Pollicott, Mark A complex Ruelle-Perron-Frobenius theorem and two counterexamples. (English) Zbl 0575.47009 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)]. Cited in 1 ReviewCited in 32 Documents 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 Citations:Zbl 0401.28016; Zbl 0316.58016; Zbl 0383.58010 PDF BibTeX XML Cite \textit{M. Pollicott}, Ergodic Theory Dyn. Syst. 4, 135--146 (1984; Zbl 0575.47009) Full Text: DOI OpenURL References: [1] Walters, An Introduction to Ergodic Theory 79 (1981) · Zbl 0475.28009 [2] Livsic, Math. USSR Izvestiza 6 pp 1276– (1972) [3] Krasnoselskii, Positive Solutions of Operator Equations (1964) [4] DOI: 10.1007/BF02392225 · Zbl 0061.16504 [5] DOI: 10.2307/1998528 · Zbl 0355.28010 [6] Gantmacher, The Theory of Matrices II (1974) · Zbl 0085.01001 [7] Gallovotti, Accad. Lincei. Rend. Sc. fismat. e nat. 61 pp 309– (1976) [8] Ferrero, Colloq. Math. Soc. János Bolyai 27 pp 333– (1979) [9] Dunford, Linear Operators (1958) [10] Corduneanu, Almost Periodic Functions (1968) [11] Taylor, An Introduction to Functional Analysis (1964) [12] Ruelle, C. R. Acad. Sci. 296 pp 191– (1983) [13] Ruelle, Thermodynamic Formalism (1978) [14] DOI: 10.1090/S0002-9904-1976-14003-7 · Zbl 0316.58016 [15] DOI: 10.1007/BF01654281 · Zbl 0165.29102 [16] Parry, Classification Problems in Ergodic Theory (1982) · Zbl 0487.28014 [17] DOI: 10.1007/BF01388488 · Zbl 0563.28008 [18] DOI: 10.2307/2006982 · Zbl 0537.58038 [19] Parry, Ergod. Th. & Dynam. Sys. 4 pp 117– (1984) [20] DOI: 10.2307/1997113 · 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.