Katrin Gelfert

The Lyapunov spectrum of some parabolic systems

We study the Hausdorff dimension spectrum for Lyapunov exponents for a class of interval maps which includes several non-hyper- bolic situations. We also analyze the level sets of points with given lower and upper Lyapunov exponents and, in particular, with zero lower Lya- punov exponent. We prove that the level set of points with zero exponent has full Hausdorff dimension, but carries no topological entropy.

Elimination of Fast Chaotic Degrees of Freedom: On the Accuracy of the Born Approximation

We apply standard projection operator techniques known from nonequilibrium statistical mechanics to eliminate fast chaotic degrees of freedom in a low-dimensional dynamical system. Through the usual perturbative approach we end up in second order with a stochastic system where the fast chaotic degrees of freedom are modelled by Gaussian white noise. The accuracy of the perturbation expansion is analysed in detail by the discussion of an exactly solvable model.

Rich phase transitions in step skew products

We present examples of partially hyperbolic and topologically transitive local diffeomorphisms defined as skew products over a horseshoe which exhibit rich phase transitions for the topological pressure. This phase transition follows from a gap in the spectrum of the central Lyapunov exponents. It is associated with the coexistence of two equilibrium states with positive entropy. The diffeomorphisms mix hyperbolic behaviour of different types. However, in some sense the expanding behaviour is not dominating which is indicated by the existence of a measure of maximal entropy with nonpositive central exponent.

Geometry of limit sets for expansive Markov systems

We describe the geometric and dynamical properties of expansive Markov systems.

Non)Invariance of Dynamical Quantities for Orbit Equivalent Flows

We study how dynamical quantities such as Lyapunov exponents, metric entropy, topological pressure, recurrence rates, and dimension-like characteristics change under a time reparameterization of a dynamical system. These quantities are shown to either remain invariant, transform according to a multiplicative factor or transform through a convoluted dependence that may take the form of an integral over the initial local values. We discuss the significance of these results for the apparent non-invariance of chaos in general relativity and explore applications to the synchronization of equilibrium states and the elimination of expansions.

Lyapunov spectrum for multimodal maps

We study the dimension spectrum of Lyapunov exponents for multimodal maps of the interval and their generalizations. We also present related results for rational maps on the Riemann sphere.

Dominated Pesin theory: convex sum of hyperbolic measures

In the uniformly hyperbolic setting it is well known that the set of all measures supported on periodic orbits is dense in the convex space of all invariant measures. In this paper we consider the converse question, in the non-uniformly hyperbolic setting: assuming that some ergodic measure converges to a convex combination of hyperbolic ergodic measures, what can we deduce about the initial measures?To every hyperbolic measure μ whose stable/unstable Oseledets splitting is dominated we associate canonically a unique class H(μ) of periodic orbits for the homoclinic relation, called its intersection class. In a dominated setting, we prove that a measure for which almost every measure in its ergodic decomposition is hyperbolic with the same index, such as the dominated splitting, is accumulated by ergodic measures if, and only if, almost all such ergodic measures have a common intersection class.We provide examples which indicate the importance of the domination assumption.

Almost complete Lyapunov spectrum in step skew-products

We study the spectrum of Lyapunov exponents of a family of partially hyperbolic and topologically transitive local diffeomorphisms that are step skew-products over a horseshoe map, continuing previous investigations. These maps are genuinely non-hyperbolic and the central Lyapunov spectrum contains negative and positive values. We show that, besides one gap, this spectrum is complete. We also investigate how Lyapunov regular points with corresponding (central) exponents are distributed in phase space. The principal ingredients of our proofs are minimality of the underlying iterated function system and shadowing-like arguments.

Dimension estimates beyond conformal and hyperbolic dynamics

We derive upper bounds for the upper box dimension of a compact invariant set in terms of the topological pressure of the exponential growth rate of k-volumes. In particular, for our approach the dynamics needs neither to be conformal nor uniformly hyperbolic.We derive upper bounds for the upper box dimension of a compact invariant set in terms of the topological pressure of the exponential growth rate of k-volumes. In particular, for our approach the dynamics needs neither to be conformal nor uniformly hyperbolic.

Exceptional sets for nonuniformly expanding maps

Given a rational map of the Riemann sphere and a subset