Subshifts on an infinite alphabet

We study transfer operators over general subshifts of sequences of an infin ite alphabet. We introduce a family of Banach spaces of functions Satisfyin g 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 measur e and topological mixing-we prove that its peripheral spectrum is reduced t o one and that this eigenvalue:is simpler We describe the consequences of t hese results in terms of existence and properties of invariant measures abs olutely continuous with respect to the conformal measure. We also give some examples of contexts in which this setting can be used-expansive maps of t he interval, statistical mechanics.