The outer billiard ball map (OBM) is defined from and to the exterior of a domain, Ω, in the plane as taking a point, q, to another point, q 1, when the line segment with endpoints q and q 1 is tangent to the boundary, ∂Ω (with a chosen orientation), and the point of tangency with the boundary divides the segment in half. Let C be an invariant circle for the OBM on Ω, with ∂Ω smooth with positive curvature. After computing the loss of derivatives between ∂Ω and C, it is shown via KAM theory that in this setting the OBM has uncountably many invariant circles in any neighborhood of the boundary. One is also led to an infinitesimal obstruction for the evolution property, an obstruction which, among closed smooth convex curves, is only removed for ellipses.
Journal of Mathematical Physics
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Copyright 1995 American Institute of Physics. The original published version of this article may be found at http://dx.doi.org/10.1063/1.531117.
Amiran, Edoh Y., "Lazutkin Coordinates and Invariant Curves for Outer Billiards" (1995). Mathematics. 14.