zbMATH — the first resource for mathematics

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.

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
Full Text: DOI
[1] {\scT. Adachi and T. Sunada}, Homology of closed geodesics in a negatively curved manifold, J. Differential Geom., to appear. · Zbl 0618.58028
[2] Bowen, R, Symbolic dynamics for hyperbolic flows, Amer. J. math., 95, 429-460, (1973) · Zbl 0282.58009
[3] Bowen, R, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, () · Zbl 0308.28010
[4] Fried, D, The zeta functions of Ruelle and Selberg, I, (1985), preprint
[5] Fried, D, Analytic torsion and closed geodesics on hyperbolic manifolds, (1985), preprint
[6] Fried, D, Fuchsian groups and Reidemeister torsion, (1985), preprint · Zbl 0597.10027
[7] Gangolli, R, Zeta functions of Selberg’s type for compact space forms of symmetric spaces of rank one, Illinois J. math., 21, 1-42, (1977) · Zbl 0354.33013
[8] Hejhal, D.A, The Selberg trace formula and the Riemann zeta function, Duke math. J., 43, 441-482, (1976) · Zbl 0346.10010
[9] {\scD. A. Hejhal}, “The Selberg Trace Formula for PSL2(R),” Springer Lecture Notes 548, Vol.; 1001, Vol. II, Springer-Verlag, New York.
[10] Ihara, Y, Discrete subgroups of PL(2, KP), (), 272-278
[11] Manning, A, Axiom A diffeomorphisms have rational zeta functions, Bull. London math. soc., 3, 215-220, (1971) · Zbl 0219.58007
[12] McKean, H.P, Selberg’s trace formula as applied to a compact Riemann surface, Comm. pure appl. math., 25, 225-246, (1972)
[13] Farry, W; Pollicott, M, An analogue of the prime number theorem for closed orbits of axiom A flows, Ann. of math., 118, 573-591, (1983) · Zbl 0537.58038
[14] {\scW. Parry and M. Pollicott}, The Chebotarev theorem for Galois coverings of Axiom A flows preprint. · Zbl 0626.58006
[15] Plante, J.F, Homology of closed orbits of Anosov flows, (), 297-300 · Zbl 0249.58012
[16] {\scM. Pollicott}, A complex Ruelle-Perron-Frobenius theorem and two counter-examples, preprint. · Zbl 0575.47009
[17] {\scM. Pollicott}, Meromorphic extensions of generalized zeta functions, preprint. · Zbl 0604.58042
[18] Ruelle, D, Thermodynamic formalism, (1978), Addison-Wesley Reading, Mass
[19] Ruelle, D, Zeta functions for expanding maps and Anosov flows, Invent. math., 34, 231-242, (1976) · Zbl 0329.58014
[20] Sarnak, P, The arithmetic and geometry of some hyperbolic three manifolds, Acta math., 151, 253-296, (1983) · Zbl 0527.10022
[21] Selberg, A, Harmonic analysis and discontinuous subgroups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian math. soc., 20, 47-48, (1956) · Zbl 0072.08201
[22] Serre, J.P, Trees, (1980), Springer-Verlag New York
[23] Smale, S, Differential dynamical systems, Bull. amer. math. soc., 73, 747-817, (1967) · Zbl 0202.55202
[24] Sunada, T, Geodesic flows and geodesic random walks, () · Zbl 0599.58037
[25] {\scT. Sunada}, Tchebotarev’s density theorem for closed geodesics in a compact locally symmetric space of negative curvature, preprint.
[26] Sunada, T, Riemannian coverings and isospectral manifolds, Ann. of. math., 121, 169-186, (1985) · Zbl 0585.58047
[27] Sunada, T, Trace formulas, Wiener integrals and asymptotics, (), 103-113
[28] {\scT. Adachi}, Closed orbits of an Anosov flow and the fundamental group, Proc. Amer. Math. Soc., to appear. · Zbl 0625.58018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.