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# Rigidity and Height Bounds for Certain Post-critically Finite Endomorphisms of $\mathbb P^N$

Published:2016-02-18
Printed: Jun 2016
• Patrick Ingram,
Colorado State University, Fort Collins, Colorado, USA
 Format: LaTeX MathJax PDF

## Abstract

The morphism $f:\mathbb{P}^N\to\mathbb{P}^N$ is called post-critically finite (PCF) if the forward image of the critical locus, under iteration of $f$, has algebraic support. In the case $N=1$, a result of Thurston implies that there are no algebraic families of PCF morphisms, other than a well-understood exceptional class known as the flexible Lattès maps. A related arithmetic result states that the set of PCF morphisms corresponds to a set of bounded height in the moduli space of univariate rational functions. We prove corresponding results for a certain subclass of the regular polynomial endomorphisms of $\mathbb{P}^N$, for any $N$.
 Keywords: post-critically finite, arithmetic dynamics, heights
 MSC Classifications: 37P15 - Global ground fields 32H50 - Iteration problems 37P30 - Height functions; Green functions; invariant measures [See also 11G50, 14G40]

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