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Dynamics of the ultra-discrete Toda lattice via Pitman’s transformation. (English) Zbl 1459.39044

Summary: 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 the parallel work D. A. Croydon and M. Sasada [J. Math. Phys. 60, No. 8, 083301, 25 p. (2019; Zbl 1426.37013)]. 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 \(\pm 1\).

MSC:

39A36 Integrable difference and lattice equations; integrability tests
37J70 Completely integrable discrete dynamical systems
37K60 Lattice dynamics; integrable lattice equations
37B15 Dynamical aspects of cellular automata

Citations:

Zbl 1426.37013
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