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On Li-Yorke measurable sensitivity


Authors: Jared Hallett, Lucas Manuelli and Cesar E. Silva
Journal: Proc. Amer. Math. Soc. 143 (2015), 2411-2426
MSC (2010): Primary 37A40; Secondary 37A05
Published electronically: February 3, 2015
MathSciNet review: 3326024
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Abstract: The notion of Li-Yorke sensitivity has been studied extensively in the case of topological dynamical systems. We introduce a measurable version of Li-Yorke sensitivity, for nonsingular (and measure-preserving) dynamical systems, and compare it with various mixing notions. It is known that in the case of nonsingular dynamical systems, a conservative ergodic Cartesian square implies double ergodicity, which in turn implies weak mixing, but the converses do not hold in general, though they are all equivalent in the finite measure-preserving case. We show that for nonsingular systems, an ergodic Cartesian square implies Li-Yorke measurable sensitivity, which in turn implies weak mixing. As a consequence we obtain that, in the finite measure-preserving case, Li-Yorke measurable sensitivity is equivalent to weak mixing. We also show that with respect to totally bounded metrics, double ergodicity implies Li-Yorke measurable sensitivity.


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Additional Information

Jared Hallett
Affiliation: Department of Mathematics, Williams College, Williamstown, Massachusetts 01267
Email: jdh4@williams.edu

Lucas Manuelli
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
Email: manuelli@mit.edu

Cesar E. Silva
Affiliation: Department of Mathematics, Williams College, Williamstown, Massachusetts 01267
Email: csilva@williams.edu

DOI: https://doi.org/10.1090/S0002-9939-2015-12430-6
Keywords: Nonsingular transformation, measure-preserving, ergodic, Li-Yorke
Received by editor(s): February 2, 2013
Received by editor(s) in revised form: August 8, 2013, and October 9, 2013
Published electronically: February 3, 2015
Additional Notes: This paper is based on research by the Ergodic Theory group of the 2011 SMALL summer research project at Williams College. Support for the project was provided by National Science Foundation REU Grant DMS - 0353634 and the Bronfman Science Center of Williams College
Communicated by: Nimish Shah
Article copyright: © Copyright 2015 American Mathematical Society