Subshifts on an infinite alphabet

Abstract

We study transfer operators over general subshifts of sequences of an infinite alphabet. We introduce a family of Banach spaces of functions satisfying a regularity condition and a decreasing condition. Under some assumptions on the transfer operator, we prove its continuity and quasi-compactness on these spaces. Under additional assumptions—existence of a conformal measure and topological mixing—we prove that its peripheral spectrum is reduced to one and that this eigenvalue is simple. We describe the consequences of these results in terms of existence and properties of invariant measures absolutely continuous with respect to the conformal measure. We also give some examples of contexts in which this setting can be used—expansive maps of the interval, statistical mechanics.