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International audience; Our aim is to quantify how complex is a Cantor set as we approximate it better and better. We formalize this by asking what is the shortest program, running on a universal Turing machine, which produces this set at the precision ε in the sense of Hausdorff distance. This is the Kolmogorov complexity of the approximated Cantor set, that we call the "ε-distortion complexity". How does this quantity behave as ε tends to 0? And, moreover, how does this behaviour relates to other characteristics of the Cantor set? This is the subject of the present work: we estimate this quantity for several types of Cantor sets on the line generated by iterated function systems (IFS's) and exhibit very different behaviours. For instance, the ε-distortion complexity of a C^k Cantor set is proven to behave like ε^{−D/k}, where D is its box counting dimension.

In this paper, we apply Devroye inequality to study various statistical estimators and fluctuations of observables for processes. Most of these observables are suggested by dynamical systems. These applications concern the co-variance function, the integrated periodogram, the correlation dimension, the kernel density estimator, the speed of convergence of empirical measure, the shadowing property and the almost-sure central limit theorem. We proved in \cite{CCS} that Devroye inequality holds for a class of non-uniformly hyperbolic dynamical systems introduced in \cite{young}. In the second appendix we prove that, if the decay of correlations holds with a common rate for all pairs of functions, then it holds uniformly in the function spaces. In the last appendix we prove that for the subclass of one-dimensional systems studied in \cite{young} the density of the absolutely continuous invariant measure belongs to a Besov space., Comment: 33 pages; companion of the paper math.DS/0412166; corrected version; to appear in Nonlinearity

Let T : X→X be a measurable map defined on a Polish space X and let Y be a non-trivial subset of X. We give conditions ensuring the existence of conditionally invariant probability measures to non-absorption in Y. For dynamics which are non-singular with respect to some fixed probability measure we supply sufficient conditions for the existence of absolutely continuous conditionally invariant measures. These conditions are satisfied for a wide class of dynamical systems including systems that are Φ-mixing and Gibbs.

We define the topological entropy per unit volume in parabolic PDE's such as the complex Ginzburg-Landau equation, and show that it exists, and is bounded by the upper Hausdorff dimension times the maximal expansion rate. We then give a constructive implementation of a bound on the inertial range of such equations. Using this bound, we are able to propose a finite sampling algorithm which allows (in principle) to measure this entropy from experimental data., 26 pages, 1 small figure

We prove that two close enough points forms a set of determining nodes for the complex Ginzburg - Landau equation on the whole real line. In dimension two, a rectangular lattice of small enough lattice size forms a set of determining nodes, and the averages over the squares of this lattice form a set of determining volume elements.

Consider an expanding map T:A contained in Rn to Rn with TA contains/implies A strictly. Pianigiani and Yorke have shown that under some suitable conditions there exists a conditionally invariant probability measure mu f, with density f satisfying Pf= alpha f for the Perron-Frobenius operator P. We prove that the conditional probability measure of staying in A when the evolution occurs with probability mu f, i.e. mu f(B intersection T-nA mod T-nA) to n to infinity nu (B) for any Borel set B contained in/implied by A, is a T-invariant probability measure nu on the limit Cantor set K= intersection n>or=0T-nA which is Gibbsian with potential log mod detDTx mod .

The authors review results concerning normal forms, fronts, pattern formation, and convergence for certain one-dimensional partial differential equations. The relationship between these results is put into perspective and is supplemented by a series of unpublished results on the convergence problem for parabolic PDE.