Dynamical degrees of automorphisms on abelian varieties

TL;DR

This paper constructs automorphisms on simple abelian varieties with specified dynamical degrees related to Salem numbers, classifies dynamical degree sequences in low dimensions, and explores ergodic properties of pullback sequences.

Contribution

It introduces a method to realize specific Salem numbers as dynamical degrees of automorphisms on abelian varieties and classifies possible degree sequences for low-dimensional cases.

Findings

Constructed automorphisms with dynamical degrees as squares of Salem numbers.

Classified all dynamical degree sequences for abelian varieties up to dimension four.

Proved an ergodic property for sequences of pullbacks of forms.

Abstract

For any given Salem number, we construct an automorphism on a simple abelian variety whose first dynamical degree is the square of the Salem number. Our construction works for both simple abelian varieties with totally indefinite quaternion multiplication and for simple abelian varieties of the second kind. We then give a complete classification of the dynamical degree sequences for abelian varieties of dimension at most four and obtain an ergodic result for sequences of pullbacks of forms.