Approximation of $Z_2$-cocycles and shift dynamical systems

Abstract

Let Gbar = G{nt, nt | nt+1, t = 0} be a subgroup of all roots of unity generated by exp(2pi/nt}, t = 0, and let t: (X, s, µ) O be an ergodic transformation with pure point spectrum Gbar. Given a cocycle f, f: X ? Z2, admitting an approximation with speed 0(1/n1+e, e>0) there exists a Morse cocycle f such that the corresponding transformations tf and t? are relatively isomorphic. An effective way of a construction of the Morse cocycle f is given. There is a cocycle f oddly approximated with an arbitrarily high speed and without roots. This note delivers examples of f's admitting an arbitrarily high speed of approximation and such that the power multiplicity function of tf is equal to one and the power rank function is oscillatory. Finally, we also prove that if f is a Morse cocycle then each proper factor of tf is rigid. In particular continuous substitutions on two symbols cannot be factors of Morse dynamical systems.