On the stability and shadowing of tree-shifts of finite type (Q6566399)

Let \(\Sigma\) be a nonempty finite set of alphabet and denote by \(\Sigma^* = \bigcup_{n \geq 0} \Sigma^n\) the set of all words over \(\Sigma\). For an alphabet \(\mathcal{A}\), a labeled tree on \(\Sigma\) over \(\mathcal{A}\) is a function \(t: \Sigma^* \rightarrow \mathcal{A}\). Denote by \(\mathcal{T}\) the space of all labeled trees on \(\Sigma\) over \(\mathcal{A}\). For \(i \in \Sigma\), the shift map \(\sigma^i\) on \(\mathcal{T}\) is defined by \(\sigma^i(t)_w = t_{iw} \hbox{ for all } t \in \mathcal{T}\). A subset \(X \subset \mathcal{T}\) is called a tree-shift if \(X\) is closed, and \(\sigma^i(X) \subset X\) for all \(i \in \Sigma\).\N\NThe author first shows that a tree-shift \(X\) is of finite type if and only if it has the pseudo-orbit-tracing property. Moreover, every tree-shift \(X\) with the pseudo-orbit-tracing property is topologically stable, and \(\sigma^i\) is open for all \(i \in \Sigma\). Finally, the author gives an example of a tree-shift for which all shift maps are open but \(X\) does not have the pseudo-orbit-tracing property.