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Thermodynamic formalism for Haar systems in noncommutative integration: transverse functions and entropy of transverse measures

Part of: Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  27 March 2020

ARTUR O. LOPES Open the ORCID record for ARTUR O. LOPES [Opens in a new window]  and
JAIRO K. MENGUE
ARTUR O. LOPES
Affiliation:
Universidade Federal do Rio Grande do Sul, Instituto de Matemática, Av. Bento Goncalves 9500, 90450-140Porto Alegre, RS, Brazil email arturoscar.lopes@gmail.com, jairokras@gmail.com
JAIRO K. MENGUE
Affiliation:
Universidade Federal do Rio Grande do Sul, Instituto de Matemática, Av. Bento Goncalves 9500, 90450-140Porto Alegre, RS, Brazil email arturoscar.lopes@gmail.com, jairokras@gmail.com
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Abstract

We consider here a certain class of groupoids obtained via an equivalence relation (the so-called subgroupoids of pair groupoids). We generalize to Haar systems in these groupoids some results related to entropy and pressure which are well known in thermodynamic formalism. We introduce a transfer operator, where the equivalence relation (which defines the groupoid) plays the role of the dynamics and the corresponding transverse function plays the role of the a priori probability. We also introduce the concept of invariant transverse probability and of entropy for an invariant transverse probability, as well as of pressure for transverse functions. Moreover, we explore the relation between quasi-invariant probabilities and transverse measures. Some of the general results presented here are not for continuous modular functions but for the more general class of measurable modular functions.

Keywords

Haar systemstransverse functionstransverse measuresentropynoncommutative integration

MSC classification

Primary: 37D35: Thermodynamic formalism, variational principles, equilibrium states
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 6 , June 2021 , pp. 1835 - 1863
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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