**
ACTA MATHEMATICA UNIVERSITATIS COMENIANAE **

Vol. 66, 2 (1997)

pp. 307-311

A NOTE ON MIXING PROPERTIES OF INVERTIBLE EXTENSIONS

G. MORRIS and T. WARD

**Abstract**.
The natural invertible extension $\tildeT$ of an $\Bbb N^d$-action $T$ has been studied by Lacroix. He showed that $\tildeT$ may fail to be mixing even if $T$ is mixing for $d\ge2$. We extend this observation by showing that if $T$ is mixing on $(k+1)$ sets then $\tildeT$ is in general mixing on no more than $k$ sets, simply because $\Bbb N^d$ has a corner. Several examples are constructed when $d=2$: (i) a mixing $T$ for which $\tildeT^(n,m)$ has an identity factor whenever $n\cdot m<0$; (ii) a mixing $T$ for which $\tildeT$ is rigid but $\tildeT^(n,m)$ is mixing for all $(n,m)\neq(0,0)$; (iii) a $T$ mixing on $3$ sets for which $\tildeT$ is not mixing on $3$ sets.

**AMS subject classification**.
28D15

**Keywords**.

**Download:** Adobe PDF Compressed Postscript

Acta Mathematica Universitatis Comenianae

Institute of Applied
Mathematics

Faculty of Mathematics,
Physics and Informatics

Comenius University

842 48 Bratislava, Slovak Republic

Telephone: + 421-2-60295111 Fax: + 421-2-65425882

e-Mail: amuc@fmph.uniba.sk
Internet: www.iam.fmph.uniba.sk/amuc
© Copyright 2001, ACTA MATHEMATICA
UNIVERSITATIS COMENIANAE