Parabolic Implosion in the Parameter Space of Cubic Polynomials

TL;DR

This paper explores how small changes in cubic polynomial families with parabolic fixed points lead to complex topological changes in their parameter spaces and Julia sets, extending known quadratic results.

Contribution

It analyzes the topology of parabolic implosion in cubic polynomial parameter spaces and connects it to quadratic Julia set enrichment, providing new insights into bifurcation phenomena.

Findings

Topological structure of parabolic implosion in cubic parameter space

Relation between cubic parameter space and quadratic Julia set enrichment

Extension of Lavaurs' quadratic results to cubic polynomials

Abstract

Parabolic implosion describes the enrichment of Julia sets when a parabolic fixed point is perturbed. It is also natural to study parabolic implosion in parameter spaces. In particular, when one perturbs properly the family of cubic polynomials having a stable parabolic fixed point into nearby families, the enrichment of the bifurcation loci occurs. We investigate the topology of such enrichment in the parameter space of cubic polynomials and relate it to the corresponding enrichment of Julia sets of quadratic polynomials, the latter of which has been studied systematically by P. Lavaurs in the 80s.