Related papers: Yet another induction scheme for non-uniformly exp…

We show that for a large class of maps on manifolds of arbitrary finite dimension, the existence of a Gibbs-Markov-Young structure (with Lebesgue as the reference measure) is a necessary as well as sufficient condition for the existence of…

We consider random perturbations of a topologically transitive local diffeomorphism of a Riemannian manifold. We show that if an absolutely continuous ergodic stationary measures is expanding (all Lyapunov exponents positive), then there is…

We construct inducing schemes for general multi-dimensional piecewise expanding maps where the base transformation is Gibbs-Markov and the return times have exponential tails. Such structures are a crucial tool in proving statistical…

We show that for every $C^\infty$ diffeomorphism of a closed Riemannian manifold, if there exists a positive volume set of points which admit some expansion with a positive Lyapunov exponent (in a weak sense) then there exists an invariant…

We consider families of transformations in multidimensional Riemannian manifolds with non-uniformly expanding behavior. We give sufficient conditions for the continuous variation (in the $L^1$-norm) of the densities of absolutely continuous…

We prove that any C^{1+} transformation, possibly with a (non-flat) critical or singular region, admits an invariant probability measure absolutely continuous with respect to any expanding measure whose Jacobian satisfies a mild distortion…

We study the statistical properties of piecewise expanding maps in the general setting of metric measure spaces. We provide sufficient conditions for exponential mixing of such systems with explicit estimates on the constants. We also…

We study partially hyperbolic sets $K$ on a Riemannian manifold $M$ whose tangent space splits as $T_K M=E^{cu}\oplus E^{s}$, for which the center-unstable direction $E^{cu}$ is non-uniformly expanding on some local unstable disk. We prove…

We consider a partially hyperbolic set $K$ on a Riemannian manifold $M$ whose tangent space splits as $T_K M=E^{cu}\oplus E^{s}$, for which the centre-unstable direction $E^{cu}$ expands non-uniformly on some local unstable disk. We show…

For a large class of nonuniformly expanding maps of $\Bbb R^m$, with indifferent fixed points and unbounded distorsion and non necessarily Markovian, we construct an absolutely continuous invariant measure. We extend to our case techniques…

In this paper we study the relationship between Lyapunov exponents and the induced map on cohomology for $C^{1}-$diffeomorphisms on compact manifolds. We show that if the induced map on cohomology has spectral radius strictly larger than 1,…

We study random dynamical systems generated by volume-preserving piecewise $C^{1}$ maps. For this class of systems, we establish an invariance principle stating that if all Lyapunov exponents vanish, then there exists a measurable family of…

For systems evolving on a Riemannian manifold, we propose converse Lyapunov theorems for asymptotic and exponential stability. The novelty of the proposed approach is that is does not rely on local Euclidean coordinate, and is thus valid on…

We show that a strengthened version of the Collet-Eckmann condition for multimodal maps is topologically invariant. In particular, if f is non-uniformly expanding and the critical points are generic with respect to the absolutely continuous…

We consider non-uniformly expanding maps on compact Riemannian manifolds of arbitrary dimension, possibly having discontinuities and/or critical sets, and show that under some general conditions they admit an induced Markov tower structure…

In this paper we study the liftability property for piecewise continuous maps of compact metric spaces, which admit inducing schemes in the sense of Pesin and Senti [PS05, PS06]. We show that under some natural assumptions on the inducing…

We construct nontrivial deformations of the standard map which preserve the symplectic actions, respectively the Lyapunov exponents, of infinitely many periodic orbits accumulating to an invariant curve. The proof uses a resonant…

We prove existence of (at most denumerable many) absolutely continuous invariant probability measures for random one-dimensional dynamical systems with asymptotic expansion. If the rate of expansion (Lyapunov exponents) is bounded away from…

This paper studies the uniformly asymptotic stability of nonautonomous systems on Riemannian manifolds. We establish corresponding Lyapunov-type theorems (Theorems 2.1 and 2.2), extending classical Euclidean results (e.g., [9, Theorems 4.9…

For a non-generic, yet dense subset of $C^1$ expanding Markov maps of the interval we prove the existence of uncountably many Lyapunov optimizing measures which are ergodic, fully supported and have positive entropy. These measures are…