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arXiv · 2404.00731

Portraits of quadratic rational maps with a small critical cycle

Tyler Dunaisky·David Krumm
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Abstract

Motivated by a uniform boundedness conjecture of Morton and Silverman, we study the graphs of pre-periodic points for maps in three families of dynamical systems, namely the collections of rational functions of degree two having a periodic critical point of period $n$, where $n\in\{2,3,4\}$. In particular, we provide a conjecturally complete list of possible graphs of rational pre-periodic points in the case $n=4$, analogous to well-known work of Poonen for $n=1$, and we strengthen earlier results of Canci and Vishkautsan for $n\in\{2,3\}$. In addition, we address the problem of determining the representability of a given graph in our list by infinitely many distinct linear conjugacy classes of maps.

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Tyler Dunaisky, David Krumm. 2024-03-31. Portraits of quadratic rational maps with a small critical cycle. https://arxiv.org/abs/2404.00731

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