Rotation numbers for planar attractors of equivariant homeomorphisms. (English) Zbl 1301.37014

Alarcón, Begoña

Topol. Methods Nonlinear Anal. 42, No. 2, 327-343 (2013).

After the explaining some notations on Denjoy maps in the circle, which will be used, the author proves the existence of \(\mathbb{Z}_m\)-equivariant Denjoy maps. For simplicity of the presentation firstly the author constructs a Cantor set, which is invariant under the rotation \(R_{\frac{1}{2}}\), then he proves the existence of \(\mathbb{Z}_2\)-equivariant Denjoy maps in the circle with the constructed Cantor set as its minimal set. A detailed proof is given for the case \(m=2\). In the general case \(m>2\), the basic steps of the proof are given.
These constructions are the basics for the proof of the statement that for \(\mathbb{Z}_m\)-equivariant homeomorphisms one can not guarantee that their rotation numbers are rational and allow to prove the existence of \(\mathbb{Z}_m\)-equivariant homeomorphisms with some complicated and interesting dynamic features.
In conclusion, the author proves the existence of homeomorphisms of the plane, which induce a symmetric Denjoy maps in the space of prime maps, at the usage of some results of the work [L. Hernández-Corbato et al., Math. Proc. Camb. Philos. Soc. 153, No. 1, 59–77 (2012; Zbl 1267.37040)].

Reviewer: Irina V. Konopleva (Ul’yanovsk)

MSC:

37C80	Symmetries, equivariant dynamical systems (MSC2010)
37E45	Rotation numbers and vectors
37C75	Stability theory for smooth dynamical systems
37C70	Attractors and repellers of smooth dynamical systems and their topological structure
34D05	Asymptotic properties of solutions to ordinary differential equations

Keywords:

planar embedding; symmetry; asymptotic stability; global attractors

Citations:

Zbl 1267.37040