On Relative Category and Morse Decompositions for Infinite-Dimensional Dynamical Systems

TL;DR

This paper introduces a method using relative category to connect Ważewski pairs and Morse decompositions in infinite-dimensional dynamical systems, enabling detection of connecting trajectories and a critical point theorem adaptation.

Contribution

It develops a new relation between Ważewski pairs and Morse decompositions using relative category, advancing analysis tools for infinite-dimensional dynamical systems.

Findings

Established a relation between Ważewski pairs and Morse decompositions.

Provided a method to detect connecting trajectories between Morse sets.

Formulated a dynamical-system version of the critical point theorem.

Abstract

We employ the relative category to develop relations between the Wa\.zewski pair $(N,E)$ and the Morse decomposition of the maximal invariant set in $\ol{N\sm E}$ for infinite-dimensional dynamical systems. Via these relations, we can detect connecting trajectories between Morse sets and obtain a dynamical-system version of critical point theorem with relative category.