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Coexistence of chaotic and elliptic behaviors among analytic, symplectic diffeomorphisms of any surface
[Coexistence de comportements chaotiques et elliptiques parmi les symplectomorphismes analytiques de toute surface]
Pierre Berger1
1 Institut de Mathématiques de Jussieu-Paris Rive Gauche, CNRS, Sorbonne Université, Université de Paris 4 place Jussieu, 75252 Paris Cedex 05, France
Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 525-547.

Nous montrons la coexistence de comportements chaotique (entropie métrique positive) et elliptique (îlots elliptiques intégrables) parmi les difféomorphismes analytiques symplectiques dans de nombreuses classes d’isotopies et toute surface fermée. En particulier nous résolvons un problème introduit par F. Przytycki (1982).

We show the coexistence of chaotic behaviors (positive metric entropy) and elliptic behaviors (integrable elliptic islands) among analytic, symplectic diffeomorphisms in many isotopy classes of any closed surface. In particular this solves a problem introduced by F. Przytycki (1982).

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DOI : 10.5802/jep.224
Classification : 37J70, 37J46, 37F80, 37E30, 37D45, 37D25
Keywords: Symplectomorphism, analytic, positive metric entropy, integrable system, elliptic island, stochastic island, coexistence
Mot clés : Symplectomorphisme, analytique, entropie métrique positive, système integrable, îlot elliptique, îlot stochastique, coexistence
Pierre Berger 1

1 Institut de Mathématiques de Jussieu-Paris Rive Gauche, CNRS, Sorbonne Université, Université de Paris 4 place Jussieu, 75252 Paris Cedex 05, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Pierre Berger. Coexistence of chaotic and elliptic behaviors among analytic, symplectic diffeomorphisms of any surface. Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 525-547. doi : 10.5802/jep.224. https://jep.centre-mersenne.org/articles/10.5802/jep.224/

[AKN88] V. I. Arnol’d, V. V. Kozlov & A. I. Neĭshtadt - Dynamical systems. III, Encyclopaedia of Math. Sciences, vol. 3, Springer-Verlag, Berlin, 1988

[AP09] A. Arroyo & E. Pujals - “C k -robust transitivity for surfaces with boundary”, 2009 | arXiv

[Arn21] M.-C. Arnaud - “La démonstration de la conjecture de l’entropie positive d’Herman d’après Berger et Turaev”, in Séminaire Bourbaki, Astérisque, vol. 430, Société Mathématique de France, Paris, 2021, p. 1-26 | DOI | Zbl

[BG89] K. Burns & M. Gerber - “Real analytic Bernoulli geodesic flows on S 2 ”, Ergodic Theory Dynam. Systems 9 (1989) no. 1, p. 27-45 | DOI | MR | Zbl

[Bou67] N. Bourbaki - Éléments de mathématique. Fasc. XXXIII. Variétés différentielles et analytiques. Fascicule de résultats (Paragraphes 1 à 7), Actualités Scientifiques et Industrielles, vol. 1333, Hermann, Paris, 1967

[BT19] P. Berger & D. Turaev - “On Herman’s positive entropy conjecture”, Adv. Math. 349 (2019), p. 1234-1288 | DOI | MR | Zbl

[DM90] B. Dacorogna & J. Moser - “On a partial differential equation involving the Jacobian determinant”, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990) no. 1, p. 1-26 | DOI | Numdam | MR | Zbl

[Don88] V. Donnay - “Geodesic flow on the two-sphere. I. Positive measure entropy”, Ergodic Theory Dynam. Systems 8 (1988) no. 4, p. 531-553 | DOI | MR | Zbl

[Ger85] M. Gerber - Conditional stability and real analytic pseudo-Anosov maps, Mem. Amer. Math. Soc., vol. 54, no. 321, American Mathematical Society, Providence, RI, 1985 | DOI | MR

[GK82] M. Gerber & A. Katok - “Smooth models of Thurston’s pseudo-Anosov maps”, Ann. Sci. École Norm. Sup. (4) 15 (1982) no. 1, p. 173-204 | DOI | Numdam | MR | Zbl

[Gor12] A. Gorodetski - “On stochastic sea of the standard map”, Comm. Math. Phys. 309 (2012) no. 1, p. 155-192 | DOI | MR | Zbl

[Her98] M. Herman - “Some open problems in dynamical systems”, Doc. Math. (1998), p. 797-808, Proceedings ICM (Berlin, 1998), Extra Vol. II | MR | Zbl

[Kat79] A. Katok - “Bernoulli diffeomorphisms on surfaces”, Ann. of Math. (2) 110 (1979) no. 3, p. 529-547 | DOI | MR | Zbl

[Kat80] A. Katok - “Lyapunov exponents, entropy and periodic orbits for diffeomorphisms”, Publ. Math. Inst. Hautes Études Sci. (1980) no. 51, p. 137-173 | DOI | Numdam | MR | Zbl

[Liv04] C. Liverani - “Birth of an elliptic island in a chaotic sea”, Math. Phys. Electron. J. 10 (2004), article ID 1, 13 pages | MR | Zbl

[Mos56] J. Moser - “The analytic invariants of an area-preserving mapping near a hyperbolic fixed point”, Comm. Pure Appl. Math. 9 (1956) no. 4, p. 673-692 | DOI | MR | Zbl

[Prz82] F. Przytycki - “Examples of conservative diffeomorphisms of the two-dimensional torus with coexistence of elliptic and stochastic behaviour”, Ergodic Theory Dynam. Systems 2 (1982) no. 3-4, p. 439-463 | DOI | MR | Zbl

[Sin94] Y. G. Sinaĭ - Topics in ergodic theory, Princeton Math. Series, vol. 44, Princeton University Press, Princeton, NJ, 1994

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