Abstract
We study the problem of finding algebraically stable models for non-invertible holomorphic fixed point germs f : ( X , x 0 ) → ( X , x 0 ) f\colon (X,x_0)\to (X,x_0) , where X X is a complex surface having x 0 x_0 as a normal singularity. We prove that as long as x 0 x_0 is not a cusp singularity of X X , then it is possible to find arbitrarily high modifications π : X π → ( X , x 0 ) \pi \colon X_\pi \to (X,x_0) such that the dynamics of f f (or more precisely of f N f^N for N N big enough) on X π X_\pi is algebraically stable. This result is proved by understanding the dynamics induced by f f on a space of valuations associated to X X ; in fact, we are able to give a strong classification of all the possible dynamical behaviors of f 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 f . Finally, we prove that in this setting the first dynamical degree is always a quadratic integer.
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