Discontinuity of straightening in anti-holomorphic dynamics: II

TL;DR

This paper rigorously defines Tricorn-like sets in real cubic polynomial parameter spaces, demonstrates the discontinuity of the straightening map, and proves rigidity theorems for polynomial parabolic germs.

Contribution

It introduces a formal definition of Tricorn-like sets as renormalization loci and establishes the discontinuity of the straightening map in this context.

Findings

Tricorn-like sets are rigorously defined as renormalization loci.

The straightening map from these sets to the Tricorn is discontinuous.

Rigidity theorems allow recovery of polynomials from parabolic germs.

Abstract

In [M3], Milnor found Tricorn-like sets in the parameter space of real cubic polynomials. We give a rigorous definition of these Tricorn-like sets as suitable renormalization loci, and show that the dynamically natural straightening map from such a Tricorn-like set to the original Tricorn is discontinuous. We also prove some rigidity theorems for polynomial parabolic germs, which state that one can recover unicritical holomorphic and anti-holomorphic polynomials from their parabolic germs.