Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-28T00:48:25.687Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-existence of sublinear diffusion for a class of torus homeomorphisms

Part of: Low-dimensional dynamical systems

Published online by Cambridge University Press:  18 January 2021

GUILHERME SILVA SALOMÃO  and
FABIO ARMANDO TAL
GUILHERME SILVA SALOMÃO
Affiliation:
Instituto de Matemática e Estatística, Rua do Matão 1010, Cidade Universitária, São Paulo, SP, Brazil, 05508-090 (e-mail: [email protected])
FABIO ARMANDO TAL*
Affiliation:
Instituto de Matemática e Estatística, Rua do Matão 1010, Cidade Universitária, São Paulo, SP, Brazil, 05508-090 (e-mail: [email protected])
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that, if f is a homeomorphism of the 2-torus isotopic to the identity whose rotation set is a non-degenerate segment and f has a periodic point, then it has uniformly bounded deviations in the direction perpendicular to the segment.

Keywords

rotation setsforcingbounded deviations

MSC classification

Primary: 37E30: Homeomorphisms and diffeomorphisms of planes and surfaces 37E45: Rotation numbers and vectors
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 4 , April 2022 , pp. 1517 - 1547
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Addas-Zanata, S.. Uniform bounds for diffeomorphisms of the torus and a conjecture of Boyland. J. Lond. Math. Soc. 91(2) (2015), 537–553.CrossRefGoogle Scholar
Addas-Zanata, S. and Liu, X.. On stable and unstable behaviours of certain rotation segments. Preprint, 2019, arXiv:1903.08703.Google Scholar
Béguin, F., Crovisier, S. and Le Roux, F.. Fixed point sets of isotopies on surfaces. J. Eur. Math. Soc., 22(6) (2020), 1971–2046.CrossRefGoogle Scholar
Brouwer, L. E. J.. Beweis des ebenen Translationssatzes. Math. Ann. 72(1) (1912), 37–54.CrossRefGoogle Scholar
Le Calvez, P.. Une version feuilletée équivariante du théorème de translation de Brouwer. Publ. Math. Inst. Hautes Études Sci. 102(1) (2005), 1–98.CrossRefGoogle Scholar
Le Calvez, P. and Tal, F. A.. Forcing theory for transverse trajectories of surface homeomorphisms. Invent. Math. 212 (2015), 619–729.CrossRefGoogle Scholar
Le Calvez, P. and Tal, F. A.. Topological horseshoes for surface homeomorphisms. Preprint, 2018, arXiv:1803.04557.Google Scholar
Conejeros, J. and Tal, F. A.. Existence of non-contractible periodic orbits for homeomorphisms of the open annulus. Math. Z. 294 (2020),1413–1439.CrossRefGoogle Scholar
Davalos, P.. On annular maps of the torus and sublinear diffusion. Inst. Math. Jussieu 17 (2018), 913–978.CrossRefGoogle Scholar
Franks, J.. Recurrence and fixed points of surface homeomorphisms. Ergod. Th. & Dynam. Sys. 8 (1988), 99–107.Google Scholar
Franks, J.. Realizing rotation vectors for torus homeomorphisms. Trans. Amer. Math. Soc. 311(1) (1989), 107–115.CrossRefGoogle Scholar
Guelman, N., Koropecki, A. and Tal, F.. A characterization of annularity for area-preserving toral homeomorphisms. Math. Z. 276(3–4) (2014), 673–689.CrossRefGoogle Scholar
Guelman, N., Koropecki, A. and Tal, F. A.. Rotation sets with non-empty interior and transitivity in the universal covering. Ergod. Th. & Dynam. Sys. 35(3) (2015), 883–894.CrossRefGoogle Scholar
Jäger, T.. Linearisation of conservative toral homeomorphisms. Invent. Math. 176(3) (2009), 601–616.CrossRefGoogle Scholar
Jäger, T. and Passeggi, A.. On torus homeomorphisms semiconjugate to irrational circle rotations. Ergod. Th. & Dynam. Sys. 35(7) (2015), 2114–2137.CrossRefGoogle Scholar
Jäger, T. and Tal, F.. Irrational rotation factors for conservative torus homeomorphisms. Ergod. Th. & Dynam. Sys. 37(5) (2017), 1537–1546.CrossRefGoogle Scholar
Kocsard, A. and Koropecki, A.. Free curves and periodic points for torus homeomorphisms. Ergod. Th. & Dynam. Sys. 28(06) (2008), 1895–1915.CrossRefGoogle Scholar
Kocsard, A.. On the dynamics of minimal homeomorphisms of t2 which are not pseudo-rotations. Ann. Sci. Éc. Norm. Supér., to appear. Preprint, 2016, arXiv:1611.03784.Google Scholar
Koropecki, A., Passeggi, A. and Sambarino, M.. The Franks–Misiurewicz conjecture for extensions of irrational rotations. Preprint, 2016, arXiv:1611.05498.Google Scholar
Kocsard, A. and Rodrigues, F. P.. Rotational deviations and invariant pseudo-foliations for periodic point free torus homeomorphisms. Math. Z. 290 (2017), 1223–1247.CrossRefGoogle Scholar
Koropecki, A. and Tal, F.. Area-preserving irrotational diffeomorphisms of the torus with sublinear diffusion. Proc. Amer. Math. Soc. 142(10) (2014), 3483–3490.CrossRefGoogle Scholar
Koropecki, A. and Tal, F.. Strictly toral dynamics. Invent. Math. 196(2) (2014), 339–381.CrossRefGoogle Scholar
Koropecki, A. and Tal, F. A.. Fully essential dynamics for area-preserving surface homeomorphisms. Ergod. Th. & Dynam. Sys. 38(5) (2018), 1791–1836.CrossRefGoogle Scholar
Llibre, J. and MacKay, R. S.. Rotation vectors and entropy for homeomorphisms of the torus isotopic to the identity. Ergod. Th. & Dynam. Sys. 11 (1991), 115–128.CrossRefGoogle Scholar
Misiurewicz, M. and Ziemian, K.. Rotation sets for maps of tori. J. Lond. Math. Soc. 2(3) (1989), 490–506.CrossRefGoogle Scholar
Passeggi, A. and Sambarino, M.. Deviations in the Franks–Misiurewicz conjecture. Ergod. Th. & Dynam. Sys. 40(9) (2020), 2533–2540.CrossRefGoogle Scholar