Spectral multiplicity for powers of weakly mixing automorphisms

Abstract

We study the behavior of maximal multiplicities $mm(R^n)$ for the powers of a weakly mixing automorphism $R$. For some special infinite set $A$ we show the existence of a weakly mixing rank-one automorphism $R$ such that $mm(R^n)=n$ and $mm(R^{n+1})=1$ for all $n\in A$. Moreover, the cardinality $cardm(R^n)$ of the set of spectral multiplicities for $R^n$ is not bounded. We have $cardm(R^{n+1})=1$ and $cardm(R^n)=2^{m(n)}$, $m(n)\to \infty$, $n\in A$. We also construct another weakly mixing automorphism $R$ with the following properties: $mm(R^n)=n$ for $n=1,2,3,...,2009,2010$ but $mm(T^{2011})=1$, all powers $(R^n)$ have homogeneous spectrum, and the set of limit points of the sequence $\{\frac{mm(R^n)}{n}:n\in \mathbb{N}\}$ is infinite.

Cited by in corpus (1)

• On spectral disjointness of powers for rank-one transformations and Möbius orthogonality