Laura DeMarco, Xander Faber
Published 2013-09-27, updated 2014-11-14Version 2
The first article in this series exhibited uniqueness of the weak limit of the equilibrium measures for a degenerating 1-parameter family of rational functions on the Riemann sphere. Here we construct a convergent countable-state Markov chain that computes the limit measure. Our technique is combinatorial in nature and may be applied to compute the location of mass for the equilibrium measure of a non-Archimedean rational function, under a certain stability hypothesis. As a byproduct, we deduce that meromorphic maps preserving the fibers of a rationally-fibered complex surface are algebraically stable after a proper modification.
Comments: Restricted to non-Archimedean fields of residue characteristic zero (a gap in the argument for positive residue characteristic was found in the previous version)
Degenerations of Complex Dynamical Systems
Distribution of points with prescribed derivative in polynomial dynamics
Equilibrium measures for the Hénon map at the first bifurcation
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