# Ring-shaped mirror chaos points

I came across this problem in the context of spectroscopy today. Because of it's simplicity, I'm sure it's a question that's been posed by some mathematician ages ago, I just can't figure out how to formulate it mathematically:

Suppose you have a mirror that forms a perfect ring and has an aperture in it with width $\delta w$. A ray of light enters the aperture at angle $\theta$ relative to the tangent at the aperture. There are several questions that I have regarding this system, some of which would be of practical use and some of which are mere curiosity:

1. Is this a chaotic system? I know a little about chaotic systems from my undergrad math major, and this looks like one to me, but I don't know where to start working on it if it is.
2. How can this problem be formulated mathematically? I would assume that there is some sort of iterated function approach, but I can't figure out what the iterated function would be (probably because as a chemist I'm more comfortable with cartesian coordinates than polar coordinates).
3. What is the optimal angle for achieving the most passes through the area within some $\epsilon$ of the center of the ring before exiting through the aperture, assuming (a) $\epsilon$ is small compared to $r$, the radius of the ring, or (b) $\epsilon$ is not small compared to the radius of the ring? (This is one of the questions of practical importance)
4. Bonus: Suppose there is a second aperture placed at an angle $\phi$ away from the first (with $\phi$ relative to the center of the circle and to the first aperture). Can this angle be optimized in conjunction with the variables in $\phi$?

To summarize, the variables are $r$, $\theta$, $\delta w$, $\epsilon$, and (if we're lucky) $\phi$. Assume that $\epsilon$ and $\delta w$ are given (obviously the optimal size for $\delta w$ is infinitesimal, but this isn't practical of course). Also (for the physicists out there), assume $\delta w$ is wide enough that the beam can be treated classically, i.e. no diffraction. Also, there needs to be a reasonable way to calculate the exit angle based on the entry angle (if this is a truly chaotic system, I realize this will be practically impossible).

Thanks in advance for any help, and let me know if I've left out any important details. I'm comfortable with any math that a typical undergraduate math major should know by the time they graduate; anything more and I'll need a reference to read up on.

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Will someone with tag adding rights add the tags chaotic-systems and iterated-functions, or something to that effect? – David Hollman Dec 14 '10 at 15:15
Also, is this the type of question that is better asked on MathOverflow? Maybe I'll give it a few days, and if I don't get a good answer, I'll move it. – David Hollman Dec 14 '10 at 15:24
A sketch would be helpful – Ross Millikan Dec 14 '10 at 19:31

The dynamics are those of a repeated constant rotation, $\theta \to \theta + c$ where $c$ is the angle on the circle (completed to fill in the aperture) cut out by the line on which the light entered. Exit will occur after $n$ reflections when $\theta_1 + nc$ is in the aperture interval, where $\theta_1$ is the point of first impact between light and mirror. This type of dynamics is integrable, not chaotic.