Eigenvaluations

We study the dynamics in C² of superattracting fixed point germs and of polynomial maps near infinity. In both cases we show that the asymptotic attraction rate is a quadratic integer, and construct a plurisubharmonic function with the adequate invariance property. This is done by finding an infinitely near point at which the map becomes rigid: the critical set is contained in a totally invariant set with normal crossings. We locate this infinitely near point through the induced action of the dynamics on a space of valuations. This space carries an R-tree structure and conveniently encodes local data: an infinitely near point corresponds to an open subset of the tree. The action respects the tree structure and admits a fixed point—or eigenvaluation—which is attracting in a certain sense. A suitable basin of attraction corresponds to the desired infinitely near point.