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On the stability and shadowing of tree-shifts of finite type

Dawid Bucki

Abstract

We investigate relations between the pseudo-orbit-tracing property, topological stability and openness for tree-shifts. We prove that a tree-shift is of finite type if and only if it has the pseudo-orbit-tracing property which implies that the tree-shift is topologically stable and all shift maps are open. We also present an example of a tree-shift for which all shift maps are open but which is not of finite type. It also turns out, that if a topologically stable tree-shift does not have isolated points then it is of finite type.

On the stability and shadowing of tree-shifts of finite type

Abstract

We investigate relations between the pseudo-orbit-tracing property, topological stability and openness for tree-shifts. We prove that a tree-shift is of finite type if and only if it has the pseudo-orbit-tracing property which implies that the tree-shift is topologically stable and all shift maps are open. We also present an example of a tree-shift for which all shift maps are open but which is not of finite type. It also turns out, that if a topologically stable tree-shift does not have isolated points then it is of finite type.
Paper Structure (6 sections, 12 theorems, 57 equations)

This paper contains 6 sections, 12 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Pseudo-orbit-tracing property
  4. Topological stability of tree-shifts
  5. Open tree-shifts
  6. Conclusion

Key Result

Lemma 2.4

A tree-shift $X$ is a tree-shift of finite type if and only if there is some $n>0$, such that for $t\in\mathcal{T}$, we have that $t\in X$ if and only if $t|_{w\Sigma^{<n}}\in\mathcal{B}_n(X)$ for every $w\in\Sigma^*$.

Theorems & Definitions (32)

  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Definition 3.3
  • Lemma 3.4
  • proof
  • ...and 22 more