On Li-Yorke measurable sensitivity
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- by Jared Hallett, Lucas Manuelli and Cesar E. Silva PDF
- Proc. Amer. Math. Soc. 143 (2015), 2411-2426 Request permission
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.References
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Additional Information
- Jared Hallett
- Affiliation: Department of Mathematics, Williams College, Williamstown, Massachusetts 01267
- MR Author ID: 999196
- 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
- MR Author ID: 251612
- Email: csilva@williams.edu
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 2411-2426
- MSC (2010): Primary 37A40; Secondary 37A05
- DOI: https://doi.org/10.1090/S0002-9939-2015-12430-6
- MathSciNet review: 3326024