Product Cocycles and the Approximate Transitivity |
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Authors: | Golodets Valentin Ya Sokhet Alexander M |
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Institution: | (1) Institute for Low Temperature Physics and Engineering, Academy of Science, 46 Lenin Avenue, 310164 Kharkov, Ukraine |
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Abstract: | Some criteria of the approximate transitivity in the terms of Mackey actions and product cocycles are proved. The Mackey action constructed by an amenable type II or III transformation group G and a 1-cocycle × , where is the Radon–Nikodym cocycle while is an arbitrary 1-cocycle with values in a locally compact separable group A, is approximately transitive (AT) if and only if the pair (G,( , )) is weakly equivalent to a product odometer supplied with a product cocycle. Besides, in the case when the given AT action from the very beginning was a range of a type II action and a nontransient cocycle, then this cocycle turns out to be cohomologous to a -product cocycle. An example is constructed that shows that it is necessary to consider the double Mackey actions since they can not be reduced to the single ones. |
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Keywords: | ergodic theory approximate transitivity product cocycle Mackey action |
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