## Homotopically unbounded disks for generic surface diffeomorphisms

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- by Salvador Addas-Zanata and Andres Koropecki PDF
- 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

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## Additional Information

**Salvador Addas-Zanata**- Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, Cidade Universitária, 05508-090 São Paulo, SP, Brazil
- MR Author ID: 690385
- ORCID: 0000-0001-5844-4868
- Email: sazanata@ime.usp.br
**Andres Koropecki**- Affiliation: Instituto de Matemática e Estatística, Universidade Federal Fluminense, Rua Professor Marcos Waldemar de Freitas Reis s/n 24210-201 Niteroi, RJ, Brazil
- MR Author ID: 856885
- ORCID: 0000-0003-1616-1215
- Email: ak@id.uff.br
- Received by editor(s): June 25, 2021
- Received by editor(s) in revised form: February 8, 2022
- Published electronically: June 3, 2022
- Additional Notes: The first author was partially supported by FAPESP

The second author was partially supported by FAPERJ

The authors were partially supported by CNPq, grants: 306348/2015-2 and 305612/2016-6. - © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**375**(2022), 5859-5888 - MSC (2020): Primary 37E30, 37C25, 37C29, 37D25
- DOI: https://doi.org/10.1090/tran/8665
- MathSciNet review: 4469239