Tropical curves and integrable piecewise linear maps

We present applications of tropical geometry to some integrable piecewise-linear maps, based on the lecture given by one of the authors (R. I.) at the workshop "Tropical Geometry and Integrable Systems" (University of Glasgow, July 2011), and on some new results obtained afterward. After a brief review on tropical curve theory, we study the spectral curves and the isolevel sets of the tropical periodic Toda lattice and the periodic Box-ball system.