Outer symplectic billiard map at infinity

We show that the second iteration $ T^2 $ of the outer symplectic billiard map with respect to a convex domain $ M $ in a symplectic vector space is approximated by an explicit Hamiltonian flow for points far away from $ M $. More precisely, denote by $ N $ the symplectic polar dual of the symmetrization $ M\ominus M $ of $ M $. If we write $ N $ as the unit level set of a 1-homogeneous function $ H $, then the difference between $ T^2 $ and the time-2-Hamiltonian flow of $ H $ applied to a point $ x $ is smaller than $ c/|x| $ for some constant $ c $ depending only on $ M $. Moreover, we show that if an orbit escapes to infinity, then its distance to the origin grows no faster than $ \sqrt{k} $ in the number of iterations. Finally, we prove that a $ k $-periodic orbit needs to be close, in terms of $ k $, to $ M $.