A Zn-symmetric local but not global attractor

Details A Zn-symmetric local but not global attractor A Zn-symmetric local but not global attractor

2012-01-23

arxiv.org/abs/1201.4818v1

Mathematics

Dynamical Systems

Scientific paper

There are many tools for studying local dynamics. An important problem is how this information can be used to obtain global information. For instance, if a map has a unique ?xed point which is a local attractor, when can one guarantee that it is a global attractor? One attempt at getting tools for global dynamics was made through the Discrete Markus-Yamabe conjecture. A counter-example (Szlenk's example) to this conjecture in dimension 2 was presented in [A. Cima, A. Gasull and F. Ma\~nosas, The Discrete Markus-Yamabe Problem Nonlinear Analysis, 35, 343-354, 1999]. In the present article we show that Szlenk's example has symmetry Z4. Based on this example we construct, for any natural n ? 3, planar maps whose symmetry group is Zn having a local attractor that is not a global attractor. 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, leading to the new question of whether Zn-symmetry implies rational rotation number.

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