Strongly mixing transformations and geometric diameters.

*(English)*Zbl 1325.37006Summary: 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 R. E. Rice [Aequationes Math. 17, 104–108 (1978; Zbl 0398.28020), 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 et al. [Commun. Math. Phys. 29, 311–317 (1973; Zbl 0269.28005)], 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 the first author [Aequationes Math. 43, No. 1, 38–44 (1992; Zbl 0760.28010); ibid. 1(13), No. 1, 49–58 (2005; Zbl 1063.28013) and ibid. 2(15), No. 2, 159–172 (2006; Zbl 1122.28010)], E. B. Saff [Surv. Approx. Theory 5, 165–200 (2010; Zbl 1285.30020)] and C. Sempi [Rad. Mat. 1, No. 1, 3–7 (1985; Zbl 0587.28015)].

We first show that the known result by R. E. Rice [Aequationes Math. 17, 104–108 (1978; Zbl 0398.28020), 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 et al. [Commun. Math. Phys. 29, 311–317 (1973; Zbl 0269.28005)], 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 the first author [Aequationes Math. 43, No. 1, 38–44 (1992; Zbl 0760.28010); ibid. 1(13), No. 1, 49–58 (2005; Zbl 1063.28013) and ibid. 2(15), No. 2, 159–172 (2006; Zbl 1122.28010)], E. B. Saff [Surv. Approx. Theory 5, 165–200 (2010; Zbl 1285.30020)] and C. Sempi [Rad. Mat. 1, No. 1, 3–7 (1985; Zbl 0587.28015)].

##### MSC:

37A25 | Ergodicity, mixing, rates of mixing |

26A18 | Iteration of real functions in one variable |

28A10 | Real- or complex-valued set functions |

39B12 | Iteration theory, iterative and composite equations |