Billiard dynamics with angle of reflection a fraction of angle of incidence 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:

 
 
 

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 $\alpha=\pi/6$.

Moreover, the filling behavior is quite different. Left below shows
standard ergodic billiard dynamics when the angle of reflection is equal to the angle of incidence, while the right image shows the path
starting with the same irrational slope,
where the angle of reflection is half the angle of incidence, both for 
$200$ reflections.

 
 
 


This remarkably predictable behavior
has made me wonder (1) what are the periodic orbits in
the square for the $\frac{1}{2}$-reflection paths illustrated above,
(2) what might be the dynamics when reflection
angles are some other fraction of the incident angles, and (3) when the 
"billiard table" is a nonsquare rectangle, or other some convex shape.
Perhaps these questions have been explored. If so, I would appreciate pointers—Thanks!
 A: This kind of billiard has been studied recently, under the name "Pinball billiards" (though this term has been used in other contexts, too) or more generally "Contracting reflection laws."  The most recent appears to be


SRB measures for polygonal billiards with contracting reflection laws G. Del Magno, J. L. Dias, P. Duarte, J. P. Gaivão and D. Pinheiro, Commun. Math. Phys. 329 687-723 (2014).


As the title indicates, the emphasis is on chaotic dynamics (in contrast to polygonal billiards with the usual reflection law, for which the entropy is always zero).  Section 5 deals with generic polygons, 6 with regular polygons, 7 with acute triangles and 8 with rectangles.  An important question is whether the polygon has parallel sides, which tend to stabilise the dynamics, as your simulations have found.  There are references to earlier studies of this model, of which the earliest is


Bifurcations of periodic and chaotic attractors in pinball billiards
    with focusing boundaries. A. Arroyo, R. Markarian and D. P. Sanders, Nonlinearity 22 1499–1522 (2009).


