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Transformations holomorphes au voisinage d’une courbe de Jordan. (Holomorphic transformations around a Jordan curve). (French) Zbl 0573.58021
Consider a Jordan curve c in the complex plane and an injective holomorphic map defined in a neighbourhood of c and preserving c. It is shown that the homeomorphism induced on that curve either has a periodic point or has dense orbits. Moreover, if the rotation number of this homeomorphism satisfies some Diophantine condition, then c turns out to be real analytic. These results are used to prove the following: let f be a rational map fixing 0 such that \(f'(0)=\exp (2\pi ia)\) where a satisfies a Diophantine condition. If the boundary of the domain of stability of 0 is a Jordan curve, then this boundary contains a critical point of f. This last result has been recently generalized by M. Herman [Commun. Math. Phys. 99, 593-612 (1985)]. Finally, some constructions are given that might produce counterexamples when a does not satisfy a Diophantine condition.

37C70 Attractors and repellers of smooth dynamical systems and their topological structure