Davor Dragičević

Nonuniform hyperbolicity and admissibility

Abstract For a nonautonomous dynamics defined by a sequence of linear operators, we consider the notion of an exponential dichotomy with respect to a sequence of norms and we characterize it completely in terms of the admissibility in pairs of spaces (ℓp, ℓq), with p and q not necessarily equal. This includes the notion of a nonuniform exponential dichotomy as a very special case. Moreover, we consider a general noninvertible dynamics and a strong nonuniform exponential behavior. The latter is the typical situation from the point of view of smooth ergodic theory.

Characterization of hyperbolicity and generalized shadowing lemma

Mather characterized uniform hyperbolicity of a discrete dynamical system as equivalent to invertibility of an operator on the set of all sequences bounded in norm in the tangent bundle of an orbit. We develop a similar characterization of nonuniform hyperbolicity and show that it is equivalent to invertibility of the same operator on a larger, Fréchet space. We apply it to obtain a condition for a diffeomorphism on the boundary of the set of Anosov diffeomorphisms to be nonuniformly hyperbolic. Finally, we generalize the Shadowing lemma in the same context.

A spectral approach for quenched limit theorems for random expanding dynamical systems

We prove quenched versions of (i) a large deviations principle (LDP), (ii) a central limit theorem (CLT), and (iii) a local central limit theorem for non-autonomous dynamical systems. A key advance is the extension of the spectral method, commonly used in limit laws for deterministic maps, to the general random setting. We achieve this via multiplicative ergodic theory and the development of a general framework to control the regularity of Lyapunov exponents of twisted transfer operator cocycles with respect to a twist parameter. While some versions of the LDP and CLT have previously been proved with other techniques, the local central limit theorem is, to our knowledge, a completely new result, and one that demonstrates the strength of our method. Applications include non-autonomous (piecewise) expanding maps, defined by random compositions of the form

Almost sure invariance principle for random piecewise expanding maps

{T_{\sigma^{n-1} \omega} \circ\cdots\circ T_{\sigma\omega}\circ T_\omega}

Tempered exponential dichotomies: admissibility and stability under perturbations

Tσn-1ω∘⋯∘Tσω∘Tω. An important aspect of our results is that we only assume ergodicity and invertibility of the random driving

On a Question by Markus Seidel

{\sigma:\Omega\to\Omega}