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The commutant is the weak closure of the powers, for rank-1 transformations

1986; Cambridge University Press; Volume: 6; Issue: 3 Linguagem: Inglês

10.1017/s0143385700003552

1469-4417

Jonathan King,

Mathematical Dynamics and Fractals

Abstract In the class of rank-1 transformations, there is a strong dichotomy. For such a T , the commutant is either irivial , consisting only of the powers of T , or is uncountable . In addition, the commutant semigroup, C ( T ), is in fact a group. As a consequence, the notion of weak isomorphism between two transformations is equivalent to isomorphism, if at least one of the transformations is rank-1. In § 2, we show that any proper factor of a rank-1 must be rigid. Hence, neither Ornstein's rank-1 mixing nor Chacón's transformation, can be a factor of a rank-1.