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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The Combinatorics of OneDimensional Endomorphisms
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absolutely continuous absolutely continuous invariant assume attracting periodic basin Beltrami coefficient Beltrami path Beltrami vector Cantor set closure combinatorially equivalent compact conjugacy conjugate continuous invariant probability contradiction converges Corollary covering map critical point cross-ratio defined definition diffeomorphism disjoint dynamical equal ergodic finite number fixed point forward invariant forward orbit function Furthermore Hence homeomorphism hyperbolic implies infinitely renormalizable integer intersection invariant measure invariant probability measure iterates Julia set kneading invariants Koebe Principle Lebesgue measure Lemma maps with negative metric Misiurewicz monotone Moreover negative Schwarzian derivative neighbourhood parameter periodic attractor periodic orbit periodic points piecewise Poincare Poincare metric point of period points of g probability measure proof of Theorem Proposition pullback quadratic differential quadratic-like map quasiconformal homeomorphism renormalizable renormalization restrictive interval result return map Riemann sphere Riemann surface Riemann surface lamination rotation number satisfies Schwarzian derivative Section sequence space Statement subset topological entropy turning point unimodal map wandering interval zero