An awkward graph

Abstract

Given a rational map $f:\overline{\mathbb C}\to \overline{\mathbb C}$ and a finite graph $G\subset \overline{\mathbb C}$ such that $f(G)\subset G$ and $f$ is expanding on some neighbourhood of $G$, we show that there is another finite graph $G'\subset \bigcup _{n\ge 0}f^{-n}(G)$ in an arbitrarily small neighbourhood of $G$ such that $f^N(G')\subset G'$ for some integer $N$ but $\bigcup _{i=0}^{N-1}f^{i}(G')$ contains accumulating {\em{plaits}} and {\em{nests}}