On invariant measures of ‘satellite’ infinitely renormalizable quadratic polynomials

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...

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Bibliographic Details
Published in:Ergodic theory and dynamical systems 2024-11, p.1-27
Main Authors: LEVIN, GENADI, PRZYTYCKI, FELIKS
Format: Article
Language:English
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Summary: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 [13], this implies that the Lyapunov exponent of such f at c is equal to zero, which partly answers a question posed by Weixiao Shen.
ISSN:0143-3857
1469-4417
DOI:10.1017/etds.2024.85