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Meromorphic extensions of generalised zeta functions. (English) Zbl 0604.58042
The author gives a full description of the spectrum of the Ruelle-Perron- Frobenius operator acting on the Banach space of Hölder continuous functions on a subshift of finite type. Particularly, the author extends the meromorphic domain of the zeta-function for a flow under a positive Hölder continuous function suspended over a shift of finite type. Then they are translated to the Axiom A case via the symbolic dynamics of Bowen and a combinatorial result of Manning. Finally, certain distribution results for orbit period, called prime orbit theorems, are proven by analogy with the prime number theorem. Here the zeta-function for an Axiom A flow assumes a similar role to the Riemann zeta-function in the analytical proof of the prime number theorem.
A special case of an Axiom A flow is a geodesic flow on the unit tangent bundle of a manifold of constant negative curvature. In this case the extensions to the zeta function come from appealing to harmonic analysis and, in particular, the Selberg trace formula.

MSC:
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37C10 Dynamics induced by flows and semiflows
11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
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References:
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