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 $f\colon (X,x_0)\to (X,x_0)$, 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 $\pi\colon X_\pi\to (X,x_0)$ such that the dynamics of $f$ (or more precisely of $f^N$ for $N$ big enough) on $X_\pi$ 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.