Statistical properties of periodic points for infinitely renormalizable unimodal maps

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Abstract

For an infinitely renormalizable negative Schwarzian unimodal map f with a non-flat critical point, we analyze statistical properties of periodic points as their periods tend to infinity. Since the standard sequence of probability measures constructed from periodic points weighted with Birkhoff sums of a given potential does not always converge to an equilibrium state, we consider another sequence of probability measures obtained by averaging over certain time windows. For a weight φ which is a continuous function or a geometric potential −β log|f′|, we obtain level-2 large deviation bounds. From the upper bound, we deduce that weighted periodic points asymptotically distribute with respect to equilibrium states for the potential φ. It follows that periodic points asymptotically distribute with respect to measures of maximal entropy, and periodic points weighted with their Lyapunov exponents asymptotically distribute with respect to the post-critical measure supported on the attracting Cantor set. In the case the pressure of φ is non-positive, we obtain the level-2 large deviation principle.

Original languageEnglish
Pages (from-to)6399-6421
Number of pages23
JournalNonlinearity
Volume35
Issue number12
DOIs
Publication statusPublished - 2022 Dec 1

Keywords

  • critical points
  • equidistribution
  • interval maps
  • large deviations
  • periodic points

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Applied Mathematics

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