Complex Dynamics: Schwarzian Derivatives and Measures of Maximal Entropy

We investigate the Schwarzian derivatives of a polynomial and its iterates, where the polyno- mial is defined over the field of complex numbers. The escape-rate function of the polynomial play an important role in the study of polynomial dynamics. The study of the sequence of Schwarzian derivatives leads to a connection with the escape-rate function. The polynomial basin of infinity admits a natural metric,which keeps a lot of polynomial dynam- ics information. The quadratic differential of the Schwarzian derivative determines a Riemannian metric on the complement of f^n’s critical points. As n tends to infinite, this sequence of metric spaces has an ultralimit, which is a complete geodesic space with non-positive curvature. And by the properties inherited from the the dynamics of the polynomial, we can naturally embed the basin of infinity isometrically to the ultralimit. We also investigate rational functions with identical measure of maximal entropy. For a given ra- tional function f : CP1→CP1 with degree d ≥ 2, there is a unique probability measure μf associated with it, which achieves maximal entropy logd among all the f -invariant probability measures. From the work of Beardon, Levin, Baker-Eremenko, Schmidt-Steinmetz, etc (1980s-90s), the set of polyno- mials with identical measure of maximal entropy has been characterized. We construct examples of non-exceptional rational functions with common measure of maximal entropy, and they won’t share an iterate up to precomposition by any M¨obius transformation. Following from Levin-Przytycki’s result (1997), we characterize the general sets of rational functions with identical measures of maximal entropy. Finally, we sum up some known results related to the set of preperiodic points and maximal entropy measure, and then provide some necessary and sufficient conditions for two rational functions sharing an iterate.