The Szymczak Functor and Shift Equivalence on the Category of Finite Sets and Finite Relations

Abstract The construction of the Conley index for dynamical systems with discrete time requires an equivalence relation between morphisms induced on index pairs. It follows from the features of the Szymczak functor that shift equivalence, whose equivalence classes are the isomorphism classes in the...

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Bibliographic Details
Published inJournal of dynamics and differential equations
Main Authors Przybylski, Mateusz, Mrozek, Marian, Wiseman, Jim
Format Journal Article
LanguageEnglish
Published 09.12.2023
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Summary:Abstract The construction of the Conley index for dynamical systems with discrete time requires an equivalence relation between morphisms induced on index pairs. It follows from the features of the Szymczak functor that shift equivalence, whose equivalence classes are the isomorphism classes in the Szymczak category, is the most general equivalence available. In the case of dynamics modeled from data, the morphisms induced on index pairs are relations. We present an algorithmizable classification of shift equivalence classes for the category of finite sets with arbitrary relations as morphisms. The research is the first step towards the construction of a Conley theory for relations.
ISSN:1040-7294
1572-9222
DOI:10.1007/s10884-023-10332-9