Related papers: Invariant measures for typical quadratic maps

We consider some classes of piecewise expanding maps in finite dimensional spaces having invariant probability measures which are absolutely continuous with respect to Lebesgue measure. We derive an entropy formula for such measures and,…

In this paper, we consider the question of existence and uniqueness of absolutely continuous invariant measures for expanding $C^1$ maps of the circle. This is a question which arises naturally from results which are known in the case of…

We prove that for certain partially hyperbolic skew-products, non-uniform hyperbolicity along the leaves implies existence of a finite number of ergodic absolutely continuous invariant probability measures which describe the asymptotics of…

We analyze a class of piecewise linear parabolic maps on the torus, namely those obtained by considering a linear map with double eigenvalue one and taking modulo one in each component. We show that within this two parameter family of maps,…

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…

Ergodic properties of rational maps are studied, generalising the work of F.\ Ledrappier. A new construction allows for simpler proofs of stronger results. Very general conformal measures are considered. Equivalent conditions are given for…

We consider the invariant measure of homogeneous random walks in the quarter-plane. In particular, we consider measures that can be expressed as an infinite sum of geometric terms. We present necessary conditions for the invariant measure…

We prove that almost every non-regular real quadratic map is Collet-Eckmann and has polynomial recurrence of the critical orbit (proving a conjecture by Sinai). It follows that typical quadratic maps have excellent ergodic properties, as…

We give a heuristic method to solve explicitly for an absolutely continuous invariant measure for a piecewise differentiable, expanding map of a compact subset $I$ of Euclidean space $R^d$. The method consists of constructing a skew product…

We consider the invariant measure of homogeneous random walks in the quarter-plane. In particular, we consider measures that can be expressed as a finite linear combination of geometric terms and present conditions on the structure of these…

We construct SRB measures for endomorphisms satisfying conditions far weaker than the non-uniformly expansion. As a consequence, the definition of non-uniformly expanding map can be weakened. We also prove the existence of an absolutely…

We study some special classes of piecewise continuous maps on a finite smooth partition of a compact manifold and look for invariant measures for such maps. We show that in the simplest one-dimensional case (so-called interval translation…

We survey the current state-of-the-art about the dynamical behavior of continuous Lebesgue measure-preserving maps on one-dimensional manifolds.

We study the invariant measures of typical $C^0$ maps on compact connected manifolds with or without boundary, and also of typical homeomorphisms. We prove that the weak$^*$ closure of the set of ergodic measurescoincides with the weak$^*$…

We describe a construction process of a relevant measure in any non-empty compact metric space. This probability measure has invariance properties with respect to isometric maps defined on open sets. These properties imply that this measure…

Let $I\subset\mathbb{R}$ be an interval and $T_a:[0,1]\to[0,1]$, $a\in I$, a one-parameter family of piecewise expanding maps such that for each $a\in I$ the map $T_a$ admits a unique absolutely continuous invariant probability measure…

I provide a proof of the existence of absolutely continuous invariant measures (and study their statistical properties) for multidimensional piecewise expanding systems with not necessarily bounded derivative or distortion. The proof uses…

We study stable conditional measures for a certain equilibrium measure for hyperbolic endomorphisms, on basic sets with overlaps; we show that these conditional measures are geometric probabilities and measures of maximal stable dimension.…

We consider a family S=S(a) of 2-valued transformations of special form on the segment [0,1] with measure $\mu=\int p(x) d\lambda$, which is absolutely continuous with respect to the Lebesgue measure $\lambda$. We endow S with a set of…

We study a random dynamical system such that one transformation is randomly selected from a family of transformations and then applied on each iteration. For such random dynamical systems, we consider estimates of absolutely continuous…