Abstract
Let \(f: \mathbb T ^2 \rightarrow \mathbb T ^2\) be a homeomorphism homotopic to the identity and \(F: \mathbb R ^2 \rightarrow \mathbb R ^2\) a lift of \(f\) such that the rotation set \(\rho (F)\) is a line segment of rational slope containing a point in \(\mathbb Q ^2\). We prove that if \(f\) is ergodic with respect to the Lebesgue measure on the torus and the average rotation vector (with respect to same measure) does not belong to \(\mathbb Q ^2\) then some power of \(f\) is an annular homeomorphism.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Addas-Zanata, S., Tal, F.A.: Homeomorphisms of the annulus with a transitive lift. Math. Z. 267, 971–980 (2011)
Addas-Zanata, S., Tal, F.A.: On generic rotationless diffeomorphisms of the annulus. Proc. Am. Math. Soc. 138, 1023 (2010)
Addas-Zanata, S., Tal, F.A., Garcia, B.A.: Dynamics of homeomorphisms of the torus homotopic to Dehn twists. Preprint arXiv:1111.5561v1 (2011)
Alpern, S., Prasad, V.S.: Topological ergodic theory and mean rotation. Proc. Am. Math. Soc. 118, 279–284 (1993)
Atkinson, G.: Recurrence of co-cycles and random walks. J. Lond. Math. Soc. 2(13), 486–488 (1976)
Dávalos, P.: On torus homeomorphisms whose rotation set is an interval. Preprint arXiv:1111.2378v1 (2011)
Franks, J.: The rotation set and periodic points for torus homeomorphisms. In: Aoki, N., Shiraiwa, K., Takahashi, Y. (eds.) Dynamical Systems and Chaos, pp. 41–48. World Scientific, Singapore (1995)
Franks, J.: Realizing rotation vectors for torus homeomorphisms. Trans. Am. Math. Soc. 311(1), 107–115 (1989)
Franks, J.: Recurrence and fixed points for surface homeomorphisms. Ergod. Theory Dyn. Syst. 8, 99–107 (1988)
Koropecki, A., Tal, F.A.: Strictly toral dynamics. Preprint arXiv:1201.1168v2 (2012)
Koropecki, A., Tal, F.A.: Area-preserving irrotational diffeomorphisms of the torus with sublinear diffusion. Preprint arXiv:1206.2409v1 (2012)
Kocsard, A., Koropecki, A.: Free curves and periodic points for torus homeomorphisms. Ergod. Theory Dyn. Syst. 28, 1895–1915 (2008)
Llibre, J., Mackay, R.: Rotation vectors and entropy for homeomorphisms of the torus isotopic to the identity. Ergod. Theory Dyn. Syst. 11, 115–128 (1991)
Misiurewicz, M., Ziemian, K.: Rotation sets for maps of Tori. J. Lond. Math. Soc. 40(2), 490–506 (1989)
Tal, F.A.: Transitivity and rotation sets with nonempty interior for homeomorphisms of the 2-torus. Proc. Am. Math. Soc. 140, 3567–3579 (2012)
Author information
Authors and Affiliations
Corresponding author
Additional information
R. B. Bortolatto supported by CNPq-Brasil. F. A. Tal supported by CNPq-Brasil, Proc. #304360/2005-8.
Rights and permissions
About this article
Cite this article
Bortolatto, R.B., Tal, F.A. Ergodicity and Annular Homeomorphisms of the Torus. Qual. Theory Dyn. Syst. 12, 377–391 (2013). https://doi.org/10.1007/s12346-012-0095-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12346-012-0095-8