Geometrical dynamics of an integrable piecewise-linear mapping

A special type of piecewise-linear mapping is discussed. It is obtained by ultradiscretizing the Quispel-Robert-Thompson system. In a special case of a parameter, it becomes a periodic mapping with a constant period for any initial data. In a general case, it becomes an integrable mapping and a period of solution is constant for each solution orbit. We show a structure of solutions discussing the dynamics in a phase plane from a viewpoint of the integrable system theory.