: Cocycles with one exponent over partially hyperbol…

In this work we study relations between regularity of invariant foliations and Lyapunov exponents of partially hyperbolic diffeomorphisms. We suggest a new regularity condition for foliations in terms of desintegration of Lebesgue measure…

In this note, we show that if all Lyapunov exponents of a matrix cocycle vanish, then it can be perturbed to become cohomologous to a cocycle taking values in the orthogonal group. This extends a result of Avila, Bochi and Damanik to…

We study the existence and regularity of invariant graphs for bundle maps (or bundle correspondences with generating bundle maps motivated by ill-posed differential equations) having some relative partial hyperbolicity on non-trivial and…

This paper is concerned with the study of linear cocycles over uniformly ergodic Markov shifts on a compact space of symbols. We establish the joint H\"older continuity of the maximal Lyapunov exponent as a function of the cocycle and the…

We give a new proof of E. Le Page's theorem on the Holder continuity of the first Lyapunov exponent in the class of irreducible Bernoulli cocycles. This suggests an algorithm to approximate the first Lyapunov exponent, as well as the…

We study nonhyperbolic and transitive partially hyperbolic diffeomorphisms having a one-dimensional center. We prove joint flexibility with respect to entropy and center Lyapunov exponent for a broad class of these systems. Flexibility…

We survey a collection of recent results on center Lyapunov exponents of partially hyperbolic diffeomorphisms. We explain several ideas in simplified setups and formulate the general versions of results. We also pose some open questions.

In several contexts the defining invariant structures of a hyperbolic dynamical system are smooth only in systems of algebraic origin (smooth rigidity), and we prove new results of this type for a class of flows. For a compact Riemannian…

We prove a rigidity theorem for dominated H\"{o}lder 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…

The purpose of these notes is to discuss the advances in the theory of Lyapunov exponents of linear $\text{SL}_2(\mathbb{R})$ cocycles over hyperbolic maps. The main focus is around results regarding the positivity of the Lyapunov exponent…

We show that every codimension one partially hyperbolic diffeomorphism must support on $\mathbb{T}^{n}$. It is locally uniquely integrable and derived from a linear codimension one Anosov diffeomorphism. Moreover, this system is…

We provide conditions which imply the continuity of the Lyapunov exponents for non-uniformly fiber-bunched cocycles in $SL(2,\mathbb{R})$. The main theorem is an extension of the result of Backes, Brown and Butler and gives a partial answer…

We derive a criterion for the positivity of the maximal Lyapunov exponent of generic mixed random-quasiperiodic linear cocycles, a model introduced in a previous work. This result is applicable to cocycles corresponding to Schr\"odinger…

We contribute to the thermodynamic formalism of H\"older continuous fiber-bunched matrix cocycles, Anosov diffeomorphisms, and hyperbolic repellers. Specifically, we prove that $1$-typical fiber-bunched cocycles $\mathcal{A}$ over…

Let $A$ be a H\"older continuous cocycle over a hyperbolic dynamical system with values in the group of diffeomorphisms of a compact manifold $M$. We consider the periodic data of $A$, i.e., the set of its return values along the periodic…

We prove the Liv\v{s}ic Theorem for arbitrary $GL(m,\mathbb R)$ cocycles. We consider a hyperbolic dynamical system $f : X \to X$ and a H\"older continuous function $A: X \to GL(m,\mathbb R)$. We show that if $A$ has trivial periodic data,…

We extend the recent progress on the cocycle rigidity of partially hyperbolic homogeneous abelian actions to the setting with rank 1 factors in the universal cover. The method of proof relies on the periodic cycle functional and analysis of…

We briefly survey some of the recent results concerning the metric behavior of the invariant foliations for a partially hyperbolic on a three-dimensional manifold and propose a conjecture to characterize atomic behavior for conservative…

We prove that the set of fiber-bunched $SL(2,\mathbb{R})$-valued H\"{o}lder cocycles with nonvanishing Lyapunov exponents over a volume preserving, accessible and center-bunched partially hyperbolic diffeomorphism is open. Moreover, we…

We show that for $n\ge2$, if a partially hyperbolic diffeomorphism $f:\mathbb T^{n+1}\to \mathbb T^{n+1}$ with $\dim E^s=\dim E^c=1$ has an invariant center-unstable foliation with a compact incompressible leaf, then this foliation has a…