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 $G^{\prime }\subset {\bigcup }_{n\ge 0}{f}^{-n}(G)$ in an arbitrarily small neighbourhood of $G$ such that $f^N(G^{\prime })\subset G^{\prime }$ for some integer $N$ but ${\bigcup }_{i=0}^{N-1}{f}^i(G^{\prime })$ contains accumulating {\em{plaits}} and {\em{nests}}

8 pages, 7 figures