Approximation of \(Z_ 2\)-cocycles and shift dynamical systems (Q1123455)

Suppose that (X,B,\(\mu)\) is a Lebesgue measure space and let \({\mathcal K}\) denote the class of all \(Z_ 2\)-cocycles of (X,B,\(\mu)\) (then \(\phi\in {\mathcal K})\) iff \(\phi\) : \(X\to Z\) is measurable). With an appropriate metric, \({\mathcal K}\) becomes a complete, separable metric space. If T is an ergodic transformation of the Lebesgue space and \(\phi\in {\mathcal K}\), let \(T_{\phi}\) denote the mapping given by \(T(x,i):=(Tx,\phi (x)+i).\) Now the main results of the paper can be formulated as follows: Theorem 1. If \(\phi\in {\mathcal K}\) is ergodic and admits an odd (or even) approximation with speed \(O(1/n^{1+\epsilon})\), \(\epsilon >0\), then there exists a Morse cycle \(\psi\) such that \(T_{\phi}\) and \(T_{\psi}\) are relatively isomorphic. (As for the definition of the mentioned approximation, we refer to the paper of \textit{A. B. Katok} and \textit{A. M. Stepin}, Usp. Mat. Nauk 22, No.5, 81-106 (1967; Zbl 0172.072).) Theorem 2. If \(\phi\) is a Morse cocycle then each proper factor of \(T_{\phi}\) is rigid. In particular continuous substitutions [see \textit{E. Coven} and \textit{M. Keane}, Trans. Am. Math. Soc. 162 (1971), 89-102 (1972; Zbl 0222.54053)] cannot be factors of Morse dynamical systems.