Infinite ergodicity that preserves the Lebesgue measure

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  • Okubo, Ken-ichi
    Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University・Present address: Department of Information and Physical Sciences, Graduate School of Information Science and Technology, Osaka University
  • Umeno, Ken
    Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University
  • 大久保, 健一
  • 梅野, 健
  • 10358872

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Description

In this study, we prove that a countably infinite number of one-parameterized one-dimensional dynamical systems preserve the Lebesgue measure and are ergodic for the measure. The systems we consider connect the parameter region in which dynamical systems are exact and the one in which almost all orbits diverge to infinity and correspond to the critical points of the parameter in which weak chaos tends to occur (the Lyapunov exponent converging to zero). These results are a generalization of the work by Adler and Weiss. Using numerical simulation, we show that the distributions of the normalized Lyapunov exponent for these systems obey the Mittag–Leffler distribution of order 1/2.

Journal

  • Chaos

    Chaos 31 (3), 2021-03

    AIP Publishing

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