Geometric stability theory methods in algebraic dynamics

Certain definable sets in difference fields, namely $\sigma$-varieties, are a natural generalization of algebraic dynamical systems, so some arithmetic questions about algebraic dynamical systems can be addressed with the model theory of difference fields. Ideas around the Zilber Trichotomy Principle, such as minimality, orthogonality, and disintegratedness (triviality) are especially useful.

In order to address two conjectures of Zhang, one about points with Zariski-dense orbits and another about subvarieties which contain a Zariski-dense set of periodic points, these notions must be extended from types to whole $\sigma$-varieties, in a somewhat unexpected way.

As a byproduct of this analysis, we find that disintegrated $\sigma$-varieties defined by $\sigma(x)=f(x)$ for a one-variable polynomial $f$ over a field of characteristic zero always have Morley rank 1 and are usually strongly minimal.

This is joint work with Thomas Scanlon.