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arXiv · 2212.13070

On invariant measures of "satellite" infinitely renormalizable quadratic polynomials

Genadi Levin·Feliks Przytycki
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Abstract

Let f(z)=z^2+c be an infinitely renormalizable quadratic polynomial and J_\infty be the intersection of forward orbits of "small" Julia sets of its simple renormalizations. We prove that if f admits an infinite sequence of satellite renormalizations, then every invariant measure of f: J_\infty\to J_\infty is supported on the postcritical set and has zero Lyapunov exponent. Coupled with [G. Levin, F. Przytycki, W. Shen, The Lyapunov exponent of holomorphic maps. Invent. Math. 205 (2016), 363-382], this implies that the Lyapunov exponent of such f at c is equal to zero, which answers partly a question posed by Weixiao Shen.

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Genadi Levin, Feliks Przytycki. 2022-12-26. On invariant measures of "satellite" infinitely renormalizable quadratic polynomials. https://arxiv.org/abs/2212.13070

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