Received: 26.12.2014 / Accepted: 26.12.2014 / Published online: 21.1.2015
Let us start with the billiard within a circle $\mathcal{C}$. The trajectories of this system are polygonal lines inscribed in $\mathcal{C}$, having all sides of the same length. A natural and easy question is whether such a line is periodic. Namely, if $\alpha$ is the central angle of $\mathcal{C}$ corresponding to a chord of the length equal to a side of a given trajectory, then the trajectory is periodic if and only if $\dfrac{\alpha}{2\pi}$ is rational.
The number $\rho=\dfrac{\alpha}{2\pi}$ is called the rotation number. It is easy to see that the period is equal to $q$ if and only if the rotation number is equal to $\frac{s}{q}$, with $(s,q)=1$.
The numerator $s$ corresponds to the winding number---the number of rounds of the billiard particle about the centre within one period.
Notice that there is a circle $\mathcal{C}_0$ concentric with $\mathcal{C}$, which is tangent to each segment of the given billiard trajectory. We will refer to $\mathcal{C}_0$ as the caustic of the trajectory.
If $R$ and $r$ are radii of $\mathcal{C}$ and $\mathcal{C}_0$ then the rotation number is: $$ \rho=\frac{1}{\pi}\arccos\frac{r}R. $$
Consider the billiard system within a domain bounded by two concentric half-circles and two segments lying on the same diameter, as shown in Fig. 1.
Each trajectory of such a billiard will also have a caustic which is concentric with the half-circles contained in the boundary.
Let $R_1$, $R_2$ be the radii of the half-circles on the boundary. For a fixed the caustic of radius $r$, denote by $\rho_1=\rho(R_1,r)$, $\rho_2=\rho(R_2,r)$ the corresponding rotation numbers.
Let us present several examples.
Now, we can reformulate the Question 1 in the following way: