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Twisted Perron-Frobenius theorem and \(L\)-functions. (English) Zbl 0658.58034
A theorem of Perron-Frobenius type and its twisted version are established in a setting of topological graphs. The applications include a partial extension of Selberg’s celebrated results on his zeta function and a recent result by Parry and Pollicot on meromorphic continuations of dynamical zeta functions to certain \(L\)-functions associated with a dynamical system of Anosov type. Well-prepared background material with a lot of new information on topological graphs and Ruelle operators for topological graphs, on \(L\)-functions of finite/profinite graphs and on \(L\)-functions of Anosov flows is also presented.

MSC:
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
57M15 Relations of low-dimensional topology with graph theory
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