Advertisement

Hausdorff Dimension versus Smoothness

  • Flávio Ferreira
  • Alberto A. Pinto
  • David A. Rand
Conference paper
  • 2.5k Downloads
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 75)

Abstract

There is a one-to-one correspondence between C 1+H Cantor exchange systems that are C 1+H fixed points of renormalization and C 1+H diffeomorphisms f on surfaces with a codimension 1 hyperbolic attractor Λ that admit an invariant measure absolutely continuous with respect to the Hausdorff measure on Λ. However, there is no such C 1+α Cantor exchange system with bounded geometry that is a C 1+α fixed point of renormalization with regularity α greater than the Hausdorff dimension of its invariant Cantor set. The proof of the last result uses that the stable holonomies of a codimension 1 hyperbolic attractor Λ are not C 1+θ for θ greater than the Hausdorff dimension of the stable leaves of f intersected with Λ.

Keywords

Hyperbolic systems attractors Hausdorff dimension 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    C. Bonatti and R. Langevin, Difféomorphismes de Smale des surfaces. Asterisque 250, 1998.Google Scholar
  2. [2]
    R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes in Mathematics 470, Springer-Verlag, New York, 1975.Google Scholar
  3. [3]
    A. Denjoy, Sur les courbes définies par les équations différentielles á la surface du tore. J. Math. Pure et Appl. 11, série 9 (1932), 333–375.zbMATHGoogle Scholar
  4. [4]
    A. Fahti, F. Laudenbach, and V. Poenaru, Travaux de Thurston sur les surfaces. Astérisque 66–67 (1991), 1–286.Google Scholar
  5. [5]
    E. de Faria, W. de Melo, and A.A. Pinto, Global hyperbolicity of renormalization for Cr unimodal mappings. Annals of Mathematics. To appear. 1–96.Google Scholar
  6. [6]
    F. Ferreira and A. A. Pinto, Explosion of smoothness from a point to everywhere for conjugacies between diffeomorphisms on surfaces. Ergod. Th. & Dynam. Sys. 23 (2003), 509–517.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    F. Ferreira and A. A. Pinto, Explosion of smoothness from a point to everywhere for conjugacies between Markov families. Dyn. Sys. 16, no. 2 (2001), 193–212.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    J. Franks, Anosov Diffeomorphisms. Global Analysis (ed. S Smale) AMS Providence (1970), 61–93.Google Scholar
  9. [9]
    J. Franks, Anosov diffeomorphisms on tori. Trans. Amer. Math. Soc. 145 (1969), 117–124.CrossRefMathSciNetGoogle Scholar
  10. [10]
    A. A. Gaspar Ruas, Atratores hiperbólicos de codimensão um e classes de isotopia em superficies. Tese de Doutoramento. IMPA, Rio de Janeiro, 1982.Google Scholar
  11. [11]
    J. Harrison, An introduction to fractals. Proceedings of Symposia in Applied Mathematics 39 American Mathematical Society (1989), 107–126.MathSciNetGoogle Scholar
  12. [12]
    M. Hirsch and C. Pugh, Stable manifolds and hyperbolic sets. Proc. Symp. Pure Math., Amer. Math. Soc. 14 (1970), 133–164.MathSciNetGoogle Scholar
  13. [13]
    H. Kollmer, On the structure of axiom-A attractors of codimension one. XI Colóquio Brasileiro de Matemática, vol. II (1997), 609–619.Google Scholar
  14. [14]
    R. Mañé, Ergodic Theory and Differentiable Dynamics. Springer-Verlag Berlin, 1987.zbMATHGoogle Scholar
  15. [15]
    A. Manning, There are no new Anosov diffeomorphisms on tori. Amer. J. Math. 96 (1974), 422.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    S. Newhouse, On codimension one Anosov diffeomorphisms. Amer. J. Math. 92 (1970), 671–762.CrossRefMathSciNetGoogle Scholar
