On some refraction billiards

Figure 1.  Examples of trajectories for $ \mathcal{E} = 2.5 $, $ \omega = 1 $, $ h = 2 $ and $ \mu = 2 $. Left: general trajectory for an elliptic domain with eccentricity $ e = 0.6 $. Right: quasi-periodic trajectory for a circular domain

Figure 2.  Orbits of the first return map $ F $ for $ \mathcal{E} = 10 $, $ \omega = 1 $, $ h = 3 $, $ \mu = 44 $ (left) and $ \mu = 55 $ (right). The shift in the stability of the homothetic equilibrium is evident, as well as the presence of invariant curves far from the $ \xi $-axis in both cases

Figure 3.  Examples of periodic and non-periodic orbits on the circle in the phase space $ \left(\xi, I\right)\in \mathbb{R}_{/2\pi\mathbb{Z}}\times\mathcal I $

Figure 4.  Example of a non-homothetic fixed point for $ \mathcal{F} $, with $ \mathcal{E} = 7, \omega^2 = 3, h = 2, \mu = 15 $. In this case, with reference to Proposition 4.10, $ \bar{\mu} = 41.6287 $

Figure 5.  Curve $ \Gamma_{\tilde \alpha} $ for the computation of the winding number in the perturbed case

Figure 6.  Sketch of the perturbed dynamics in the region described by Proposition 5.6 and Theorem 5.12 in the phase plane $ (\xi, I) $. Red: the invariant curves of Diophantine rotation numbers $ \rho_0 $ and $ \rho_1 $, which are deformations of the unperturbed invariant straight lines $ I = \bar I_{\rho_0} $, $ I = \bar I_{\rho_1} $ (green) such that $ \bar\theta\left(\bar I_{\rho_0}\right) = \rho_0 $ and $ \bar\theta\left(\bar I_{\rho_1}\right) = \rho_1 $. In the striped region the map $ \mathcal F(\xi_{0}, I_0; \epsilon) $ is area-preserving and twist. The blue dashed lines denote two singular action values for the unperturbed dynamics (i.e. $ I\in\bar{\mathcal{I}} $)