Conjugations, joinings, and direct products of locally rank one dynamical systems
The work is devoted to studying properties of the finite rank automorphism defined on a standard Borel nonatomic probability space. It is shown that if this automorphism is ergodic, has uniform rank \(n\) and \(S\) is an automorphism conjugate to its inverse, then \(S^{2m}=I\) (where \(I\) is the identity automorphism) for some positive integer \(0< m\leq n\). Related results for an ergodic map having positive \(\beta\)-rank with no partial rigidity are given. Further the article is concerned with the question of whether the Cartesian square and its square root can have finite rank. This question is answered in certain cases, in particular, it is proved that in the cases when \(T\) is mixing, \(T\) is not partially rigid, \(T\) is isomorphic to its inverse the Cartesian square and its square root cannot be of finite rank.
scientific article; zbMATH DE number 1431840
scientific article