Symbolic extensions in intermediate smoothness on surfaces

Burguet, David

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 45 (2012) no. 2, pp. 337-362.

Nous montrons que toute dynamique de classe ${𝒞}^r$ avec $r>1$ sur une surface compacte admet une extension symbolique, i.e. une extension topologique qui est un sous-décalage à alphabet fini. Nous donnons plus précisément une borne (optimale) sur l’infimum de l’entropie topologique de toutes les extensions symboliques. Ceci répond positivement à une conjecture de S. Newhouse and T. Downarowicz en dimension deux et améliore un résultat précédent de l’auteur [11].

We prove that ${𝒞}^r$ maps with $r>1$ on a compact surface have symbolic extensions, i.e., topological extensions which are subshifts over a finite alphabet. More precisely we give a sharp upper bound on the so-called symbolic extension entropy, which is the infimum of the topological entropies of all the symbolic extensions. This answers positively a conjecture of S. Newhouse and T. Downarowicz in dimension two and improves a previous result of the author [11].

Classification : 37C05, 37C40, 37A35
Keywords: entropy structure, symbolic extension, Yomdin's theory
Mot clés : structure d'entropie, extension symbolique, théorie de Yomdin

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