Renormalization, Zygmund Smoothness and the Epstein Class

  • D. Sullivan
Part of the NATO ASI Series book series (NSSB, volume 280)


To randomize a deck of n-cards one may turn over one of the split stacks before shuffling. The resulting permutation of order n if irreducible is called a folding permutation because it may be accomplished by a continuous mapping f of the real line to itself which folds the line once. The orbit of the turning point is finite and f restricted to this finite orbit is the folding permutation.




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  1. [CT]
    P. Coullet and C. Tresser. “Iteration d’endomorphismes et groupe de renormalisation”. J. de Physique Colloque C 539, C5–25 (1978). CRAS Paris 287 A, (1978).Google Scholar
  2. [CEK]
    P. Collet, J.-P. Eckmann and H. Koch. “Period-doubling bifurcations for families of maps on R n”. J. Stat. Phys. 25, 1–14 (1980).MathSciNetADSCrossRefGoogle Scholar
  3. [E]
    H. Epstein. “New proofs of the existence of the Feigenbaum functions”. Commun. Math. Phys., 106, 395–426 (1986).ADSMATHCrossRefGoogle Scholar
  4. [F1]
    M.J. Feigenbaum. “Quantitative universality for a class of non-linear transformation”. J. Stat. Phys. 19. 25–52 (1978).MathSciNetADSMATHCrossRefGoogle Scholar
  5. [F2]
    M.J. Feigenbaum. “Universal metric properties of non-linear transformations”. J. Stat. Phys. 21, 669–706 (1979).MathSciNetADSMATHCrossRefGoogle Scholar
  6. [MV]
    W de Melo and S. Van Strien. “Schwarzian derivative and beyond”. Bull Amer Math Soc 18, 159–162 (1988).MathSciNetMATHCrossRefGoogle Scholar
  7. [S1]
    D. Sullivan D. “Bounds, quadratic, differentials, and renormalization conjectures”. To appear in AMS volume (2) (1991) celebrating the Centennial of the American Mathematical Society.Google Scholar
  8. [S2]
    D. Sullivan. “Quasiconformal homeomorphisms in dynamics, topology, and geometry”. ICM Berkeley 1986.Google Scholar
  9. [S3]
    D. Sullivan. “Differentiable structures on fractal like sets”. In: Non-linear Evolution and Chaotic Phenomena, G. Gallavotti and P. Zweifel, eds., New-York, Plenum (1988). See also Herman Weyl Centenary Volume.Google Scholar
  10. [SW]
    G. Swiatek “Critical Circle Maps”. Commun. Math. Phys. 119.109–128 (1988).MathSciNetADSMATHCrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1991

Authors and Affiliations

  • D. Sullivan
    • 1
  1. 1.THESBures sur YvetteFrance

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