Reductions and simplifications of orbital sums in a Hamiltonian repeller

We solve the “center of mass puzzle”, namely the empirical observation that in the presence of a complete Smale horseshoe, the sum of the coordinates of orbital points for low periodic orbits of the Hamiltonian Hénon map reduces to simple rational numbers every so often. We show that such reductions occur systematically in phase-space and arise from specific factorizations of the equations defining orbital coordinates and their sums. The factorizations manifest sets of correlated orbital clusterings and imply an algebraic way of ordering orbits.