Related papers: Invariant measures for interval maps without Lyapu…
We consider families of geometries of D--dimensional space, described by a finite number of parameters. Starting from the De Witt metric we extract a unique integration measure which turns out to be a geometric invariant, i.e. independent…
We consider complex polynomials $f(z) = z^\ell+c_1$ for $\ell \in 2\N$ and $c_1 \in \R$, and find some combinatorial types and values of $\ell$ such that there is no invariant probability measure equivalent to conformal measure on the Julia…
An invariant measure for a flow is, of course, an invariant measure for any of its time-t maps. But the converse is far from being true. Hence, one may naturally ask: What is the obstruction for an invariant measure for the time-one map to…
We study the entropy and Lyapunov exponents of invariant measures $\mu$ for smooth surface diffeomorphisms $f$, as functions of $(f,\mu)$. The main result is an inequality relating the discontinuities of these functions. One consequence is…
We study the pointwise dimension for a new class of projection measures on arbitrary fractal limit sets without separation conditions. We prove that the pointwise dimension exists a.e. for this class of measures associated to equilibrium…
We consider perturbations of interval maps with indifferent fixed points, which we refer to as wobbly interval intermittent maps, for which stable laws for general H\"older observables fail. We obtain limit laws for such maps and H\"older…
Expanding maps with indifferent fixed points, a.k.a. intermittent maps, are popular models in nonlinear dynamics and infinite ergodic theory. We present a simple proof of the exactness of a wide class of expanding maps of [0,1], with…
We propose a composite Lyapunov framework for nonlinear autonomous systems that ensures strict decay through a pair of differential inequalities. The approach yields integral estimates, quantitative convergence rates, vanishing of…
In this paper we shall consider the connections between Lyapunov integral operators and Gibbs measures for four competing interactions of models with uncountable (i.e. $[0,1]$) set of spin values on a Cayley tree. And we shall prove the…
It follows from Oseledec Multiplicative Ergodic Theorem (or Kingman's Sub-additional Ergodic Theorem) that the set of `non-typical' points for which the Oseledec averages of a given continuous cocycle diverge has zero measure with respect…
An integral criterion for the existence of an invariant measure of an It\^{o} process is developed. This new criterion is based on the probabilistic symbol of the It\^{o} process. In contrast to the standard integral criterion for invariant…
We prove that infinitely renormalizable contracting Lorenz maps with bounded geometry or the so-called {\it a priori bounds} satisfies the slow recurrence condition to the singular point $c$ at its two critical values $c_1^-$ and $c_1^+$.…
We show how to construct a topological Markov map of the interval whose invariant probability measure is the stationary law of a given stochastic chain of infinite order. In particular we caracterize the maps corresponding to stochastic…
We introduce the \emph{metric spectrum}, which measures the exponential rate of approximation to an isolated invariant set of points starting in its stable set, and relate it to the Lyapunov spectrum. We determine the metric spectrum of…
Let $f$ be a piecewise continuous and monotonic map on the interval with at most finitely many discontinuities and turning points. In this paper we study properties about this class of maps and show its main difference from the continuous…
We study several new invariants associated to a holomorphic projective structure on a Riemann surface of finite analytic type: the Lyapunov exponent of its holonomy which is of probabilistic/dynamical nature and was introduced in our…
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.
This paper is concerned with relationships of Lyapunov exponents with sensitivity and stability for non-autonomous discrete systems. Some new concepts are introduced for non-autonomous discrete systems, including Lyapunov exponents, strong…
We describe a framework in which is possible to develop and implement algorithms for the approximation of invariant measures of dynamical systems with a given bound on the error of the approximation. Our approach is based on a general…
We consider piecewise deterministic Markov processes with degenerate transition kernels of the "house-of-cards"-type. We use a splitting scheme based on jump times to prove the absolute continuity, as well as some regularity, of the…