We consider the robust family of geometric Lorenz attractors. These attractors are chaotic, in the sense that they are transitive and have sensitive dependence on initial conditions. Moreover, they support SRB measures whose ergodic basins cover a full Lebesgue measure subset of points in the topological basin of attraction. Here we prove that the SRB measures depend continuously on the dynamics in the weak* topology.

Soit $r\ge 1$ un réel. Ici, on étudie les homéomorphismes du cercle qui sont de classe $P$$C^r$ par morceaux et de nombres de rotation irrationnels. On caractérise ceux qui sont $C^r$ par morceaux conjugués à des $C^r$-difféomorphismes. Comme conséquence, on obtient un critère de conjugaison...

A nonuniformly entropy expanding map is any ¹ map defined on a compact manifold whose ergodic measures with positive entropy have only nonnegative Lyapunov exponents. We prove that a ${}^r$ nonuniformly entropy expanding map T with r > 1 has a symbolic extension and we give an explicit upper bound of the symbolic extension entropy in terms of the positive Lyapunov exponents by following the approach of T. Downarowicz and A. Maass [Invent. Math. 176 (2009)].

We prove that ${𝒞}^r$ maps with $r\>1$ on a compact surface have symbolic extensions, i.e., topological extensions which are subshifts over a finite alphabet. More precisely we give a sharp upper bound on the so-called symbolic extension entropy, which is the infimum of the topological entropies of all the symbolic extensions. This answers positively a conjecture of S. Newhouse and T. Downarowicz in dimension two and improves a previous result of the author [11].

We consider iterated function systems on the interval with random perturbation. Let $Y_{\epsilon }$ be uniformly distributed in [1-ε,1+ ε] and let $f_i\in C^{1+\alpha }$ be contractions with fixpoints $a_i$. We consider the iterated function system $Y_{\epsilon }{f}_i+a_i(1-Y_{\epsilon }){ⁿ}_{i=1}$, where each of the maps is chosen with probability $p_i$. It is shown that the invariant density is in L² and its L² norm does not grow faster than 1/√ε as ε vanishes. The proof relies on defining a piecewise hyperbolic dynamical system on the cube with an SRB-measure whose projection is the...

We show upper estimates of the concentration and thin dimensions of measures invariant with respect to families of transformations. These estimates are proved under the assumption that the transformations have a squeezing property which is more general than the Lipschitz condition. These results are in the spirit of a paper by A. Lasota and J. Traple [Chaos Solitons Fractals 28 (2006)] and generalize the classical Moran formula.