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Explosion of Smoothness for Conjugacies Between Unimodal Maps

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Dynamics, Games and Science II

Part of the book series: Springer Proceedings in Mathematics ((PROM,volume 2))

Abstract

Let f and g be C r unimodal maps, with r ≥ 3, topologically conjugated by h and without periodic attractors. If h is differentiable at a point p in the expanding set E(f), with h′(p)≠0, then, there is an open renormalization interval J such that h is a C r diffeomorphism in the basin B(J) of J, and h is not differentiable at any point in I ∖ B(J). The expanding set E(f) contains all points with positive Lyapunov exponent, and if f has a Milnor’s interval cycle attractor A then E(f) has full Lebesgue measure.

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Acknowledgements

Alberto Pinto would like to thank LIAAD-INESC Porto LA, Calouste Gulbenkian Foundation, PRODYN-ESF, POCTI and POSI by FCT and Ministério da Ciência e da Tecnologia, and the FCT Pluriannual Funding Program of the LIAAD-INESC Porto LA and of the Research Centre of Mathematics of University of Minho, for their financial support.

Author information

Authors and Affiliations

  1. Departamento de Matemática Pura, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre 687, 4169-007, Porto, Portugal

    José F. Alves

  2. Departamento de Matemática, Universidade Federal da Bahia, Av. Ademar de Barros s/n, 40170-110, Salvador, Brazil

    Vilton Pinheiro

  3. LIAAD-INESC Porto LA, Department of Mathemátics, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre 687, 4169-007, Porto, Portugal

    Alberto A. Pinto

  4. Centro de Matemática e Departamento de Matemática e Aplicações, Escola de Ciências, Universidade do Minho, Campus de Gualtar, 4710-057, Braga, Portugal

    Alberto A. Pinto

Authors
  1. José F. Alves
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  2. Vilton Pinheiro
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  3. Alberto A. Pinto
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Corresponding author

Correspondence to José F. Alves .

Editor information

Editors and Affiliations

  1. Pura e Aplicada (IMPA), Instituto de Matemática, Estrada Dona Castorina 110, Rio de Janeiro, 22460-320, Rio de Janeiro, Brazil

    Mauricio Matos Peixoto

  2. Faculdade de Ciências, Departamento de Matemática, Universidade do Porto, Rua do Campo Alegre 687, Porto, 4169-007, Portugal

    Alberto Adrego Pinto

  3. Warwick Systems Biology, University of Warwick, Coventry House, Coventry, CV4 7AL, United Kingdom

    David A. Rand

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Alves, J.F., Pinheiro, V., Pinto, A.A. (2011). Explosion of Smoothness for Conjugacies Between Unimodal Maps. In: Peixoto, M., Pinto, A., Rand, D. (eds) Dynamics, Games and Science II. Springer Proceedings in Mathematics, vol 2. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14788-3_8

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