Sanju Velani (University of York) – Shrinking targets versus recurrence

Let (X, d) be a compact metric space and (X, A, μ, T) a measure preserving dynamical system. Furthermore, given a real, positive function ψ, let W (T, ψ) and R(T, ψ) respectively denote the shrinking target set and the recurrent set associated with the dynamical system. Under certain mixing properties it is known that if the natural measure sum diverges then the recurrent and shrinking target sets are of full μ-measure. The purpose of this talk is to give a brief overview of these results, to discuss possible quantitative refinements of the full-measure conclusions, and to highlight important structural differences between the shrinking-target and recurrence theories. These differences lead to a natural and strikingly simple problem that, perhaps surprisingly, remains unresolved.