Research article
Entropie et classification en dynamique
Jérôme Buzzi1
1 C.N.R.S., Institut de Mathématique d’Orsay
Séminaire Laurent Schwartz — EDP et applications (2022-2023), Talk no. 13, 23 p.

Cet exposé présente quelques aspects de la théorie des systèmes dynamiques topologiques ou différentiables munis de l’ensemble de leurs mesures invariantes - ce qu’on pourrait qualifier de façon un peu provocatrice de théorie ergodique sans mesure distinguée. Nous expliquons comment les entropies topologiques et de Kolmogorov-Sinaï structurent l’ensemble des mesures invariantes et distinguent des mesures maximisant l’entropie. En fait, pour les difféomorphismes de surfaces, le nombre et le type de ces mesures suffisent à décrire une large part de la dynamique.

Dans un travail récent avec Sylvain Crovisier et Omri Sarig (2022), nous avons démontré une conjecture de Newhouse (1990) et analysé ces mesures dans le cas des difféomorphismes lisses et bi-dimensionnels. Ces résultats ont pour corollaire une classification des difféomorphismes C ∞ de surfaces compactes et d’entropie non nulle. En conclusion, nous indiquons quelques perspectives et travaux en cours.

Published online:
DOI: 10.5802/slsedp.163
Jérôme Buzzi 1

1 C.N.R.S., Institut de Mathématique d’Orsay 
@article{SLSEDP_2022-2023____A7_0,
     author = {J\'er\^ome Buzzi},
     title = {Entropie et classification en dynamique},
     journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
     note = {talk:13},
     pages = {1--23},
     publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2022-2023},
     doi = {10.5802/slsedp.163},
     language = {fr},
     url = {https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.163/}
}
TY  - JOUR
AU  - Jérôme Buzzi
TI  - Entropie et classification en dynamique
JO  - Séminaire Laurent Schwartz — EDP et applications
N1  - talk:13
PY  - 2022-2023
SP  - 1
EP  - 23
PB  - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
UR  - https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.163/
DO  - 10.5802/slsedp.163
LA  - fr
ID  - SLSEDP_2022-2023____A7_0
ER  - 
%0 Journal Article
%A Jérôme Buzzi
%T Entropie et classification en dynamique
%J Séminaire Laurent Schwartz — EDP et applications
%Z talk:13
%D 2022-2023
%P 1-23
%I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
%U https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.163/
%R 10.5802/slsedp.163
%G fr
%F SLSEDP_2022-2023____A7_0
Jérôme Buzzi. Entropie et classification en dynamique. Séminaire Laurent Schwartz — EDP et applications (2022-2023), Talk no. 13, 23 p. doi : 10.5802/slsedp.163. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.163/

[1] R. Adler, A. Konheim, M. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309–319. | DOI | MR | Zbl

[2] S. Ben Ovadia, Symbolic dynamics for non-uniformly hyperbolic diffeomorphisms of compact smooth manifolds J. Mod. Dyn. 13 (2018), 43–113. | DOI | MR | Zbl

[3] M. Boyle, T. Downarowicz, The entropy theory of symbolic extensions, Invent. Math. 156 (2004), no. 1, 119–161. | DOI | MR | Zbl

[4] M. Boyle, J. Buzzi, The almost Borel structure of surface diffeomorphisms, Markov shifts and their factors, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 9, 2739–2782. | DOI | MR | Zbl

[5] J. Buzzi, Intrinsic ergodicity of smooth interval maps, Israel J. Math. 100(1997), 125–161. | DOI | MR | Zbl

[6] J. Buzzi, C r surface diffeomorphisms with no maximal entropy measure, Ergodic Theory Dynam. Systems 34 (2014), no. 6, 1770–1793. | DOI | MR | Zbl

[7] J. Buzzi, The almost Borel structure of diffeomorphisms with some hyperbolicity, Proc. Sympos. Pure Math. 89, American Mathematical Society, Providence, RI, 2015, 9–44. | DOI | Zbl

[8] J. Buzzi, The degree of Bowen factors and injective codings of diffeomorphisms, J. Mod. Dyn. 16 (2020), 1–36. | DOI | MR | Zbl

[9] J. Buzzi, S. Crovisier, O. Sarig, Measures of maximal entropy for surface diffeomorphisms, Ann. of Math. (2) 195 (2022), no. 2, 421–508. | DOI | MR | Zbl

