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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.

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)
Full Text: DOI EuDML
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