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

Tropical curves and integrable piecewise linear maps