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On conservative sequences and their application to ergodic multiplier problems

Volume 151 / 2018

Madeleine Elyze, Alexander Kastner, Juan Ortiz Rhoton, Vadim Semenov, Cesar E. Silva Colloquium Mathematicum 151 (2018), 123-145 MSC: Primary 37A40; Secondary 37A05, 37A50. DOI: 10.4064/cm7127-6-2017 Published online: 24 November 2017

Abstract

The conservative sequence of a set $A$ under a transformation $T$ is the set of all $n \in \mathbb {Z}$ such that $T^n A \cap A \not = \emptyset $. By studying these sequences, we prove that given any countable collection of nonsingular transformations with no finite invariant measure $\{T_i\}$, there exists a rank-one transformation $S$ such that $T_i \times S$ is not ergodic for any $i$. Moreover, $S$ can be chosen to be rigid or have infinite ergodic index. We establish similar results for $\mathbb {Z}^d$-actions and flows. Then, we find sufficient conditions on rank-one transformations $T$ that guarantee the existence of a rank-one transformation $S$ such that $T \times S$ is ergodic, or alternatively, conditions that guarantee that $T \times S$ is conservative but not ergodic. In particular, the infinite Chacón transformation satisfies both conditions. Finally, for a given ergodic transformation $T$, we study the Baire categories of the sets $E(T)$, $\bar{E}C(T)$ and $\bar{C}(T)$ of transformations $S$ such that $T \times S$ is ergodic, conservative but not ergodic, and not conservative, respectively.

Authors

  • Madeleine ElyzeDepartment of Mathematics and Statistics
    Williams College
    Williamstown, MA 01267, U.S.A.
    e-mail
  • Alexander KastnerDepartment of Mathematics and Statistics
    Williams College
    Williamstown, MA 01267, U.S.A.
    e-mail
  • Juan Ortiz RhotonMIT
    Cambridge, MA 02139, U.S.A.
    e-mail
  • Vadim SemenovCourant Institute
    New York University
    New York, NY 10012, U.S.A.
    e-mail
  • Cesar E. SilvaDepartment of Mathematics and Statistics
    Williams College
    Williamstown, MA 01267, U.S.A.
    e-mail

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