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Rigidity of joinings for some measure-preserving systems

Part of: Ergodic theory

Published online by Cambridge University Press:  30 April 2021

CHANGGUANG DONG ,
ADAM KANIGOWSKI  and
DAREN WEI
CHANGGUANG DONG
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD20742, USA (e-mail: dongchg@math.umd.edu, adkanigowski@gmail.com)
ADAM KANIGOWSKI
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD20742, USA (e-mail: dongchg@math.umd.edu, adkanigowski@gmail.com)
DAREN WEI*
Affiliation:
Einstein Institute of Mathematics, Edmond J. Safra Campus, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, 9190401, Israel
*
e-mail: Daren.Wei@mail.huji.ac.il
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Abstract

We introduce two properties: strong R-property and $C(q)$ -property, describing a special way of divergence of nearby trajectories for an abstract measure-preserving system. We show that systems satisfying the strong R-property are disjoint (in the sense of Furstenberg) with systems satisfying the $C(q)$ -property. Moreover, we show that if $u_t$ is a unipotent flow on $G/\Gamma $ with $\Gamma $ irreducible, then $u_t$ satisfies the $C(q)$ -property provided that $u_t$ is not of the form $h_t\times \operatorname {id}$ , where $h_t$ is the classical horocycle flow. Finally, we show that the strong R-property holds for all (smooth) time changes of horocycle flows and non-trivial time changes of bounded-type Heisenberg nilflows.

Keywords

rigidity of joiningstime changeshorocycle flowunipotent flowsHeisenberg flows

MSC classification

Primary: 37A35: Entropy and other invariants, isomorphism, classification
Secondary: 37A10: One-parameter continuous families of measure-preserving transformations
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 2: Anatole Katok Memorial Issue Part 1: Special Issue of Ergodic Theory and Dynamical Systems , February 2022 , pp. 665 - 690
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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