For mixing transformations \(rank(T^ k)=k\cdot rank(T)\).

*(English)*Zbl 0626.47012For 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.]

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.]

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##### References:

[1] | J. L. King,The commutant is the weak closure of the powers, for rank-1transformations, Ergodic Theory and Dynamical Systems6 (1986), 363–384. · Zbl 0595.47005 · doi:10.1017/S0143385700003552 |

[2] | J. L. King,Remarks on the commutant and factors of finite rank mixing transformations, in preparation. |

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