The Hamiltonian with parameter and kicks represented by a sequence of Dirac delta functions is

,

where , , and are the position, momentum, and time. The map then takes the form

.

References

[1] A. J. Scott, C. A. Holmes, and G. J. Milburn, "Hamiltonian Mappings and Circle Packing Phase Spaces," Physica D, 155(1–2), 2001 pp. 34–50. doi:10.1016/S0167-2789(01)00263-9.

[2] G. Hughes, "Outer Billiards, Digital Filters and Kicked Hamiltonians," 2012. arxiv.org/abs/1206.5223.