Invariant graphs of rational maps

TL;DR

This paper proves that for postcritically finite rational maps, there exists an invariant finite graph containing the postcritical set for sufficiently large iterates, aiding in understanding the map's dynamics.

Contribution

It establishes the existence of invariant graphs containing the postcritical set for large iterates of postcritically finite rational maps, a new structural insight.

Findings

Existence of invariant graphs for large n

Invariant graphs contain the postcritical set

Supports structural understanding of rational maps

Abstract

Let $f$ be a postcritically finite rational map. We prove that, as $n$ large enough, there exists an $f^n$-invariant (finite connected) graph on $\widehat{\mathbb{C}}$ such that it contains the postcritical set of $f$.