We study the billiard dynamics in annular tables between two eccentric circles. As the center and the radius of the inner circle changes, 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.

This article concerns a class of open billiards consisting of a finite number of strictly convex, non-eclipsing obstacles $K$. The non-wandering set ${{M}_{0}}$ of the billiard ball map is a topological Cantor set, and its Hausdorff dimension has been previously estimated for billiards in ${{\mathbb{R}}^{2}}$ using well-known techniques. We extend these estimates to billiards in ${{\mathbb{R}}^{n}}$ and make various refinements to the estimates. These refinements also allow improvements to other results. We also show that in many cases, the non-wandering set is confined to a particular subset of ${{\mathbb{R}}^{n}}$ formed by the convex hull of points determined by period 2 orbits. This allows more accurate bounds on the constants used in estimating Hausdorff dimension.

A body moves in a rarefied medium composed of point particles at rest. The particles make elastic reflections when colliding with the body surface and do not interact with each other. We consider a generalization of Newton’s minimal resistance problem: given two bounded convex bodies ${{C}_{1}}$ and ${{C}_{2}}$ such that ${{C}_{1}}\,\subset \,{{C}_{2}}\,\subset \,{{\mathbb{R}}^{3}}$ and $\partial {{C}_{1}}\,\cap \,\partial {{C}_{2}}\,=\,\varnothing $, minimize the resistance in the class of connected bodies $B$ such that ${{C}_{1}}\,\subset \,B\,\subset \,{{C}_{2}}$. We prove that the infimum of resistance is zero; that is, there exist “almost perfectly streamlined” bodies.

Newton's problem of the body of minimal aerodynamic resistance is traditionallystated in the class of convex axially symmetric bodies withfixed length and width. We state and solve the minimal resistanceproblem in the wider class of axially symmetric but generallynonconvex bodies. The infimum in this problem is not attained. Weconstruct a sequence of bodies minimizing the resistance. Thissequence approximates a convex body with smooth front surface, whilethe surface of approximating bodies becomes more and morecomplicated. The shape of the resulting convex body and the value ofminimal resistance are compared with the corresponding results forNewton's problem and for the problem in the intermediate class ofaxisymmetric bodies satisfying the single impact assumption[Comte and Lachand-Robert, J. Anal. Math.83 (2001) 313–335]. In particular, the minimal resistance in our class issmaller than in Newton's problem; the ratio goes to 1/2 as(length)/(width of the body) → 0, and to 1/4 as(length)/(width) → +∞.