On the rate of mixing of Axiom A flows

Abstract

w 0. IntroductionAxiom A systems were originally introduced by Smale in his seminal paper on dynamical systems [28]. One of their main purposes was to generalise Anosov systems (both diffeomorphisms and flows). Perhaps the most significant feature of this generalisation was that it further divorced the purely dynamical aspects of the system from the underlying geometry of the manifold. Even in such generality remarkably powerful results can still be obtained for Axiom A diffeomorphisms. For example, the rate of mixing of an Axiom A diffeomorphism is always exponential ([2], p. 38) and the zeta function for the diffeomorphism is rational [16]. However, for Axiom A flows the corresponding results are not always valid. For example, the rate of mixing for Axiom A flows need not be exponentially fast [22] and the zeta functions for these flows need not be meromorphic in the entire complex plane [12].The purpose of this paper is to actually relate the rate of mixing of an Axiom A flow to the meromorphic domain of its zeta function. In particular, we shall give necessary conditions for exponential mixing (which we also refer to as exponential decay of correlations). We shall also exhibit examples of Axiom A flows for which the rate of mixing can be chosen to be arbitrarily slow. In particular, this answers a question of Bowen regarding the possibility of polynomial rates of mixing for Axiom A flows ([-3], p. 31). Following the program advanced by Bowen and Ruelle [6] our first step is to reduce the problem to the case of suspended flows using the powerful and useful symbolic dynamics of Bowen [4]. The rate of decay of the so-called correlation function is then reflected in the analytic domain of its Fourier transform. (In particular, for exponential decay this is governed by the Paley-Wiener theorem ([15], p. 174)). Our approach is to relate domain of the Fourier Transform (for the case of the suspended flow) to the spectrum of an associated Ruelle operator acting on an appropriate Banach space. We then use previous work by the author to relate this spectrum to the domain of the zeta function for the flow (for both suspended flows and Axiom A flows) [-20]. We should point out that Parry and the author originally studied this zeta function in connection with Prime Orbit Theorems for Axiom A flows [19].In Sect. 1 we introduce our principle tool, the Ruelle operator. In Sect. 2 we recall some results on suspended flows and in Sect. 3 we introduce the zeta function and correlation functions for these flows. In the fourth section we prove our main result in the context of suspended flows. In the fifth section we introduce Axiom A flows and rephrase our results in this stronger setting. In Sect. 6 we give a simple counter-example to show that in general there is no order of mixing common to the class of all Axiom A flows. This paper was written while the author was enjoying the hospitality and financial support of the Institut des Hautes Etudes Scientifiques. I would like to express my gratitude to David Ruelle for his consid...

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