Homotopically unbounded disks for generic surface diffeomorphisms

Addas-Zanata, Salvador; Koropecki, Andres

doi:10.1090/tran/8665

Homotopically unbounded disks for generic surface diffeomorphisms
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Trans. Amer. Math. Soc. 375 (2022), 5859-5888 Request permission

Abstract:

In this paper we consider closed orientable surfaces $S$ of positive genus and $C^r$-diffeomorphisms $f:S\rightarrow S$ isotopic to the identity ($r\geq 1)$. The main objective is to study periodic open topological disks which are homotopically unbounded (i.e. which lift to unbounded connected sets in the universal covering). We show that these disks are not uncommon, and are related to important dynamical phenomena. We also study the dynamics on these disks under certain generic conditions. Our first main result implies that for the torus (or for arbitrary surfaces, with an additional condition) if the rotation set of a map has nonempty interior and is not locally constant, then the map is $C^r$-accumulated by diffeomorphisms exhibiting periodic homotopically unbounded disks. Our second result shows that $C^r$-generically, if the rotation set has nonempty interior (plus an additional hypothesis if the genus of $S$ is greater than $1$) a maximal periodic disk which is unbounded and has a rational prime ends rotation number must be the basin of some compact attractor or repeller contained in the disk. As a byproduct we obtain results describing certain periodic components of the complement of the closure of stable or unstable manifolds of a periodic orbit in the $C^r$-generic setting.

References

Salvador Addas-Zanata, Uniform bounds for diffeomorphisms of the torus and a conjecture of Boyland, J. Lond. Math. Soc. (2) 91 (2015), no. 2, 537–553. MR 3355114, DOI 10.1112/jlms/jdu081
Salvador Addas-Zanata, Area-preserving diffeomorphisms of the torus whose rotation sets have non-empty interior, Ergodic Theory Dynam. Systems 35 (2015), no. 1, 1–33. MR 3294290, DOI 10.1017/etds.2013.44
Salvador Addas-Zanata and Bruno de Paula Jacoia, A condition that implies full homotopical complexity of orbits for surface homeomorphisms, Ergodic Theory Dynam. Systems 41 (2021), no. 1, 1–47. MR 4190050, DOI 10.1017/etds.2019.62
Salvador Addas-Zanata, A consequence of the growth of rotation sets for families of diffeomorphisms of the torus, Ergodic Theory Dynam. Systems 40 (2020), no. 6, 1441–1458. MR 4092853, DOI 10.1017/etds.2018.120
Kathleen T. Alligood and James A. Yorke, Accessible saddles on fractal basin boundaries, Ergodic Theory Dynam. Systems 12 (1992), no. 3, 377–400. MR 1182653, DOI 10.1017/S0143385700006842
Pavol Brunovský, On one-parameter families of diffeomorphisms, Comment. Math. Univ. Carolinae 11 (1970), 559–582. MR 279827
Jack Carr, Applications of centre manifold theory, Applied Mathematical Sciences, vol. 35, Springer-Verlag, New York-Berlin, 1981. MR 635782
M. L. Cartwright and J. E. Littlewood, Some fixed point theorems, Ann. of Math. (2) 54 (1951), 1–37. With appendix by H. D. Ursell. MR 42690, DOI 10.2307/1969308
Crovisier S., Pujals E. and Tresser C., Mildly dissipative diffeomorphisms of the disk with zero entropy, Preprint, 2020
Pablo Dávalos, On annular maps of the torus and sublinear diffusion, J. Inst. Math. Jussieu 17 (2018), no. 4, 913–978. MR 3835526, DOI 10.1017/S1474748016000268
John Franks, Realizing rotation vectors for torus homeomorphisms, Trans. Amer. Math. Soc. 311 (1989), no. 1, 107–115. MR 958891, DOI 10.1090/S0002-9947-1989-0958891-1
John Franks and Patrice Le Calvez, Regions of instability for non-twist maps, Ergodic Theory Dynam. Systems 23 (2003), no. 1, 111–141. MR 1971199, DOI 10.1017/S0143385702000858
Andres Koropecki, Patrice Le Calvez, and Meysam Nassiri, Prime ends rotation numbers and periodic points, Duke Math. J. 164 (2015), no. 3, 403–472. MR 3314477, DOI 10.1215/00127094-2861386
Andres Koropecki and Fabio Armando Tal, Strictly toral dynamics, Invent. Math. 196 (2014), no. 2, 339–381. MR 3193751, DOI 10.1007/s00222-013-0470-3
Andres Koropecki and Fabio Armando Tal, Fully essential dynamics for area-preserving surface homeomorphisms, Ergodic Theory Dynam. Systems 38 (2018), no. 5, 1791–1836. MR 3820002, DOI 10.1017/etds.2016.110
Andres Koropecki and Alejandro Passeggi, A Poincaré-Bendixson theorem for translation lines and applications to prime ends, Comment. Math. Helv. 94 (2019), no. 1, 141–183. MR 3941469, DOI 10.4171/CMH/457
P. Le Calvez and F. A. Tal, Forcing theory for transverse trajectories of surface homeomorphisms, Invent. Math. 212 (2018), no. 2, 619–729. MR 3787834, DOI 10.1007/s00222-017-0773-x
John N. Mather, Topological proofs of some purely topological consequences of Carathéodory’s theory of prime ends, Selected studies: physics-astrophysics, mathematics, history of science, North-Holland, Amsterdam-New York, 1982, pp. 225–255. MR 662863
MichałMisiurewicz and Krystyna Ziemian, Rotation sets for maps of tori, J. London Math. Soc. (2) 40 (1989), no. 3, 490–506. MR 1053617, DOI 10.1112/jlms/s2-40.3.490
J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1962 (1962), 1–20. MR 147741
J. Palis, On Morse-Smale dynamical systems, Topology 8 (1968), 385–404. MR 246316, DOI 10.1016/0040-9383(69)90024-X
Jacob Palis and Floris Takens, Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations, Cambridge Studies in Advanced Mathematics, vol. 35, Cambridge University Press, Cambridge, 1993. Fractal dimensions and infinitely many attractors. MR 1237641
Dennis Pixton, Planar homoclinic points, J. Differential Equations 44 (1982), no. 3, 365–382. MR 661158, DOI 10.1016/0022-0396(82)90002-X
J. Sotomayor, Generic one-parameter families of vector fields on two-dimensional manifolds, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 5–46. MR 339279