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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
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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 Latt├Ęs 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 post-critically finite, arithmetic dynamics, heights
MSC Classifications: 37P15, 32H50, 37P30 show english descriptions Global ground fields
Iteration problems
Height functions; Green functions; invariant measures [See also 11G50, 14G40]
37P15 - Global ground fields
32H50 - Iteration problems
37P30 - Height functions; Green functions; invariant measures [See also 11G50, 14G40]
 

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