I will discuss work with Jeffrey Diller concerning rational surface maps that we call `toric'. These include all monomial maps and some recent examples by Bell, Diller, and Jonsson of rational maps that have transcendental first dynamical degrees. Our main theorem is an equidistribution result for preimages of curves that works precisely when the map in question is "bad" in the sense that it cannot be birationally conjugated to an algebraically stable map. I will also describe the work we are now doing in an effort to study the ergodic properties of these mappings.
