On critical renormalization of complex polynomials

TL;DR

This paper explores conditions under which complex polynomials can be renormalized to simpler forms, even when they lack polynomial-like Julia sets, by introducing generalized holomorphic renormalization concepts.

Contribution

It extends the theory of holomorphic renormalization and polynomial-like maps to include cases with invariant continua containing extra critical points, broadening the applicability of these methods.

Findings

Established conditions for invariant continua with conjugacy to lower degree polynomials.

Extended holomorphic renormalization notions to more general settings.

Provided a framework for topological conjugacies involving additional critical points.

Abstract

Holomorphic renormalization plays an important role in complex polynomial dynamics. We consider certain conditions guaranteeing that a polynomial which does not admit a polynomial-like connected Julia set still admits an invariant continuum on which it is topologically conjugate to a lower degree polynomial. This invariant continuum may contain extra critical points of the original polynomial that are not visible in the dynamical plane of the conjugate polynomial. Thus, we extend the notions of holomorphic renormalization and polynomial-like maps and describe a setup where new generalized versions of these notions are applicable and yield useful topological conjugacies.