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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
MathSciNet review: 1348856
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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. Pierre Collet and Stefano Isola, On the essential spectrum of the transfer operator for expanding Markov maps, Comm. Math. Phys. 139 (1991), no. 3, 551–557. MR 1121133
  • 3. H. R. Dowson, Spectral theory of linear operators, London Mathematical Society Monographs, vol. 12, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1978. MR 511427
  • 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. Yun Ping Jiang, Takehiko Morita, and Dennis Sullivan, Expanding direction of the period doubling operator, Comm. Math. Phys. 144 (1992), no. 3, 509–520. MR 1158758
  • 7. Gerhard Keller, On the rate of convergence to equilibrium in one-dimensional systems, Comm. Math. Phys. 96 (1984), no. 2, 181–193. MR 768254
  • 8. O.E. Lanford III, Essential norms of some operators on spaces of Hölder continuous functions, Unpublished notes, 1992.
  • 9. Olli Lehto, Univalent functions and Teichmüller spaces, Graduate Texts in Mathematics, vol. 109, Springer-Verlag, New York, 1987. MR 867407
  • 10. Welington de Melo and Sebastian van Strien, One-dimensional dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 25, Springer-Verlag, Berlin, 1993. MR 1239171
  • 11. Roger D. Nussbaum, The radius of the essential spectrum, Duke Math. J. 37 (1970), 473–478. MR 0264434
  • 12. William Parry and Mark Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque 187-188 (1990), 268 (English, with French summary). MR 1085356
  • 13. Mark Pollicott, Meromorphic extensions of generalised zeta functions, Invent. Math. 85 (1986), no. 1, 147–164. MR 842051,
  • 14. H. M. Reimann, Ordinary differential equations and quasiconformal mappings, Invent. Math. 33 (1976), no. 3, 247–270. MR 0409804,
  • 15. David Ruelle, The thermodynamic formalism for expanding maps, Comm. Math. Phys. 125 (1989), no. 2, 239–262. MR 1016871
  • 16. David Ruelle, An extension of the theory of Fredholm determinants, Inst. Hautes Études Sci. Publ. Math. 72 (1990), 175–193 (1991). MR 1087395
  • 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. Dennis Sullivan, Bounds, quadratic differentials, and renormalization conjectures, American Mathematical Society centennial publications, Vol. II (Providence, RI, 1988) Amer. Math. Soc., Providence, RI, 1992, pp. 417–466. MR 1184622
  • 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

Yunping Jiang
Affiliation: Department of Mathematics, Queens College, The City University of New York, Flushing, New York 11367-1597

Oscar E. Lanford III
Affiliation: ETH Zurich, CH-8092 Zurich, Switzerland

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