Matteo Ruggiero, gave a lecture titled“Local dynamics of non-invertible selfmaps on complex surfaces”at AMSS on 26, October 2017. 

In this talk, they considered the local dynamical system induced by a non-invertible selfmap f of C^2 fixing the origin.Given a modification (composition of blow-ups) over the origin, the lift of f on the modified space X defines a meromorphic map F. They said that F is algebraically stable if, for every compact curve E in X, its image F^n(E) through the iterates of F did not belong to the indeterminacy set of F for all n big enough. They showed that, starting from any modification, they could blow-up some more and obtain another modification for which the lift F is algebraically stable.The proof relied on the study of the action f_* induced by f on a suitable space of valuations V.In particular they constructed a distance on V for which f_* is non-expanding. This allowed them to deduce fixed point theorems for f_*.If time allowed, they would comment on the recent developments about local dynamics on normal surface singularities.  

Matteo Ruggiero is from University Paris Diderot (Paris VII), French.