Immediate renormalization of complex polynomials

Abstract

A cubic polynomial $P$ with a non-repelling fixed point $b$ is said to be \emph{immediately renormalizable} if there exists a (connected) quadratic-like invariant filled Julia set $K^{\ast }$ such that $b\in K^{\ast }$. In that case exactly one critical point of $P$ does not belong to $K^{\ast }$. We show that if, in addition, the Julia set of $P$ has no (pre)periodic cutpoints then this critical point is recurrent.

34 pages, 2 figures