On The Support of The Bifurcation Measure of Cubic Polynomials

TL;DR

This paper constructs examples of cubic polynomials with parabolic fixed points that challenge existing notions of bifurcation support, showing some maximal bifurcations are outside the measure's support.

Contribution

It introduces new cubic polynomial examples with parabolic fixed points that are not approximable by Misiurewicz polynomials, expanding understanding of bifurcation measure support.

Findings

Certain cubic polynomials with parabolic fixed points are outside the support of the bifurcation measure.

Maximal bifurcations can occur without being in the measure's support.

New examples challenge previous assumptions about bifurcation measure support.

Abstract

We construct new examples of cubic polynomials with a parabolic fixed point that cannot be approximated by Misiurewicz polynomials. In particular, such parameters admit maximal bifurcations, but do not belong to the support of the bifurcation measure.