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Katrin Gelfert

Publications and source records attributed to Katrin Gelfert.

At least 37 records · Page 2Linked to original sources
arXiv2017

Entropy spectrum of Lyapunov exponents for nonhyperbolic step skew-products and elliptic cocycles

We study the fiber Lyapunov exponents of step skew-product maps over a complete shift of $N$, $N\ge2$, symbols and with $C^1$ diffeomorphisms of the circle as fiber maps. The systems we study are transitive and genuinely nonhyperbolic, exhibiting simultaneously ergodic measures with positive, negative, and zero exponents. Examples of such systems arise from the projective action of $2\times 2$ matrix cocycles and our results apply to an open and dense subset of elliptic $\mathrm{SL}(2,\bR)$ cocycles. We derive a multifractal analysis for the topological entropy of the level sets of Lyapunov exponent. The results are formulated in terms of Legendre-Fenchel transforms of restricted variational pressures, considering hyperbolic ergodic measures only, as well as in terms of restricted variational principles of entropies of ergodic measures with a given exponent. We show that the entropy of the level sets is a continuous function of the Lyapunov exponent. The level set of the zero exponent has positive, but not maximal, topological entropy. Under the additional assumption of proximality, as for example for skew-products arising from certain matrix cocycles, there exist two unique ergodic measures of maximal entropy, one with negative and one with positive fiber Lyapunov exponent.

math.DS↗
arXiv2017

Horseshoes for diffeormorphisms preserving hyperbolic measures

We give extensions of Katok's horseshoe constructions, comment on related results, and provide a self-contained proof. We consider either a $C^{1+α}$ diffeomorphism preserving a hyperbolic measure or a $C^1$ diffeomorphism preserving a hyperbolic measure whose support admits a dominated splitting.

math.DS↗
arXiv2017

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 \emph{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.

math.DS↗
arXiv2017

Geodesic flows modeled by expansive flows

Given a smooth compact surface without focal points and of higher genus, it is shown that its geodesic flow is semi-conjugate to a continuous expansive flow with a local product structure such that the semi-conjugation preserves time-parametrization. It is concluded that the geodesic flow has a unique measure of maximal entropy.

math.DS↗
arXiv2017

Hyperbolic graphs: critical regularity and box dimension

We study fractal properties of invariant graphs of hyperbolic and partially hyperbolic skew product diffeomorphisms in dimension three. We describe the critical (either Lipschitz or at all scales Hölder continuous) regularity of such graphs. We provide a formula for their box dimension given in terms of appropriate pressure functions. We distinguish three scenarios according to the base dynamics: Anosov, one-dimensional attractor, or Cantor set. A key ingredient for the dimension arguments in the latter case will be the presence of a so-called fibered blender.

math.DS↗
arXiv2016

Exceptional sets for nonuniformly expanding maps

Given a rational map of the Riemann sphere and a subset $A$ of its Julia set, we study the $A$-exceptional set, that is, the set of points whose orbit does not accumulate at $A$. We prove that if the topological entropy of $A$ is less than the topological entropy of the full system then the $A$-exceptional set has full topological entropy. Furthermore, if the Hausdorff dimension of $A$ is smaller than the dynamical dimension of the system then the Hausdorff dimension of the $A$-exceptional set is larger than or equal to the dynamical dimension, with equality in the particular case when the dynamical dimension and the Hausdorff dimension coincide. We discuss also the case of a general conformal $C^{1+α}$ dynamical system and, in particular, certain multimodal interval maps on their Julia set.

