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"We study the problem of finding algebraically stable models for non-invertible holomorphic fixed point germs f : (X, x0) (X, x0), where X is a complex surface having x0 as a normal singularity. We prove that as long as x0 is not a cusp singularity of X, then it is possible to find arbitrarily high modifications : X (X, x0) such that the dynamics of f (or more precisely of f N for N big enough) on X 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"--

Singularities (Mathematics) --- Holomorphic mappings. --- Germs (Mathematics) --- Holomorphic functions. --- Several complex variables and analytic spaces -- Singularities -- Local singularities. --- Several complex variables and analytic spaces -- Holomorphic mappings and correspondences -- Iteration problems. --- Commutative algebra -- General commutative ring theory -- Valuations and their generalizations. --- Dynamical systems and ergodic theory -- Arithmetic and non-Archimedean dynamical systems -- Dynamical systems on Berkovich spaces. --- Several complex variables and analytic spaces -- Singularities -- Modifications; resolution of singularities.