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Rational billiards and flat structures. (English) Zbl 1057.37034
Hasselblatt, B. (ed.) et al., Handbook of dynamical systems. Volume 1A. Amsterdam: North-Holland (ISBN 0-444-82669-6/hbk). 1015-1089 (2002).
This is a survey of many relatively recent developments in the study of rational billiards and related subjects. It is divided into 6 main parts, beginning with an overview of rational billiards and its connections to other studies, including flat structures on surfaces, interval exchange maps, and quadratic differentials. The second part deals with the structure of the space of quadratic differentials and Teichmüller space. The third and fourth parts deal with ergodic properties and properties related to periodic orbits, respectively. Part 5 describes Veech groups and surfaces and some of their properties, and part 6 studies interval exchange maps in more detail. A seventh part contains a few brief descriptions of several related matters. Proofs of a number of results are given; for the others there are detailed references.
For the entire collection see [Zbl 1013.00016].

37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
37A25 Ergodicity, mixing, rates of mixing
37F99 Dynamical systems over complex numbers
28D05 Measure-preserving transformations
30F60 Teichmüller theory for Riemann surfaces