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A characterization of $ \mu-$equicontinuity for topological dynamical systems


Author: Felipe García-Ramos
Journal: Proc. Amer. Math. Soc. 145 (2017), 3357-3368
MSC (2010): Primary 37B05, 37A50, 54H20, 28A75
DOI: https://doi.org/10.1090/proc/13404
Published electronically: April 26, 2017
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Abstract: Two different notions of measure theoretical equicontinuity
($ \mu -$equicontinuity) for topological dynamical systems with respect to Borel probability measures appeared in works by Gilman (1987) and Huang, Lee and Ye (2011). We show that if the probability space satisfies Lebesgue's density theorem and Vitali's covering theorem (for example a Cantor set or a subset of $ \mathbb{R}^{d}$), then both notions are equivalent. To show this we characterize Lusin measurable maps using $ \mu -$continuity points. As a corollary we also obtain a new characterization of $ \mu -$mean equicontinuity.


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Additional Information

Felipe García-Ramos
Affiliation: Catedras Conacyt, Av. Insurgentes Sur 1582, Benito Juarez CDMX, 03940, Mexico – and – Instituto de Física, Universidad Autónoma de San Luis Potosí, Av. Manuel Nava #6, Zona Universitaria San Luis Potosí, SLP, 78290, México
Email: felipegra@yahoo.com

DOI: https://doi.org/10.1090/proc/13404
Received by editor(s): July 31, 2014
Received by editor(s) in revised form: August 17, 2015, May 13, 2016, and July 28, 2016
Published electronically: April 26, 2017
Additional Notes: While writing and correcting this paper the author was supported by IMPA, CAPES and NSERC. The author would like to thank Brian Marcus for his support and comments.
Communicated by: Nimish Shah
Article copyright: © Copyright 2017 American Mathematical Society

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