degeneration of complex rational maps

Overview & Synopsis

We develop non-Archimedean techniques to analyze the degeneration of the dynamics of a sequence of holomorphic maps of the Riemann sphere.Given any sequence of rational maps f_n of degree d at least 2, we construct in the first chapter all its possible dynamical limits as rational maps over complete metrized (possibly non-Archimedean) fields. To do so, we avoid the use of pointed ultralimits of the hyperbolic 3-space as done by Y. Luo, and rely in a more systematic way on Berkovich theory. This allows us to prove a general statement on the convergence of the equilibrium measures and to control in a precise way the blow-up ofthe Lyapunov exponents of f_n. A key observation is that the scale at which f_n has to be renormalized to define a non-trivial limit of degree d is given by the Rumely’s minimal resultant function.In the second chapter, we focus more specifically on the link between f_n and the limit along a fixed ultrafilter. We establish a remarkable dichotomy on the blow-up of the multipliers of the periodic points of f_n: either almost all periodic points have uniformly bounded multipliers; or almost all multipliers diverge at a uniform rate given by the inverse of the minimal resultant.We then consider the set of all possible scales at which a multiplier of a periodic cycle of f_n explodes, and show that this set is finite of cardinality at most 2d-2. Finally we develop a complete theory of rescaling limits, expanding on the foundational work of J. Kiwi. By definition such an object is a (possibly non-Archimedean) rational map of degree at least 2 and of positive entropy obtained as a limit of a sequence of iterate of conjugates of f_n. Each of these limits comes with associated coordinate changes and a well-defined scales of zooming. We prove that the set of all rescaling limits can be naturally organized into a tree structure, and describe this tree for special families of rational maps, including polynomials, Bernoulli maps in the sense of Favre-Rivera-Letelier and quadratic maps. We conclude by comparing the set of scales associated with multipliers and with the set of scales arising from rescaling limits in the case of polynomial sequences.