Universal Hitting Time Statistics for Integrable Flows

Carl P Dettmann, Jens Marklof, Andreas Strömbergsson*

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

5 Citations (Scopus)
302 Downloads (Pure)


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.

Original languageEnglish
Pages (from-to)714-749
Number of pages36
JournalJournal of Statistical Physics
Issue number3
Early online date25 Aug 2016
Publication statusPublished - Feb 2017


  • Hitting time statistics
  • Integrable flows
  • Unipotent flows


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