Abstract
The essential spectrum of the transfer operator for expanding markov maps of the interval is studied in detail. To this end we construct explicityly an infinite set of eigenfunctions which allows us to prove that the essential spectrum inC k is a disk whose radius is related to the free energy of the Liapunov exponent.
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[Ba] Baladi, V.: Fonctions zêta, fonctions de corrélation et mesures d'équilibre pour quelques systèmes dynamiques non axiomeA. Thesis presented at the Faculty of Sciences, University of Geneva, 1989
[BPPV] Benzi, R., Paladin, G., Parisi, G., Vulpiani, A.: J. Phys. A17, 513 (1984)
[BR] Bohr, T., Rand, D.: Physica D25, 387 (1986)
[Bo] Bowen, R.: Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lectures Notes in Mathematics, Vol.470. Berlin, Heidelberg, New York: Springer 1975
[CL] Collet, P., Lesne, A.: J. Stat. Phys.57, 967 (1989)
[Ec] Eckmann, J.-P.: Resonances in dynamical systems. In: IXth International Congress on Mathematical Physics. July 1988, Swansea, Wales, Simon, B., Truman, A., Davies, I.M. (eds.). Bristol, New York: Adam Hilger 1989
[El] Ellis, R.: Entropy, Large Deviations and Statistical Mechanics. Berlin, Heidelberg, New York: Springer 1985
[GH] Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Berlin, Heidelberg, New York: Springer 1983
[H] Haydn, N.T.A.: Meromorphic extension of the zeta function for Axiom A flows, Preprint (1989). To appear in Ergod. Th. Dynam. Sys.
[K1] Keller, G.: On the rate of convergence to equilibrium in one-dimensional systems. Commun. Math. Phys.96, 181–193 (1984)
[K2] Keller, G.: Markov extensions, zeta-functions, and Fredholm theory for piecewise invertible dynamical systems. Trans. Am. Math. Soc.314, 433–499 (1989)
[Ka] Kato, T.: Perturbation theory of linear operators, II Ed. Berlin, Heidelberg, New York: Springer 1976
[M] Mañe, R.: Ergodic theory and differentiable dynamics. Berlin, Heidelberg, New York: Springer 1987
[N] Nussbaum, R.D.: The radius of the essential spectrum. Duke Math. J.37, 473–478 (1970)
[P1] Pollicott, M.: Meromorphic extensions of generalized zeta functions. Invent. Math.85, 147–164 (1986)
[P2] Pollicott, M.: The differential zeta function for Axiom A attractors. Ann. Math.131, 331–354 (1990)
[R1] Ruelle, D.: Thermodynamic formalism. Reading, MA: Addison-Wesley 1978
[R2] Ruelle, D.: Locating resonances for Axiom A dynamical systems. J. Stat. Phys.44, 281–292 (1987)
[R3] Ruelle, D.: The thermodynamic formalism for expanding maps. Commun. Math. Phys.125, 239–262 (1989)
[R4] Ruelle, D.: An extension of the theory of Fredholm determinants. Preprint IHES/P/89/38 (1989)
[S] Sinai, Ya.G.: Gibbs measures in ergodic theory. Russ. Math. Surv.166, 21–69 (1972)
[T] Tangerman, F.: Meromorphic continuation of Ruelle zeta functions. Boston University thesis (1986)
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Communicated by J.-P. Eckmann
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Collet, P., Isola, S. On the essential spectrum of the transfer operator for expanding markov maps. Commun.Math. Phys. 139, 551–557 (1991). https://doi.org/10.1007/BF02101879
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DOI: https://doi.org/10.1007/BF02101879