Multicorns are not Path Connected

2012/09/08 by John H. Hubbard, Hubbard, John, Dierk Schleicher +2 · 3 citations
Mathematics · #37F10 #37F20 #37F45 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.1209.1753

openalex publication_date 2012/09/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The "multicorn" is the connectedness locus of unicritical antiholomorphic polynomials z↦ rz^d+c; the special case d=2 was named "tricorn" by Milnor. It appears as a natural local configuration in spaces of real cubic polynomials. We prove that no multicorn for d≥ 2 is pathwise connected, confirming a classical prediction based on numerical observations.

Citations

• Hyperbolic components in spaces of polynomial maps
• ON MULTICORNS AND UNICORNS I: ANTIHOLOMORPHIC DYNAMICS, HYPERBOLIC COMPONENTS AND REAL CUBIC POLYNOMIALS

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• ON MULTICORNS AND UNICORNS I: ANTIHOLOMORPHIC DYNAMICS, HYPERBOLIC COMPONENTS AND REAL CUBIC POLYNOMIALS
• Accessible hyperbolic components in anti-holomorphic dynamics
• Discontinuity of Straightening in Anti-holomorphic Dynamics: I
• Self-similarity for the tricorn
• Geography of the cubic connectedness locus I: Intertwining surgery