Binding groups for algebraic dynamics
Abstract
A binding group theorem is proved in the context of quantifierfree internality to the fixed field in differenceclosed 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 DixmierMoeglin 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 algebraicdynamical analogues of recent results in differentialalgebraic geometry.
 Publication:

arXiv eprints
 Pub Date:
 May 2024
 DOI:
 10.48550/arXiv.2405.06092
 arXiv:
 arXiv:2405.06092
 Bibcode:
 2024arXiv240506092K
 Keywords:

 Mathematics  Logic;
 Mathematics  Algebraic Geometry;
 03C45;
 12H10;
 12L12;
 14E07
 EPrint:
 40 pages