We consider sufficient conditions to determine the global dynamics for equivariant maps of the plane with a unique fixed point which is also hyperbolic. When the map is equivariant under the action of a subgroup of O(2), it is possible to describe the local dynamics and -from this- also the global dynamics. In particular, if the group contains a reflection, there is a line invariant by the map. This invariant line allows us to describe the global dynamical behaviour. Otherwise, in the absence of reflections, equivariant examples can be construct to show that global dynamics may not follow from local dynamics near the unique fixed point. We will see in this seminar an example based on the construction of a symmetric Cantor set.
This is joint work with Begoña Alarcón (Department of Mathematics, University of Oviedo) and Isabel S. Labouriau (Centro de Matemática, Universidade do Porto), reported in [1] and [2].
Referencias
[1] B. Alarcón, S. B. S. D. Castro and I. S. Labouriau, Global Dynamics for Symmetric Planar Maps, Preprint 2012.
[2] B. Alarcón, Rotation numbers for planar attractors of equivariant homeomorphisms, Preprint 2012.