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

Available as a pdf file file

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

Available as a pdf file file

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

Available as a pdf file file

The following is a nice discussion of Veech surfaces

Available as a pdf file file

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

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