Cubic Siegel polynomials and the bifurcation measure

TL;DR

This paper proves that certain cubic polynomials with a fixed Siegel disk are in the support of the bifurcation measure, revealing new insights into the structure of parameter spaces and answering a previously open question.

Contribution

It establishes the presence of these polynomials in the bifurcation measure support and demonstrates the existence of holomorphic disks therein, advancing understanding of cubic polynomial dynamics.

Findings

Cubic polynomials with a fixed Siegel disk are in the bifurcation measure support.

The set of rigid parameters is not closed in the moduli space.

Existence of holomorphic disks in the support of the bifurcation measure.

Abstract

We prove that cubic polynomial maps with a fixed Siegel disk and a critical orbit eventually landing inside that Siegel disk lie in the support of the bifurcation measure. This answers a question of Dujardin in positive. Our result implies the existence of holomorphic disks in the support of the bifurcation measure, and also implies that the set of rigid parameters is not closed in the moduli space of cubic polynomials.