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.