Dispersion Under Iteration of Strongly Mixing Transformations on Metric Spaces
DOI:
https://doi.org/10.18100/ijamec.90047Keywords:
Dispersion under iteration, diameter, harmonic n-diameter, measurability-preserving dynamical systems, measure, metric space, mixing transformations, probability space.Abstract
In this paper, we investigate metric properties and dispersive effects of strongly mixing transformations on general metric spaces endowed with a probability measure; in particular, we investigate their connections with the theory of generalized (α-harmonic) diameters on general metric spaces. We first show that the known result by R. E. Rice ([Aequationes Math. 17(1978), 104-108], 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 T. Erber, B. Schweizer and A. Sklar [Comm. Math. Phys., 29 (1973), 311 – 317], can be generalized in such a way that this result remains valid when "ordinary diameter" is replaced by "α-harmonic diameter of any finite order". Next we show that "ordinary essential diameter" in the mentioned Rice's result can be replaced by the" essential α-harmonic diameter of any finite order". These results also complement the previous results (on dynamical systems with discrete time and/or generalised diameters) of N. Faried and M. Fathey, H. Fatkić, E. B. Saff, S. Sekulović and V. Zakharyuta.Downloads
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