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Séminaires et Congrès - Parutions - 20 (2010) 39-82

Parutions < 20

École de théorie ergodique
Yves Lacroix, Pierre Liardet, Jean-Paul Thouvenot, éds.
Séminaires et Congrès 20 (2010), xxii+266 pages
Acheter l'ouvrage
Présentation, Sommaire

Entropy Theory on the Interval
Jérôme Buzzi
Séminaires et Congrès 20 (2010), 39-82
Télécharger le document

Résumé :
Theorie de l'entropie sur un intervalle
Nous présentons un survol de la théorie de l'entropie des applications de l'intervalle. Nous nous plaçons du point de vue de la théorie ergodique et considérons notamment les mesures d'entropie maximale et les points périodiques. Les outils principaux sont (i) une forme adaptée du diagramme de Markov introduit par Hofbauer, (ii) la propriété de pistage et ses conséquences (borne sur l'entropie, propriété de rang 1 faible), (iii) les sous-décalages markoviens à espace d'états dénombrable fortement positivement récurrents. Les preuves ne sont données que pour une sélection de résultats. Cet article est basé sur les conférences prononcées à l'occasion de l'École thématique de théorie ergodique qui s'est tenue au C.I.R.M. en avril 2006.

Mots-clefs : Systèmes dynamiques en topologie et en combinatoire, théorie ergodique, systèmes dynamiques symboliques, entropie, principe variationnel, applications d'intervalles, applications monotones par morceaux, applications fer à cheval, mesures d'entropie maximale, orbites périodiques, zêta fonction de Artin-Mazur, invariants par tricotage, décalages markoviens fortement positivement récurrents, diagramme de Markov, pistage, rang 1 faible

Abstract:
We give a survey of the entropy theory of interval maps as it can be analyzed using ergodic theory, especially measures of maximum entropy and periodic points. The main tools are (i) a suitable version of Hofbauer's Markov diagram, (ii) the shadowing property and the implied entropy bound and weak rank one property, (iii) strongly positively recurrent countable state Markov shifts. Proofs are given only for selected results. This article is based on the lectures given at the École thématique de théorie ergodique at the C.I.R.M., Marseille, in April 2006.

Keywords: Topological and combinatorial dynamics, ergodic theory, symbolic dynamics, entropy, variational principle, interval maps, piecewise monotone maps, horseshoes, measures maximizing entropy, periodic orbits, Artin-Mazur zeta function, kneading invariants, strongly positive recurrent Markov shifts, Markov diagram, shadowing, weak rank one

Class. math. : 37Axx, 37Bxx


ISBN : 978-2-85629-312-6
ISSN : 1285-2783

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