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The following is a very nice survey of the subject of translation surfaces, Teichmuller flow

Available as a pdf file
file

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The following is a survey of rational billiards, translation surfaces.

Available as a pdf file
file

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The following is a more detailed survey of the study of translation surfaces.

Available as a pdf file
file

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The following is a nice discussion of Veech surfaces

Available as a pdf file
file

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There are many books on ergodic theory.
I like the book by Peter Walters called
Introduction to Ergodic Theory

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A classical book on the subject of quadratic differentials on Riemann surfaces is the book by K. Strebel called
Quadratic differentials