Dynamics of quadratic polynomials, III: Parapuzzle and SBR measures
Dynamical Systems
2016-09-06 v1
Abstract
This is a continuation of notes on dynamics of quadratic polynomials. In this part we transfer the our prior geometric result to the parameter plane. To any parameter value c in the Mandelbrot set (which lies outside of the main cardioid and little Mandelbrot sets attached to it) we associate a ``principal nest of parapuzzle pieces'' and show that the moduli of the annuli grow at least linearly. The main motivation for this work was to prove the following: Theorem B (joint with Martens and Nowicki). Lebesgue almost every real quadratic polynomial which is non-hyperbolic and at most finitely renormalizable has a finite absolutely continuous invariant measure.
Keywords
Cite
@article{arxiv.math/9606219,
title = {Dynamics of quadratic polynomials, III: Parapuzzle and SBR measures},
author = {Mikhail Lyubich},
journal= {arXiv preprint arXiv:math/9606219},
year = {2016}
}