Non-degenerate near-parabolic renormalization

Abstract: Invariant classes under parabolic and near-parabolic renormalization have proved extremely useful for studying the dynamics of polynomials. The first such class was introduced by Inou-Shishikura to study quadratic polynomials; their argument has been extended to the unicritical cubic case by Ch\'eritat and Yang. However, all of these classes are only applicable to maps which have a fixed point with multiplier close to one, though it is well-known that similar phenomena occur when the multiplier is close to any root of unity. In this paper we define the parabolic and near-parabolic renormalization operators in the general setting and construct invariant classes. Additionally, we compare the corresponding near-parabolic renormalizations when one root of unity is close to another.