The ergodic properties of a rational map \(f\colon \mathbb{CP}^2 \longrightarrow \mathbb{CP}^2\) are tied to its dynamical degrees \(\lambda_1(f)\) and \(\lambda_2(f)\). Maps with \(\lambda_1(f) > \lambda_2(f)\) share many properties of the Hénon maps, having a measure of maximal entropy of saddle type. Maps with \(\lambda_2(f) > \lambda_1(f)\) share many properties of holomorphic endomorphisms, having a measure of maximal entropy that is repelling. In both cases it is believed (and often proved) that \(f\) has a unique measure of maximal entropy.

Early examples of maps with \(\lambda_1(f) = \lambda_2(f)\) were skew products, having an invariant fibration. Guedj asked whether this happens in general. We show that there is a simple way to produce many rational maps of \(\mathbb{CP}^2\) with equal dynamical degrees, no invariant foliation, and two measures of maximal entropy, one of saddle-type and one repelling. Many of the techniques are geometric. This is joint work with Jeff Diller and Han Liu and it builds on previous joint work with Scott Kaschner and Rodrigo Pérez.