Properties of the flow on a polygonal Andreev billiard

A formal definition of a (mathematical) polygonal Andreev billiard and a construction of an equivalence relation that captures the dynamics described in physical toy model of Andreev reflection are given. The continuous flow and discrete flow on the respective phase spaces. It is then shown that the continuous flow preserves the absolute value of the volume element $dx\wedge dy\wedge d\theta$ and the billiard (collision) map preserves the measure $\cos \phi dr d\phi$, respectively. One can then characterize the dynamics of a rational polygonal Andreev billiard table. Finally, a discussions of the effect of a fractal perturbation of the toy model of a rectangular nanowire lying upon a superconducting medium is given.