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Smooth transformations of intervals

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Smooth transformations of intervals SÉMINAIRE N. BOURBAKI OSCAR E. LANFORD III Smooth transformations of intervals Séminaire N. Bourbaki, 1980-1981, exp. no 563, p. 36-54. <> © Association des collaborateurs de Nicolas Bourbaki, 1980-1981, tous droits réservés. L’accès aux archives du séminaire Bourbaki (http://www.bourbaki. implique l’accord avec les conditions générales d’utilisa- tion ( Toute utilisation commer- ciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques 36 Séminaire BOURBAKI 33e année, 1980/81, n° 563 Novembre 1980 SMOOTH TRANSFORMATIONS OF INTERVALS par Oscar E. LANFORD III § 1. Introduction The theory of differentiable dynamical systems investigates the orbit struc- tures of one parameter groups - discrete or continuous - of diffeomorphisms. For many applications, the principal questions are : What is the asymptotic behavior of typical orbits ? (theory of attractors) How does this behavior change as the generator of the group changes conti- nuously ? (bifurcation theory) This report is an introduction to what is known about these questions in the case, which ought to be the simplest possible, where the space on which the transformations act is a compact interval. Even in this simple context, compli- cated behavior is not only possible but inevitable.There is, however, relatively simple theory which accounts for much of the complexity. This is a subject with the charm of concreteness ; all the phenomena to be described occur in the one- parameter family of mappings on [--1,1] , where the parameter p is in [0,2] . Although limits of time and space prevent elaboration on this point, it should be mentioned that much of wha

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