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

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 Gurevic pressure of the potential.

References in corpus (2)

• An extension of Kesten's criterion for amenability to topological Markov chains
• Fractal Models for Normal Subgroups of Schottky Groups

Cited by in corpus (1)

• Conformal Fractals for Normal Subgroups of Free Groups