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Published online by Cambridge University Press: 08 March 2005
We consider a real-analytic compact surface diffeomorphism $f$, for which the tangent space over the non-wandering set $\varOmega$ admits a dominated splitting. We study the dynamical determinant
$$ d_f(z)=\exp-\sum_{n\ge1}\frac{z^n}{n}\sum_{x\in\textrm{Fix}^*f^n}|\textrm{Det}(Df^n(x)-\textrm{Id})|^{-1}, $$
where $\textrm{Fix}^*f^n$ denotes the set of fixed points of $f^n$ with no zero Lyapunov exponents. By combining previous work of Pujals and Sambarino on $C^2$ surface diffeomorphisms with, on the one hand, results of Rugh on hyperbolic analytic maps and, on the other, our two-dimensional version of the same author’s analysis of one-dimensional analytic dynamics with neutral fixed points, we prove that $d_f(z)$ is either an entire function or a holomorphic function in a (possibly multiply) slit plane.
AMS 2000 Mathematics subject classification: Primary 37C30; 37D30; 37E30
