Ergodic Theory and Dynamical Systems



Mixing on a class of rank-one transformations


DARREN CREUTZ a1 and CESAR E. SILVA a1
a1 Department of Mathematics, Williams College, Williamstown, MA 01267, USA (e-mail: dcreutz@wso.williams.edu, csilva@williams.edu)

Article author query
creutz d   [Google Scholar] 
silva c   [Google Scholar] 
 

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.

(Received January 14 2002)
(Revised August 4 2003)