2024/01/24 by Holland, Mark, Todd, Mike · 1 citation
#37A50 #37B20 #37E05 #60G55 #60G70 #Dynamical Systems (math.DS) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2401.13300
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 Xn(x)=dist(fn(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 Rn(x) associated to the number recurrences below a certain radii sequence rn(τ) follows an averaged Poisson distribution G(τ). Furthermore, we obtain quantitative results on almost sure rates for the recurrence statistics of the process Xn.