Let Gbar = G{nt, nt | nt+1, t ≥ 0} be a subgroup of all roots of unity generated by exp(2πi/nt}, t ≥ 0, and let τ: (X, β, μ) O be an ergodic transformation with pure point spectrum Gbar. Given a cocycle φ, φ: X → Z2, admitting an approximation with speed 0(1/n1+ε, ε>0) there exists a Morse cocycle φ such that the corresponding transformations τφ and τψ are relatively isomorphic. An effective way of a construction of the Morse cocycle φ is given. There is a cocycle φ oddly approximated with an arbitrarily high speed and without roots.
This note delivers examples of φ's admitting an arbitrarily high speed of approximation and such that the power multiplicity function of τφ is equal to one and the power rank function is oscillatory. Finally, we also prove that if φ is a Morse cocycle then each proper factor of τφ is rigid. In particular continuous substitutions on two symbols cannot be factors of Morse dynamical systems.
@article{urn:eudml:doc:41043,
title = {Aproximation of Z2-cocycles and shift dynamical systems.},
journal = {Publicacions Matem\`atiques},
volume = {32},
year = {1988},
pages = {91-110},
mrnumber = {MR0939773},
zbl = {0677.58018},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41043}
}
Filipowicz, I.; Kwiatkowski, J.; Lemanczyk, M. Aproximation of Z2-cocycles and shift dynamical systems.. Publicacions Matemàtiques, Tome 32 (1988) pp. 91-110. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41043/