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We apply the (C, F)-construction from [Da] to produce a number of funny rank one infinite measure preserving actions of Abelian groups G with " unusual " multiple recurrence properties. In particular, we construct the following for each p ∈ N ∪ {∞}: (i) a p-recurrent action T = (T g) g∈G such that (if p = ∞) no one transformation T g is (p + 1)-recurrent… (More)

Glossary 1 1. Definition of the subject and its importance 2 2. Basic Results 2 3. Panorama of Examples 8 4. Mixing notions and multiple recurrence 10 5. Topological group Aut(X, µ) 13 6. Orbit theory 15 7. Smooth nonsingular transformations 21 8. Spectral theory for nonsingular systems 22 9. Entropy and other invariants 25 10. Nonsingular Joinings and… (More)

- Alexandre I. Danilenko, Valery V. Ryzhikov
- 2009

Each subset E ⊂ N is realized as the set of essential values of the multi-plicity function for the Koopman operator of an ergodic conservative infinite measure preserving transformation.

- Alexandre I. Danilenko, Valery V. Ryzhikov
- 2009

We introduce high staircase infinite measure preserving transformations and prove that they are mixing under a restricted growth condition. This is used to (i) realize each subset E ⊂ N as the set of essential values of the multiplicity function for the Koopman operator of a mixing ergodic infinite measure preserving transformation, (ii) construct mixing… (More)

It is shown that each subset of positive integers that contains 2 is re-alizable as the set of essential values of the multiplicity function for the Koopman operator of some weakly mixing transformation. Let (X, B, µ) be a standard non-atomic probability space. Given a µ-preserving (invertible) transformation T , we denote by U T the corresponding Koopman… (More)

Let X and Y be Polish spaces with non-atomic Borel measures µ and ν of full support. Suppose that T and S are ergodic non-singular homeomorphisms of (X, µ) and (Y, ν) with continuous Radon-Nikodym derivatives. Suppose that either they are both of type III 1 or that they are both of type III λ , 0 < λ < 1 and, in the III λ case, suppose in addition that both… (More)

Using techniques related to the (C, F)-actions we construct explicitly mixing rank-one (by cubes) actions T of G = R d 1 × Z d 2 for any pair of non-negative integers d 1 , d 2. It is also shown that h(T g) = 0 for each g ∈ G. Mixing rank-one transformations (and actions of more general groups) have been of interest in ergodic theory since 1970 when… (More)

- Alexandre I. Danilenko, Cesar E. Silva
- Encyclopedia of Complexity and Systems Science
- 2009

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