Combinatorial rigidity for some infinitely renormalizable unicritical polynomials

TL;DR

This paper proves combinatorial rigidity for certain infinitely renormalizable unicritical polynomials, leading to local connectivity results for their parameter spaces, including the Mandelbrot set, under specific conditions.

Contribution

It establishes combinatorial rigidity for a class of infinitely renormalizable unicritical polynomials, extending understanding of their parameter space structure.

Findings

Proves combinatorial rigidity under a priori bounds and a combinatorial condition.

Shows local connectivity of the Mandelbrot set at specific parameters.

Extends rigidity results to higher degree unicritical polynomials.

Abstract

We prove combinatorial rigidity of infinitely renormalizable unicritical polynomials, P_c :z \mapsto z^d+c, with complex c, under the a priori bounds and a certain "combinatorial condition". This implies the local connectivity of the connectedness loci (the Mandelbrot set when d = 2) at the corresponding parameters.