Symmetrization of Rational Maps: Arithmetic Properties and Families of Lattès Maps of $\mathbb P^k$

Abstract

In this paper we study properties of endomorphisms of P k using a symmetric product construction (P 1 ) k /S k ∼ = P k . Symmetric products have been used to produce examples of endomorphisms of P k with certain characteristics, k ≥ 2. 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 Lattès maps.

Dynamics of post-critically finite maps in higher dimension

“…In [Jon98], Jonsson remarks that PCF endomorphisms of P 2 are always post-critically finite all the way down. In [GHK16], the authors prove that the same is true for a very specific subclass of PCF endomorphisms of P k , and ask if the same holds for any PCF endomorphism of P k . They also give examples of rational maps f ∶ P k ⇢ P k that are PCF but not PCF all the way down.…”

“…One can then define inductively what it means to be post-critically finite of order m, with m ≤ k (see Definition 4 for a precise definition). Post-critically finite endomorphisms of P k of order k are somewhat colloquially referred to as post-critically finite all the way down (the authors of [GHK16] use the term strongly post-critically finite).…”

Multipliers and invariants of endomorphisms of projective space in dimension greater than 1

“…One way to generate isospectral maps in higher dimensions is to apply a construction to a family of Lattès maps. For example, symmetrization [8], cartesian products, and Segre embeddings can be used to construct isotrivial families starting with a Lattès family.…”

“…Example 5.4. We compute with an example from Gauthier-Hutz-Kaschner [8]. Starting with the Lattès family…”