## 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].
Reviewer: L.Maté (Budapest)

### MSC:

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

### Keywords:

cubic polynomials; Julia set; Cantor set; turning curve

Zbl 0668.30008