Smooth symmetric and Lorenz models for unimodal maps

Ming-Chia Li*, Mikhail Malkin

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

Abstract

For a given unimodal map F : I → I on the interval I, we consider symmetric unimodal maps (models) so that they are conjugate to F. The question motivated by [Gambaudo & Tresser, 1992] is the following: whether it is possible for symmetric model to preserve smoothness of the initial map F? We construct a symmetric model which is proved to be as smooth as F provided F has a nonflat turning point with sufficient "reserve of local evenness" at the turning point (in terms of one-sided higher derivatives at the turning point, see Definition 2.4 and Theorem 2.7). We also consider from different points of view the relationship between dynamical and ergodic properties of unimodal maps and of symmetric Lorenz maps. In particular, we present a one-to-one correspondence preserving the measure theoretic entropy, between the set of invariant measures of a symmetric unimodal map F and the set of symmetric invariant measures of the Lorenz model of F (Theorem 3.5), where by Lorenz model of F we mean the discontinuous map obtained from F by reversing its decreasing branch. Finally we extend for nonsymmetric unimodal maps, the result of Gambaudo and Tresser [1992] on Ck structural instability of the maps whose rotation interval has irrational end point (answering a question from [Gambaudo & Tresser, 1992]).

Original languageEnglish
Pages (from-to)3353-3371
Number of pages19
JournalInternational Journal of Bifurcation and Chaos in Applied Sciences and Engineering
Volume13
Issue number11
DOIs
StatePublished - 1 Jan 2003

Keywords

  • Lorenz maps
  • Maximal measures
  • One dimensional maps
  • Rotation intervals
  • Topological congugacy
  • Unimodal maps

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