Memory Maps with Elliptical Trajectories

A family of maps with memory, parameterized by [Formula: see text], is shown to have either periodic trajectories or dense trajectories on ellipses which support absolutely continuous invariant measures. Furthermore, for [Formula: see text], i.e. [Formula: see text] with [Formula: see text] and [Formula: see text], all points except [Formula: see text] either go into a polygonal region centered at [Formula: see text] if [Formula: see text] is rational, or are attracted to an elliptical region having the same center, if [Formula: see text] is irrational. In the polygonal case, we examine a mechanism for the appearance of islands supporting absolutely continuous invariant measures.