Dynamical degrees of affine-triangular automorphisms of affine spaces

We study the possible dynamical degrees of automorphisms of the affine space $\mathbb {A}^n$ . In dimension $n=3$ , we determine all dynamical degrees arising from the composition of an affine automorphism with a triangular one. This generalizes the easier case of shift-like automorphisms which can be studied in any dimension. We also prove that each weak Perron number is the dynamical degree of an affine-triangular automorphism of the affine space $\mathbb {A}^n$ for some n, and we give the best possible n for quadratic integers, which is either $3$ or $4$ .

Affine almost automorphic actions on compact nilmanifolds

We discuss conditions under which an affine automorphism of a compact nilmanifold is almost automorphic, and the structure of such automorphisms from dynamical as well as algebraic points of view.