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Universal Hitting Time Statistics for Integrable Flows.

Carl P Dettmann1, Jens Marklof1, Andreas Strömbergsson2

  • 11School of Mathematics, University of Bristol, Bristol, BS8 1TW UK.

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|April 10, 2020
PubMed
Summary
This summary is machine-generated.

Hitting times in chaotic dynamical systems follow universal laws. Even without mixing, integrable flows exhibit non-Poisson limit distributions for hitting times, revealing new probabilistic patterns in chaotic dynamics.

Keywords:
Hitting time statisticsIntegrable flowsUnipotent flows

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Area of Science:

  • Mathematics
  • Dynamical Systems
  • Probability Theory

Background:

  • Chaotic dynamical systems exhibit perceived randomness in their time evolution.
  • Universal probabilistic limit laws, like the Poisson law for hitting times, characterize these systems, particularly those with strong mixing properties.

Purpose of the Study:

  • To investigate the probabilistic limit laws governing hitting times in integrable flows, specifically in the absence of mixing properties.
  • To describe the non-Poisson limit distributions for hitting times in generic integrable flows and specific target sets.

Main Methods:

  • Development of a new equidistribution theorem in the space of lattices.
  • Exploitation of Ratner's measure classification theorem for unipotent flows.
  • Extension of prior work by Elkies and McMullen on related mathematical concepts.

Main Results:

  • Demonstrated that hitting times of integrable flows, despite lacking mixing, satisfy universal limit laws.
  • Identified these limit distributions as non-Poisson, differing from systems with strong mixing properties.
  • Illustrated findings with examples from central force fields and ellipse billiards.

Conclusions:

  • The study reveals novel universal limit laws for hitting times in integrable dynamical systems, expanding beyond the traditional Poisson law.
  • The findings are supported by a new equidistribution theorem, highlighting its significance in understanding lattice spaces and unipotent flows.