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Homoclinic orbits for area preserving diffeomorphisms of surfaces

Part of: Low-dimensional dynamical systems Smooth dynamical systems: general theory

Published online by Cambridge University Press:  04 May 2021

PATRICE LE CALVEZ  and
MARTÍN SAMBARINO
PATRICE LE CALVEZ*
Affiliation:
Institut de Mathématiques de Jussieu-Paris Rive Gauche, IMJ-PRG, Sorbonne Université, Université Paris-Diderot, CNRS, F-75005, Paris, France Institut Universitaire de France, 1 rue Descartes, 75231Paris Cedex 05, France
MARTÍN SAMBARINO
Affiliation:
CMAT, Facultad de Ciencias, Universidad de la República, Montevideo11400, Uruguay (e-mail: [email protected])
*
e-mail: [email protected]
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Abstract

We show that $C^r $ generically in the space of $C^r$ conservative diffeomorphisms of a compact surface, every hyperbolic periodic point has a transverse homoclinic orbit.

Keywords

low dimensional dynamicssmooth dynamicsHamiltonian dynamichomoclinic intersections

MSC classification

Primary: 37C20: Generic properties, structural stability 37C25: Fixed points, periodic points, fixed-point index theory 37C29: Homoclinic and heteroclinic orbits 37E30: Homeomorphisms and diffeomorphisms of planes and surfaces
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 3: Anatole Katok Memorial Issue Part 2: Special Issue of Ergodic Theory and Dynamical Systems , March 2022 , pp. 1122 - 1165
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

