Work

Morse Predecomposition of an Invariant Set

Year: 2024

Type: article

Abstract: Motivated by the study of recurrent orbits and dynamics within a Morse set of a Morse decomposition we introduce the concept of Morse predecomposition of an isolated invariant set within the setting of both combinatorial and classical dynamical systems. While Morse decomposition summarizes solely the gradient part of a dynamical system, the developed generalization extends to the recurrent component as well. In particular, a chain recurrent set, which is indecomposable in terms of Morse decomposition, can be represented more finely in the Morse predecomposition framework. This generalization is achieved by forgoing the poset structure inherent to Morse decomposition and relaxing the notion of connection between Morse sets (elements of Morse decomposition) in favor of what we term 'links'. We prove that a Morse decomposition is a special case of Morse predecomposition indexed by a poset. Additionally, we show how a Morse predecomposition may be condensed back to retrieve a Morse decomposition.

Source: Qualitative Theory of Dynamical Systems

DOI: 10.1007/s12346-024-01144-3

Authors: Michał Lipiński, Konstantin Mischaikow, Marian Mrozek

Institutions: Institute of Science and Technology Austria, Institute of Mathematics, Polish Academy of Sciences, Rutgers, The State University of New Jersey, Jagiellonian University

Language: English

Topic: Mathematical Dynamics and Fractals

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,190