A complex Ruelle-Perron-Frobenius theorem and two counterexamples. (English) Zbl 0575.47009

Pollicott, Mark

Ergodic Theory Dyn. Syst. 4, 135-146 (1984).

This paper generalizes a result of D. Ruelle [’Thermodynamic formalism’ (1978; Zbl 0401.28016)] on the spectrum of Perron-Frobenius type operator acting on real Hölder continuous functions (on a one- sided subshift of finite type) to the analogous case for complex Hölder continuous functions. The spectrum is shown to be quasi-compact. We also give an alternative proof of the Ruelle operator theorem. The main application of this result is to extending the domain of a generalized zeta-function introduced by D. Ruelle [Bull. Am. Math. Soc. 82, 153-156 (1976; Zbl 0316.58016)].
The remainder of the paper is devoted to two examples which answer questions about the domains of such zeta-functions (particularly in the context of suspended flows) raised by Ruelle [Thermodynamic formalism, p. 173] and R. Bowen [’On axiom A diffeomorphisms’, Reg. Conf. No.35, Am. Math. Soc. VII, p. 45 (1978; Zbl 0383.58010)].

MSC:

47A35	Ergodic theory of linear operators
28D05	Measure-preserving transformations
47A10	Spectrum, resolvent

Keywords:

spectrum of Perron-Frobenius type operator acting on real Hölder continuous functions; zeta-functions

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