Statistical properties of periodic points for infinitely renormalizable unimodal maps

2020/02/27 by Hiroki Takahasi, Takahasi, Hiroki
Mathematics · Physics and Astronomy · #Chaos control and synchronization #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #math.DS

paper · pdf · doi:10.48550/arxiv.2002.12000

Withdrawn due to an error in the proof

arxiv created 2021/03/30 · arxiv updated 2021/04/01

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=-βlog|f'| (β∈\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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