On semi-unimodal maps of the plane and the structure of their sets of non-wandering points

We consider a class of semi-unimodal endomorphisms of the plane and study their sets of non-wandering points. It is proved that there are parameter values such that these sets consist of several components, one of which is non-trivially non-compact (that is, has a trajectory receding to infinity), whereas the other components are compact and include a set that is an equivariant image of the Cantor set and part of its boundary is composed of self-similar elements (that is, has a fractal type structure). Furthermore, it turns out that there are parameter values such that the compact and non-compact components intertwine on the coordinate axes.