We construct ${}^{\infty }$ maps T on the interval and on the circle which are Lebesgue exact preserving an absolutely continuous infinite measure μ ≪ λ, such that for any probability measure ν ≪ λ the sequence ${\left(n^{-1}{\sum }_{k=0}^{n-1}\nu \circ T^{-k}\right)}_{n\ge 1}$ of arithmetical averages of image measures does not converge weakly.

We give an elementary proof for the uniqueness of absolutely continuous invariant measures for expanding random dynamical systems and study their mixing properties.

The rate of growth of an operator T satisfying the mean ergodic theorem (MET) cannot be faster than linear. It was recently shown (Kornfeld-Kosek, Colloq. Math. 98 (2003)) that for every γ > 0, there are positive L¹[0,1] operators T satisfying MET with $li{m}_{n\to \infty }\left|\right|Tⁿ\left|\right|/n^{1-\gamma }=\infty$. In the class of positive L¹ operators this is the most one can hope for in the sense that for every such operator T, there exists a γ₀ > 0 such that $limsup\left|\right|Tⁿ\left|\right|/n^{1-\gamma ₀}=0.$ In this note we construct an example of a nonpositive L¹ operator with the highest possible...

The example is constructed of the C1-smooth skew product of interval maps possessing the one-dimensional ramified continuum (containing no arcs homeomorphic to the circle) with an infinite set of ramification points as the global attractor.

A bifurcation problem for variational inequalities $$U\left(t\right)\in K,(\dot{U}\left(t\right)-B_{\lambda }U\left(t\right)-G(\lambda ,U\left(t\right)),Z-U\left(t\right))\ge 0\text{for}\text{all}Z\in K,\text{a.a.}t\ge 0$$ is studied, where $K$ is a closed convex cone in ${ℝ}^{\kappa }$, $\kappa \ge 3$, $B_{\lambda }$ is a $\kappa \times \kappa$ matrix, $G$ is a small perturbation, $\lambda$ a real parameter. The main goal of the paper is to simplify the assumptions of the abstract results concerning the existence of a bifurcation of periodic solutions developed in the previous paper and to give examples in more than three dimensional case.

We prove that for every ϵ > 0 there exists a minimal diffeomorphism f: ² → ² of class $C^{3-ϵ}$ and semiconjugate to an ergodic translation with the following properties: zero entropy, sensitivity to initial conditions, and Li-Yorke chaos. These examples are obtained through the holonomy of the unstable foliation of Mañé’s example of a derived-from-Anosov diffeomorphism on ³.

Lorsqu'on observe des orbites de certains automates cellulaires, on peut penser qu'elles apparaissent comme des mélanges d'orbites d'autres automates (composants). Dans cet article, nous tentons de comprendre ce phénomène en construisant un hybride de deux automates au moyen d'un troisième. Deux types d'automates cellulaires sont introduits : les captifs et les foulards. Nous comparons des propriétés de ces hybrides dans le cadre des classifications algébriques introduites par [B. Martin...

This note addresses the Cauchy problem for the gradient flow equation in a Hilbert space $H$$$\left\{\begin{array}{c}{u}^{}\hfill \end{array}\right.$$