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One-dimensional maps. (English) Zbl 1054.37016

Hasselblatt, B. (ed.) et al., Handbook of dynamical systems. Volume 1A. Amsterdam: North-Holland (ISBN 0-444-82669-6/hbk). 599-664 (2002).
This is an in-depth survey article on one-dimensional dynamics. It includes classical results such as the Sharkovskii theorem and the kneading theory, as well as some of the most recent developments and current directions. Extensive references are provided.
The following is how the paper is organized: Section 1 on Topological theory includes continuous maps, circle homeomorphisms, piecewise monotone maps, topological theory of unimodal maps, and tent maps. Section 2 on Distortion theory includes distortion estimates, cross-ratio distortion, and cross-ratio distortion under iterates. Section 3 on Ergodic theory includes expanding maps, absolutely continuous invariant measures for non-expanding maps, stochastic properties, and typical behavior of points for S-unimodal maps. Section 4 on Hyperbolic theory includes symbolic dynamics and topological Markov chains, Axiom A and stability, and hyperbolic sets. Section 5 on Geometric theory includes wandering intervals, circle automorphisms, topological invariance of the Collet-Eckmann condition, and the Feigenbaum universality law. Section 6 on Complex analytic theory includes universality, inducing and box mappings, the real Fatou conjecture, and analytic linearization.
Other detailed survey articles and monographs in one-dimensional dynamics include: L. S. Block and W. A. Coppel [Dynamics in one dimension, Lecture Notes in Mathematics. 1513. Berlin etc.: Springer-Verlag (1992; Zbl 0746.58007)], J. Graczyk and G. Swiatek [Ergodic Theory Dyn. Syst. 19, 263–287 (1999; Zbl 0941.37024)], and E. de Faria and W. de Melo [One dimensional dynamics: The mathematical tools. 23rd Brazilian mathematics colloquium, IMPA Mathematical Publications. Rio de Janeiro: Instituto de Matemática Pura e Aplicada (2001; Zbl 0990.37035)].
For the entire collection see [Zbl 1013.00016].

MSC:

37E05 Dynamical systems involving maps of the interval
37E10 Dynamical systems involving maps of the circle
37F25 Renormalization of holomorphic dynamical systems
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
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