Parapuzzle of the Multibrot set and typical dynamics of unimodal maps

TL;DR

This paper investigates the parameter space of unicritical polynomials, showing that for almost all parameters, the dynamics are either hyperbolic, renormalizable, or Collet-Eckmann, with results based on parapuzzle analysis.

Contribution

It establishes measure-theoretic properties of unicritical polynomial dynamics, extending understanding of typical behaviors in complex and real parameter spaces.

Findings

Almost all complex parameters yield hyperbolic or infinitely renormalizable maps.

Almost all real parameters yield hyperbolic, Collet-Eckmann, or infinitely renormalizable maps.

Analysis is based on controlling spacing in the principal nest of parapuzzle pieces.

Abstract

We study the parameter space of unicritical polynomials $f_c:z\mapsto z^d+c$. For complex parameters, we prove that for Lebesgue almost every $c$, the map $f_c$ is either hyperbolic or infinitely renormalizable. For real parameters, we prove that for Lebesgue almost every $c$, the map $f_c$ is either hyperbolic, or Collet-Eckmann, or infinitely renormalizable. These results are based on controlling the spacing between consecutive elements in the ``principal nest'' of parapuzzle pieces.