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


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
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