The aim of this book is to present a self-contained account of discrete weak KAM theory. Putting aside its intrinsic elegance, this theory also provides a toy model for classical weak KAM theory, where many technical difficulties disappear, but where the central ideas and results persist. It therefore serves as a good introduction to (continuous) weak KAM theory. The first three chapters give a general exposition of the general abstract theory, concluding with a discussion of the relations between the results proved in the discrete setting and the analogous theorems of classical weak KAM theory. Several examples are studied and some key differences between the discrete and classical theory are highlighted. The final chapter is devoted to the historical problem of conservative twist maps of the annulus.

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Maxime Zavidovique studied mathematics at Ecole Normale Supérieure in Lyon, France. He completed his PhD in 2011, under the supervision of Albert Fathi. Since 2011 he has held an Assistant Professor position at Sorbonne Université (formerly Jussieu) in the IMJ-PRG laboratory. His research focuses on various versions of weak KAM theory (including the discrete and the classical ones), and convergence problems of solutions to approximations of the Hamilton-Jacobi equation.