Variational approach to homoclinic orbits in twist maps and an application to billiard systems (Q701754)

The paper is devoted to the study of topological entropy of billiard systems on a convex domain of the Euclidean plane whose boundary curve has positive curvature. It is shown that for generic billiard ball systems of this kind, the topological entropy is positive. More precisely, the main result is as follows: For any \(C^r\), \(r\geq 3\), closed curve \(\gamma\) in \(\mathbb R^2\) with positive curvature function \(k(s)\) and for any \(\varepsilon >0\), there is a \(C^r\) closed curve \(\gamma '\) in the \(C^r\) \(\varepsilon\)-neighborhood of \(\gamma\) with positive curvature function such that the billiard map induced by the curve \(\gamma '\) has positive topological entropy. Results obtained relate to the conjecture by \textit{G.~Knieper} and \textit{H.~Weiss} [J. Differ. Geom. 39, 229--249 (1994; Zbl 0809.53043)] which says that for a generic metric on \(S^2\), the corresponding geodesic flow has positive topological entropy.