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The Lamination of Infinitely Renormalizable Dissipative Gap Maps: Analyticity, Holonomies and Conjugacies

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Abstract

Motivated by return maps near saddles for three-dimensional flows and also by return maps in the torus associated to Cherry flows, we study gap maps with derivative positive and smaller than one outside the discontinuity point. We prove that the lamination of infinitely renormalizable maps (or else maps with irrational rotation numbers) has analytic leaves in a natural subset of a Banach space of analytic maps of this kind. With maps having Hölder continuous derivative and derivative bounded away from zero, we also prove Hölder continuity of holonomies of the lamination and also of conjugacies between maps having the same combinatorics.

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References

  1. Aranson S.Kh., Zhuzhoma E.V.: Qualitative theory of flows on surfaces (a review). J. Math. Sci. (New York) 90(3), 2051–2110 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aranson S.Kh., Zhuzhoma E.V., Medvedev T.V.: Classification of Cherry transformations on a circle and of Cherry flows on a torus. Izv. Vyssh. Uchebn. Zaved. Mat. 4, 7–17 (1996)

    MathSciNet  Google Scholar 

  3. Berry D., Mestel B.D.: Wandering intervals for Lorenz maps with bounded nonlinearity. Bull. Lond. Math. Soc. 23(2), 183–189 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  4. Boyd C.: On the structure of the family of Cherry fields on the torus. Ergodic Theory Dyn. Syst. 5(1), 27–46 (1985)

    Article  MATH  Google Scholar 

  5. Brette R.: Rotation numbers of discontinuous orientation-preserving circle maps. Set-Valued Anal. 11(4), 359–371 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chae, S.B.: Holomorphy and calculus in normed spaces. Monographs and Textbooks in Pure and Applied Mathematics, vol. 92. With an appendix by Angus E. Taylor. Marcel Dekker Inc., New York (1985)

  7. Cherry T.M.: Analytic quasi-periodic curves of discontinuous type on the torus. Proc. Lond. Math. Soc. 44, 175–215 (1938)

    Article  MathSciNet  Google Scholar 

  8. de Melo W.: On the cyclicity of recurrent flows on surfaces. Nonlinearity 10(2), 311–319 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. dos Anjos A.G.: Polynomial vector fields on the torus. Bol. Soc. Brasil. Mat. 17(2), 1–22 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  10. Guckenheimer J., Williams R.F.: Structural stability of Lorenz attractors. Inst. Hautes Études Sci. Publ. Math. 50, 59–72 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gutiérrez, C.: Smoothability of Cherry flows on two-manifolds. In: Geometric Dynamics (Rio de Janeiro, 1981). Lecture Notes in Math., vol. 1007, pp. 308–331. Springer, Berlin (1983)

  12. Keller, G., St. Pierre, M.: Topological and measurable dynamics of Lorenz maps. In: Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, pp. 333–361. Springer, Berlin (2001)

  13. Krantz S.G.: Function theory of several complex variables. The Wadsworth & Brooks/Cole Mathematics Series, 2nd edn. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove (1992)

    Google Scholar 

  14. Lorenz E.N.: Deterministic non-periodic flow. J. Atmos. Sci. 20, 130–141 (1963)

    Article  Google Scholar 

  15. Martens M., de Melo W., van Strien S.: Julia-Fatou-Sullivan theory for real one-dimensional dynamics. Acta Math. 168(3–4), 273–318 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  16. Martens M., van Strien S., de Melo W., Mendes P.: On Cherry flows. Ergodic Theory Dyn. Syst. 10(3), 531–554 (1990)

    Article  MATH  Google Scholar 

  17. Moreira P.C., Ruas A.A.G.: Metric properties of Cherry flows. J. Differ. Equ. 97(1), 16–26 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  18. Mujica, J.: Complex analysis in Banach spaces. In: North-Holland Mathematics Studies, vol. 120. Holomorphic functions and domains of holomorphy in finite and infinite dimensions, Notas de Matemática [Mathematical Notes], 107. North-Holland, Amsterdam. (1986)

  19. Nikolaev, I., Zhuzhoma, E.V.: Flows on 2-dimensional manifolds. In: Lecture Notes in Mathematics, vol. 1705. Springer, Berlin (1999)

  20. Palis J., de Melo W.: An Introduction. Translated from the Portuguese by A. K. Manning. Springer, New York (1982)

    Google Scholar 

  21. St. Pierre M.: Topological and measurable dynamics of Lorenz maps. Dissertationes Math. (Rozprawy Mat.) 382, 134 (1999)

    MathSciNet  Google Scholar 

  22. van Strien S.: One-dimensional dynamics in the new millennium. Discrete Contin. Dyn. Syst. 27(2), 557–588 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. van Strien, S., Vargas E.: Real bounds, ergodicity and negative Schwarzian for multimodal maps. J. Am. Math. Soc. 17(4), 749–782 (2004)

    Article  MATH  Google Scholar 

  24. Williams, R.F.: The structure of Lorenz attractors. In: Turbulence Seminar (Univ. Calif., Berkeley, Calif., 1976/1977), pp. 94–112. Lecture Notes in Math., vol. 615. Springer, Berlin (1977)

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Authors and Affiliations

  1. Faculdade de Ciências e Tecnologia da Universidade Estadual Paulista, Presidente Prudente, São Paulo, Brazil

    Márcio Ricardo Alves Gouveia

  2. Instituto de Matemática e Estatística da Universidade de São Paulo, São Paulo, Brazil

    Eduardo Colli

Authors
  1. Márcio Ricardo Alves Gouveia
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  2. Eduardo Colli
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Correspondence to Eduardo Colli.

Additional information

M. R. A. Gouveia was partially supported by brazilian agency Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and Programa Primeiros Projectos - PROPe/UNESP, and Eduardo Colli by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP).

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Gouveia, M.R.A., Colli, E. The Lamination of Infinitely Renormalizable Dissipative Gap Maps: Analyticity, Holonomies and Conjugacies. Qual. Theory Dyn. Syst. 11, 231–275 (2012). https://doi.org/10.1007/s12346-011-0058-5

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