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In this paper we study dynamical systems of intervals moving on incommensurable metric graphs. They can be generally viewed as congruent intervals moving around some graph with unit velocity and propagating on all respective incident edges whenever sliding through any vertex. In particular, we analyse the phenomenon of saturation: a state of the system when the entire graph is covered by these moving intervals. Our main contributions are the following: (1) we prove the existence of the finite moment of permanent saturation for any incommensurable metric graph and any positive length of these intervals; (2) we present an upper bound for the moment of permanent saturation. To show the validity of our results, we reduce the system of moving intervals to the system of dispersing moving points for the analysis of which we primarily use methods of discrepancy theory and number theory, specifically Kronecker sequences and the celebrated Three Gap Theorem. |
