Singularity and stratification theory applied to dynamical systems

Abstract

Transversality theory is a basic technical and theoretical tool in the study of smooth mappings, dynamical systems and bifurcation theory. In this paper we describe a version of transversality theory applicable to the study of maps and vector fields which are equivaxiant under the smooth action of a compact Lie group G. From a local point of view, we will be outlining a theory for the analysis of solutions of symmetric equations. From the global point of view, we will be describing an intersection theory for G-manifolds. The foundational theory of transversality for G-manifolds was developed in the mid 1970’s by Bierstone ’ and the author lo. The focus of Bierstone’s work was on extending Mather’s theory of stable mappings to smooth equivariant maps. As part of that program, Bierstone extended Thom’s jet transversality theorem to equivariant maps ’. On the other hand, Field’s motivation was to extend parts of the Smale program to equivariant dynamical systems ll. Much later it turned out that techniques of equivariant transversality had powerful applications to equivariant bifurcation theory 16712-14. Very recently there have also been applications to equivariant reversible systems * and equivariant Hamiltonian systems 6 . In this paper, we emphasize applications of equivariant transversality to the