MIXING IN THE ABSENCE OF THE SHRINKING TARGET PROPERTY

Abstract

Using reparametrizations of linear flows, we show that there exist area-preserving real analytic maps of the three-dimensional torus that are ‘mixing of all orders’ and do not enjoy the monotone shrinking target property. Prior to that, we give a short proof of a result of Kurzweil from 1955: namely, that a translation $T_\alpha$ of the torus $\mathbb{T}^d$ has the monotone shrinking target property if and only if the vector $\alpha$ is badly approximable (that is, of constant type).