Generic spectral properties of measure-preserving maps and applications

Abstract

Let $\mathcal {K}$ denote the group of all automorphisms of a finite Lebesgue space equipped with the weak topology. For $T \in \mathcal {K}$, let ${\sigma _T}$ denote its maximal spectral type. Theorem 1. There is a dense ${G_\delta }$ subset $G \subset \mathcal {K}$ such that, for each $T \in G$ and $k(1), \ldots ,k(l) \in {\mathbb {Z}^ + },k’(1), \ldots ,k’(l’) \in {\mathbb {Z}^ + }$, the convolutions \[ {\sigma _{{T^{k(1)}}}}* \cdots *{\sigma _{{T^{k(l)}}}}\quad and\quad {\sigma _{{T^{k’(1)}}}}* \cdots *{\sigma _{{T^{k’(l’)}}}}\] are mutually singular, provided that ($(k(1), \ldots ,k(l))$) is not a rearrangement of $(k’(1), \ldots ,k’(l’))$. Theorem 1 has the following consequence. Theorem 2. $\mathcal {K}$ has a dense ${G_\delta }$ subset $F \subset G$ such that for $T \in F$ the following holds: For any ${\mathbf {k}}:\mathbb {N} \to \mathbb {Z} - \{ 0\}$ and $l \in \mathbb {Z} - \{ 0\}$, the only way that ${T^l}$, or any factor thereof, can sit as a factor in ${T^{{\mathbf {k}}(1)}} \times {T^{{\mathbf {k}}(2)}} \times \cdots$ is inside the $i$th coordinate $\sigma$-algebra for some $i$ with ${\mathbf {k}}(i) = l$. Theorem 2 has applications to the construction of certain counterexamples, in particular nondisjoint automorphisms having no common factors and weakly isomorphic automorphisms that are not isomorphic.