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
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