# zbMATH — the first resource for mathematics

Matings of quadratic polynomials. (English) Zbl 0756.58024
We apply Thurston’s equivalence theory between dynamical systems of postcritically finite branched coverings and rational maps to try to construct, from a pair of polynomials, a rational map. We prove that given two postcritically finite quadratic polynomials $$f_ c: z\mapsto z^ 2+c$$ and $$f_{c'}: z\mapsto z^ 2+c'$$ one can get a rational map if and only if $$c$$, $$c'$$ are not in conjugate limbs of the Mandelbrot set.
Reviewer: T.Lei (Lyon)

##### MSC:
 37B99 Topological dynamics 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
Full Text:
##### References:
 [1] Gantmacher, Matrix theory (1960) [2] Douady, A proof of Thurston’s topological characterization of rational functions · Zbl 0806.30027 · doi:10.1007/BF02392534 [3] Douady, Publ. Math. d’Orsay 84 pp none– (1984) [4] Douady, Séminaire Bourbaki 599 pp none– (1982) [5] Thurston, The combinatorics of iterated rational maps (1985) [6] Tan, Accouplements des polynômes complexes (1987) [7] Hirsch, Differential Topology (1976) · Zbl 0356.57001 · doi:10.1007/978-1-4684-9449-5 [8] Shishikura, A family of cubic rational maps and matings of cubic polynomials 88?50 (1988) [9] Shishikura, On a theorm of M. Rees for matings of polynomials (1990) · Zbl 1062.37039 [10] Rees, A partial description of parameter space of rational maps of degree two: Part II (1991) [11] Rees, A partial description of parameter space of rational maps of degree two: Part I (1990) [12] Lavaurs, C. R. Acad. Sc. 303 pp none– (1986) [13] Tan, C. R. Acad. Sc. 302 pp 635– (1986)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.