Dynamics of the ultra-discrete Toda lattice via Pitman's transformation

[1904.13185] Dynamics of the ultra-discrete Toda lattice via Pitman's transformation Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematical Physics arXiv:1904.13185 (math-ph) [Submitted on 30 Apr 2019 ( v1 ), last revised 1 May 2019 (this version, v2)] Title: Dynamics of the ultra-discrete Toda lattice via Pitman's transformation Authors: David A. Croydon , Makiko Sasada , Satoshi Tsujimoto View a PDF of the paper titled Dynamics of the ultra-discrete Toda lattice via Pitman's transformation, by David A. Croydon and 2 other authors View PDF HTML (experimental) Abstract: 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 $\pm1$. Subjects: Mathematical Physics (math-ph) MSC classes: 37B15 (primary), 82B99 (secondary) Cite as: arXiv:1904.13185 [math-ph] (or arXiv:1904.13185v2 [math-ph] for this version) https://doi.org/10.48550/arXiv.1904.13185 Focus to learn more arXiv-issued DOI via DataCite Journal reference: RIMS Kokyuroku Bessatsu78 (2020), 235-250 Submission history From: David Croydon [ view email ] [v1] Tue, 30 Apr 2019 12:21:57 UTC (26 KB) [v2] Wed, 1 May 2019 04:49:42 UTC (26 KB) Full-text links: Access Paper: View a PDF of the paper titled Dynamics of the ultra-discrete Toda lattice via Pitman's transformation, by David A. Croydon and 2 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math-ph < prev | next > new | recent | 2019-04 Change to browse by: math math.MP References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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