math.DS↗
arXiv2015

On density of ergodic measures and generic points

We provide conditions which guarantee that ergodic measures are dense in the simplex of invariant probability measures of a dynamical system given by a continuous map acting on a Polish space. Using them we study generic properties of invariant measures and prove that every invariant measure has a generic point. In the compact case, density of ergodic measures means that the simplex of invariant measures is either a singleton of a measure concentrated on a single periodic orbit or the Poulsen simplex. Our properties focus on the set of periodic points and we introduce two concepts: close\-ability with respect to a set of periodic points and linkability of a set of periodic points. Examples are provided to show that these are independent properties. They hold, for example, for systems having the periodic specification property. But they hold also for a much wider class of systems which contains, for example, irreducible Markov chains over a countable alphabet, all $β$-shifts, all $S$-gap shifts, ${C}^1$-generic diffeomorphisms of a compact manifold $M$, and certain geodesic flows of a complete connected negatively curved manifold.

math.DS↗
arXiv2014

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.

math.DS↗
arXiv2014

Pressures for geodesic flows of rank one manifolds

We study the geodesic flow on the unit tangent bundle of a rank one manifold and we give conditions under which all classical definitions of pressure of a Hölder continuous potential coincide. We provide a large deviation statement, which allows to neglect (periodic) orbits that lack sufficient hyperbolic behavior. Our results involve conditions on the potential, that take into consideration its properties in the nonhyperbolic part of the manifold. We draw some conclusions for the construction of equilibrium states.

math.DS↗
arXiv2011

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 to the coexistence of two equilibrium states with positive entropy. The diffeomorphisms mix hyperbolic behavior of different types. However, in some sense the expanding behavior is not dominating which is indicated by the existence of a measure of maximal entropy with nonpositive central exponent.

math.DS↗
arXiv2010

Lyapunov spectrum for exceptional rational maps

We study the dimension spectrum for Lyapunov exponents for rational maps acting on the Riemann sphere and characterize it by means of the Legendre-Fenchel transform of the hidden variational pressure. This pressure is defined by means of the variational principle with respect to non-atomic invariant probability measures and is associated to certain $σ$-finite conformal measures. This allows to extend previous results to exceptional rational maps.

math.DS↗
arXiv2010

(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.

math.DS↗
arXiv2010

Repellers for non-uniformly expanding maps with singular or critical points

Given an ergodic measure with positive entropy and only positive Lyapunov exponents, its dynamical quantifiers can be approximated by means of quantifiers of some family of uniformly expanding repellers. Here non-uniformly expanding maps are studied that are $C^{1+β}$ smooth outside a set of possibly critical or singular points.

math.DS↗
arXiv2009

Time-metric equivalence and dimension change under time reparameterizations

We study the behavior of dynamical systems under time reparameterizations, which is important not only to characterize chaos in relativistic systems but also to probe the invariance of dynamical quantities. We first show that time transformations are locally equivalent to metric transformations, a result that leads to a transformation rule for all Lyapunov exponents on arbitrary Riemannian phase spaces. We then show that time transformations preserve the spectrum of generalized dimensions D_q except for the information dimension D_1, which, interestingly, transforms in a nontrivial way despite previous assertions of invariance. The discontinuous behavior at q=1 can be used to constrain and extend the formulation of the Kaplan-Yorke conjecture.

nlin.CD↗
arXiv2009

On the distribution of periodic orbits

Let $f:M\to M$ be a $C^{1+ε}$-map on a smooth Riemannian manifold $M$ and let $Λ\subset M$ be a compact $f$-invariant locally maximal set. In this paper we obtain several results concerning the distribution of the periodic orbits of $f|Λ$. These results are non-invertible and, in particular, non-uniformly hyperbolic versions of well-known results by Bowen, Ruelle, and others in the case of hyperbolic diffeomorphisms. We show that the topological pressure $P_{\rm top}(φ)$ can be computed by the values of the potential $φ$ on the expanding periodic orbits and also that every hyperbolic ergodic invariant measure is well-approximated by expanding periodic orbits. Moreover, we prove that certain equilibrium states are Bowen measures. Finally, we derive a large deviation result for the periodic orbits whose time averages are apart from the space average of a given hyperbolic invariant measure.

math.DS↗