Invariant Complete Transversals and the classification of symmetric bifurcation problems
by
Sofia Castro
Faculdade de Economia do Porto and Centro de Matematica Aplicada da Universidade do Porto
Coauthors: Andrew du Plessis (Aarhus University)

Let G be a Lie group acting smoothly on an affine space A and let W be a vector subspace of V_A. A vector subspace T of W is a complete transversal if it is transversal to the orbit of x₀ in A and meets each orbit through the affine space x₀+W of A.

Complete transversals were used by Bruce, Kirk and du Plessis (Complete transversals and the classification of singularities, Nonlinearity 10, 253-275 (1997)), together with some properties of unipotent algebraic groups (see Bruce, du Plessis and Wall, Determinacy and unipotency, Invent. Math. 88, 521-554 (1987)), to classify singularities of map-germs with respect to a range of equivalence relations.

We show that the concept of complete transversal can be used to classify bifurcation problems, with or without symmetry. In many cases a complete transversal can be chosen, which is invariant under a large subgroup of the group of equivalences. This allows the recognition problem to be solved systematically. We illustrate this with well-known and new examples.

Date received: May 7, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # calh-17.