Strongly Mixing Transformations and Geometric Diameters
DOI:
https://doi.org/10.5644/SJM.08.2.06Keywords:
Abstract dynamical system, essential diameter, geometric diameter of finite order, strongly mixing dynamical system, strongly mixing transformation, weakly mixing transformationAbstract
In this paper, we investigate metric properties and dispersive effects of strongly mixing transformations on general metric spaces endowed with a finite measure; in particular, we investigate their connections with the theory of generalized (geometric) diameters on general metric spaces. We first show that the known result by Rice [17,Theorem 2] (motivated by some physical phenomena and offer some clarifications of these phenomena), which is a substantial improvement of Theorems 1 and 2 due to Erber, Schweizer and Sklar [4], can be generalized in such a way that this result remains valid when ``ordinary diameter" is replaced by ``geometric diameter of any finite order". Next we show that ``ordinary essential diameter" in the mentioned Rice's result can be replaced by ``essential geometric diameter of any finite order". These results also complement the previous results of Fatkić , Saff and Sempi.
2000 Mathematics Subject Classification. 37A25, 26A18, 28A10
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