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Meromorphic extension of \(L\)-functions of Anosov flows and profinite graphs. (English) Zbl 0668.58041
Let \(X\) be a compact Riemannian manifold and let \(\phi_ t: X\to X\) be an Anosov flow the non-wandering set of which coincides with \(X\). The notion of an \(L\)-function \(L_{\phi_ t}(s;\rho)\) of \(\phi_ t\) associated with a unitary representation \(\rho: \pi_ 1(X)\to U(N)\) was defined and studied by the author and T. Sunada [J. Funct. Anal. 71, 1–46 (1987; Zbl 0658.58034)]. It is shown in the paper under review that \(L_{\phi_ t}(s;\rho)\) extends non-zero meromorphically to some domain \(\text{Re}(s)>h-\delta\), \(\delta >0\), where \(h=h(\phi_ t)\) denotes the topological entropy of the flow \(\phi_ t\). The proof is an extension of the proof given by M. Pollicott [Invent. Math. 85, 147–164 (1986; Zbl 0604.58042)] for the case when \(\rho\) is the trivial representation. Similar result is established for the \(L\)-function of a profinite graph associated with a unitary representation of the fundamental group of the graph.
Reviewer: L.N.Stoyanov

MSC:
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
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