We study dynamical systems induced by birational automorphisms on smooth cubic surfaces defined over a number field K. In particular we are interested in the product of noncommuting birational Geiser involutions of the cubic surface. We present results describing the sets of K and \bar{K}-periodic points of the system, and give a necessary and sufficient condition for a dynamical local-global property called strong residual periodicity. Finally, we give a dynamical result relating to the Mordell-Weil problem on cubic surfaces.

The author's research was supported by the Israel Science Foundation, grants 657/09 and 1207/12, and by the Skirball Foundation via the Center for Advanced Studies in Mathematics at Ben-Gurion University of the Negev.