Fixed points of composition operators

Epstein, Henri



Abstract : This extended version of lectures given at eht NATO advanced Study Institute on Non-Linear Evolution and Chaotic Phenomena held in June 1987 in Noto (Italy), and directed by G. Gallovotti, A. M. Anile and P. Zweifel, will appear in the proceedings of that institute. It gives a review of the proofs of the existence of fixed points of composition operators (of Feigenbaum's type) for interval and circle maps obtained by J.-P. Eckmann and the author [E], [EE]. In addition, the fixed-r method is shown to word for all r > 1 in the case of the interval (r characterizez the order of the critical point of solutions) ; the solutions are shown to have inverses univalent in the upper and lower half-planes, and, in the case of the interval, for even integer r, to be polynomial-like in the sense of Douady and Hubbard [DH].




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EPSTEIN, “Fixed points of composition operators,” Archives de l'IHES, consulté le 22 juin 2024,