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!

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    $\begingroup$ For any rectangular table, if the fraction is $\lambda<1$ then the angles tend to $\frac{\lambda}{\lambda+1}\frac{\pi}{2}$ and its complement alternately. This follows upon noting that $|\theta_n-\frac{\lambda}{\lambda+1}\frac{\pi}{2}|$ becomes the fraction $\lambda$ of its value at the next step. Note that the usual billiards for a rectangular table similarly conserves the first given angle and its complement. $\endgroup$ – Lucia Jun 7 '14 at 14:32
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    $\begingroup$ reflection from a rotating billiard would capture some the effect you are seeking; see, for example, diegofregolent.com/wp-content/uploads/2014/04/… $\endgroup$ – Carlo Beenakker Jun 7 '14 at 14:53
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    $\begingroup$ an entire Ph.D. thesis devoted to the dynamics in a rotating circular billiard: etheses.dur.ac.uk/6228/1/6228_3583.PDF?UkUDh:CyT $\endgroup$ – Carlo Beenakker Jun 7 '14 at 15:41
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    $\begingroup$ When you time-reverse the dynamical law governing this system, you get $\lambda\rightarrow 1/\lambda$, in Lucia's notation. What happens when you extrapolate backwards? Lucia's analysis only applies to $\lambda<1$. For $\lambda>1$, does the ball get trapped sliding along two walls? It's not clear to me how the time-reversed dynamics are to be defined if, say, $\lambda=2$ and $\theta_n=\pi/2$; in this example, the ball has to break a symmetry. $\endgroup$ – Ben Crowell Jun 8 '14 at 0:50
  • $\begingroup$ @BenCrowell: Nice question! Maybe the right model is that, for incident $\theta \in [0,\pi/2]$, the reflected ray has $\theta' = \alpha \theta + (1-\alpha) 0$ for smaller reflected angles, and $\theta' = \alpha \theta + (1-\alpha) \pi/2$ for larger reflected angles. So $\theta=88^\circ$ could reflect to $\theta' = 89^\circ$ with $\alpha=\frac{1}{2}$. $\endgroup$ – Joseph O'Rourke Jun 8 '14 at 1:27

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).


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