Symmetrization of rational maps: Arithmetic properties and families of Lattès maps of ℙ^{𝕜}

Gauthier, Thomas; Hutz, Benjamin; Kaschner, Scott

doi:10.1090/ecgd/382

Symmetrization of rational maps: Arithmetic properties and families of Lattès maps of $\mathbb {P}^k$
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by Thomas Gauthier, Benjamin Hutz and Scott Kaschner

Conform. Geom. Dyn. 27 (2023), 98-117

DOI: https://doi.org/10.1090/ecgd/382

Published electronically: February 14, 2023

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Abstract:

In this paper we study properties of endomorphisms of $\mathbb {P}^k$ using a symmetric product construction $(\mathbb {P}^1)^k/\mathfrak {S}_k \cong \mathbb {P}^k$. Symmetric products have been used to produce examples of endomorphisms of $\mathbb {P}^k$ with certain characteristics, $k\geq 2$. In the present note, we discuss the use of these maps to enlighten stability phenomena in parameter spaces. In particular, we study $k$-deep post-critically finite maps and characterize families of Lattès maps.

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