Local dynamics of non-invertible maps near normal surface singularities

2017/04/16 by Gignac, William, Ruggiero, Matteo
#13A18 #32H50 #32S05 #32S45 #37P50 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1704.04726

Abstract

We study the problem of finding algebraically stable models for non-invertible holomorphic fixed point germs f\colon (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 π\colon X_π→ (X,x₀) such that the dynamics of f (or more precisely of f^N for N big enough) on X_π 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.

Related

• Holomorphic self-maps of singular rational surfaces
• Dynamics of surface homeomorphisms Topological versions of the Leau-Fatou flower theorem and the stable manifold theorem
• On the neighborhood of a torus leaf and dynamics of holomorphic foliations
• Normal del Pezzo surfaces containing a nonrational singularity
• Global dynamics and perspectives on singularities of holomorphic foliations