The natural invertible extension T* of an Nd-action T has been studied by Lacroix. He showed that T* may fail to be mixing even if T is mixing for d ³ 2. We extend this observation by showing that if T is mixing on (k+1) sets then T* is in general mixing on no more than k sets, simply because Nd has a corner. Several examples are constructed when d = 2: (i) a mixing T for which T*(n,m) has an identity factor whenever n·m < 0; (ii) a mixing T for which T* is rigid but T*(n,m) is mixing for all (n,m) ¹ (0,0); (iii) a T mixing on 3 sets for which T* is not mixing on 3 sets.
|Number of pages||5|
|Journal||Acta Mathematica Universitatis Comenianae|
|Publication status||Published - 1997|