An Extended Fatou-Shishikura inequality and wandering branch continua for polynomials

TL;DR

This paper extends the classical Fatou-Shishikura inequality for polynomials by incorporating wandering continua and weakly recurrent critical points, providing a more comprehensive bound on the number of non-repelling cycles.

Contribution

It introduces an improved inequality that accounts for wandering branch continua and weak recurrence, enhancing understanding of polynomial dynamics.

Findings

Extended inequality includes wandering continua and weakly recurrent critical points.

Provides a new bound: -1 + N_{irr} + hi + \u2200i ( ext{eval}(Q_i)-2) -1.

Relates individual cycles to critical points with weak recurrence.

Abstract

Let $P$ be a polynomial of degree $d$ with Julia set $J_P$. Let $\widetilde N$ be the number of non-repelling cycles of $P$. By the famous Fatou-Shishikura inequality $\widetilde N\le d-1$. The goal of the paper is to improve this bound. The new count includes \emph{wandering collections of non-precritical branch continua}, i.e., collections of continua or points $Q_i\subset J_P$ \emph{all} of whose images are pairwise disjoint, contain no critical points, and contain the limit sets of $\mathrm{eval}(Q_i)\ge 3$ external rays. Also, we relate individual cycles, which are either non-repelling or repelling with no periodic rays landing, to individual critical points that are recurrent in a weak sense. A weak version of the inequality reads \[ \widetilde N+N_{irr}+\chi+\sum_i (\mathrm{eval}(Q_i)-2) \le d-1 \] where $N_{irr}$ counts repelling cycles with no periodic rays landing at points in the cycle, $\{Q_i\}$ form a wandering collection $\mathcal{B}_\mathbb{C}$ of non-precritical branch continua, $\chi=1$ if $\mathcal{B}_\mathbb{C}$ is non-empty, and $\chi=0$ otherwise.