Spectral multiplicity for powers of weakly mixing automorphisms

TL;DR

This paper investigates the spectral multiplicities of powers of weakly mixing automorphisms, revealing complex behaviors such as unbounded multiplicity sets and specific spectral properties for constructed examples.

Contribution

It constructs explicit weakly mixing automorphisms with prescribed spectral multiplicity behaviors and analyzes their spectral properties in detail.

Findings

Existence of weakly mixing automorphisms with multiplicity sequences like n and 1.

Unbounded cardinality of spectral multiplicities for powers of automorphisms.

Construction of automorphisms with specific spectral and multiplicity properties.

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 $\{\frac{mm (R^n)}{n} : n\in \N \}$ is infinite.