Multicorns are not Path Connected

TL;DR

This paper proves that multicorns, the connectedness loci of certain antiholomorphic polynomials, are not path-connected for degrees two and higher, confirming previous numerical predictions.

Contribution

It establishes the non-path-connectedness of multicorns for all degrees greater than or equal to two, a significant theoretical result in complex dynamics.

Findings

Multicorns are not path-connected for all degrees d ≥ 2.

Confirmed classical predictions based on numerical observations.

Provides a rigorous proof of the topological structure of multicorns.

Abstract

The "multicorn" is the connectedness locus of unicritical antiholomorphic polynomials $z\mapsto \bar r{z}^d+c$; the special case $d=2$ was named "tricorn" by Milnor. It appears as a natural local configuration in spaces of real cubic polynomials. We prove that no multicorn for $d\ge 2$ is pathwise connected, confirming a classical prediction based on numerical observations.