Hostname: page-component-7c8c6479df-94d59 Total loading time: 0 Render date: 2024-03-27T01:48:39.567Z Has data issue: false hasContentIssue false

Mixing on a class of rank-one transformations

Published online by Cambridge University Press:  09 March 2004

DARREN CREUTZ
Affiliation:
Department of Mathematics, Williams College, Williamstown, MA 01267, USA (e-mail: dcreutz@wso.williams.edu, csilva@williams.edu)
CESAR E. SILVA
Affiliation:
Department of Mathematics, Williams College, Williamstown, MA 01267, USA (e-mail: dcreutz@wso.williams.edu, csilva@williams.edu)

Abstract

We prove that a rank-one transformation satisfying a condition called restricted growth is a mixing transformation if and only if the spacer sequence for the transformation is uniformly ergodic. Uniform ergodicity is a generalization of the notion of ergodicity for sequences, in the sense that the mean ergodic theorem holds for a family of what we call dynamical sequences. The application of our theorem shows that the class of polynomial rank-one transformations, rank-one transformations where the spacers are chosen to be the values of a polynomial with some mild conditions on the polynomials, that have restricted growth are mixing transformations, implying, in particular, Adams' result on staircase transformations. Another application yields a new proof that Ornstein's class of rank-one transformations constructed using ‘random spacers’ are almost surely mixing transformations.

Type
Research Article
Copyright
2004 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)