Spectra of Self-Similar Ergodic Actions

Abstract

Answering Vershik's question we show that quasi-similarity does not conserve the entropy, proving quasisimilarity of all Bernoulli actions of the groups with an element of infinite order. We produce joining-stable invariants considering zero P -entropy systems and prove an analog of Pinsker's theorem establishing the disjointness of such systems with the actions of comletely positive P -entropy. Then we apply Poisson suspensions to get a class of the corresponding examples.