Homoclinic points of symplectic partially hyperbolic systems with 2D centre

References

• Asaoka M. and Irie K., A C∞-closing lemma for Hamiltonian diffeomorphisms of closed surfaces, Geom. Funct. Anal. 26(5) (2016), pp. 1245–1254.
   Web of Science ®Google Scholar
• Birkhoff G., Surface transformations and their dynamical applications, Acta. Math. 43 (1922), pp. 1–119.
   Google Scholar
• Birkhoff G., Nouvelles recherches sur les systemes dynamiques, Memoriae Pont Acad. Sci. Novi Lyncaei1 (1935), pp. 85–216.
   Google Scholar
• Burns K. and Wilkinson A., Dynamical coherence and center bunching, Discrete Contin. Dyn. Syst.22 (1/2) (2008), pp. 89–100.
   Web of Science ®Google Scholar
• Burns K. and Wilkinson A., On the ergodicity of partially hyperbolic systems, Ann. Math. 171(1) (2010), pp. 451–489.
   Web of Science ®Google Scholar
• Contreras G. and Oliveira F., Homoclinics for geodesic flows of surfaces, arXiv:2205.14848.
   Google Scholar
• Cristofaro-Gardiner D., Prasad R., and Zhang B., Periodic Floer homology and the smooth closing lemma for area-preserving surface diffeomorphisms, arXiv:2110.02925.
   Google Scholar
• Edtmair O. and Hutchings M., PFH spectral invariants and C∞ closing lemmas, arXiv:2110.02463.
   Google Scholar
• Franks J. and Le Calvez P., Regions of instability for non-twist maps, Ergod. Th. Dynam. Syst. 23(1) (2003), pp. 111–141.
   Google Scholar
• Gan S. and Shi Y., Cr-Closing lemma for partially hyperbolic diffeomorphisms with 1D-center bundle, arXiv:2004.06855.
   Google Scholar
• Gourmelon N., Adapted metrics for dominated splittings, Ergod. Th. Dynam. Syst. 27(6) (2007), pp. 1839–1849.
   Google Scholar
• Hayashi S., Connecting invariant manifolds and the solution of the C1-stability and stability conjectures for flows, Ann Math. 145 (1) (1997), pp. 81–137.
   Web of Science ®Google Scholar
• Hirsch M., Pugh C., and Shub M., Invariant Manifolds, Vol. 583, Lect. Notes Math, Springer-Verlag, Berlin, 1977.
   Google Scholar
• Irie K., Dense existence of periodic reeb orbits and ECH spectral invariants, J. Mod. Dyn. 9(1) (2015), pp. 357–363.
   Google Scholar
• Koropecki A., Le Calvez P., and Nassiri M., Prime ends rotation numbers and periodic points, Duke Math. J. 164(3) (2015), pp. 403–472.
   Web of Science ®Google Scholar
• Le Calvez P. and Sambarino M., Homoclinic orbits for area preserving diffeomorphisms of surfaces, Ergod. Th. Dynam. Syst. 42(3) (2022), pp. 1122–1165.
   Web of Science ®Google Scholar
• Marin K., Cr-density of (non-uniform) hyperbolicity in partially hyperbolic symplectic diffeomorphisms, Comment. Math. Helv. 91(2) (2016), pp. 357–396.
   Web of Science ®Google Scholar
• Mather J., Invariant subsets of area-preserving homeomorphisms of surfaces, Adv. Math. Suppl. Stud.7B (1981), pp. 531–62.
   Google Scholar
• Mather J., Topological proofs of some purely topological consequences of caratheodory's theory of prime ends, in Selected Studies, T. M. Rassias, G. M. Rassias, eds., North-Holland Publishing Co., Amsterdam, 1982, pp. 225–255.
   Google Scholar
• McDuff D. and Salamon D., Introduction to Symplectic Topology, 3rd ed., Oxford Grad. Texts Math., Oxford University Press, Oxford, 2017.
   Google Scholar
• Moser J., On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1962 (1962), pp. 1–20.
   Google Scholar
• Moser J., Stable and Random Motions in Dynamical Systems. Annals Math. Studies, Vol. 77, Princeton University Press, Princeton, NJ, 1973.
   Google Scholar
• Niţic aˇ V. and Török A., An open dense set of stably ergodic diffeomorphisms in a neighborhood of a non-ergodic one, Topology 40(2) (2001), pp. 259–278.
