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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Varieties of mixing


Authors: Ethan Akin and Jim Wiseman
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 37B20; Secondary 37B05, 37B35
DOI: https://doi.org/10.1090/tran/7681
Published electronically: November 5, 2018
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Abstract: We consider extensions of the notion of topological transitivity for a dynamical system $ (X,f)$. In addition to chain transitivity, we define strong chain transitivity and vague transitivity. Associated with each, there is a notion of mixing, defined by transitivity of the product system $ (X \times X, f \times f)$. These extend the concept of weak mixing which is associated with topological transitivity. Using the barrier functions of Fathi and Pageault, we obtain for each of these extended notions a dichotomy result in which a transitive system of each type either satisfies the corresponding mixing condition or else factors into an appropriate type of equicontinuous minimal system. The classical dichotomy result for minimal systems follows when it is shown that a minimal system is weak mixing if and only if it is vague mixing.


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Additional Information

Ethan Akin
Affiliation: Department of Mathematics, The City College, 137 Street and Convent Avenue, New York, New York 10031
Email: ethanakin@earthlink.net

Jim Wiseman
Affiliation: Department of Mathematics, Agnes Scott College, 141 East College Avenue, Decatur, Georgia 30030
Email: jwiseman@agnesscott.edu

DOI: https://doi.org/10.1090/tran/7681
Keywords: Chain transitivity, chain mixing, strong chain transitivity, strong chain mixing, vague transitivity, vague mixing, barrier functions
Received by editor(s): October 13, 2017
Received by editor(s) in revised form: June 29, 2018, and August 23, 2018
Published electronically: November 5, 2018
Additional Notes: The second author was supported by a grant from the Simons Foundation (282398, JW)
Article copyright: © Copyright 2018 American Mathematical Society