Upper Bound for Permanent Saturation of Metric Graphs using Interval Exchange Transformations
We study upper bounds for the moment of permanent $\varepsilon$-saturation in finite metric graphs. The dynamics is generated by moving points travelling with unit speed along edges and branching into all outgoing directions whenever they reach a vertex. We first reformulate this branched dynamics in terms of birth times at vertices and prove a sufficient same-time criterion for permanent $\varepsilon$-saturation. The main rigorous estimate is obtained from a rotation, regarded as a two-interval exchange transformation. More precisely, if the graph contains two closed walks based at the initial vertex whose lengths have irrational ratio, then the covering properties of the corresponding rotation imply an explicit upper bound for the permanent saturation time. In particular, bounded-type rotations yield a bound of order $\varepsilon^{-1}$. We also construct a more general auxiliary interval exchange transformation on the set of oriented edges. This construction depends on cyclic orders at the vertices and organizes the ordered edge-state data of the graph. Since the branched graph dynamics is non-invertible, whereas an interval exchange transformation is invertible away from discontinuities, this auxiliary IET is not identified with the full graph dynamics. Instead, we formulate the additional birth-time transfer property required for recurrence estimates of the auxiliary IET to imply saturation bounds. We also discuss rotation-type and non-rotation examples of graph-induced self-similar IETs, together with numerical illustrations for star and complete graphs.