DEGENERATIONS of COMPLEX DYNAMICAL SYSTEMS

We show that the weak limit of the maximal measures for any degenerating sequence of rational maps on the Riemann sphere must be a countable sum of atoms. For a one-parameter family of rational maps, we refine this result by showing that the measures of maximal entropy have a unique limit on as the family degenerates. The family may be viewed as a single rational function on the Berkovich projective line over the completion of the field of formal Puiseux series in and the limiting measure on is the 'residual measure' associated with the equilibrium measure on. For the proof, we introduce a new technique for quantizing measures on the Berkovich projective line and demonstrate the uniqueness of solutions to a quantized version of the pullback formula for the equilibrium measure on.