Non-landing parameter rays of the multicorns

TL;DR

This paper investigates the boundary behavior of parameter rays in multicorns, revealing that many do not land but accumulate on arcs, and identifies the existence of undecorated arcs and the non-density of Misiurewicz parameters.

Contribution

It provides a complete description of accumulation properties of rational parameter rays in multicorns, highlighting differences from the Mandelbrot set and revealing new boundary structures.

Findings

Parameter rays accumulating on odd period hyperbolic components do not land.

Existence of undecorated real-analytic arcs on multicorn boundaries.

Misiurewicz parameters are not dense on the boundary.

Abstract

It is well known that every rational parameter ray of the Mandelbrot set lands at a single parameter. We study the rational parameter rays of the multicorn $\mathcal{M}_d^*$, the connectedness locus of unicritical antiholomorphic polynomials of degree $d$, and give a complete description of their accumulation properties. One of the principal results is that the parameter rays accumulating on the boundaries of odd period (except period $1$) hyperbolic components of the multicorns do not land, but accumulate on arcs of positive length consisting of parabolic parameters. We also show the existence of undecorated real-analytic arcs on the boundaries of the multicorns, which implies that the centers of hyperbolic components do not accumulate on the entire boundary of $\mathcal{M}_d^*$, and the Misiurewicz parameters are not dense on the boundary of $\mathcal{M}_d^*$.