Spectra of Self-Similar Ergodic Actions

Abstract

Self-similar constructions of transformations preserving a sigma-finite measure are considered and their properties and the spectra of the induced Gaussian and Poisson dynamical systems are studied. The orthogonal operator corresponding to such a transformation has the property that some power of this operator is a nontrivial direct sum of operators isomorphic to the original one. The following results are obtained. For any subset $M$ of the set of positive integers, in the class of Poisson suspensions, sets of spectral multiplicities of the form $M\cup\{\infty\}$ are realized. A Gaussian flow $S_t$ is presented such that the set of spectral multiplicities of the automorphisms $S_{p^{n}}$ is $\{1,\infty\}$ if $n\le 0$ and $\{p^n,\infty\}$ if $n> 0$. A Gaussian flow $T_t$ such that the automorphisms $T_{p^{n}}$ have distinct spectral types for $n\le 0$ but all automorphisms $T_{p^{n}}$, $n>0$, are pairwise isomorphic is constructed.