Loading…
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_\i...
Saved in:
Published in: | arXiv.org 2024-09 |
---|---|
Main Authors: | , |
Format: | Article |
Language: | English |
Subjects: | |
Online Access: | Get full text |
Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
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 [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. |
---|---|
ISSN: | 2331-8422 |