Analyticity properties of the Feigenbaum function
RESEAUX CEREBRAUX
PHYSIQUE STATISTIQUE
SYSTEMES COMPLEXES
THEORIES NON LINEAIRES
DYNAMIQUE
INFORMATIQUE QUANTIQUE
Absract : Analyticity properties of the Feigenbaum function [a solution ofg(x)=???1g(g(?x)) withg(0)=1,g?(0)=0,g?(0)<0] are investigated by studying its inverse function which turns out to be Herglotz or anti-Herglotz on all its sheets. It is found thatg is analytic and uniform in a domain with a natural boundary.
EPSTEIN
LASCOUX
P/81/27
IHES
05/1981
A4
27 f.
EN
TEXTE
PREPUBLICATION
P_81_27.pdf
1981
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
Fixed points of composition operators II
TOPOLOGIE
THEOREME DU POINT FIXE
OPERATEURS DE COMPOSITION
THEORIE DES POINTS CRITIQUES
Abstract : Analytic unicritical fixed points of composition operators of Feigenbaum's type for inteval and circle maps are shown to exist for every value of r > 1, where r is the order of the critical point.
EPSTEIN
P/88/21
IHES
04/1988
A4
5 f.
EN
TEXTE
PREPUBLICATION
P_88_21.pdf
1988
On the Structure of infinitely many dynamical systems nested inside or outside a given one
SYSTEMES DYNAMIQUES
RENORMALISATION
PHYSIQUE
Abstract : In the content of smooth folding mappings we show bounded return time renormalization is topologically hyperbolic and find the stable and unstable manifolds. The main consequence is the asymptotic geometric rigidity of the Cantor sets defined by the critical orbits. We use the Teichmüller Contraction Principle to prove renormalization contraction in a space of holomorphic dynamical systems that contains the limit set of renormalization.
SULLIVAN
M/90/75
IHES
09/1990
A4
25 f.
EN
TEXTE
PREPUBLICATION
M_90_75.pdf
1990
Bounds, quadratic differentials and renormalization conjectures
RENORMALISATION
PHYSIQUE
THEOREME DU POINT FIXE
ANALYSE VECTORIELLE
Abstract : In the context of smooth folding mappings we verifie certain conjecture by showing bounded return time renormalization is topologically hyperbolic and find the stable and unstable manifolds. The main consequence is the asymptotic geometric rigidity of the Cantor sets defined by the critical orbits. We use the bounds and quadratic differentials on Riemann surface lamination to prove exponential renormalization contraction in a space of holomorphic dynamical systems that contains the limit set of renormalization.
SULLIVAN
M/91/25
IHES
03/1991
A4
25 f.
EN
TEXTE
PREPUBLICATION
M_91_25.pdf
1991