Computer assisted proofs in analysis
INFORMATIQUE
ORDINATEURS
DEMONSTRATION
CONFERENCES ET CONGRES
NOMBRES REELS
ANALYSE MATHEMATIQUE
LANFORD
P/87/16
IHES
05/1985
A4
5 f.
EN
TEXTE
PREPUBLICATION
P_87_16.pdf
1987
Existence and properties of p-tupling fixed points
RENORMALISATION
CARTOGRAPHIE
THEOREME DU POINT FIXE
Abstract : We prove the existence of fixed points of p-tupling renormalization operators for interval and circle mappings having a critical point of arbitrary real degree r > 1. Some properties of the resulting maps are studied: analyticity, univalence, behavior as r tends to infinity.
EPSTEIN
P/00/25
IHES
03/2000
A4
19 f.
EN
TEXTE
PREPUBLICATION
P_00_25.pdf
2000
Fixed points of composition operators
TOPOLOGIE
THEOREME DU POINT FIXE
OPERATEURS DE COMPOSITION
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].
EPSTEIN
P/87/36
IHES
09-1987
A4
17 f.
EN
TEXTE
PREPUBLICATION
P_87_36.pdf
1987
News proofs of the existence of the Feigenbaum functions
EQUATIONS FONCTIONNELLES
SYSTEMES DYNAMIQUES
Abstract : A new proof of the existence of analytic, unimodal soutions of the Cvitanovic-Feigenbaum functional equation ?g (x) = -g(g-?x)), g(x) ? 1-const. |x| r at 0, walid for all ? in (0,1), is given, and the existence of the Eckmann-Wittwer functions [8] is recovered. The method also provides the existence of solutions for certain given values of r, and in particular, for r=2, a proof requiring no computer.
EPSTEIN
P/85/55
IHES
10/1985
A4
22 f.
EN
TEXTE
PREPUBLICATION
P_85_55.pdf
1985
On the Existence of fixed points of the composition operator for circle maps
RESEAUX CEREBRAUX
PHYSIQUE STATISTIQUE
SYSTEMES COMPLEXES
THEORIES NON LINEAIRES
DYNAMIQUE
INFORMATIQUE QUANTIQUE
Abstract : In the theory of circle maps with golden ratio rotation number formulated by Feigenbaum, Kadanoff, and Shenker [FKS], and by Ostlund, Rand, Sethna, and Siggia [ORSS], a central role is played by fixed points of a certain composition operator in map space. We define a common setting for the problem of proving the existence of these fixed points and of those occurring in the theory of maps of the interval. We give a proof of the existence of the fixed points for a wide range of the parameters on which they depend.
EPSTEIN
ECKMANN
P/86/29
IHES
05/1986
A4
13 f.
EN
TEXTE
PREPUBLICATION
P_86_29.pdf
1986