The Index Bundle and Multiparameter Bifurcation for Discrete Dynamical Systems

TL;DR

This paper introduces a K-theoretic framework for analyzing multiparameter bifurcations in discrete dynamical systems, providing new theoretical tools and relaxing previous assumptions for bifurcation analysis.

Contribution

It develops a novel K-theoretic approach to multiparameter bifurcation theory and establishes a family index theorem for asymptotically hyperbolic systems, improving upon prior results.

Findings

Established a new K-theoretic bifurcation framework.

Proved a family index theorem for hyperbolic systems.

Weakened assumptions for single-parameter bifurcation theorems.

Abstract

We develop a $K$-theoretic approach to multiparameter bifurcation theory of homoclinic solutions of discrete non-autonomous dynamical systems from a branch of stationary solutions. As a byproduct we obtain a family index theorem for asymptotically hyperbolic linear dynamical systems which is of independent interest. In the special case of a single parameter, our bifurcation theorem weakens the assumptions in previous work by Pejsachowicz and the first author.