Generalizations of a Result on Symmetry Groups of Attractors

Pattern Formation: Symmetry Methods and Applications (J. Chadam et al. eds.) Fields Institute Communications 5, Amer. Math. Soc., Providence, RI, 1996

Ian Melbourne

Abstract

The admissible symmetry groups of attractors for continuous equivariant mappings were classified in Ashwin and Melbourne and Melbourne, Dellnitz and Golubitsky . We consider extensions of these results to include attractors in fixed-point subspaces, attractors for equivariant diffeomorphisms and flows [1], and attractors in the presence of a continuous symmetry group [2]. Our results lead to surprising (if somewhat speculative) implications for both theory and applications [3] of equivariant dynamical systems.

[1] The results on attractors for diffeomorphisms and flows are very preliminary and are completed in Field, Melbourne and Nicol

[2] The results on attractors in the presence of a continuous symmetry group are made more precise and extended in Melbourne and Stewart (see also Dellnitz and Melbourne ).

[3] For more on the implications for applications, see Melbourne