  On a non-compact, smooth, connected, boundaryless, complete Riemannian
manifold $(M,g)$, one can define its ideal boundary by rays (or equivalently,
Busemann functions). From the viewpoint of Mather theory, boundary elements
could be regarded as the static classes of Aubry sets, and thus lines should be
think as the semi-statics curves connecting different static classes. In Mather
theory, one core property is Lipschitz graph property for Aubry sets and for
some kind of semi-static curves. In this article, we prove a such kind of
result for a set of lines which connect the same pair of boundary elements.

Dynamical Systems