Local Dynamics of Non-Invertible Maps Near Normal Surface Singularities [Pehme köide]

"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"--

Gignac and Ruggiero study the problem of finding algebraically stable models for non-invertible holomorphic fixed point germs f: (X,x0) to (X,x0), where X is a complex surface having x0 as a normal singularity. Their topics include normal surface singularities and their valuation spaces, dynamics on valuation spaces, dynamics of non-invertible finite germs, algebraic stability, attraction rates, and examples and remarks. Annotation ©2022 Ringgold, Inc., Portland, OR (protoview.com)