In [R. E. Rice, On mixing transformations, Aequationes Math. 17 (1978), 104 – 108; Theorem 2 (motivated by some physical phenomena and offer some clarifications of these phenomena)] it is shown that, under iteration, all (strongly) mixing transformations tend to spread sets out in (ordinary) diameter. Thus, even though some set A may not spread out in “volume” (measure), there is a very definite sense in which A does not remain small.
Even though a large set cannot have a small diameter and a set with a large diameter cannot be truly small, it is still true that the diameter of a set need not be a good measure of its size and shape. A better measure is furnished by the harmonic diameter, (geometric) transfinite diameter and the generalized diameters.
Dear Colleague Jasmina ,
The answer is: YES! This follows from Theorem 2.1. in [Hana Fatkić, Abe Sklar, Huse Fatkić: Dispersion Under Iteration of Strongly Mixing Transformations on Metric Spaces; In: International Journal of Applied Mathematics, Electronics and Computers, (IJAMEC), 2015, pp. 150-154] (see https://www.researchgate.net/publication/279750914_Dispersion_Under_Iteration_of_Strongly_Mixing_Transformations_on_Metric_Spaces).
Dear Colleague Sekulovic, I think that you correctly answered to Jasmina to her question: "Does, under iteration, all strongly mixing transformations tend to spread sets out, not only in (ordinary) diameter, but also in harmonic diameter?"
Dear Colleague Sekulovic, Thank you very much for the precise and useful answer to my question about the strongly mixing transformations and the harmonic diameter! Jasmina
Dear Dr. Jack Son,
Thank you very much for your comment on my question: “Does, under iteration, all strongly mixing transformations tend to spread sets out, not only in (ordinary) diameter, but also in harmonic diameter?”.
With the best regards,
Jasmina Fatkić
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