Shrinking Targets versus Recurrence: a brief survey

Let $(X,d)$ be a compact metric space and $(X,\mathcal{A},\mu ,T)$ a measure preserving dynamical system. Furthermore, given a real, positive function $\psi$, let $W(T,\psi )$ and $R(T,\psi )$ 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 $\mu$-measure. The purpose of this survey is to provide a brief overview of such results, to discuss the potential quantitative strengthening of the full measure statements and to bring to the forefront key differences in the theory.