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Mixing actions of the rationals

Published online by Cambridge University Press:  07 September 2006

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

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.

Type
Research Article
Copyright
2006 Cambridge University Press

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