Local dynamics of holomorphic maps on normal surface singularities

To describe the behavior of the iterates of a holomorphic germ f on a complex surface X fixing a (possibly singular) point x0, we are led to study the lifts fπ to birational models Xπ over (X,x0). In general fπ has indeterminacy points: when the fπ-orbits eventually avoid these indeterminacy points, we say that X_π is algebraically stable. In a joint work with William Gignac, we show the existence of algebraically stable models in this setting (but for one class of exceptions, where no such models exist). The proof relies on fixed point theorems for the dynamics induced on suitable valuation spaces, following Favre and Jonsson.