   Google Scholar
• Oliveira F., On the generic existence of homoclinic points, Ergod. Th. Dynam. Syst. 7(4) (1987), pp. 567–595.
   Google Scholar
• Oliveira F., On the C∞ genericity of homoclinic orbits, Nonlinearity 13(3) (2000), pp. 653–662.
   Web of Science ®Google Scholar
• Pesin Y., Lectures on partial hyperbolicity and stable ergodicity, Zurich Lectures in Advanced Mathematics, EMS, Zürich, 2004.
   Google Scholar
• Pixton D., Planar homoclinic points, J. Differ. Equ. 44(3) (1982), pp. 365–382.
   Web of Science ®Google Scholar
• Poincaré H., Sur le probléme des trios corps et les équations de la dynamique, Acta Math.13 (1890), pp. 211–270. (French) [On the Three-Body Problem and the Equations of Dynamics.].
   Google Scholar
• Poincaré H., Les methodes nouvelles de la mecanique celeste (French). [New Methods of Celestial Mechanics]. Gauthier-Villars, Paris, vol. 1 in 1892; vol. 2 in 1893; vol. 3 in 1899.
   Google Scholar
• Pugh C., The closing lemma, Amer. J. Math. 89(4) (1967), pp. 956–1009.
   Web of Science ®Google Scholar
• Pugh C., An improved closing lemma and a general density theorem, Amer. J. Math. 89(4) (1967), pp. 1010–1021.
   Web of Science ®Google Scholar
• Pugh C. and Robinson C., The C1 closing lemma including Hamiltonians, Ergod. Theor. Dyn. Sys. 3(2) (1983), pp. 261–313.
   Google Scholar
• Pugh C., Shub M., and Wilkinson A., Hölder foliations, revisited, J. Mod. Dyn 6(1) (2012), pp. 79–120.
   Web of Science ®Google Scholar
• Qu H. and Xia Z., A C∞ closing lemma on torus, arXiv:2106.08844.
   Google Scholar
• Robinson C., Generic properties of conservative systems, Amer. J. Math. 92(3) (1970), pp. 562–603.
   Web of Science ®Google Scholar
• Robinson C., Closing stable and unstable manifolds in the two-sphere, Proc. Am. Math. Soc. 41(1) (1973), pp. 299–299.
   Web of Science ®Google Scholar
• Saghin R. and Xia Z., Partial hyperbolicity or dense elliptic periodic points for C1-generic symplectic diffeomorphisms, Trans. Amer. Math. Soc. 358(11) (2006), pp. 5119–5138.
   Web of Science ®Google Scholar
• Smale S., Diffeomorphisms with Many Periodic Points, Differential and Combinatorial Topology, Princeton Univ. Press, Princeton, NJ, 1965, pp. 63–80.
   Google Scholar
• Takens F., Homoclinic points in conservative systems, Invent. Math. 18(3–4) (1972), pp. 267–292.
   Web of Science ®Google Scholar
• Xia Z., Homoclinic points in symplectic and volume-preserving diffeomorphisms, Comm. Math. Phys.177(2) (1996), pp. 435–449.
   Web of Science ®Google Scholar
• Xia Z., Area-preserving surface diffeomorphisms, Comm. Math. Phys. 263(3) (2006), pp. 723–735.
   Web of Science ®Google Scholar
• Xia Z., Homoclinic points for area-preserving surface diffeomorphisms, arxiv:math/0606291.
   Google Scholar
• Xia Z. and Zhang H., A Cr closing lemma for a class of symplectic diffeomorphisms, Nonlinearity 19(2) (2006), pp. 511–516.
   Web of Science ®Google Scholar
• Xia Z. and Zhang P., Homiclinic points for convex billiards, Nonlinearity 27(6) (2014), pp. 1181–1192.
   Web of Science ®Google Scholar
• Xia Z. and Zhang P., Homoclinic intersections for geodesic flows on convex spheres, Contemp. Math.698 (2017), pp. 221–238.
   Google Scholar
• Zehnder E., Note on Smoothing Symplectic and Volume-Preserving Diffeomorphisms, Geometry and Topology, Lecture Notes in Math, Vol. 597, Springer, Berlin, 1977, pp. 828–854.
   Google Scholar
• Zhang P., Convex billiards on convex spheres, Ann. Inst. H. Poincaré Anal. Non Linéaire 34(4) (2017), pp. 793–816.
   Web of Science ®Google Scholar