Quartic Julia sets including any two copies of quadratic Julia sets
Let \(P\) be a polynomial of degree \(d\geqslant 2\) and denote by \(K(P)\) the filled-in Julia set of \(P\), which is the set of points that have bounded orbit under iteration by \(P\). It is well-known that if all finite critical points of \(P\) escape to infinity, then \(K(P)\) is totally disconnected, while \(K(P)\) is connected if and only if all finite critical points have a bounded orbit. This paper focuses on the remaining intermediate cases, where only some of the critical points escape to infinity, and hence \(K(P)\) is disconnected but not totally disconnected. In this situation, if we pick an equipotential curve surrounding \(K(P)\) such that it passes through an escaping critical value, there is a preimage \(\gamma\) of this curve that contains the critical point and has a figure-eight shape. The restriction of \(P\) to each bounded complementary component \(U\) of \(\gamma\) defines what is known as a polynomial-like map \((f,U,U')\) of degree \(d'<d\), where \(f=P|_{U}\) and \(U'=P(U)\). Then, one can define the filled-in Julia set of such map by considering the points that never leave \(U\), which is a subset of \(K(P)\). \textit{A. Douady} and \textit{J. H. Hubbard} [Ann. Sci. Éc. Norm. Supér. (4) 18, 287--343 (1985; Zbl 0587.30028)] introduced the notion of polynomial-like mapping to explain the appearance of sets resembling quadratic filled-in Julia sets in the dynamical planes of higher degree polynomials and transcendental entire functions, as well as copies of the Mandelbrot set in the parameter spaces of families of such functions. They showed that every polynomial-like map \(f\) is hybrid equivalent to a polynomial \(P\) of the same degree and, in particular, the filled Julia set of \(f\) is a quasiconformal image of \(K(P)\). This paper looks at the converse problem: given two filled-in Julia sets of quadratic polynomials \(P_1,P_2\), can we find a polynomial \(P\) of degree \(4\) such that \(K(P)\) contains quasiconformal copies of \(K(P_1)\) and \(K(P_2)\)? The author provides a positive answer to this question by constructing the polynomial \(P\) using quasiconformal surgery.
scientific article; zbMATH DE number 6498605