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Ergodic universality of some topological dynamical systems


Authors: Anthony Quas and Terry Soo
Journal: Trans. Amer. Math. Soc. 368 (2016), 4137-4170
MSC (2010): Primary 37A35
DOI: https://doi.org/10.1090/tran/6489
Published electronically: July 10, 2015
MathSciNet review: 3453367
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Abstract: The Krieger generator theorem says that every invertible ergodic measure-preserving system with finite measure-theoretic entropy can be embedded into a full shift with strictly greater topological entropy. We extend Krieger's theorem to include toral automorphisms and, more generally, any topological dynamical system on a compact metric space that satisfies almost weak specification, asymptotic entropy expansiveness, and the small boundary property. As a corollary, one obtains a complete solution to a natural generalization of an open problem in Halmos's 1956 book regarding an isomorphism invariant that he proposed.


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Additional Information

Anthony Quas
Affiliation: Department of Mathematics and Statistics, University of Victoria, P.O. Box 3060 STN CSC, Victoria, BC V8W 3R4, Canada
Email: aquas@uvic.ca

Terry Soo
Affiliation: Department of Statistics, University of Warwick, Coventry, CV4 7AL, United Kingdom
Address at time of publication: Department of Mathematics, University of Kansas, 405 Snow Hall, 1460 Jayhawk Boulevard, Lawrence, Kansas 66045-7594
Email: t.soo@warwick.ac.uk, tsoo@ku.edu

DOI: https://doi.org/10.1090/tran/6489
Keywords: Specification, universality, toral automorphism, Burton--Rothstein
Received by editor(s): November 2, 2012
Received by editor(s) in revised form: April 9, 2014
Published electronically: July 10, 2015
Additional Notes: Both authors were funded in part by NSERC and MSRI
Article copyright: © Copyright 2015 American Mathematical Society