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

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, $k\geq2$. In the present note, we discuss the use of these maps to enlighten arithmetic phenomena and stability phenomena in parameter spaces. In particular, we study notions of uniform boundedness of rational preperiodic points via good reduction information, $k$-deep postcritically finite maps, and characterize families of Latt\`es maps.Comment: Added more background and references; repaired a small gap in Lemma 3.1; reordered some statements in Propositions 1.1 and 1.2; 26 page