# zbMATH — the first resource for mathematics

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)
Full Text:
##### References:
 [1] Anosov, D.V., Sinai, Ya.G.: Some smooth ergodic systems. Russ. Math. Surv.22, 103-167 (1967) · Zbl 0177.42002 [2] Beurling, A.: Analyse de la loi asymptotic de la distribution des nombres premiers généralisés I. Acta Math.68, 255-291 (1937) · Zbl 0017.29604 [3] Bhatia, R., Mukheryea, K.K.: On the rate of change of Spectra of operators. Linear Algebra Appl.27, 147-157 (1979) · Zbl 0421.15019 [4] Bhatia, R., Parthasarathy, K.R.: Lectures on functional analysis I. MacMillan, Dehli, 1977 · Zbl 0451.47020 [5] Bowen, R.: Symbolic dynamics for hyperbolic flows. Am. J. Math.95, 429-459 (1973) · Zbl 0282.58009 [6] Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lect. Notes Math.470, Berlin-Heidelberg-New York: Springer 1975 · Zbl 0308.28010 [7] Bowen, R.: One-dimensional hyperbolic sets for flows. J. Differ. Equations12, 173-179 (1972) · Zbl 0242.58005 [8] Bowen, R.: Periodic orbits for hyperbolic flows. Am. J. Math.94, 1-30 (1972) · Zbl 0254.58005 [9] Bowen, R., Ruelle, D.: The ergodic theory of Axiom A flows. Invent. math.29, 181-202 (1975) · Zbl 0311.58010 [10] Browder, F.E.: On the spectral theory of elliptic differential operators I. Math. Ann.142, 22-130 (1961) · Zbl 0104.07502 [11] Davenport, H.: Multiplicative number theory. G.T.M.74, Berlin-Heidelberg-New York: Springer 1980 · Zbl 0453.10002 [12] Erdelyi, I., Lange, R.: Spectral decompositions on Banach spaces. Lect. Notes Math.623, Berlin-Heidelberg-New York: Springer 1977 · Zbl 0381.47001 [13] Gallovotti, G.: Funzioni zeta ed insiemi basilar. Accad. Lincei. Rend. Sc. fismat. e nat.61, 309-317 (1976) [14] Hejhal, D.A.: The Selberg trace formula and the Riemann zeta function. Duke Math. J.43, 441-482 (1976) · Zbl 0346.10010 [15] Hejhal, D.A.: The Selberg trace formula for PSL(2,?). Vol. I, Lect. Notes Math.548, Berlin-Heidelberg-New York: Springer 1976 · Zbl 0347.10018 [16] Ingham, A.E.: The distribution of prime numbers. C.M.T.30, Cambridge, 1932 · Zbl 0006.39701 [17] Kato, T.: Perturbation theory for linear operators. Berlin-Heidelberg-New York: Spinger (1966) · Zbl 0148.12601 [18] Manning, A.: Axiom A diffeomorphisms have rational zeta functions. Bull. Lond. Math. Soc.3, 215-220 (1971) · Zbl 0219.58007 [19] Margulis, G.A.: Applications of ergodic theory to the investigation of manifolds of negative curvature. Funktional Analizi Ego Prilozhen3, 89-90 (1969) [20] Marinescu, G., Ionescu Tulcea, C.T.: Theorie ergodique pour des classes d’operations non completement continues. Ann. Math.52, 140-147 (1950) · Zbl 0040.06502 [21] Mayer, D.H.: The Ruelle-Araki transfer operator in classical statistical mechanics. Lect. Notes Math. (in physics)123, Berlin-Heidelberg-New York: Springer 1980 · Zbl 0454.60079 [22] Nussbaum, R.D.: The radius of the essential spectrum. Duke Math. J.37, 473-478 (1970) · Zbl 0216.41602 [23] Parry, W.: An analogue of the prime number theorem for shifts of finite type and their suspensions. Isr. J. Math.45, 41-52 (1983) · Zbl 0552.28020 [24] Parry, W.: Bowen’s equidistribution theory and the Dirichlet density theorem. Ergodic Theory Dyn. Syst.4, 117-134 (1984) · Zbl 0567.58014 [25] Parry, W., Pollicott, M.: An analogue of the prime number theorem for closed orbits of Axiom A flows. Ann. Math.118, 573-592 (1983) · Zbl 0537.58038 [26] Parry, W., Pollicott, M.: The Chebotarov theorem for Galois coverings of Axiom A flows (preprint) · Zbl 0626.58006 [27] Pollicott, M.: A complex Ruelle-Perron-Frobenius theorem and two counter-examples. Ergodic Theory Dyn. Syst.4, 135-146 (1984) · Zbl 0575.47009 [28] Pollicott, M.: Asymptotic distribution of closed geodesics. Isr. J. Math.52, 209-224 (1985) · Zbl 0582.58029 [29] Ruelle, D.: Zeta functions for expanding maps and Anosov flows. Invent. math.34, 231-242 (1976) · Zbl 0329.58014 [30] Ruelle, D.: Generalised zeta functions for Axiom A basic sets. Bull. Am. Math. Soc.82, 153-156 (1976) · Zbl 0316.58016 [31] Ruelle, D.: Equilibrium statistical mechanics of one-dimensional classical lattice systems, from International Symposium on mathematical problems in theoretical physics. Lect. Notes Phys.39, 449-457 (1975) [32] Ruelle, D.: Thermodynamic formalism. Reading Mass.: Addison-Wesley 1978 · Zbl 0401.28016 [33] Sarnak, P.: The arithmetic and geometry of some hyperbolic three manifolds. Acta Math.151, 253-296 (1983) · Zbl 0527.10022 [34] Sinai, Ya.G.: Gibbs measures in ergodic theory. Russ. Math. Surv.27(4) 21-69 (1972) · Zbl 0246.28008 [35] Sunada, T.: Geodesic flows and geodesic random walks. Adv. Stud. Pure Math.3, 47-85 (1984) · Zbl 0599.58037 [36] Taylor, A.E.: Introduction to functional analysis. New York: Wiley 1958 · Zbl 0081.10202 [37] Tuncel, S.: Conditional pressure and coding. Isr. J. Math.39, 101-112 (1981) · Zbl 0472.28016 [38] Viswanthan, K.S.: Statistical mechanics of a one-dimensional lattice gas with exponentialpolynomial interactions. Commun. Math. Phys.47, 131-141 (1976) [39] Walters, P.: Ruelle’s operator theorem andg-measures. Trans. Am. Math. Soc.214, 375-387 (1975) · Zbl 0331.28013 [40] Wiener, N.: The Fourier integral and certain of its applications. Cambridge: C.U.P. 1967
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.