Jens Marklof (Bristol)
Universal hitting time statistics for integrable flows
Abstract:
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. Joint work with C. Dettmann and A.
Strombergsson.