# Statistical properties of periodic points for infinitely renormalizable unimodal maps

## 抄録

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.

本文言語 English 6399-6421 23 Nonlinearity 35 12 https://doi.org/10.1088/1361-6544/ac9a96 Published - 2022 12月 1

## ASJC Scopus subject areas

• 統計物理学および非線形物理学
• 数理物理学
• 物理学および天文学（全般）
• 応用数学

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