Canonical heights for plane polynomial maps of small topological degree

TL;DR

This paper investigates canonical heights in plane polynomial maps with small topological degree, showing that points with zero canonical height have arithmetic degrees bounded by the topological degree, which is less than the dynamical degree.

Contribution

It establishes a bound on the arithmetic degree for points with zero canonical height in the context of plane polynomial maps of small topological degree.

Findings

Points with canonical height zero have arithmetic degree bounded by the topological degree.

The topological degree is strictly smaller than the first dynamical degree for these maps.

Existence of specific compactifications of the plane adapted to the dynamics was used in the proof.

Abstract

We study canonical heights for plane polynomial mappings of small topological degree. In particular, we prove that for points of canonical height zero, the arithmetic degree is bounded by the topological degree and hence strictly smaller than the first dynamical degree. The proof uses the existence, proved by Favre and the first author, of certain compactifications of the plane adapted to the dynamics.