Tischler graphs of critically fixed rational maps and their applications

A rational map $f:\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ on the Riemann sphere $\widehat{\mathbb{C}}$ is called critically fixed if each critical point of $f$ is fixed under $f$. In this article, we study the properties of a combinatorial invariant, called the Tischler graph, associated with such a map. We show that the Tischler graph of a critically fixed rational map is always connected, establishing a conjecture made by Kevin Pilgrim. This result allows us to solve two classical open problems in rational dynamics in the setting of critically fixed rational maps, namely the combinatorial classification problem and the global curve attractor problem. In particular, we prove that there is a canonical one-to-one correspondence between the conjugacy classes of critically fixed rational maps and the isomorphism classes of connected planar embedded graphs.