A result in global bifurcation theory using the Conley index

Abstract

We study a saddle-node bifurcation in a Lipschitz family of diffeomorphisms on a manifold, in the case that the stable set and unstable set of the fixed point intersect transversally in a countable collection of one-dimensional manifolds diffeomorphic to circles. We formulate generic conditions on the circles stated in terms of standard coordinates, a recently defined tool for the study of saddle-node bifurcations. Under the conditions, it is shown that there is a decreasing sequence of intervals $[\underline{\mu_j},\overline{\mu_j}]$ of parameter values for which the diffeomorphism is semi-conjugated to shift dynamics on the space of binary sequences. The semi-conjugacy is implied by a recent result in the Conley index theory.