One-Dimensional Dynamics
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Beschreibung
One-dimensional dynamics has developed in the last decades into a subject in its own right. Yet, many recent results are inaccessible and have never been brought together. For this reason, we have tried to give a unified ac count of the subject and complete proofs of many results. To show what results one might expect, the first chapter deals with the theory of circle diffeomorphisms. The remainder of the book is an attempt to develop the analogous theory in the non-invertible case, despite the intrinsic additional difficulties. In this way, we have tried to show that there is a unified theory in one-dimensional dynamics. By reading one or more of the chapters, the reader can quickly reach the frontier of research. Let us quickly summarize the book. The first chapter deals with circle diffeomorphisms and contains a complete proof of the theorem on the smooth linearizability of circle diffeomorphisms due to M. Herman, J.-C. Yoccoz and others. Chapter II treats the kneading theory of Milnor and Thurstonj also included are an exposition on Hofbauer's tower construction and a result on fuB multimodal families (this last result solves a question posed by J. Milnor).
This monograph gives an account of the state of the art in one-dimensional dynamical systems. The subject is studied from a combinatorial, continuous, ergodic and smooth point of view. Several results in this book are new; moreover, the exciting new developments on universality and renormalization due to D. Sullivan, are presented here in full detail for the first time. The results are presented in a unified way and with complete and thorough proofs. The study of circle maps, interval and holomorphic maps of the Riemann sphere are all shown to be based on similar principles. With this book, the reader is able to quickly get to the frontier of this exciting subject without studying many inaccessible papers.
Autor*in
Welington de Melo
Themen in »One-Dimensional Dynamics«
Dynamical Systems Dynamische Systeme Eindimensionale dynamische Systeme Interval dynamics Intervalldynamik Iterationen im Intervall Iterations on the interval One-dimensional dynamics Power Universality Universalität ergodic theory ergodicity functional analysis measure theory