We discuss a stochastic operator method in ergodic theory and its application to the well-known Rokhlin higher-order mixing problem. In this paper invariants of dynamical systems which guarantee multiple mixing property are considered. These invariants, which are expressed in terms of operators intertwining Cartesian products of systems, are some analogs of known properties of joinings. A typical result: any mixing flow (an action of the group ℝn) with a simple stochastic centralizer is mixing of all orders.
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The research was supported by the G. Soros International Science Foundation, Grant M1E000.
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Ryzhikov, V.V. Stochastic intertwinings and multiple mixing of dynamical systems. Journal of Dynamical and Control Systems 2, 1–19 (1996). https://doi.org/10.1007/BF02259620
1991 Mathematics Subject Classification
Key words and phrases
- Simple dynamical system
- higher-order mixing
- intertwining operators