Unmating of rational maps: Sufficient criteria and examples

Abstract

Abstract. Douady and Hubbard introduced the operation of mating of polynomials. This identifies two filled Julia sets and the dynamics on them via external rays. In many cases one obtains a rational map. Here the opposite question is tackled. Namely we ask when a given (postcritically finite) rational map f arises as a mating. A sufficient condition when this is possible is given. If this condition is satisfied, we present a simple explicit algorithm to unmate the rational map. This means we decompose f into polynomials, that when mated yield f . Several examples of unmatings are presented.

On The Notions of Mating

“…On the other hand, regarding the question whether a rational map arises as a mating of polynomials, we have the following result obtained in [8], see also [7] and [9]. Theorem 4.19 (Meyer).…”

Thurston's algorithm and rational maps from quadratic polynomial matings

“…Theorem 2.4 expresses when a mating can be viewed as equivalent to a rational map-but what about when a rational map can be viewed as a mating? In [9], it is shown that a postcritically finite rational map with Julia set the 2-sphere can be expressed as a mating if the map possesses a pseudo-equator :…”

“…Unmating of rational maps. In [9], Meyer comments that the existence of a pseudo equator is sufficient but not necessary to be indicative of a mating. The methods of this paper work in the quadratic cases that a pseudo equator can in some manner be found.…”

Boundary values of the Thurston pullback map

“…Another way to produce examples of Thurston maps is by mating two postcritically finite polynomials of the same degree, yielding a Thurston map of equal degree [19,20].…”