The generic transformation has roots of all orders. (English) Zbl 0972.37001

King, Jonathan L. F.

Colloq. Math. 84-85, Pt. 2, 521-547 (2000).

D. S. Ornstein [Proc. 6th Berkeley Sympos. math. Statist. Probab., Univ. Calif. 1970, 2, 347-356 (1972; Zbl 0262.28009)] constructed a measure-preserving transformation with no roots under composition. In this paper it is shown that the generic transformation (in the sense of Baire category) has roots of all orders, by showing that the image of the squaring, cubing,... maps on the group of all invertible measure-preserving transformations are not meager sets. This implies the result via a 0-1 law for ‘dynamical’ sets, considered in this context by E. Glasner and the author [Contemp. Math. 215, 231-242 (1998; Zbl 0909.28014)], which states that such sets are residual if they are not meager. Several interesting questions are raised at the end of the paper, and one of the most important of these has already been answered: de la Rue and de Sam Lazaro have applied the methods of this paper to show that the generic transformation embeds in a flow.

Reviewer: Thomas Ward (Norwich)

MSC:

37A05	Dynamical aspects of measure-preserving transformations
28D05	Measure-preserving transformations

Keywords:

generic measure-preserving transformation; roots of all orders