Symbolic extensions in intermediate smoothness on surfaces
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 45 (2012) no. 2, p. 337-362

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].

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].

DOI : https://doi.org/10.24033/asens.2167
Classification:  37C05,  37C40,  37A35
Keywords: entropy structure, symbolic extension, Yomdin's theory
@article{ASENS_2012_4_45_2_337_0,
     author = {Burguet, David},
     title = {Symbolic extensions in intermediate smoothness on surfaces},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 45},
     number = {2},
     year = {2012},
     pages = {337-362},
     doi = {10.24033/asens.2167},
     zbl = {1282.37015},
     mrnumber = {2977622},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2012_4_45_2_337_0}
}
Burguet, David. Symbolic extensions in intermediate smoothness on surfaces. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 45 (2012) no. 2, pp. 337-362. doi : 10.24033/asens.2167. http://www.numdam.org/item/ASENS_2012_4_45_2_337_0/

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