On invariant measures of 'satellite' infinitely renormalizable quadratic polynomials

Genadi Levin; Feliks Przytycki

doi:10.1017/etds.2024.85

On invariant measures of 'satellite' infinitely renormalizable quadratic polynomials

Genadi Levin, Feliks Przytycki

Einstein Institute of Mathematics

The Hebrew University of Jerusalem

Research output: Contribution to journal › Article › peer-review

Abstract

Let be an infinitely renormalizable quadratic polynomial and 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 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.

Original language	English
Pages (from-to)	1843-1869
Number of pages	27
Journal	Ergodic Theory and Dynamical Systems
Volume	45
Issue number	6
DOIs	https://doi.org/10.1017/etds.2024.85
State	Published - 1 Jun 2025

Keywords

Julia sets
Lyapunov exponent
infinitely renormalizable
invariant measures
iteration of complex polynomials

All Science Journal Classification (ASJC) codes

General Mathematics
Applied Mathematics

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10.1017/etds.2024.85

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N2 - Let be an infinitely renormalizable quadratic polynomial and 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 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.

AB - Let be an infinitely renormalizable quadratic polynomial and 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 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.

KW - Julia sets

KW - Lyapunov exponent

KW - infinitely renormalizable

KW - invariant measures

KW - iteration of complex polynomials

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DO - 10.1017/etds.2024.85

M3 - مقالة

SN - 0143-3857

VL - 45

SP - 1843

EP - 1869

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

IS - 6

ER -

On invariant measures of 'satellite' infinitely renormalizable quadratic polynomials

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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