A ug 2 00 3 Dynamical zeta functions for analytic surface diffeomorphisms with dominated splitting

We consider a real-analytic compact surface diffeomorphism f , for which the tangent space over the nonwandering set Ω admits a dominated splitting. We study the dynamical determinant df (z) = exp− ∑ n≥1 z n ∑ x∈Fix ∗fn |Det (Df (x)− Id )|, where Fix fn denotes the set of fixed points of f with no zero Lyapunov exponents. By combining previous work of Pujals and Sambarino [PS1, PS2] on C surface diffeomorphisms with, on the one hand, results of Rugh [Ru1] on hyperbolic analytic maps, and on the other, our two-dimensional version of the same author’s [Ru3] analysis of one-dimensional analytic dynamics with neutral fixed points, we prove that df (z) is either an entire function or a holomorphic function in a (possibly multiply) slit plane.