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A sequence of integers {n i : i = 0, 1 .... } is an exhaustive weakly wandering sequence for a transformation T if for some measurable set W, X = U~o Tni W(disj). We introduce a hereditary Property (H) for a sequence of integers associated with an infinite ergodic transformation T, and show that it is a sufficient condition for the sequence to be an exhaustive weakly wandering sequence for T. We then show that every infinite ergodic transformation admits sequences that possess Property (H), and observe that Property (H) is inherited by all subsequences of a sequence that possess it. As a corollary, we obtain an application to tiling the set of integers Z with infinite subsets. ## Introduction Let T be a measurable transformation defined on a non-atomic, a-finite measure space (X,/3, m). In the sequel, all sets mentioned will be assumed to be measurable (belong to B), and statements will be understood to mean "up to sets of measure zero" even when not explicitly stated. All transformations T : X --+ X are 1-1 and onto. We say that T preserves the measure m, or m is an invariant measure for T, ifm(TA) = m(T-1A) = m(A) for every A ~/3, and the transformation T is ergodic if T A = A im

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Author

S. Eigen; A. Hajian

Publisher

Elsevier Science; Elsevier ; Elsevier BV (ISSN 0019-3577)

Published

2007

Field

Mathematics (Physical Sciences)