On Distribution Limit Laws for Recurrence.
Ergodic Theory and Dynamical Systems Seminar
5th December 2024, 2:00 pm – 3:00 pm
Fry Building, 2.04
For a probability measure preserving dynamical system $f:X\to X$, the Poincare Recurrence Theorem asserts that almost every orbit is recurrent with respect to its initial condition. This motivates study of the statistics of the process $X_n(x)=d(f^n x, x)$ and real-valued functions thereof. For a wide class of non-uniformly expanding dynamical systems, we show that the time-n counting process associated to the number recurrences below a certain radii sequence $r_n(t)$ follows an averaged Poisson distribution $G(t)$. This is a different limit law relative to standard Poisson laws obtained for hitting time counting processes. Furthermore, we obtain quantitative results on almost sure rates for the recurrence statistics of the process $X_n$.
This is based on a joint work with M. Todd at St Andrews.
Comments are closed.