This monograph gives an account of the state of the art in one-dimensional dynamical systems. It presents the theory in a unified way emphasizing the similarities and differences between invertible and non-invertible dynamics (i.e., between diffeomorphisms and endomorphisms).It starts with the invertible case: the combinatorial topological, ergodic and smooth structures are analysed extensively. Then it proceeds by showing that endomorphisms have a much richer dynamics, but that the theory for these endomorphisms can still be developed along the same lines and with similar tools. Moreover, holomorphic dynamical systems are shown to be based on similar principles. In fact, it is shown that complex analytic tools are very powerful even for the study of real one-dimensional systems. 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.
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3 of 3 people found the following review helpful:
5.0 out of 5 stars
A subject in its own right,
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This review is from: One-Dimensional Dynamics (Ergebnisse Der Mathematik Und Ihrer Grenzgebiete 3 Folge) (Hardcover)
This book is a mathematically beautiful and rigorous treatment of one dimensional dynamics. Most of the subject matter, proofs and techniques were new to me, and difficult or impossible to find elsewhere. The information seems to have been culled from a great number of sources, but is nicely tied together. An indespensable resource for anyone interested in researching dynamics from a mathematical standpoint.
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