2025/08/21 by Albers, Peter, Caliz, Ana Chavez, Tabachnikov, Serge · 2 citations
#Dynamical Systems (math.DS) #FOS: Mathematics #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.2508.15142
We show that the second iteration T2 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 T2 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 not faster than √(k) in the number of iterations. Finally, we prove that a k-periodic orbit needs to be close, in terms of k, to M.