Transition Matrix Theory

TL;DR

This paper introduces a unified generalized transition matrix theory that encompasses algebraic, singular, topological, and directional transition matrices, extending their applications to Conley Index theory.

Contribution

The paper develops a generalized transition matrix framework unifying previous types and demonstrates its applicability to Conley Index theory.

Findings

Unified transition matrix encompasses four previous types.

Existence results for generalized transition matrices.

Applications to Conley Index theory are extended.

Abstract

In this article we present a unification of the theory of algebraic, singular, topological and directional transition matrices by introducing the (generalized) transition matrix which encompasses each of the previous four. Some transition matrix existence results are presented as well as verification that each of the previous transition matrices are cases of the generalized transition matrix. Furthermore we address how applications of the previous transition matrices to the Conley Index theory carry over to the generalized transition matrix.