Group Extended Markov Systems, Amenability, and the Perron-Frobenius Operator

TL;DR

This paper links the amenability of countable groups to the spectral properties of the Perron-Frobenius operator in group-extended Markov systems, extending previous results to a broader dynamical context.

Contribution

It characterizes group amenability via the spectral radius of the Perron-Frobenius operator for group extensions of Markov shifts, generalizing earlier work on random walks and dynamical systems.

Findings

Amenability characterized by spectral radius of Perron-Frobenius operator.

Logarithm of spectral radius equals Gurevic pressure under symmetry.

Extension of Day's and Stadlbauer's results to broader dynamical systems.

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 H\"older 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 Gurevic pressure of the potential.