Cubic Siegel Polynomials and 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 $$\mu _{\textrm{bif}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>μ</mml:mi> <mml:mtext>bif</mml:mtext> </mml:msub> </mml:math> . This answers a question of Dujardin in positive. Our result implies the existence of holomorphic disks in the support of $$\mu _{\textrm{bif}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>μ</mml:mi> <mml:mtext>bif</mml:mtext> </mml:msub> </mml:math> , and also implies that the set of rigid parameters is not closed in the moduli space of cubic polynomials.