Upper bound on the number of collisions of pinned billiard balls

We consider systems of “pinned balls”—that is, balls that have fixed positions and pseudo-velocities. Pseudo-velocities change according to the same rules as those for velocities of totally elastic collisions between moving balls. The times of possible pseudo-collisions for different pairs of pinned balls are chosen in an exogenous way. We give an explicit upper bound for the maximum number of pseudo-collisions for a system of n pinned balls in a d-dimensional space. The proof is based on analysis of foldings—that is, mappings that formalize the idea of folding a piece of paper along a crease. We prove an upper bound for the size of an orbit of a point subjected to foldings.