Relatively finite measure-preserving extensions and lifting multipliers by Rokhlin cocycles

Lifting mixing properties by Rokhlin cocycles

We study the problem of lifting various mixing properties from a base automorphism $T\in {\rm Aut}\xbm$ to skew products of the form $\tfs$, where $\va:X\to G$ is a cocycle with values in a locally compact Abelian group $G$, $\cs=(S_g)_{g\in G}$ is a measurable representation of $G$ in ${\rm Aut}\ycn$ and $\tfs$ acts on the product space $(X\times Y,\cb\ot\cc,μ\otν)$ by $$\tfs(x,y)=(Tx,S_{\va(x)}(y)).$$ It is also shown that whenever $T$ is ergodic (mildly mixing, mixing) but $\tfs$ is not ergodic (is not mildly mixing, not mixing), then on a non-trivial factor $\ca\subset\cc$ of $\cs$ the cor…

The classification of Rokhlin flows on C*-algebras

We study flows on C*-algebras with the Rokhlin property. We show that every Kirchberg algebra carries a unique Rokhlin flow up to cocycle conjugacy, which confirms a long-standing conjecture of Kishimoto. We moreover present a classification theory for Rokhlin flows on C*-algebras satisfying certain technical properties, which hold for many C*-algebras covered by the Elliott program. As a consequence, we obtain the following further classification theorems for Rokhlin flows. Firstly, we extend the statement of Kishimoto's conjecture to the non-simple case: Up to cocycle conjugacy, a Rokhlin fl…

Correlation of sequences and of measures, generic points for joinings and ergodicity of certain cocycles

The main subject of the paper, motivated by a question raised by Boshernitzan, is to give criteria for a bounded complex-valued sequence to be uncorrelated to any strictly ergodic sequence. As a tool developed to study this problem we introduce the notion of correlation between two shift-invariant measures supported by the symbolic space with complex symbols. We also prove a "lifting lemma" for generic points: given a joining $ξ$ of two shift-invariant measures $μ$ and $ν$, every point $x$ generic for $μ$ lifts to a pair $(x,y)$ generic for $ξ$ (such $y$ exists in the full symbolic space). Thi…

确定性$[T,T^{-1}]$系统的同构与慢熵

Isomorphisms and slow entropy of deterministic $[T,T^{-1}]$ systems

本文研究确定性$[T,T^{-1}]$系统的同构刚性，证明斜积同构条件，并构造任意低慢熵及违反慢熵变分原理的新例子。