Hostname: page-component-6b989bf9dc-jks4b Total loading time: 0 Render date: 2024-04-15T04:44:19.371Z Has data issue: false hasContentIssue false
  • English
  • Français

Rotation sets with non-empty interior and transitivity in the universal covering

Published online by Cambridge University Press:  28 August 2013

NANCY GUELMAN ,
ANDRES KOROPECKI  and
FABIO ARMANDO TAL
NANCY GUELMAN
Affiliation:
IMERL, Facultad de Ingeniería, Universidad de la República, C.C. 30, Montevideo, Uruguay email nguelman@fing.edu.uy
ANDRES KOROPECKI
Affiliation:
Universidade Federal Fluminense, Instituto de Matemática e Estatística, Rua Mário Santos Braga S/N, 24020-140 Niteroi, RJ, Brasil email ak@id.uff.br
FABIO ARMANDO TAL
Affiliation:
Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, Cidade Universitária, 05508-090 São Paulo, SP, Brasil email fabiotal@ime.usp.br
Get access Rights & Permissions [Opens in a new window]

Abstract

Let $f$ be a transitive homeomorphism of the two-dimensional torus in the homotopy class of the identity. We show that a lift of $f$ to the universal covering is transitive if and only if the rotation set of the lift contains the origin in its interior.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 3 , May 2015 , pp. 883 - 894
Copyright
© Cambridge University Press, 2013 

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. and Tal, F. A.. Homeomorphisms of the annulus with a transitive lift. Math. Z. 267 (3–4) (2011), 971–980.Google Scholar
Addas-Zanata, S. and Tal, F. A.. Homeomorphisms of the annulus with a transitive lift II. Discrete Contin. Dyn. Syst. 31 (3) (2011), 651–668.CrossRefGoogle Scholar
Boyland, P.. Transitivity of surface dynamics lifted to abelian covers. Ergod. Th. & Dynam. Sys. 29 (5) (2009), 1417–1449.Google Scholar
Brown, M.. Homeomorphisms of two-dimensional manifolds. Houston J. Math. 11 (4) (1985), 455–469.Google Scholar
Enrich, H., Guelman, N., Larcanché, A. and Liousse, I.. Diffeomorphisms having rotation sets with non-empty interior. Nonlinearity 22 (8) (2009), 1899–1907.Google Scholar
Franks, J.. Realizing rotation vectors for torus homeomorphisms. Trans. Amer. Math. Soc. 311 (1) (1989), 107–115.Google Scholar
Koropecki, A. and Tal, F. A.. Strictly toral dynamics. Invent. Math., to appear, doi:10.1007/s00222-013-0470-3.Google 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 (1) (1991), 115–128.Google Scholar
Misiurewicz, M. and Ziemian, K.. Rotation sets for maps of tori. J. Lond. Math. Soc. (2) 40 (3) (1989), 490–506.CrossRefGoogle Scholar
Tal, F. A.. Transitivity and rotation sets with non-empty interior for homeomorphisms of the 2-torus. Proc. Amer. Math. Soc. 140 (2012), 3567–3579.Google Scholar