Ergodic Theory and Dynamical Systems



Mixing actions of the rationals


RICHARD MILES a1 and TOM WARD a1
a1 School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK (e-mail: r.miles@uea.ac.uk, t.ward@uea.ac.uk)

Article author query
miles r   [Google Scholar] 
ward t   [Google Scholar] 
 

Abstract

We study mixing properties of algebraic actions of $\mathbb Q^d$, showing in particular that prime mixing $\mathbb Q^d$ actions on connected groups are mixing of all orders, as is the case for $\mathbb Z^d$-actions. This is shown using a uniform result on the solution of $S$-unit equations in characteristic zero fields due to Evertse, Schlickewei and W. Schmidt. In contrast, algebraic actions of the much larger group $\mathbb Q^*$ are shown to behave quite differently, with finite order of mixing possible on connected groups.

(Published Online September 7 2006)
(Received January 13 2005)
(Revised May 9 2006)