I'm numerically studying a 2D billiard system whose domain is a unit outer circle with an inner elliptical scatterer of variable geometry. EDIT: Both the circle and ellipse share a common centre.
From my readings, I understand that the 'twist condition' is an important assumption to understand the evolution of the systems phase section, and its behaviour with varying geometrical parameters.
For the case where the trajectory only hits the outer circular boundary it's easy to show, by the boundary's convexity, that this condition is obeyed.
However, I'm at a loss as to a mathematically rigourous way to demonstrate that it is obeyed (or disobeyed!) in the case where the particle hits the inner ellipse (mixed dynamics manifest in this case)
Any suggestions or pointers would be gratefully received.
Thankyou in advance !