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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)

37B99 Topological dynamics
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
Full Text: DOI
[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)
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