  17. [17]
    S. Newhouse and J. Palis, Hyperbolic nonwandering sets on two-dimension manifolds. Dynamical systems (ed. M Peixoto), 1973.Google Scholar
  18. [18]
    A. Norton, Denjoy’s Theorem with exponents. Proceedings of the American Mathematical Society 127, no. 10 (1999), 3111–3118.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    A. A. Pinto and D. A. Rand, Solenoid Functions for Hyperbolic sets on Surfaces. A survey paper to be published by MSRI book series (published by Cambridge University Press) (2006), 1–29.Google Scholar
  20. [20]
    A. A. Pinto and D. A. Rand, Rigidity of hyperbolic sets surfaces. Journal London Math. Soc. 71,2 (2004), 481–502.CrossRefMathSciNetGoogle Scholar
  21. [21]
    A. A. Pinto and D. A. Rand, Smoothness of holonomies for codimension 1 hyperbolic dynamics. Bull. London Math. Soc. 34 (2002), 341–352.zbMATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    A. A. Pinto and D. A. Rand, Teichmüller spaces and HR structures for hyperbolic surface dynamics. Ergod. Th. Dynam. Sys. 22 (2002), 1905–1931.zbMATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    A. A. Pinto and D. A. Rand, Classifying C 1+ structures on hyperbolical fractals: 1 The moduli space of solenoid functions for Markov maps on train-tracks. Ergod. Th. Dynam. Sys. 15 (1995), 697–734.zbMATHMathSciNetCrossRefGoogle Scholar
  24. [24]
    A. A. Pinto and D. A. Rand, Classifying C 1+ structures on hyperbolical fractals: 2 Embedded trees. Ergod. Th. Dynam. Sys. 15 (1995), 969–992.zbMATHMathSciNetGoogle Scholar
  25. [25]
    A. A. Pinto, D. A. Rand, and F. Ferreira, Hausdorff dimension bounds for smoothness of holonomies for codimension 1 hyperbolic dynamics, J. Differential Equations. Special issue dedicated to A. Cellina and J. A. Yorke. Accepted for publication (2007).Google Scholar
  26. [26]
    A. A. Pinto, D. A. Rand, and F. Ferreira, Cantor exchange systems and renormalization. Submitted (2006).Google Scholar
  27. [27]
    A. A. Pinto and D. Sullivan, The circle and the solenoid. Dedicated to Anatole Katok On the Occasion of his 60th Birthday. DCDS-A 16, no. 2 (2006), 463–504.zbMATHMathSciNetGoogle Scholar
  28. [28]
    T. Sauer, J. A. Yorke, and M. Casdagli, Embedology, Journal of Statistical Physics 65, no. 3–4 (1991), 579–616.zbMATHCrossRefMathSciNetGoogle Scholar
  29. [29]
    M. Schub, Global Stability of Dynamical Systems. Springer-Verlag, 1987.Google Scholar
  30. [30]
    Ya Sinai, Markov partitions and C-diffeomorphisms. Anal. and Appl. 2 (1968), 70–80.MathSciNetGoogle Scholar
  31. [31]
    D. Sullivan, Differentiable structures on fractal-like sets determined by intrinsic scaling functions on dual Cantor sets. Proceedings of Symposia in Pure Mathematics 48 American Mathematical Society, 1988.Google Scholar
  32. [32]
    W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc. 19 (1988), 417–431.zbMATHCrossRefMathSciNetGoogle Scholar
  33. [33]
    R. F. Williams, Expanding attractors. Publ. I.H.E.S. 43 (1974), 169–203.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2007

Authors and Affiliations

  • Flávio Ferreira
    • 1
  • Alberto A. Pinto
    • 2
  • David A. Rand
    • 3
  1. 1.E.S.E.I.G.Instituto Politécnico do PortoVila do CondePortugal
  2. 2.Departamento de MatemáticaUniversidade do MinhoBragaPortugal
  3. 3.Mathematics InstituteUniversity of WarwickCoventryUK

Personalised recommendations

Cite paper

Buy options