Related papers: Lyapunov optimization for non-generic one-dimensio…
We study quadratic skew-products with parameters driven over piecewise expanding and Markov interval maps with countable many inverse branches, a generalization of the class of maps introduced by Viana. In particular we construct a class of…
We present several new easy ways of generating smooth one-dimensional maps displaying robust chaos, i.e., chaos for whole intervals of the parameter. Unlike what happens with previous methods, the Lyapunov exponent of the maps constructed…
In this paper we prove that for sufficiently large parameters the standard map has a unique measure of maximal entropy (m.m.e.). Moreover, we prove: the m.m.e. is Bernoulli, and the periodic points with Lyapunov exponents bounded away from…
We prove a finite smooth version of the entropic continuity of Lyapunov exponents of Buzzi-Crovisier-Sarig for $C^\infty$ surface diffeomorphisms [9]. As a consequence we show that any $C^r$, $r > 1$, smooth surface diffeomorphism $f$ with…
We prove that a class of one-dimensional maps with an arbitrary number of non-degenerate critical and singular points admits an induced Markov tower with exponential return time asymptotics. In particular the map has an absolutely…
We obtain a dichotomy for $C^1$-generic, volume-preserving diffeomorphisms: either all the Lyapunov exponents of almost every point vanish or the volume is ergodic and non-uniformly Anosov (i.e. nonuniformly hyperbolic and the splitting…
In this work we present a theoretical and numerical study of the behaviour of the maximum Lyapunov exponent for a generic coupled-map-lattice in the weak-coupling regime. We explain the observed results by introducing a suitable…
We introduce a weak transversality condition for piecewise C^{1+\alpha} and piecewise hyperbolic maps which admit a C^{1+\alpha} stable distribution. We show good bounds on the essential spectral radius of the associated transfer operators…
For a class of piecewise hyperbolic maps in two dimensions, we propose a combinatorial definition of topological entropy by counting the maximal, open, connected components of the phase space on which iterates of the map are smooth. We…
We study the measure theoretic properties of typical C 0 maps of the interval. We prove that any ergodic measure is pseudo-physical, and conversely, any pseudo-physical measure is in the closure of the ergodic measures, as well as in the…
We show that there is a residual subset $\mathcal{R}$ of $Diff^1(M)$ such that for any $f\in\mathcal{R}$ and any partially hyperbolic homoclinic class $H(p,f)$ with one dimensional center direction, the set of central Lyapunov exponents…
Despite the prominent importance of the Lyapunov exponents for characterizing chaos, it still remains a challenge to measure them for large experimental systems, mainly because of the lack of recurrences in time series analysis. Here we…
In this paper we establish a dichotomy for the ergodic measures of maximal entropy for partially hyperbolic diffeomorphisms with one-dimensional compact center leaves which are virtually skew products over (transitive) Anosov homeomorphism.…
Hierarchy of one-parameter families of chaotic maps with an invariant measure have been introduced, where their appropriate coupling has lead to the generation of some coupled chaotic maps with an invariant measure. It is shown that these…
We study the optimization of ergodic averages for multi-valued dynamical systems, i.e. where points may have multiple different forward orbits. Under upper semi-continuity assumptions, we show that the maximum space average with respect to…
For dynamical systems satisfying the approximate $\mathbb{Z}^{d}$ or $\mathbb{Z}_+^{d}$-product property and asymptotically entropy expansiveness, we establish a precise description of the structure of their space of invariant measures. In…
We give both sufficient conditions and necessary conditions for the stochastic stability of non-uniformly expanding maps either with or without critical sets. We also show that the number of probability measures describing the statistical…
In this article we prove that for a $C^{1+\alpha}$ diffeomorphism on a compact Riemannian manifold, if there is a hyperbolic ergodic measure whose support is not uniformly hyperbolic, then the topological entropy of the set of irregular…
Lyapunov exponents of a hyperbolic ergodic measure are approximated by Lyapunov exponents of hyperbolic atomic measures on periodic orbits.
We obtain large deviation results for non-uniformly expanding maps with non-flat singularities or criticalities and for partially hyperbolic non-uniformly expanding attracting sets. That is, given a continuous function we consider its space…