We consider pinching cocycles taking values in the space of homeomorphisms of the circle over an hyperbolic base. Using the Invariance Principle of Malicet, we prove that the cocycles having non-zero exponents of contraction are dense. In this article we generalize some common notions an results known of linear cocycles and cocycles of diffeomorphisms, to the non-linear non-differentiable case.
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.
Given a linear cocycle over an ergodic homeomorphism on a compact metric space, we show that the existence of a uniform gap between the $p$-th and $(p+1)$-th Lyapunov exponent on a $C^0$-neighbourhood implies the existence of a dominated splitting of index $p$.
Given an ergodic flow $φ^t\colon M\rightarrow M$ defined on a probability space $M$ we study a family of continuous-time kinetic linear cocycles associated to the solutions of the second order linear homogeneous differential equations $\ddot x +α(φ^t(ω))\dot x+β(φ^t(ω))x=0$, where the parameters $α,β$ evolve along the $φ^t$-orbit of $ω\in M$. Our main result states that for a generic subset of kinetic continuous-time linear cocycles, where generic means a Baire second category with respect to an $L^p$-like topology on the infinitesimal generator, the Lyapunov spectrum is trivial.
Consider the space of analytic, quasi-periodic cocycles on the higher dimensional torus. We provide examples of cocycles with nontrivial Lyapunov spectrum, whose homotopy classes do not contain any cocycles satisfying the dominated splitting property. This shows that the main result in the recent work "Complex one-frequency cocycles" by A. Ávila, S. Jitomirskaya and C. Sadel does not hold in the higher dimensional torus setting.
We consider the problem of approximating a linear cocycle (or, more generally, a vector bundle automorphism) over a fixed base dynamics by another cocycle admitting a dominated splitting. We prove that the possibility of doing so depends only on the homotopy class of the cocycle, provided that the base dynamics is a minimal diffeomorphism and the fiber dimension is least 3. This result is obtained by means of a general theorem on the existence of almost invariant sections for fiberwise isometries of bundles with compact fibers and finite fundamental group. The main novelty of the proofs is the…
We consider group-valued cocycles over dynamical systems. The base system is a homeomorphism $f$ of a metric space satisfying a closing property, for example a hyperbolic dynamical system or a subshift of finite type. The cocycle $A$ takes values in the group of invertible bounded linear operators on a Banach space and is Hölder continuous. We prove that upper and lower Lyapunov exponents of $A$ with respect to an ergodic invariant measure $μ$ can be approximated in terms of the norms of the values of $A$ on periodic orbits of $f$. We also show that these exponents cannot always be approximate…
This paper proves that there are no compact forms for a large class of homogeneous spaces admitting actions by higher-rank semisimple Lie groups. It builds on Zimmer's approach for studying such spaces using cocycle superrigidity. The proof involves cocycle superrigidity, measure rigidity for unipotent flows, techniques from partially hyperbolic dynamics, and the geometry and pseudo-Riemannian structure of the homogeneous space.
Given a hyperbolic homeomorphism on a compact metric space, consider the space of linear cocycles over this base dynamics which are Hölder continuous and whose projective actions are partially hyperbolic dynamical systems. We prove that locally near any typical cocycle, the Lyapunov exponents are Hölder continuous functions relative to the uniform topology. This result is obtained as a consequence of a uniform large deviations type estimate in the space of cocycles. As a byproduct of our approach, we also establish other statistical properties for the iterates of such cocycles, namely a centra…
In the present paper we prove that densely, with respect to an $L^p$-like topology, the Lyapunov exponents associated to linear continuous-time cocycles $Φ:\mathbb{R}\times M\to \text{GL}(2,\mathbb{R})$ induced by second order linear homogeneous differential equations $\ddot x+α(φ^t(ω))\dot x+β(φ^t(ω))x=0$ are almost everywhere distinct. The coefficients $α,β$ evolve along the $φ^t$-orbit for $ω\in M$ and $φ^t: M\to M$ is an ergodic flow defined on a probability space. We also obtain the corresponding version for the frictionless equation $\ddot x+β(φ^t(ω))x=0$ and for a Schrödinger equation $…
