Dispersion Under Iteration of Strongly Mixing Transformations on Metric Spaces

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

Öz

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