# Topological conjugacy between induced non-autonomous set-valued systems and subshifts of finite type

## TL;DR

This paper explores the conditions under which non-autonomous set-valued systems are topologically conjugate or semiconjugate to subshifts of finite type, extending existing results and providing criteria for chaos and entropy estimation.

## Contribution

It establishes new conditions for conjugacy and semiconjugacy between non-autonomous systems and subshifts, extending autonomous system results to non-autonomous cases.

## Key findings

- Derived conditions for topological conjugacy and semiconjugacy.
- Provided entropy estimates and chaos criteria.
- Extended results from autonomous to non-autonomous systems.

## Abstract

This paper establishes topological (equi-)semiconjugacy and (equi-)conjugacy between induced non-autonomous set-valued systems and subshifts of finite type. First, some necessary and sufficient conditions are given for a non-autonomous discrete system to be topologically semiconjugate or conjugate to a subshift of finite type. Further, several sufficient conditions for it to be topologically equi-semiconjugate or equi-conjugate to a subshift of finite type are obtained. Consequently, estimations of topological entropy and several criteria of Li-Yorke chaos and distributional chaos in a sequence are derived. Second, the relationships of several related dynamical behaviors between the non-autonomous discrete system and its induced set-valued system are investigated. Based on these results, the paper furthermore establishes the topological (equi-)semiconjugacy and (equi-)conjugacy between induced set-valued systems and subshifts of finite type. Consequently, estimations of the topological entropy for the induced set-valued system are obtained, and several criteria of Li-Yorke chaos and distributional chaos in a sequence are established. Some of these results not only extend the existing related results for autonomous discrete systems to non-autonomous discrete systems, but also relax the assumptions of the counterparts in the literature. Two examples are finally provided for illustration.