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Free ergodic $ \mathbb{Z}^2$-systems and complexity


Authors: Van Cyr and Bryna Kra
Journal: Proc. Amer. Math. Soc. 145 (2017), 1163-1173
MSC (2010): Primary 28D05, 37A25, 37A35
DOI: https://doi.org/10.1090/proc/13279
Published electronically: September 15, 2016
MathSciNet review: 3589316
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Abstract: Using results relating the complexity of a two dimensional subshift to its periodicity, we obtain an application to the well-known conjecture of Furstenberg on a Borel probability measure on $ [0,1)$ which is invariant under both $ x\mapsto px \pmod 1$ and $ x\mapsto qx \pmod 1$, showing that any potential counterexample has a nontrivial lower bound on its complexity.


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Additional Information

Van Cyr
Affiliation: Department of Mathematics, Bucknell University, Lewisburg, Pennsylvania 17837
Email: van.cyr@bucknell.edu

Bryna Kra
Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208
Email: kra@math.northwestern.edu

DOI: https://doi.org/10.1090/proc/13279
Received by editor(s): January 25, 2016
Received by editor(s) in revised form: May 5, 2016
Published electronically: September 15, 2016
Additional Notes: The second author was partially supported by NSF grant 1500670.
Communicated by: Nimish Shah
Article copyright: © Copyright 2016 American Mathematical Society