Surgery on complex polynomials.

*(English)*Zbl 0668.58026
Holomorphic dynamics, Proc. 2nd Int. Colloq. Dyn. Syst., Mexico City/Mex. 1986, Lect. Notes Math. 1345, 11-72 (1988).

[For the entire collection see Zbl 0653.00010.]

We present two results concerning the parameter spaces for quadratic and cubic polynomials, considered as dynamical systems. The results are obtained by surgery in the dynamical planes. The limb \(M_{}\) of the Mandelbrot set M is shown to be homeomorphic to a subset of the limb \(M_{1/3}\) of M and to a subset of the connectedness locus of cubic polynomials. A consequence of the first homeomorphism is that the principal vein of \(M_{1/3}\) is a topological arc. This result can be viewed as a step towards the local connectivity conjecture for M, since any connected compact metric space which is locally connected is arcwise connected. It was pointed out to us by Ben Bielefeld that the proof of lemma 5 in section 11 is incorrect. A new proof of the lemma is available.

We present two results concerning the parameter spaces for quadratic and cubic polynomials, considered as dynamical systems. The results are obtained by surgery in the dynamical planes. The limb \(M_{}\) of the Mandelbrot set M is shown to be homeomorphic to a subset of the limb \(M_{1/3}\) of M and to a subset of the connectedness locus of cubic polynomials. A consequence of the first homeomorphism is that the principal vein of \(M_{1/3}\) is a topological arc. This result can be viewed as a step towards the local connectivity conjecture for M, since any connected compact metric space which is locally connected is arcwise connected. It was pointed out to us by Ben Bielefeld that the proof of lemma 5 in section 11 is incorrect. A new proof of the lemma is available.

Reviewer: B.Branner

##### MSC:

37F45 | Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010) |

30D05 | Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable |