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Transfer operators acting on Zygmund functions


Authors: Viviane Baladi, Yunping Jiang and Oscar E. Lanford III
Journal: Trans. Amer. Math. Soc. 348 (1996), 1599-1615
MSC (1991): Primary 47A10, 47B38, 58F03, 26A16
DOI: https://doi.org/10.1090/S0002-9947-96-01599-1
MathSciNet review: 1348856
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Abstract | References | Similar Articles | Additional Information

Abstract: We obtain a formula for the essential spectral radius $\rho _{\text{ess}}$ of transfer-type operators associated with families of $C^{1+\delta }$ diffeomorphisms of the line and Zygmund, or Hölder, weights acting on Banach spaces of Zygmund (respectively Hölder) functions. In the uniformly contracting case the essential spectral radius is strictly smaller than the spectral radius when the weights are positive.


References [Enhancements On Off] (What's this?)

  • 1. V. Baladi and D. Ruelle, Sharp determinants, IHES preprint (1994); to appear Invent. Math.
  • 2. P. Collet and S. Isola, On the essential spectrum of the transfer operator for expanding Markov maps, Comm. Math. Phys. 139 (1991), 551--557. MR 92h:58157
  • 3. H.R. Dowson, Spectral theory of linear operators, Academic Press, London, 1978. MR 80c:47022
  • 4. D. Fried, The flat-trace asymptotics of a uniform system of contractions, Preprint, 1993.
  • 5. F. Gardiner, Infinitesimal earthquaking and bending in universal Teichmüller space, Preprint, 1993.
  • 6. Y. Jiang, T. Morita, and D. Sullivan, Expanding direction of the period doubling operator, Comm. Math. Phys. 144 (1992), 509--520. MR 93c:58169
  • 7. G. Keller, On the rate of convergence to equilibrium in one-dimensional systems, Comm. Math. Phys. 96 (1984), 181--193. MR 86k:58071
  • 8. O.E. Lanford III, Essential norms of some operators on spaces of Hölder continuous functions, Unpublished notes, 1992.
  • 9. O. Lehto, Univalent functions and Teichmüller spaces, Springer-Verlag, New York Berlin, 1987. MR 88f:30073
  • 10. W. de Melo and S. van Strien, One-dimensional dynamics, Springer-Verlag, Berlin, 1993. MR 95a:58035
  • 11. R.D. Nussbaum, The radius of the essential spectrum, Duke Math. J. 37 (1970), 473--478. MR 41:9028
  • 12. W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Société Mathématique de France (Astérisque, vol. 187-188), Paris, 1990. MR 92f:58141
  • 13. M. Pollicott, Meromorphic extensions of generalized zeta functions, Invent. Math. 85 (1986), 147--164. MR 87k:58218
  • 14. M. Reimann, Ordinary differential equations and quasiconformal mappings, Invent. Math. 33 (1976), 247--270. MR 53:13556
  • 15. D. Ruelle, The thermodynamic formalism for expanding maps, Comm. Math. Phys. 125 (1989), 239--262. MR 91a:58149
  • 16. D. Ruelle, An extension of the theory of Fredholm determinants, Inst. Hautes Etudes Sci. Publ. Math. 72 (1990), 175--193. MR 92b:58187
  • 17. D. Ruelle, Dynamical zeta functions for piecewise monotone maps of the interval, CRM Monograph Series, Vol. 4, Amer. Math. Soc., Providence, RI, 1994. CMP 94:12
  • 18. H.H. Rugh, Generalized Fredholm determinants and Selberg zeta functions for Axiom A dynamical systems, Preprint (1994), to appear Ergodic Theory Dynamical Systems.
  • 19. D. Sullivan, Bounds, quadratic differentials and renormalization conjectures, A.M.S. Centennial Publications, vol. 2, Mathematics into the Twenty-first Century (1988), 1992, pp. 417--466. MR 93k:58194
  • 20. A. Zygmund, Smooth functions, Duke Math. J. 12 (1945), 47--76. MR 7:60b

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Additional Information

Viviane Baladi
Affiliation: ETH Zurich, CH-8092 Zurich, Switzerland (on leave from CNRS, UMR 128, ENS Lyon, France) \phantom{vb}
Address at time of publication: Mathématiques, Université de Genève, 1211 Geneva 24, Switzerland
Email: baladi@sc2a.unige.ch

Yunping Jiang
Affiliation: Department of Mathematics, Queens College, The City University of New York, Flushing, New York 11367-1597
Email: yunqc@qcunix.acc.qc.edu

Oscar E. Lanford III
Affiliation: ETH Zurich, CH-8092 Zurich, Switzerland
Email: lanford@math.ethz.ch

DOI: https://doi.org/10.1090/S0002-9947-96-01599-1
Received by editor(s): March 30, 1995
Additional Notes: Y. Jiang is partially supported by an NSF grant (contract DMS-9400974), and PSC-CUNY awards.
Article copyright: © Copyright 1996 American Mathematical Society

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