Classification of Positive Rational Reflection Maps for the Relativistic Collision Yang--Baxter Map

Abstract

We classify positive jointly rational parametric reflection maps for the one-dimensional relativistic collision Yang--Baxter map, without imposing a degree or birationality hypothesis. There are exactly three families. Two preserve one of the light-cone momenta and allow an arbitrary positive rational action on the mass; the third preserves rapidity and inverts the mass. Their completeness follows from a semiconjugacy degree argument, endpoint divisor transport, and exact elimination. In particular, the mass-preserving case is rigid: the identity is the only jointly rational reflection map.