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Universal Hitting Time Statistics for Integrable Flows
Univ Bristol, Sch Math, Bristol BS8 1TW, Avon, England..
Univ Bristol, Sch Math, Bristol BS8 1TW, Avon, England..
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
2017 (English)In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 166, no 3-4, 714-749 p.Article in journal (Refereed) Published
Abstract [en]

The perceived randomness in the time evolution of "chaotic" dynamical systems can be characterized by universal probabilistic limit laws, which do not depend on the fine features of the individual system. One important example is the Poisson law for the times at which a particle with random initial data hits a small set. This was proved in various settings for dynamical systems with strong mixing properties. The key result of the present study is that, despite the absence of mixing, the hitting times of integrable flows also satisfy universal limit laws which are, however, not Poisson. We describe the limit distributions for "generic" integrable flows and a natural class of target sets, and illustrate our findings with two examples: the dynamics in central force fields and ellipse billiards. The convergence of the hitting time process follows from a new equidistribution theorem in the space of lattices, which is of independent interest. Its proof exploits Ratner's measure classification theorem for unipotent flows, and extends earlier work of Elkies and McMullen.

Place, publisher, year, edition, pages
SPRINGER , 2017. Vol. 166, no 3-4, 714-749 p.
Keyword [en]
Integrable flows, Hitting time statistics, Unipotent flows
National Category
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-320774DOI: 10.1007/s10955-016-1604-yISI: 000394366800014OAI: oai:DiVA.org:uu-320774DiVA: diva2:1090771
Funder
EU, European Research Council, 291147Göran Gustafsson Foundation for promotion of scientific research at Uppala University and Royal Institute of TechnologySwedish Research Council, 621-2011-3629
Available from: 2017-04-25 Created: 2017-04-25 Last updated: 2017-04-25Bibliographically approved

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