On invariant measures of "satellite" infinitely renormalizable quadratic polynomials

2022/12/26 by Genadi Levin, Levin, Genadi, Feliks Przytycki +1
Mathematics · #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.2212.13070

openalex publication_date 2022/12/26 · openalex created_date 2023/01/06 · openalex updated_date 2026/07/28

Abstract

Let f(z)=z²+c be an infinitely renormalizable quadratic polynomial and J_∞ 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_∞→ J_∞ 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.

Related

• No hyperbolic sets in J_∞ for infinitely renormalizable quadratic polynomials
• Satellite renormalization of quadratic polynomials
• On the Lebesgue measure of the Julia set of a quadratic polynomial
• Polynomial diffeomorphisms of C²: V. Critical points and Lyapunov exponents
• Rigidity and non local connectivity of Julia sets of some quadratic\n polynomials