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3 Colloquium Mathematicum

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On v-positive type transformations in infinite measure

100%

Pădurariu T. , Silva C. , Zachos E.

nr 2

149-170

For each vector v we define the notion of a v-positive type for infinite-measure-preserving transformations, a refinement of positive type as introduced by Hajian and Kakutani. We prove that a positive type transformation need not be (1,2)-positive type. We study this notion in the context of Markov shifts and multiple recurrence, and give several examples.

On weakly mixing and doubly ergodic nonsingular actions

88%

Iams S. , Katz B. , Silva C. , Street B. , Wickelgren K.

tom 103

nr 2

247-264

We study weak mixing and double ergodicity for nonsingular actions of locally compact Polish abelian groups. We show that if T is a nonsingular action of G, then T is weakly mixing if and only if for all cocompact subgroups A of G the action of T restricted to A is weakly mixing. We show that a doubly ergodic nonsingular action is weakly mixing and construct an infinite measure-preserving flow that is weakly mixing but not doubly ergodic. We also construct an infinite measure-preserving flow whose cartesian square is ergodic.

Ergodicity and conservativity of products of infinite transformations and their inverses

75%

Clancy J. , Friedberg R. , Kasmalkar I. , Loh I. , Pădurariu T. , Silva C. , Vasudevan S.

nr 2

271-291

We construct a class of rank-one infinite measure-preserving transformations such that for each transformation T in the class, the cartesian product T × T is ergodic, but the product $T × T^{-1}$ is not. We also prove that the product of any rank-one transformation with its inverse is conservative, while there are infinite measure-preserving conservative ergodic Markov shifts whose product with their inverse is not conservative.