We introduce the notion of metric entropy for a nonautonomous dynamical system given by a sequence (Xn, μn) of probability spaces and a sequence of measurable maps fn : Xn → Xn+1 with fnμn = μn+1. This notion generalizes the classical concept of metric entropy established by Kolmogorov and Sinai, and is related via a variational inequality to the topological entropy of nonautonomous systems as defined by Kolyada, Misiurewicz, and Snoha. Moreover, it shares several properties with the classical notion...

Let f be a continuous self-map of a smooth compact connected and simply-connected manifold of dimension m ≥ 3 and r a fixed natural number. A topological invariant $D_r^m\left[f\right]$, introduced by the authors [Forum Math. 21 (2009)], is equal to the minimal number of r-periodic points for all smooth maps homotopic to f. In this paper we calculate $D{³}_r\left[f\right]$ for all self-maps of S³.

The problem of description of the set Per(f) of all minimal periods of a self-map f:X → X is studied. If X is a rational exterior space (e.g. a compact Lie group) then there exists a description of the set of minimal periods analogous to that for a torus map given by Jiang and Llibre. Our approach is based on the Haibao formula for the Lefschetz number of a self-map of a rational exterior space.

Let f be a smooth self-map of m-dimensional, m ≥ 4, smooth closed connected and simply-connected manifold, r a fixed natural number. For the class of maps with periodic sequence of Lefschetz numbers of iterations the authors introduced in [Graff G., Kaczkowska A., Reducing the number of periodic points in smooth homotopy class of self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers, Ann. Polon. Math. (in press)] the topological invariant J[f] which is equal to the...

We show that in normalized families of polynomial or rational maps, Misiurewicz maps (critically finite or infinite) unfold generically. For example, if $f_{{\lambda }_0}$ is critically finite with non-degenerate critical point $c_1\left({\lambda }_0\right),...,c_n\left({\lambda }_0\right)$ such that $f_{{\lambda }_0}^{k_i}\left(c_i\left({\lambda }_0\right)\right)=p_i\left({\lambda }_0\right)$ are hyperbolic periodic points for i = 1,...,n, then  IV-1. Age impartible......................................................................................................................................................................... 31  $\lambda \mapsto (f_{\lambda }^{k_1}\left(c_1\left(\lambda \right)\right)-p_1\left(\lambda \right),...,f_{\lambda }^{k_{d-2}}\left(c_{d-2}\left(\lambda \right)\right)-p_{d-2}\left(\lambda \right))$ is a local diffeomorphism...

We propose a new framework for the study of continuous time dynamical systems on networks. We view such dynamical systems as collections of interacting control systems. We show that a class of maps between graphs called graph fibrations give rise to maps between dynamical systems on networks. This allows us to produce conjugacy between dynamical systems out of combinatorial data. In particular we show that surjective graph fibrations lead to synchrony subspaces in networks. The injective graph fibrations,...

We consider germs of one-parameter generic families of resonant analytic diffeomorphims and we give a complete modulus of analytic classification by means of the unfolding of the Écalle modulus. We describe the parametric resurgence phenomenon. We apply this to give a complete modulus of orbital analytic classification for the unfolding of a generic resonant saddle of a 2-dimensional vector field by means of the unfolding of its holonomy map. Here again the modulus is an unfolding of the Martinet-Ramis...

In this article we give an obstruction to integrability by quadratures of an ordinary differential equation on the differential Galois group of variational equations of any order along a particular solution. In Hamiltonian situation the condition on the Galois group gives Morales-Ramis-Simó theorem. The main tools used are Malgrange pseudogroup of a vector field and Artin approximation theorem.

En utilisant le théorème de Ruelle d'opérateur de transfert, nous démontrons que la moyenne 2-k Σn=02k-1 ||^wn||L1 de la localisation fréquentielle pour les paquets d'ondelettes admet un équivalent de la forme cρk (c > 0, 1 < ρ < √2). Cela améliore une inégalité antérieurement obtenue par Coifman, Meyer et Wickerhauser. Des estimations numériques de ρ sont obtenues pour des filtres de Daubechies.

We consider the multifractal analysis for Birkhoff averages of continuous potentials on a class of non-conformal repellers corresponding to the self-affine limit sets studied by Lalley and Gatzouras. A conditional variational principle is given for the Hausdorff dimension of the set of points for which the Birkhoff averages converge to a given value. This extends a result of Barral and Mensi to certain non-conformal maps with a measure dependent Lyapunov exponent.