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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Varieties of mixing
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by Ethan Akin and Jim Wiseman PDF
Trans. Amer. Math. Soc. 372 (2019), 4359-4390 Request permission

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.
References
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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
  • MR Author ID: 24025
  • Email: ethanakin@earthlink.net
  • Jim Wiseman
  • Affiliation: Department of Mathematics, Agnes Scott College, 141 East College Avenue, Decatur, Georgia 30030
  • MR Author ID: 668909
  • Email: jwiseman@agnesscott.edu
  • 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)
  • © Copyright 2018 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 372 (2019), 4359-4390
  • MSC (2010): Primary 37B20; Secondary 37B05, 37B35
  • DOI: https://doi.org/10.1090/tran/7681
  • MathSciNet review: 4009391