[10] J. Buzzi, S. Crovisier, O. Sarig, Continuity properties of Lyapunov exponents for surface diffeomorphisms, Invent. Math. 230 (2022), no. 2, 767–849. | DOI | MR | Zbl

[11] W. de Melo, S. van Strien, One-dimensional dynamics, Ergeb. Math. Grenzgeb. (3), 25 Springer-Verlag, Berlin, 1993, xiv+605 pp. | DOI | Zbl

[12] T. Downarowicz, B. Weiss, Pure strictly uniform models of non-ergodic measure automorphisms, Discrete Contin. Dyn. Syst. 42 (2022), no. 2, 863–884. | DOI | MR | Zbl

[13] S. Eigen, A. Hajian, B. Weiss, Borel automorphisms with no finite invariant measure. Proc. Amer. Math. Soc. 126 (1998), no.12, 3619–3623. | DOI | MR | Zbl

[14] M. Foreman, A. Gorodetski, Anti-classification results for smooth dynamical systems. Preliminary Report. 2022. | arXiv

[15] C. González-Tokman, A. Quas, A concise proof of the multiplicative ergodic theorem on Banach spaces, J. Mod. Dyn. 9 (2015), 237–255. | DOI | Zbl

[16] M. Hochman, Isomorphism and embedding of Borel systems on full sets, Acta Appl. Math. 126 (2013), 187–201. | DOI | MR | Zbl

[17] M. Hochman, Every Borel automorphism without finite invariant measures admits a two-set generator, J. Eur. Math. Soc. (JEMS) 21 (2019), no. 1, 271–317. | DOI | MR | Zbl

[18] E. Jeandel, Strong shift equivalence as a category notion. 2021. | arXiv

[19] W. Krieger, On entropy and generators of measure-preserving transformations, Trans. Amer. Math. Soc. 149 (1970), 453–464. | DOI | MR | Zbl

[20] V. Kaloshin, Generic diffeomorphisms with superexponential growth of number of periodic orbits, Comm. Math. Phys. 211 (2000), no. 1, 253–271. | DOI | MR | Zbl

[21] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. (1980), no. 51, 137–173. | DOI | Zbl

[22] A. Kolmogorov, Entropy per unit time as a metric invariant of automorphisms, Dokl. Akad. Nauk SSSR 124 (1959), 754–755. | Zbl

[23] D. Lind, B. Marcus, An introduction to symbolic dynamics and coding, Cambridge Math. Lib. Cambridge University Press, Cambridge, 2021, xix+550 pp.

[24] M. Misiurewicz, Diffeomorphism without any measure with maximal entropy, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 21 (1973), 903–910. | Zbl

[25] S. Newhouse, Entropy in smooth dynamical systems, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), 1285–1294. Mathematical Society of Japan, Tokyo, 1991. | Zbl

[26] S. Newhouse, Continuity properties of entropy, Ann. of Math. (2) 129 (1989), no. 2, 215–235. | DOI | MR | Zbl

[27] C. Robinson, An introduction to dynamical systems—continuous and discrete, Pure Appl. Undergrad. Texts, 19 American Mathematical Society, Providence, RI, 2012, xx+733 pp. | Zbl

[28] F. Rodriguez-Hertz, M.-A. Rodriguez-Hertz, A. Tazhibi, R. Urès,

[29] M. Shub, Stabilité globale des systèmes dynamiques, Astérisque, 56. Société Mathématique de France, Paris, 1978, iv+211 pp. | Zbl

[30] Ya. Sinai, On the concept of entropy for a dynamic system, Dokl. Akad. Nauk SSSR 124(1959), 768–771. | Zbl

[31] B. Weiss, Measurable dynamics. Conference in modern analysis and probability (New Haven, Conn., 1982), 395–421. Contemp. Math., 26 American Mathematical Society, Providence, RI, 1984 | DOI

[32] B. Weiss, Countable generators in dynamics—universal minimal models. in Measure and measurable dynamics (Rochester, NY, 1987), 321–326. | DOI | Zbl

[33] R. Williams, Strong shift equivalence of matrices in GL(2,Z), Symbolic dynamics and its applications (New Haven, CT, 1991), 445–451. Contemp. Math., 135 American Mathematical Society, Providence, RI, 1992 | DOI | Zbl

[34] Y. Yomdin, Volume growth and entropy, Israel J. Math. 57 (1987), no. 3, 285–300. | DOI | MR | Zbl

Cited by Sources: