Binding groups for algebraic dynamics
Moshe Kamensky, Rahim Moosa
TL;DR
The paper develops a quantifier-free binding-group theorem for the theory $ ext{ACFA}$, linking isotriviality and internality to the fixed field with birational dynamics on $ ext{V}$. It proves that the natural binding group is realized by an algebraic group of birational transformations, and it connects this to a precise correspondence between model-theoretic automorphisms and birational symmetries. The results yield new cases of the Zariski Dense Orbit Conjecture, establish a Dixmier-Moeglin-type equivalence for isotrivial $\sigma$-varieties, and bound nonorthogonality to the fixed field, with implications for differential-algebraic geometry analogues. The work also recovers known autonomous-case results and provides a robust framework for understanding isotrivial dynamics via algebraic groups and their actions.
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 $σ$-varieties. It asserts that if $(V,φ)$ is an isotrivial $σ$-variety then a certain subgroup of the group of birational transformations of $V$, namely those that preserve all the relations between $(V,φ)$ 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 $σ$-varieties. These applications give algebraic-dynamical analogues of recent results in differential-algebraic geometry.