The paper develops an existence theory for solutions of the Abreu equation, which include extremal metrics on toric surfaces. The technique employed is a continuity method, combined with “blow-up” arguments. General existence results are obtained, assuming a hypothesis (the “M-condition”) on the solutions, which is shown to be related to the injectivity radius.

Compact locally maximal hyperbolic sets are studied via geometrically defined functional spaces that take advantage of the smoothness of the map in a neighborhood of the hyperbolic set. This provides a self-contained theory that not only reproduces all the known classical results, but also gives new insights on the statistical properties of these systems.