We consider sufficient conditions which guarantee that an embedding from the plane R 2 into itself has a unique fixed point. We study sufficient conditions which imply the appearing of a globally attracting fixed point for such an embedding.
Abstract. There are many tools for studying local dynamics. An important problem is how this information can be used to obtain global information. We present examples for which local stability does not carry on globally. To this purpose we construct, for any natural n ≥ 2, planar maps whose symmetry group is Z n having a local attractor that is not a global attractor. The construction starts from an example with symmetry group Z 4 . We show that although this example has codimension 3 as a Z 4 -symmetric map-germ, its relevant dynamic properties are shared by two 1-parameter families in its universal unfolding. The same construction can be applied to obtain examples that are also dissipative. The symmetry of these maps forces them to have rational rotation numbers.
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 compact Lie group, it is possible to describe the local dynamics. In particular, if the group contains a reflection, there is a line invariant by the map. This allows us to use results based on the theory of free homeomorphisms to describe the global dynamical behaviour. We briefly discuss the case when reflections are absent, for which global dynamics may not follow from local dynamics near the unique fixed point.
Let ǫ > 0, F : R 2 → R 2 be a differentiable (not necessarily C 1 ) map and Spec(F ) be the set of (complex) eigenvalues of the derivative DF p when p varies in R 2 .
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