Recurrence for semi-decompositions

TL;DR

This paper develops a unified framework for analyzing semi-group actions, group actions, filtrations, and decompositions through the introduction of semi-decompositions, highlighting differences in recurrence concepts.

Contribution

It introduces semi-decompositions as a generalization to unify the analysis of various actions and filtrations, addressing limitations of existing decomposition methods.

Findings

Semi-decompositions generalize decompositions and filtrations.

Recurrence concepts differ between group and semi-group actions.

The framework unifies analysis of actions and filtrations.

Abstract

This paper constructs a foundation to analyze semi-group actions, group actions, filtrations, and decompositions in a unified manner. In fact, though the studies of decomposition can be applied to foliated spaces and group actions, they can not be applied to semi-group actions and filtrations in general because filtration and the set of orbits of a semi-group need not be decompositions of the base spaces. To analyze these concepts in a unified manner, we introduce a concept of a semi-decomposition which is a natural generalization of these concepts because similar relations among recurrence and their variants to group actions and decompositions hold for semi-decompositions. On the other hand, we demonstrate the difference between the recurrent concepts for group actions and those even for semi-group actions.