On a theorem of M. Rees for matings of polynomials.

*(English)*Zbl 1062.37039
Lei, Tan, The Mandelbrot set, theme and variations. Cambridge: Cambridge University Press (ISBN 0-521-77476-4/pbk). Lond. Math. Soc. Lect. Note Ser. 274, 289-305 (2000).

From the introduction: M. Rees [Acta Math. 168, 11–87 (1992; Zbl 0774.58035)] proved: If the formal mating of two hyperbolic polynomials of Thurston’s type is Thurston equivalent to a rational map, then the topological mating is conjugate to the rational map.

The main theorem of this paper claims that this is also the case even if the polynomials are merely of Thurston’s type and the formal mating can be replaced by the degenerate mating. This generalization has an interesting consequence. If we start with two polynomials of Thurston’s type which have empty filled-in Julia sets (consequently with Lebesgue measure zero), when they are “matable”, by gluing these Julia sets (which are “dendrites”), we obtain not only a topological 2-sphere but also a complex structure on it such that the induced dynamics is analytic with respect to this structure. For example, this is the case when \(z^2+ i\) is mated with itself.

The main idea of the proof is similar to Rees (loc. cit.). However, the critical points in the Julia set cause a lot of complications. The reader should notice that many of our arguments are trivial in hyperbolic case. M. Rees says she has also a proof for this generalization, so the author does not insist on the priority for it. This paper is aimed simply to provide a proof available for the sake of future development of the theory of matings.

In §1, we define the matings and state the main theorem. In §2, when two polynomials are matable, we construct a semi-conjugacy \(h\) from the formal mating to the rational map. In §3, we study the properties of \(h\). In §4, we conclude that two points are mapped by \(h\) to the same point if and only if they are ray-equivalent. Here, we use a generalization of Douady’s lemma instead of a lemma by Rees (loc. cit.). Together with this, the use of the map \(G\) is the new idea, which made it possible to prove the theorem in nonhyperbolic case.

For the entire collection see [Zbl 0935.00019].

The main theorem of this paper claims that this is also the case even if the polynomials are merely of Thurston’s type and the formal mating can be replaced by the degenerate mating. This generalization has an interesting consequence. If we start with two polynomials of Thurston’s type which have empty filled-in Julia sets (consequently with Lebesgue measure zero), when they are “matable”, by gluing these Julia sets (which are “dendrites”), we obtain not only a topological 2-sphere but also a complex structure on it such that the induced dynamics is analytic with respect to this structure. For example, this is the case when \(z^2+ i\) is mated with itself.

The main idea of the proof is similar to Rees (loc. cit.). However, the critical points in the Julia set cause a lot of complications. The reader should notice that many of our arguments are trivial in hyperbolic case. M. Rees says she has also a proof for this generalization, so the author does not insist on the priority for it. This paper is aimed simply to provide a proof available for the sake of future development of the theory of matings.

In §1, we define the matings and state the main theorem. In §2, when two polynomials are matable, we construct a semi-conjugacy \(h\) from the formal mating to the rational map. In §3, we study the properties of \(h\). In §4, we conclude that two points are mapped by \(h\) to the same point if and only if they are ray-equivalent. Here, we use a generalization of Douady’s lemma instead of a lemma by Rees (loc. cit.). Together with this, the use of the map \(G\) is the new idea, which made it possible to prove the theorem in nonhyperbolic case.

For the entire collection see [Zbl 0935.00019].

##### MSC:

37F10 | Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets |