  We study the billiard dynamics in annular tables between two excentric
circles. As the center and the radius of the inner circle change, a two
parameters map is defined by the first return of trajectories to the obstacle.
We obtain an increasing family of hyperbolic sets, in the sense of the
Hausdorff distance, as the radius goes to zero and the center of the obstacle
approximates the outer boundary. The dynamics on each of these sets is
conjugate to a shift with an increasing number of symbols. We also show that
for many parameters the system presents quadratic homoclinic tangencies whose
bifurcation gives rise to elliptical islands (Conservative Newhouse
Phenomenon). Thus, for many parameters we obtain the coexistence of a "large"
hyperbolic set with many elliptical islands.
