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Group-extended Markov systems, amenability, and the Perron-Frobenius operator


Author: Johannes Jaerisch
Journal: Proc. Amer. Math. Soc. 143 (2015), 289-300
MSC (2010): Primary 37D35, 43A07, 37C30; Secondary 37C85
Published electronically: August 25, 2014
MathSciNet review: 3272754
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Abstract: We characterise amenability of a countable group in terms of the spectral radius of the Perron-Frobenius operator associated to a group extension of a countable Markov shift and a Hölder continuous potential. This extends a result of Day for random walks and recent work of Stadlbauer for dynamical systems. Moreover, we show that if the potential satisfies a symmetry condition with respect to the group extension, then the logarithm of the spectral radius of the Perron-Frobenius operator is given by the Gurevič pressure of the potential.


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Additional Information

Johannes Jaerisch
Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, 1-1 Machikaneyama Toyonaka, Osaka, 560-0043 Japan
Email: jaerisch@cr.math.sci.osaka-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-2014-12237-4
Keywords: Thermodynamic formalism, group extension, amenability, Perron-Frobenius operator, Gurevi\v{c} pressure
Received by editor(s): April 1, 2013
Published electronically: August 25, 2014
Additional Notes: The author was supported by the research fellowship JA 2145/1-1 of the German Research Foundation (DFG)
Communicated by: Nimish Shah
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.