Sanju Velani

Shrinking Targets versus Recurrence

Let (X,d) be a compact metric space and (X,A,\mu,T) be a probability measure preserving system. Furthermore, given a real, positive function \psi:\N\to\R_{\ge 0}$ let

R(\psi) := \{ x \in X : d(T^nx, x) < \psi(n) for infinitely many n\in \N \} denote the associated recurrent set, and given a point x_0 \in X let W( \psi): := \{ x \in X : d(T^nx, x_0) < \psi(n) for infinitely many n\in \N \} denote the associated shrinking target set. Under certain mixing properties it known that if \sum_{n \in \N} \psi(n) diverges then both the recurrent and shrinking target sets are of full \mu-measure. The purpose of this talk is discuss the potential quantitative strengthening of these full measure statements. This is joint work with Jason Levesley (York), Bing Li (SCUT) and David Simmons (York), and ongoing work with Junjie Huang (SCUT) and Bing Li (SCUT)