Conjugacy of symbolic dynamical systems with at most countable phase space (Q1280376)

The author treats symbolic dynamical systems \((M,T)\) over the finite alphabet \(A\) (e.g. \(A=\{0,1\}\)). Besides, \(T(\{x_i\}^{+\infty}_{i=-\infty}) = \{y_i\}^{+\infty}_{i=-\infty}\), where \(y_i=x_{i-1}\) for any integer \(i\), \(M\) is not more than a countable closed set invariant with respect to the shift. For the dynamical system \((M,T)\) it is assumed that \[ M^{(0)}:=M;\quad M^{(1)}:=M';\quad M^{(2)}:=M^{(1)'};\quad\dots;\quad M^{(n)}:=M^{(n-1)'} \] and \[ M^{(\alpha)}:=\Big(\bigcap\limits_{\beta<\alpha}M^{(\beta)}\Big)'. \] The minimal ordinal number \(\alpha\) with the property \(M^{(\alpha)}=0\) is called the depth of the set \(M\) (of the dynamical system \((M,T)\)) \[ \text{depth} (M):=\min \{\alpha\: M^{(\alpha)}=0\}. \] Basic results of the paper are: 1. For any nonlimit order number \(\alpha\) not larger than the second class a dynamical system \((M,T)\) is found such that \(\text{depth} (M)=\alpha\). 2. There are no dynamical systems whose depth is in the limit the transfinite number.