Exponential Decay of Correlations for Surface Semi-flows without Finite Markov Partitions

We extend Dolgopyat’s bounds on iterated transfer operators to suspensions of interval maps with infinitely many intervals of monotonicity. 1. Statement of results Let 0 < c1 < . . . < cm < cm+1 < . . . < 1 be a finite or countable partition of I = [0, 1] into subintervals, and let T : I → I be so that T |(cm,cm+1) is C and extends to a homeomorphism from [cm, cm+1] to I. We assume that T is piecewise uniformly expanding: there exist C ≥ 1 and ρ̂ < 1 so that |h(x)−h(y)| ≤ Cρ̂|x−y| for every inverse branch h of T n and all n. Let H be the set of inverse branches h : I → [cm, cm+1] of T . We suppose (Renyi’s condition) that there is a K > 0 so that every h ∈ H satisfies |h′′| ≤ K|h′|. Let r : I → R+ be so that r|(cm,cm+1) is C, and inf r > 0. Assume that there is σ0 < 0 so that ∑ h∈H sup exp(−σ(r◦h))|h′| < ∞ for all σ > σ0, and that |r′ ◦ h| · |h′| ≤ K for all h ∈ H. For n ≥ 1, write r(x) = ∑n−1 k=0 r(T )(x). We study the transfer operators, indexed by s = σ + it,