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Transactions of the American Mathematical Society

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

 
 

 

Entropy and drift for Gibbs measures on geometrically finite manifolds


Authors: Ilya Gekhtman and Giulio Tiozzo
Journal: Trans. Amer. Math. Soc. 373 (2020), 2949-2980
MSC (2010): Primary 37D35, 60G50; Secondary 53D25
DOI: https://doi.org/10.1090/tran/8036
Published electronically: January 23, 2020
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Abstract: We prove a generalization of the fundamental inequality of Guivarc'h relating entropy, drift and critical exponent to Gibbs measures on geometrically finite quotients of $ CAT(-1)$ metric spaces. For random walks with finite superexponential moment, we show that the equality is achieved if and only if the Gibbs density is equivalent to the hitting measure. As a corollary, if the action is not convex cocompact, any hitting measure is singular to any Gibbs density.


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Additional Information

Ilya Gekhtman
Affiliation: University of Toronto, 40 St George Street, Toronto, Ontario M5S, Canada
Email: ilyagekh@gmail.com

Giulio Tiozzo
Affiliation: University of Toronto, 40 St George Street, Toronto, Ontario M5S, Canada
Email: tiozzo@math.utoronto.ca

DOI: https://doi.org/10.1090/tran/8036
Received by editor(s): May 20, 2019
Received by editor(s) in revised form: September 29, 2019
Published electronically: January 23, 2020
Additional Notes: The second author was partially supported by NSERC and the Alfred P. Sloan Foundation
Article copyright: © Copyright 2020 American Mathematical Society