Local Dynamics of Non-Invertible Maps Near Normal Surface Singularities (Memoirs of the American Mathematical Society, 272)

By Matteo Ruggiero and William Gignac

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 fN 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.

• Format: Paperback
• Language: English
• ISBN: 1470449587
• ISBN13: 9781470449582
• Release Date: November 2021
• Publisher: American Mathematical Society
• Length: 100 Pages
• Weight: 0.49 lbs.
• Dimensions: 0.8" x 7.0" x 10.0"