Puzzles of Quasi-Finite Type, Zeta Functions and Symbolic Dynamics for Multi-Dimensional Maps

Buzzi, Jérôme

Annales de l'Institut Fourier, Tome 60 (2010) no. 3, pp. 801-852.

Les transformations entropie-dilatantes forment une classe de systèmes dynamiques différentiables généralisant les applications de l’intervalle d’entropie non-nulle et les applications dilatantes. Dans ce travail, on construit une représentation symbolique de ces dynamiques en termes de puzzles (au sens de Yoccoz), évitant ainsi une condition de connexité difficile à satisfaire en dimension supérieure. Ces puzzles sont contrôlés par une « entropie de contrainte » bornée par l’entropie d’hypersurface des transformations précédentes.

L’analyse de ces puzzles repose sur un graphe dénombrable « stablement positif récurrent ». Plus précisément on introduit une « entropie à l’infini » du graphe, contrôlée par l’entropie de contrainte du puzzle, qui permet de généraliser des propriétés classiques des sous-décalages de type fini : multiplicité finie des mesures d’entropie maximale, classification presque topologique, extension méromorphe de fonctions zéta d’Artin-Mazur comptant les points périodiques.

Ces résultats sont enfin appliqués aux puzzles et aux applications entropie-dilatantes « non-dégénérées ».

Entropy-expanding transformations define a class of smooth dynamics generalizing interval maps with positive entropy and expanding maps. In this work, we build a symbolic representation of those dynamics in terms of puzzles (in Yoccoz’s sense), thus avoiding a connectedness condition, hard to satisfy in higher dimensions. Those puzzles are controled by a «constraint entropy» bounded by the hypersurface entropy of the aforementioned transformations.

The analysis of those puzzles rests on a «stably positively recurrent» countable graph. More precisely, we introduce an «entropy at infinity» for such graphs, bounded by the constraint entropy of the puzzle. This allows the generalization of classical properties of subshifts of finite type: finite multiplicity of maximal entropy measures, almost topological classification, meromorphic extension of Artin-Mazur zeta functions counting periodic points.

These results are finally applied to puzzles and non-degenerate entropy-expanding maps.

Classification : 37B10, 37A35, 37D25, 37C30, 37B40
Keywords: Symbolic dynamics, topological dynamics, ergodic theory, entropy, measures of maximal entropy, periodic points, Artin-Mazur zeta function, puzzle, non-uniform hyperbolicity, entropy-expanding transformations, countable state topological Markov chains, stable positive recurrence, meromorphic extensions, entropy-conjugacy, complexity
Mot clés : dynamique symbolique, dynamique topologique, théorie ergodique, entropie, mesures d’entropie maximale, points périodiques, fonction zéta d’Artin-Mazur, puzzle, hyperbolicité non-uniforme, transformations entropie-dilatantes, chaînes de Markov topologiques à ensemble d’états dénombrable, récurrence stablement positive, extension méromorphe, conjugaison du point de vue de l’entropie, complexité

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