Multiplier scales of a sequence of rational maps

Abstract

We analyze the behavior of multipliers of a degenerating sequence of complex rational maps. We show either most periodic points have uniformly bounded multipliers, or most of them have exploding multipliers at a common scale. We further explore the set of scales induced by the growth of multipliers. Using Ahbyankar's theorem, we prove that there can be at most 2d-2 such non-trivial multiplier scales.