Anosov Geodesic Flows, Billiards and Linkages

Abstract

Any smooth surface in R 3 may be flattened along the z-axis, and the flattened surface becomes close to a billiard table in R 2 . We show that, under some hypotheses, the geodesic flow of this surface converges locally uniformly to the billiard flow. Moreover, if the billiard is dispersive and has finite horizon, then the geodesic flow of the corresponding surface is Anosov. We apply this result to the theory of mechanical linkages and their dynamics: we provide a new example of a simple linkage whose physical behavior is Anosov. For the first time, the edge lengths of the mechanism are given explicitly.

Embedded surfaces with Anosov geodesic flows, approximating spherical billiards

“…Later, Arnold [Arn63] writes that smooth Sinaï billiards could be approximated by surfaces with nonpositive curvature, with Anosov geodesic flows. In [Kou16a], we proved this fact for a large class of flattened surfaces.…”

Rolling systems and their billiard limits

“…Notice that this correspondence between rolling on surfaces and no-slip billiards on domains in the plane is similar to the relation between geodesic flows on surfaces and ordinary billiard systems on domains obtained by a flattening of the surface. (An early mention of this relation is [1], page 184; see also [15]. This fact is also a corollary of Theorem 1.…”

Uniform hyperbolicity in nonflat billiards

“…Theorem 2.1 was mentioned in [DP03] and [MP13], without details about the proof. In [Kou16a], we apply Theorem 2.1 to give new examples of surfaces whose geodesic flow is Anosov while their curvature is not negative everywhere. The genus of such surfaces is necessarily at least 2 [Kli74].…”