Mixing in the Absence of the Shrinking Target Property

Using reparametrizations of linear flows, we show that thereexist area-preserving real analytic maps of the three-dimensionaltorus that are ‘mixing of all orders’ and do notenjoy 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 of the torus T^d has the monotone shrinkingtarget property if and only if the vector is badly approximable(that is, of constant type). 2000 Mathematics Subject Classification37E45, 37A25, 11J13. 2000 Mathematics Subject Classification37E45, 37A25, 11J13.