Multicorns are not Path Connected
John Hubbard, Dierk Schleicher

TL;DR
This paper proves that multicorns, the connectedness loci of certain antiholomorphic polynomials, are not path-connected for degrees two and higher, confirming previous numerical predictions.
Contribution
It establishes the non-path-connectedness of multicorns for all degrees greater than or equal to two, a significant theoretical result in complex dynamics.
Findings
Multicorns are not path-connected for all degrees d ≥ 2.
Confirmed classical predictions based on numerical observations.
Provides a rigorous proof of the topological structure of multicorns.
Abstract
The "multicorn" is the connectedness locus of unicritical antiholomorphic polynomials ; the special case was named "tricorn" by Milnor. It appears as a natural local configuration in spaces of real cubic polynomials. We prove that no multicorn for is pathwise connected, confirming a classical prediction based on numerical observations.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Differential Equations and Dynamical Systems · Mathematical Dynamics and Fractals · Algebraic Geometry and Number Theory
