Symmetrization of Rational Maps: Arithmetic Properties and Families of Latt\`es Maps of $\mathbb P^k$

TL;DR

This paper explores the use of symmetric product constructions to analyze endomorphisms of projective spaces, focusing on stability phenomena and characterizing families of Lattès maps, with implications for arithmetic properties.

Contribution

It introduces a symmetric product approach to study endomorphisms of projective spaces and characterizes families of Lattès maps, advancing understanding of stability and arithmetic properties.

Findings

Symmetric product construction illuminates stability phenomena.

Characterization of families of Lattès maps.

Analysis of $k$-deep post-critically finite maps.

Abstract

In this paper we study properties of endomorphisms of $\p^k$ using a symmetric product construction $(\p^1)^k/\mathfrak{S}_k \cong \p^k$. Symmetric products have been used to produce examples of endomorphisms of $\p^k$ with certain characteristics, $k\geq2$. 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 Latt\`es maps.