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Upper bound on the number of collisions of pinned billiard balls

We consider systems of "pinned balls," i.e., 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, i.e., 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.