Limit sets, internal chain transitivity and orbital shadowing of tree-shifts defined on Markov-Cayley trees

TL;DR

This paper extends the theory of tree-shifts on Markov-Cayley trees by defining omega-limit sets, pseudo orbits, and shadowing properties, revealing new relationships and concepts in symbolic dynamics on complex tree structures.

Contribution

It introduces omega-limit sets and pseudo orbits for tree-shifts on Markov-Cayley trees, expanding previous results and establishing new shadowing properties and relationships.

Findings

Established relationships between omega-limit sets.

Introduced a modified omega-limit set based on complete prefix sets.

Investigated the shadowing property for projected pseudo orbits.

Abstract

In this paper, we introduce the concepts of $\omega$-limit sets and pseudo orbits for a tree-shift defined on a Markov-Cayley tree, extending the results of tree-shifts defined on $d$-trees [5,6]. Firstly, we establish the relationships between $\omega$-limit sets and we introduce a modified definition of $\omega$-limit set based on complete prefix sets (Theorems 1.4 and 1.9). Secondly, we introduce the concept of projected pseudo orbits and investigate the concept of the shadowing property (Theorems 1.12 and 1.14).