Tischler graphs of critically fixed rational maps and their applications

Mikhail Hlushchanka

Research output: Working paperAcademic


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 properties of a combinatorial invariant, called Tischler graph, associated with such a map. More precisely, we show that the Tischler graph of a critically fixed rational map is always connected, establishing a conjecture made by Kevin Pilgrim. We also discuss the relevance of this result for classical open problems in holomorphic dynamics, such as combinatorial classification problem and global curve attractor problem.
Original languageEnglish
Number of pages14
Publication statusPublished - 9 Apr 2019
Externally publishedYes


  • math.DS


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