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Despite the popularity of this question, the only good answer that it can ever have is to point to the vast literature of hard sphere interactions. If you go to Google scholar and ask for "hard sphere interaction" with quotes, then you will see thousands of papers, mostly in physics but some in dynamical systems. Also "lorentz gas", again with quotes, matches thousands of papers. Of course you can't expect very many explicit orbit results that you sometimes get for one circle. Much of the rigorous attention is concerned with proving ergodicity without intermediate conjectures. This is still apparently open in many cases in which you would expect it to be true. Anyway, the literature is so vast that I don't think that you can reasonably expect a survey of it as an MO answer.