Topologically Anosov plane homeomorphisms

Gonzalo Cousillas, Jorge Groisman, Juliana Xavier

DOI: http://dx.doi.org/10.12775/TMNA.2019.050

Abstract


This paper deals with classifying the dynamics of {\it topologically Anosov} plane homeomorphisms. We prove that a topologically Anosov homeomorphism $f\colon\mathbb{R}^2 \to \mathbb{R}^2$ is conjugate to a homothety if it is the time one map of a flow. We also obtain results for the cases when the nonwandering set of $f$ reduces to a fixed point, or if there exists an open, connected, simply connected proper subset $U$ such that $\overline {f(U)} \subset \rom{Int} (U)$, and such that $$ \bigcup\limits_{n\leq 0} f^n (U)= \mathbb{R}^2.$$% In the general case, we prove a structure theorem for the $\alpha$-limits of orbits with empty $\omega$-limit (or the $\omega$-limits of orbits with empty $\alpha$-limit).

Keywords


Topologically expansive homeomorphism; topological shadowing property; topologically Anosov plane homeomorphism; homothety

Full Text:

PREVIEW FULL TEXT

References


N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems, North-Holland Math. Library, vol. 52, 1994.

L.E.J. Brouwer, Beweis des ebenen Translationssatzes, Math. Ann. 72 (1912), 37–54.

B.F. Bryant, Unstable self-homeomorphisms of a compact space, thesis, Vanderbilt University, 1954.

G. Cousillas, A fixed point theorem for topologically Anosov plane homeomorphisms, preprint, arXiv:1804.02244

E. Coven and M. Keane, Every compact space that supports a positively expansive homeomorphism is finite, IMS Lecture Notes – Monograph Series, Vol. 48, 2006, 304–305.

T. Das, K. Lee, D. Richeson and J. Wiseman, Spectral decomposition for topologically Anosov homeomorphisms on non-compact and non-metrizable spaces, Topology Appl. 160 (2013), 149–158.

A. Gasull, J. Groisman and F. Mañosas, Linearization of Planar Homeomorphisms, Topol. Methods Nonlinear Anal. 48 (2016), no. 2, 493–506.

B. Kerékjártó, Sur le caractère topologique des representations conformes, Acad. Sci. Paris Sér. 198 (1934), 317–320.

B. Kerékjártó, Topologische Charakterisierung der linearen, Acta Litt. Acad. Sei. Szeged. 6 (1934), 235–262.

J. Lewowicz, Expansive homeomorphisms of surfaces, Bol. Soc. Bras. Mat. 20 (1989), Fasc. 1, 113–133.


Refbacks

  • There are currently no refbacks.

Partnerzy platformy czasopism