Binding groups for algebraic dynamics

TL;DR

This paper establishes a binding group theorem for algebraic dynamical systems in difference-closed fields, linking birational geometry with algebraic groups and providing new results in algebraic dynamics.

Contribution

It introduces a binding group theorem for isotrivial $\sigma$-varieties, connecting internality to the fixed field with algebraic groups and applications in algebraic dynamics.

Findings

Proves the binding group theorem in difference-closed fields.

Provides new cases of the Zariski Dense Orbit Conjecture.

Establishes finiteness of invariant rational functions on $\sigma$-varieties.

Abstract

A binding group theorem is proved in the context of quantifier-free internality to the fixed field in difference-closed fields of characteristic zero. This is articulated as a statement about the birational geometry of isotrivial algebraic dynamical systems, and more generally isotrivial $\sigma$-varieties. It asserts that if $(V,\phi)$ is an isotrivial $\sigma$-variety then a certain subgroup of the group of birational transformations of $V$, namely those that preserve all the relations between $(V,\phi)$ and the trivial dynamics on the affine line, is in fact an algebraic group. Several application are given including new special cases of the Zariski Dense Orbit Conjecture and the Dixmier-Moeglin Equivalence Problem in algebraic dynamics, as well as finiteness results about the existence of nonconstant invariant rational functions on cartesian powers of $\sigma$-varieties. These applications give algebraic-dynamical analogues of recent results in differential-algebraic geometry.