Generalized topological transition matrix. (English) Zbl 1362.37038

Franzosa, Robert; de Rezende, Ketty A.; Vieira, Ewerton R.

Topol. Methods Nonlinear Anal. 48, No. 1, 183-212 (2016).

Summary: This article represents a major step in the unification of the theory of algebraic, topological and singular transition matrices by introducing a definition which is a generalization that encompasses all of the previous three. When this more general transition matrix satisfies the additional requirement that it covers flow-defined Conley-index isomorphisms, one proves algebraic and connection-existence properties. These general transition matrices with this covering property are referred to as generalized topological transition matrices and are used to consider connecting orbits of Morse-Smale flows without periodic orbits, as well as those in a continuation associated to a dynamical spectral sequence.

MSC:

37B30	Index theory for dynamical systems, Morse-Conley indices
37D15	Morse-Smale systems
70K70	Systems with slow and fast motions for nonlinear problems in mechanics
70K50	Bifurcations and instability for nonlinear problems in mechanics

Keywords:

Conley index; connection matrices; transition matrices; Morse-Smale system; sweeping method; spectral sequence