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Cubic polynomials: Turning around the connectedness locus. (English) Zbl 0801.58024
Goldberg, Lisa R. (ed.) et al., Topological methods in modern mathematics. Proceedings of a symposium in honor of John Milnor’s sixtieth birthday, held at the State University of New York at Stony Brook, USA, June 14-June 21, 1991. Houston, TX: Publish or Perish, Inc. 391-427 (1993).
By the aid of the natural isomorphism \(z^ 3+ c_ 1z+ c_ 0\to [c_ 1,c_ 0]\) we can identify the points of \(\mathbb{C}^ 2\) with cubic polynomials. The paper contains a review of some known results concerning the topological properties of certain subsets of \(\mathbb{C}^ 2\) and the dynamical behaviours of polynomials corresponding, via the above mapping, to these subsets.
There are new results concerning cubic polynomials \(p\) with Julia set which is a Cantor set. The main device of the investigations is the wring construction defined in Acta Math. 160, No. 3/4, 143-206 (1988; Zbl 0668.30008) by the author and J. H. Hubbard. The result of this construction applied to \(p\) and the line parallel to the imaginary axes through 1 is defined as the turning curve \({\mathcal T}(p)\) through \(p\).
Most of the new results of the paper deal with the connections between certain dynamical properties of cubic polynomials \(p\) and the topological properties of the turning curve \({\mathcal T}(p)\). The paper ends with the open question: “We do not know if the turning curve through a critically recurrent polynomial \(p\) is infinite in general”.
For the entire collection see [Zbl 0780.00031].

MSC:
37B99 Topological dynamics
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
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