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Yoccoz puzzles for mappings with rational allure. (Puzzles de Yoccoz pour les applications à allure rationnelle.) (French) Zbl 0977.37023
The author extracts very interesting results by extending Yoccoz’s techniques on puzzles on rational functions. He also proves an interesting result concerning the boundary of the immediate basin of attraction of a fixed point which is also critical point of a polynomial. In particular, he proves the following: Suppose that we have a polynomial of degree \(d+1\), with \(d \geq 2\) and suppose that a fixed point is also a critical point of multiplicity \(d-1\). Then this polynomial is conjugate to a polynomial \[ f(z)=\alpha + \Biggl(z+\frac{(d+2)\alpha}{d}\Biggr) (z- \alpha)^d, \] where \(\alpha\) is the fixed point which is also critical of multiplicity \(d-1\). Let \(B(\alpha)\) be the immediate basin of attraction of \(\alpha\). Then he proves that the \(\partial B(\alpha)\) is a Jordan curve.
In the sequel the author proves some interesting lemmas on the separation of the plane by using external rays.

37F20 Combinatorics and topology in relation with holomorphic dynamical systems
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets