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Rigidity and Height Bounds for Certain Post-critically Finite Endomorphisms of ℙN

Published online by Cambridge University Press:  20 November 2018

Patrick Ingram
Patrick Ingram*
Affiliation:
Colorado State University, Fort Collins, CO, USA e-mail: pingram@rams.colostate.edu
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Abstract

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The morphism $f\,:\,{{\mathbb{P}}^{N}}\,\to \,{{\mathbb{P}}^{N}}$ is called post-critically finite $\left( \text{PCF} \right)$ 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 Lattés 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

37P1532H5037P30post–critically finitearithmetic dynamicsheights
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 68 , Issue 3 , 01 June 2016 , pp. 625 - 654
Copyright
Copyright © Canadian Mathematical Society 2016

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