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Frontiers in Complex DynamicsIn Celebration of John Milnor's 80th Birthday$
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Araceli Bonifant, Misha Lyubich, and Scott Sutherland

Print publication date: 2014

Print ISBN-13: 9780691159294

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691159294.001.0001

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Multicorns are not path connected

Multicorns are not path connected

(p.73) Multicorns are not path connected
Frontiers in Complex Dynamics

John Hamal Hubbard

Dierk Schleicher

, Araceli Bonifant, Mikhail Lyubich, Scott Sutherland
Princeton University Press

This chapter proves that the tricorn is not locally connected and not even pathwise connected, confirming an observation of John Milnor from 1992. The tricorn is the connectedness locus in the space of antiholomorphic quadratic polynomials z ↦ ̄z² + c. The chapter extends this discussion more generally for antiholomorphic unicritical polynomials of degrees d ≥ 2 and their connectedness loci, known as multicorns. The multicorn M*subscript d is the connectedness locus in the space of antiholomorphic unicritical polynomials psubscript c(z) = ̄zsubscript d + c of degree d, i.e., the set of parameters for which the Julia set is connected. The special case d = 2 is the tricorn, which is the formal antiholomorphic analog to the Mandelbrot set.

Keywords:   tricorn, multicorn, antiholomorphic quadratic polynomials, antiholomorphic unicritical polynomials, connectedness loci, Maldelbrot set

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