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Arcwise connectedness of the set of ergodic measures of hereditary shifts


Authors: Jakub Konieczny, Michal Kupsa and Dominik Kwietniak
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 37B05; Secondary 37A35, 37B10, 37B40, 37D20
DOI: https://doi.org/10.1090/proc/14029
Published electronically: March 30, 2018
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Abstract: We show that the set of ergodic invariant measures of a shift space with a safe symbol (this includes all hereditary shifts) is arcwise connected when endowed with the $ d$-bar metric. As a consequence the set of ergodic measures of such a shift is also arcwise connected in the weak-star topology, and the entropy function over this set attains all values in the interval between zero and the topological entropy of the shift (inclusive). The latter result is motivated by a conjecture of A. Katok.


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Additional Information

Jakub Konieczny
Affiliation: Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG United Kingdom
Email: jakub.konieczny@gmail.com

Michal Kupsa
Affiliation: The Czech Academy of Sciences, Institute of Information Theory and Automation, Prague 8, CZ-18208 Czech Republic
Email: kupsa@utia.cas.cz

Dominik Kwietniak
Affiliation: Faculty of Mathematics and Computer Science, Jagiellonian University in Kraków, ul. Łojasiewicza 6, 30-348 Kraków, Poland – and – Institute of Mathematics, Federal University of Rio de Janeiro, Cidade Universitaria - Ilha do Fundão, Rio de Janeiro 21945-909, Brazil
Email: dominik.kwietniak@uj.edu.pl

DOI: https://doi.org/10.1090/proc/14029
Keywords: Kolmogorov-Sinai (metric) entropy, hereditary shift space, Poulsen simplex, Besicovitch pseudometric, $d$-bar metric
Received by editor(s): October 13, 2016
Received by editor(s) in revised form: October 18, 2017, and November 19, 2017
Published electronically: March 30, 2018
Additional Notes: The first author was supported by the National Science Centre (NCN) under grant 2012/07/A/ST1/00185
The third author was supported by the National Science Centre (NCN) grant 2013/08/A/ST1/00275 and partially supported by CAPES/Brazil grant no. 88881.064927/2014-01.
Communicated by: Nimish Shah
Article copyright: © Copyright 2018 American Mathematical Society

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