2019/04/30 by David A. Croydon, Croydon, David A., Makiko Sasada +3
Mathematics · Physics and Astronomy · #37B15 (primary) #82B99 (secondary) #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.1904.13185
openalex publication_date 2019/04/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
By encoding configurations of the ultra-discrete Toda lattice by piecewise linear paths whose gradient alternates between -1 and 1, we show that the dynamics of the system can be described in terms of a shifted version of Pitman's transformation (that is, reflection in the past maximum of the path encoding). This characterisation of the dynamics applies to finite configurations in both the non-periodic and periodic cases, and also admits an extension to infinite configurations. The latter point is important in the study of invariant measures for the ultra-discrete Toda lattice, which is pursued in a parallel work. We also describe a generalisation of the result to a continuous version of the box-ball system, whose states are described by continuous functions whose gradient may take values other than ±1.