The depth invariant for group actions on countable phase spaces

The construction of the depth of a countable symbolic system, which was studied earlier by the author for actions of the group , is generalized to the case of an arbitrary finitely generated Abelian group action. A new property, which is called the uniformity of the depth, is studied. The depth, which takes values in the set of at most countable ordinals, is a topological invariant of countable symbolic systems. In the paper we describe the set of possible values of the depth invariant and the method of constructing dynamical systems with an arbitrary admissible depth. Translated fromMatematicheskie Zametki, Vol. 65, No. 6, pp. 893–907, June, 1999.