Dispersion Under Iteration of Strongly Mixing Transformations on Metric Spaces

Year 2015, Volume: 3 Issue: 3, 150 - 154, 29.06.2015

Hana Fatkić Abe Sklar Huse Fatkić

https://doi.org/10.18100/ijamec.90047

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.

Keywords

Dispersion under iteration, diameter, harmonic n-diameter, measurability-preserving dynamical systems, measure, metric space, mixing transformations, probability space.

Original Research Paper

Abstract

References

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