Dynamical degrees of automorphisms of complex simple abelian varieties and Salem numbers

TL;DR

This paper demonstrates that every Salem number can be realized as the first dynamical degree of an automorphism of a complex simple abelian variety, revealing new connections between Salem numbers and complex dynamics.

Contribution

It establishes the realization of all Salem numbers as dynamical degrees and identifies minimal and near-one dynamical degrees for automorphisms of complex simple abelian varieties.

Findings

Every Salem number is a first dynamical degree of some automorphism.

The set of first dynamical degrees has a minimum value for fixed dimension.

Existence of automorphisms with dynamical degrees arbitrarily close to 1.

Abstract

We prove that every Salem number can be realized as the first dynamical degree of an automorphism of a complex simple abelian variety. Also by using the similar technique, we prove that the set of first dynamical degrees of automorphisms of complex simple abelian varieties except 1 has the minimum value when fixing the dimension of complex simple abelian varieties. Moreover, we prove that there is an automorphism of a complex simple abelian variety, whose first dynamical degree is as close as possible to 1. These results are inspired by the work of Nguyen-Bac Dang and Thorsten Herrig.