Suppose that a billiard ball bouncing in a unit square (or a lightray reflecting in a mirrored square) has the property that the angle of reflection is a fraction of the angle of incidence, rather than equal to it. What are the dynamics?

I explored this a bit with the angle of reflection $\frac{1}{2}$ the angle of incidence, and was surprised to discover that, regardless of the starting angle, the path quickly converges to follow $30^\circ{-}60^\circ{-}90^\circ$ triangles:

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     ![RayInSquareSkewed][1]

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This is because (a) the ray alternates hitting a vertical and a horizontal square side, i.e., it never bounces twice in a row on opposing horizontal or opposing vertical sides, and (b) $\alpha = (\pi/2 - \alpha)/2$ solves to $\alpha=\pi/6$.

This remarkably predictable behavior has made me wonder what might be the dynamics when reflection angles are some other fraction of the incident angles, and when the "billiard table" is a nonsquare rectangle, or other some convex shape. Very likely these questions have been explored. If so, I would appreciate pointers—Thanks!