On distributional limit laws for recurrence

For a probability measure preserving dynamical system (𝒳,f,μ), the Poincaré Recurrence Theorem asserts that μ-almost every orbit is recurrent with respect to its initial condition. This motivates study of the statistics of the process X_n(x)=dist(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 R_n(x) associated to the number recurrences below a certain radii sequence r_n(τ) follows an averaged Poisson distribution G(τ). Furthermore, we obtain quantitative results on almost sure rates for the recurrence statistics of the process X_n.