  We study the dynamics in C^2 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 real-tree structure and conveniently encodes
local data: an infinitely near point corresponds to a 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.
