I will discuss a newly identified example of a plane rational map that has constant Jacobian determinant relative to the natural holomorphic volume form on the algebraic torus. A distinguishing feature is that it `tropicalizes' as a self-map of the circle that is topologically conjugate to angle doubling. Using this and post-composing with an appropriate monomial map we get a new and rather different example of a rational surface map that is not birationally conjugate to any algebraically stable map. This work, joint with Chase Bender and Holly Krieger, relies on some AI input, but maybe not where you'd guess it does.