In the memory of Anatole Katok

References

Addas-Zanata, S.. Some extensions of the Poincaré–Birkhoff theorem to the cylinder and a remark on mappings of the torus homotopic to Dehn twists. Nonlinearity 18(5) (2005), 2243–2260.CrossRefGoogle Scholar
Asaoka, M. and Irie, K.. A ${C}^{\infty }$ closing lemma for Hamiltonian diffeomorphisms of closed surfaces. Geom. Funct. Anal. 26(5) (2016), 1245–1254.CrossRefGoogle Scholar
Béguin, F., Crovisier, S. and Le Roux, F.. Fixed point sets of isotopies on surfaces J. Eur. Math. Soc. (JEMS) 22(6) (2020), 1971–2046.CrossRefGoogle Scholar
Birkhoff, G.-D.. An extension of Poincaré’s last geometric theorem. Acta Math. 47 (1926), 297–311.CrossRefGoogle Scholar
Crovisier, S., Birth of homoclinic intersections: a model for the central dynamics of partially hyperbolic systems. Ann. of Math. 172 (2010), 1641–1677.CrossRefGoogle Scholar
Casson, A. and Bleiler, S.. Automorphisms of Surfaces after Nielsen and Thurston (London Mathematical Society Student Texts, 9). Cambridge University Press, Cambridge, 1988.CrossRefGoogle Scholar
Conley, C. C. and Zehnder, E.. The Birkhoff-Lewis fixed point theorem and a conjecture of V. I. Arnol’d. Invent. Math. 73(1) (1983), 33–49.CrossRefGoogle Scholar
Floer, A.. Proof of the Arnol’d conjecture for surfaces and generalizations to certain Kähler manifolds. Duke Math. J. 53(1) (1986), 1–32.CrossRefGoogle Scholar
Fathi, A., Laudenbach, F. and Poénaru, V.. Travaux de Thurston sur les surfaces. Astérisque. Vols. 66–67. Société Mathématique de France, Paris, 1979.Google Scholar
Franks, J. and Le Calvez, P.. Regions of instability for non-twist maps. Ergod. Th. & Dynam. Sys. 23(1) (2003), 111–141.CrossRefGoogle Scholar
Handel, M.. Global shadowing of pseudo-Anosov homemorphisms. Ergod. Th. & Dynam. Sys. 5(3) (1985), 373–377.CrossRefGoogle Scholar
Hall, M.. A topology for free groups and related topics. Ann. of Math. 52(1950),127–139.CrossRefGoogle Scholar
Jaulent, O.. Existence d’un feuilletage positivement transverse à un homéomorphisme de surface. Ann. Inst. Fourier (Grenoble) 64(4) (2014), 1441–1476.CrossRefGoogle Scholar
Katok, A.. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 51 (1980), 137–173.CrossRefGoogle Scholar
Koropecki, A., Le Calvez, P. and Nassiri, M.. Prime ends rotation numbers and periodic points. Duke Math. J. 164(3) (2015), 403–472.CrossRefGoogle Scholar
Le Calvez, P.. Une version feuilletée équivariante du théorème de translation de Brouwer. Publ. Math. Inst. Hautes Études Sci. 102 (2005), 1–98.CrossRefGoogle Scholar
Le Calvez, P.. Periodic orbits of Hamiltonian homeomorphisms of surfaces. Duke Math. J. 133(1), (2006), 125-184.CrossRefGoogle Scholar
Le Calvez, P. and Tal, F.. Forcing theory for transverse trajectories of surface homeomorphisms. Invent. Math. 212(2) (2018), 619–729.CrossRefGoogle Scholar
Lellouch, G.. Sur les ensembles de rotation des homéomorphismes de surface en genre ≥ 2. PhD Thesis, Sorbonne Université, 2019.Google Scholar
Le Roux, F.. L’ensemble de rotation autour d’un point fixe. Astérisque 350 (2013), 109.Google Scholar
Mather, J. N.. Invariant subsets for area preserving homeomorphisms of surfaces. Mathematical Analysis and Applications , Part B (Advances in Mathematics, Supplementary Studies, 7). Academic Press, New York, (1981), pp. 531–562.Google Scholar
Oliveira, F.. On the generic existence of homoclinic points. Ergod. Th. & Dynam. Sys. 7(4) (1987), 567–595.CrossRefGoogle Scholar
Oliveira, F.. On the ${C}^{\infty }$ genericity of homoclinic orbits. Nonlinearity 13(3) (2000), 653–662.CrossRefGoogle Scholar
Pixton, D.. Planar homoclinic point. J. Differential Equations 44(3) (1982), 365–382.CrossRefGoogle Scholar
Poincaré, H.. Les Méthodes Nouvelles de la Mécanique Céleste. Vols. 1–3. Gauthier Villars, Paris, 1899.Google Scholar
Pugh, C.. The closing lemma. Amer. J. Math. 89 (1967) 956–1009.CrossRefGoogle Scholar
Robinson, C.. Generic properties of conservative systems. Amer. J. Math. 92 (1970), 562–603.CrossRefGoogle Scholar
Robinson, C.. Closing stable and unstable manifolds on the two sphere. Proc. Amer. Math. Soc. 41 (1973), 299–303.CrossRefGoogle Scholar
Smale, S.. Diffeomorphisms with many periodic points. Differential and Combinatorial Topology. Princeton University Press, Princeton, NJ, 1964, pp. 63–80.Google Scholar
Schwartzman, S.. Asymptotic cycles. Ann. of Math. 68 (1957), 270–284.CrossRefGoogle Scholar
Sikorav, J.-C.. Points fixes d’une application symplectique homologue à l’identité. J. Differential Geom. 22(1) (1985), 4–79.CrossRefGoogle Scholar
Takens, F.. Homoclinic points in conservative systems. Invent. Math. 18 (1972), 267–292.CrossRefGoogle Scholar
Thurston, W.. On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc. (N.S.) 19(2) (1988), 417–431.CrossRefGoogle Scholar
Weiss, H.. A remark on papers by Pixton and Oliveira: genericity of symplectic diffeomorphisms of ${S}^2 $ with positive topological entropy. J. Stat. Phys. 80 (1995), 481–485.CrossRefGoogle Scholar
Xia, Z.. Area-preserving surface diffeomorphisms. Comm. Math. Phys. 263(3) (2006), 723–735.CrossRefGoogle Scholar
Xia, Z.. Homoclinic points for area-preserving surface diffeomorphisms. Preprint, 2006, arXiv:math/0606291.Google Scholar
Zehnder, E.. Homoclinic points near elliptic fixed points. Comm. Pure Appl. Math. 26 (1973), 131–182.CrossRefGoogle Scholar