I will describe a construction of maps of the sphere that are piecewise given by matrices in SL(2,Z). They expand/contract a pair of positive currents, analogously to pseudo-Anosov maps, but are non-uniformly hyperbolic. The construction of these currents involves some ideas from convex and Berkovich geometry. The objects that appear have concrete, elementary descriptions, and they also encode a degenerating family of holomorphic automorphisms of K3 surfaces.