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Infinite ergodic index \(\mathbb{Z}^d\)-actions in infinite measure. (English) Zbl 0940.28014

In this paper certain interesting examples of ergodic infinite-measure preserving (and non-singular free) \(\mathbb{Z}^d\)-actions on the reals are constructed using cutting and stacking methods. The first example is ergodic but has non-ergodic infinite conservative index basis transformations. The next examples are staircase rank-one, for these it is shown that the individual basis transformations have conservative ergodic Cartesian products of all orders (and therefore infinite ergodic index), and this condition is generalized to show power weak-mixing (a group action is power weak-mixing if all Cartesian products of non-trivial elements of the action are ergodic). Finally, non-singular examples are built for each Krieger ratio set type with individual basis transformations with similar properties.

MSC:

28D15 General groups of measure-preserving transformations
37A40 Nonsingular (and infinite-measure preserving) transformations
22D40 Ergodic theory on groups
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