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# On invariant measures of "satellite" infinitely renormalizable quadratic polynomials

Dec 2022

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

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