Local dynamics of non-invertible maps near normal surface singularities

Abstract

We study the problem of finding algebraically stable models for non-invertible holomorphic fixed point germs f : (X , x 0 ) → (X , x 0 ), where X is a complex surface having x 0 as a normal singularity. We prove that as long as x 0 is not a cusp singularity of X , then it is possible to find arbitrarily high modifications π: X π → (X , x 0 ) such that the dynamics of f (or more precisely of f N for N big enough) on X π is algebraically stable. This result is proved by understanding the dynamics induced by f on a space of valuations associated to X ; in fact, we are able to give a strong classification of all the possible dynamical behaviors of f on this valuation space. We also deduce a precise description of the behavior of the sequence of attraction rates for the iterates of f . Finally, we prove that in this setting the first dynamical degree is always a quadratic integer.