For mixing transformations \(rank(T^ k)=k\cdot rank(T)\)
For the class of finite-rank transformations having no partial rigidity (this contains the class of mixing maps) the rank behaves like a logarithm on positive powers in that \(rk(T^ k)=k\cdot rk(T)\). The proof proceeds via a coding argument on generic names. Combined with a group-theoretic argument, this theorem yields a structure theorem for the commutant group of a mixing T: It is a type of twisted product (a ``carry product'') of the integers with a certain finite group called the essential commutant. A consequence of this is that any mixing map with rank no greater than five must have an abelian commutant. [The algebraic argument is done in a more general setting in the article: Joining-rank and the structure of finite rank mixing transformations, which will appear in J. d'Analyse.]
scientific article; zbMATH DE number 4017678