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

Abstract

Using a recent result of Stadlbauer for group extensions of Holder continuous potentials on a countable Markov shift, we characterise amenability in terms of the spectral radius of the Perron-Frobenius operator associated to the group extended Markov system. This generalises a result of Day for random walks on groups. Moreover, we show that if the potential has symmetry with respect to the group, then the spectral radius is given by the Gurevipressure of the potential.