An awkward graph

TL;DR

This paper studies the dynamics of rational maps on the Riemann sphere, demonstrating the existence of invariant graphs with complex accumulation structures like plaits and nests near a given expanding graph.

Contribution

It introduces a method to find invariant graphs close to an initial expanding graph that exhibit intricate accumulation phenomena.

Findings

Existence of invariant graphs with complex accumulation structures.

Construction of graphs with plaits and nests near a given expanding graph.

Demonstration of intricate dynamical behavior in rational map systems.

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}}