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arXiv · 2002.12000

Statistical properties of periodic points for infinitely renormalizable unimodal maps

Hiroki Takahasi
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Abstract

For an infinitely renormalizable negative Schwarzian unimodal map $f$ with non-flat critical point, we analyze statistical properties of periodic points as the periods tend to infinity. Introducing a weight function $φ$ which is a continuous or a geometric potential $φ=-β\log|f'|$ ($β\in\mathbb R$), we establish the level-2 Large Deviation Principle for weighted periodic points. From this, we deduce that all weighted periodic points equidistribute with respect to equilibrium states for the potential $φ$. In particular, it follows that all periodic points are equidistributed with respect to measures of maximal entropy, and all periodic points weighted with their Lyapunov exponents are equidistributed with respect to the post-critical measure supported on the attracting Cantor set.

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Hiroki Takahasi. 2021-03-30. Statistical properties of periodic points for infinitely renormalizable unimodal maps. https://arxiv.org/abs/2002.12000

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