On invariant measures of "satellite" infinitely renormalizable quadratic polynomials

TL;DR

This paper investigates the invariant measures of certain infinitely renormalizable quadratic polynomials, showing they are supported on the postcritical set with zero Lyapunov exponent, thus advancing understanding of their dynamical behavior.

Contribution

It proves that for satellite infinitely renormalizable quadratic polynomials, all invariant measures on a specific set are supported on the postcritical set with zero Lyapunov exponent, answering a question about their dynamical properties.

Findings

Invariant measures supported on postcritical set

Lyapunov exponent is zero for these measures

Supports partial answer to Shen's question

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.