Almost polynomial-like maps in polynomial dynamics

Alexander Blokh, Lex Oversteegen, Vladlen Timorin

Published 2020-08-15Version 1

Holomorphic renormalization plays an important role in complex polynomial dynamics. We consider a situation, when a polynomial is not immediately renormalizable but admits an invariant continuum on which it is topologically conjugate to a lower degree polynomial. This invariant continuum may contain extra critical or parabolic 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 these generalized notions are applicable and yield useful topological conjugacies.