An analogue of the Variational Principle for group and pseudogroup actions

Biś, Andrzej

Annales de l'Institut Fourier, Tome 63 (2013) no. 3, pp. 839-863.

On généralise au cas des groupes d’homéomorphismes de type fini la notion d’entropie mesure locale introduite par Brin et Katok [7] pour une seule transformation. On applique la théorie des caractéristiques de type dimension d’un système dynamique élaborée par Pesin [25] pour obtenir une relation entre l’entropie topologique d’un pseudogroupe et d’un groupe d’homéomorphismes d’un espace métrique, définie par Ghys, Langevin et Walczak dans [12], et ses entropies mesure locale. On prouve un analogue du principe variationnel pour les actions de groupe et de pseudogroupe qui nous permet d’étudier les dynamiques locales des feuilletages.

We generalize to the case of finitely generated groups of homeomorphisms the notion of a local measure entropy introduced by Brin and Katok [7] for a single map. We apply the theory of dimensional type characteristics of a dynamical system elaborated by Pesin [25] to obtain a relationship between the topological entropy of a pseudogroup and a group of homeomorphisms of a metric space, defined by Ghys, Langevin and Walczak in [12], and its local measure entropies. We prove an analogue of the Variational Principle for group and pseudogroup actions which allows us to study local dynamics of foliations.

Classification : 37C85, 28D20, 37B40
Keywords: variational principle, topological entropy, Carathéodory structures, Carathéodory measures and dimensions, local measure entropy, pseudogroups, foliations, Hausdorff measure, homogeneous measure
Mot clés : principe variationnel, l’entropie topologique, structures Carathéodory, des mesures de Carathéodory et de dimensions de Carathéodory, l’entropie mesure locale, pseudogroups, feuilletages, mesure de Hausdorff, mesure homogene

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