# zbMATH — the first resource for mathematics

On the rate of mixing of Axiom A flows. (English) Zbl 0591.58025
The rate of mixing of an Axiom A diffeomorphism is always exponential [R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lect. Notes Math. 470 (1975; Zbl 0308.28010)] and has a zeta function which is ration [A. Manning, Bull. Lond. Math. Soc. 3, 215–220 (1971; Zbl 0219.58007)]. However, for Axiom A flows the situation is different, their rate of mixing need not be exponentially fast [D. Ruelle, C. R. Acad. Sci., Paris, Sér. A 296, 191–193 (1983; Zbl 0531.58040)] and their zeta functions need not be meromorphic in the entire complex plane [G. Gallavotti, Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 61(1976), 309–317 (1977; Zbl 0378.58007)].
In this paper the author relates the rate of mixing an Axiom A flow to the meromorphic domain of its zeta function. Necessary and sufficient conditions for exponential mixing are given and also examples of these flows for which the rate of mixing can be chosen arbitrarily slowly. This answers a question of Bowen, regarding the possibility of polynomial rates of mixing for Axiom A flows.
These results are obtained by first applying the methods of R. Bowen and D. Ruelle [Invent. Math. 29, 181–202 (1975; Zbl 0311.58010)] to reduce the problem to that of suspended flows so that the symbolic dynamics of Bowen can be applied. The Ruelle operator and Fourier analysis play an important role, enabling the author to apply his earlier work which relates the spectrum of the Ruelle operator to the domain of the zeta function.
Reviewer: G.Goodson

##### MSC:
 37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc. 37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 37B10 Symbolic dynamics
##### Keywords:
Axiom A flow; symbolic dynamics; mixing
Full Text:
##### References:
 [1] Arnold, V.I., Avez, A.: Ergodic problems of classical mechanics. New York: Benjamin 1968 · Zbl 0167.22901 [2] Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. S.L.N.470, Berlin-Heidelberg-New York: Springer 1975 · Zbl 0308.28010 [3] Bowen, R.: On Axiom A diffeomorphisms. Am. Math. Soc. Regional Conf. Proc. No.35, Providence 1978 · Zbl 0383.58010 [4] Bowen, R.: Symbolic dynamics for hyperbolic flows. Am J. Math.95, 429-459 (1973) · Zbl 0282.58009 [5] Bowen, R.: Periodic orbits for hyperbolic flows. Am. J. Math.94, 1-30 (1972) · Zbl 0254.58005 [6] Bowen, R., Ruelle, D.: The ergodic theory of Axiom A flows. Invent. Math.29, 181-202 (1975) · Zbl 0311.58010 [7] Bowen, R., Walters, P.: Expansive one-parameter flows. J. Differ. Equations12, 180-193 (1972) · Zbl 0242.54041 [8] Bowen, R., Series, C.: Markov maps associated with Fuchsian groups. Publ. Math. Inst. Hautes Etud. Sci.50, 153-170 (1979) · Zbl 0439.30033 [9] Collet, P., Epstein, H., Gallavotti, G.: Perturbations of geodesic flows on surfaces of constant negative curvature and their mixing properties. Commun. Math. Phys.95, 61-112 (1984) · Zbl 0585.58022 [10] Cornfeld, I.P., Fomin, S.V., Sinai, Y.G.: Ergodic Theory. Berlin-Heidelberg-New York: Springer 1982 · Zbl 0493.28007 [11] Fomin, S.V., Gelfand, I.M.: Geodesic flows on manifolds of constant negative curvature. Transl. Am. Math. Soc.1, 49-65 (1955) · Zbl 0066.36101 [12] Gallavotti, G.: Funzioni zeta ed insiemi basilar. Accad. Lincei. Rend. Sc. fismat. e mat.61, 309-317 (1976) · Zbl 0378.58007 [13] Hardy, G.H., Wright, E.M.: An introduction to the theory of numbers. Oxford: O.U.P. 1983 · Zbl 0020.29201 [14] Hejhal, D.A.: The Selberg trace formula and the Riemann zeta function. Duke Math. J.43 441-482 (1976) · Zbl 0346.10010 [15] Katznelson, Y.: An introduction to harmonic analysis. Dover: New York 1976 · Zbl 0352.43001 [16] Manning, A.: Axiom A diffeomorphisms have rational zeta functions. Bull. London Math. Soc.3, 215-220 (1971) · Zbl 0219.58007 [17] Parry, W.: Intrinsic Markov Chains. Trans. Am. Math. Soc.112, 55-65 (1964) · Zbl 0127.35301 [18] Parry, W.: Topics in Ergodic theory. Cambridge: C.U.P. 1981 · Zbl 0449.28016 [19] Parry, W., Pollicott, M.: An analogue of the prime number theorem for closed orbits of Axiom A flows. Ann. Math.118, 573-591 (1983) · Zbl 0537.58038 [20] Pollicott, M.: Meromorphic extensions of generalised zeta functions (Preprint) · Zbl 0604.58042 [21] Pollicott, M.: Asymptotic distribution of closed geodesics (To appear in Isr. J. Math.) [22] Ruelle, D.: Flows which do not exponentially mix. C.R. Acad. Sci. Paris296, 191-194 (1983) [23] Ruelle, D.: Thermodynamic formalism. Reading: Addison-Weley 1978 · Zbl 0401.28016 [24] Ruelle, D.: Zeta functions for expanding maps and Anosov flows. Invent. Math.34, 23L-242 (1976) · Zbl 0329.58014 [25] Series, C.: Symbolic dynamics for geodesic flows. Acta Math.146, 103-128 (1981) · Zbl 0488.58016 [26] Schoen, R., Wolpert, S., Yau, S.T.: Geometric bounds on the low eigenvalues of a compact surface. In: Proc. Symp. Pure Math.36 (1980) · Zbl 0446.58018 [27] Sinai, Y.G.: Gibbs measures in ergodic. theory. Russ. Math. Surv.27, 21-69 (1972) · Zbl 0246.28008 [28] Smale, S.: Differentiable dynamical systems. Bull. Am. Math. Soc.73, 747-817 (1967) · Zbl 0202.55202 [29] Tuncel, S.: Conditional pressure and coding. Isr. J. Math.39, 101-112 (1981) · Zbl 0472.28016 [30] Venkov, A.B.: Spectral theory of automorphic functions, the selberg zeta function, and some problems of analytic number theory and mathematical physics. Russ. Math. Surv.34, 79-153 (1979) · Zbl 0437.10012 [31] Walters, P.: A variational principle for the pressure of continuous transformations. Am. J. Math.97, 937-971 (1976) · Zbl 0318.28007 [32] Walters, P.: An introduction to ergodic theory. G.T.M.79, Berlin-Heidelberg-New York: Springer 1981 · Zbl 0475.28009 [33] Walters, P.: Ruelle’s operator theorem andg-measures. Trans. Am. Math. Soc.214, 375-387 (1975) · Zbl 0331.28013
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.