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

Abstract

In this paper we study properties of endomorphisms of P k \mathbb {P}^k using a symmetric product construction ( P 1 ) k / S k ≅ P k (\mathbb {P}^1)^k/\mathfrak {S}_k \cong \mathbb {P}^k . Symmetric products have been used to produce examples of endomorphisms of P k \mathbb {P}^k with certain characteristics, k ≥ 2 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 k -deep post-critically finite maps and characterize families of Lattès maps.