Morse Predecomposition of an Invariant Set.

Area of Science:

• Dynamical Systems Theory
• Topology and Geometry in Dynamics

Background:

• Morse decomposition is a standard tool for analyzing the gradient dynamics of isolated invariant sets.
• Existing methods struggle to fully capture the recurrent dynamics within these sets, particularly chain recurrent sets.
• A need exists for a more comprehensive framework to understand the full dynamics of isolated invariant sets.

Purpose of the Study:

• To introduce and define the concept of Morse predecomposition for isolated invariant sets.
• To generalize Morse decomposition by incorporating recurrent dynamics beyond purely gradient behavior.
• To provide a framework capable of finer analysis for chain recurrent sets.

Main Methods:

• Developed a new framework, Morse predecomposition, by relaxing the poset structure of Morse decomposition.
• Introduced the concept of 'links' to represent connections between set elements, replacing strict ordering.
• Utilized both combinatorial and classical dynamical systems settings for the generalization.

Main Results:

• Morse predecomposition successfully extends analysis to include recurrent components of dynamical systems.
• Proved that standard Morse decomposition is a special case of Morse predecomposition.
• Demonstrated that Morse predecomposition can be condensed to recover a Morse decomposition.

Conclusions:

• Morse predecomposition offers a more refined and comprehensive analysis of isolated invariant sets in dynamical systems.
• This generalization enhances the study of recurrent orbits and complex dynamics.
• The framework provides a valuable tool for understanding systems not fully described by gradient dynamics alone.