Local connectivity of the Julia set of real polynomials. (English) Zbl 0941.37031

Levin, Genadi; van Strien, Sebastian

Ann. Math. (2) 147, No. 3, 471-541 (1998).

The paper deals with an answer to J. Milnor. One of the important questions in complex dynamics is indeed the question whether the Mandelbrot set is locally connected, and related to this, for which maps the Julia set is locally connected. The authors prove that the Julia set of real polynomials of the form \(f(z)=z^\ell+c_1\) with \(\ell\) an even integer and \(c_1\) real is either totally disconnected or locally connected. In particular, the Julia set of \(f(z)=z^\ell+c_1\) is locally connected for \(c_1\in R\) if and only if the critical point does not escape the infinity.

Reviewer: Messoud Efendiev (Berlin)

MSC:

37F50	Small divisors, rotation domains and linearization in holomorphic dynamics
37F45	Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)

Keywords:

local connectivity; Mandelbrot set; Julia set; real polynomials; critical point