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Dynamical directions in numeration

Guy Barat, Valérie Berthé, Pierre Liardet, Jörg Thuswaldner (2006)

Annales de l’institut Fourier

This survey aims at giving a consistent presentation of numeration from a dynamical viewpoint: we focus on numeration systems, their associated compactification, and dynamical systems that can be naturally defined on them. The exposition is unified by the fibred numeration system concept. Many examples are discussed. Various numerations on rational integers, real or complex numbers are presented with special attention paid to β -numeration and its generalisations, abstract numeration systems and...

La mesure d’équilibre d’un endomorphisme de k ( )

Xavier Buff (2004/2005)

Séminaire Bourbaki

Soit f un endomorphisme holomorphe de k ( ) . Je présenterai une construction géométrique, due à Briend et Duval, d’une mesure de probabilité μ ayant les propriétés suivantes : μ reflète la distribution des préimages des points en dehors d’un ensemble exceptionnel algébrique, les points périodiques répulsifs de f s’équidistribuent par rapport à μ et μ est l’unique mesure d’entropie maximale de f .

Results and open questions on some invariants measuring the dynamical complexity of a map

Jaume Llibre, Radu Saghin (2009)

Fundamenta Mathematicae

Let f be a continuous map on a compact connected Riemannian manifold M. There are several ways to measure the dynamical complexity of f and we discuss some of them. This survey contains some results and open questions about relationships between the topological entropy of f, the volume growth of f, the rate of growth of periodic points of f, some invariants related to exterior powers of the derivative of f, and several invariants measuring the topological complexity of f: the degree (for the case...

Rosen fractions and Veech groups, an overly brief introduction

Thomas A. Schmidt (2009)

Actes des rencontres du CIRM

We give a very brief, but gentle, sketch of an introduction both to the Rosen continued fractions and to a geometric setting to which they are related, given in terms of Veech groups. We have kept the informal approach of the talk at the Numerations conference, aimed at an audience assumed to have heard of neither of the topics of the title.The Rosen continued fractions are a family of continued fraction algorithms, each gives expansions of real numbers in terms of elements of a corresponding algebraic...

Stability and gradient dynamical systems.

Jack K. Hale (2004)

Revista Matemática Complutense

The objective in these notes is to present an approach to dynamical systems in infinite dimensions. It does not seem reasonable to make a comparison of all of the orbits of the dynamics of two systems on non locally compact infinite dimensional spaces. Therefore, we choose to compare them on the set of globally defined bounded solutions. Fundamental problems are posed and several important results are stated when this set is compact. We then give results on the dynamical system which will ensure...

Statistical stability of geometric Lorenz attractors

José F. Alves, Mohammad Soufi (2014)

Fundamenta Mathematicae

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.

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