Measurability – Preserving Weakly Mixing Transformations
DOI:
https://doi.org/10.5644/SJM.02.2.04Keywords:
Measurability-preserving weakly mixing transformations, measure-preserving transformations, abstract dynamical system, weakly mixing dynamical systems, continuous time, ergodicity, strongly mixing, metric space, Banach space, Hilbert spaceAbstract
In this paper we investigate measure-theoretic properties of the class of all weakly mixing transformations on a finite measure space which preserve measurability. The main result in this paper is the following theorem: If $\phi $ is a weakly mixing transformation on a finite measure space $( S, \mathcal A , \mu )$ with the property that $\phi (\mathcal A ) \subseteq \mathcal A ,$ then for every $A, B $ in $\mathcal A$ there is a subset $J(A,B)$ of the set of non-negative integers of density zero such that $\lim _{m \to \infty ,m \notin J(A,B)} \mu (A \cap \phi ^m(B)) = (\mu (A) / \mu (S))\lim _{n \to \infty } \mu \,(\phi^n(B)).$ Furthermore, we show that for most useful measure spaces we can strengthen the preceding statement to obtain a set of density zero that works for all pairs of measurable sets $A$ and $ B.$ As corollaries we obtain a number of inclusion theorems. The results presented here extend the well-known classical results (for invertible weakly mixing transformations), results of R. E. Rice [17] (for strongly mixing), a result of C. Sempi [19] (for weakly mixing) and previous results of the author [8, 10] (for weakly mixing and ergodicity).
2000 Mathematics Subject Classification. Primary: 28D05, 37A25; Secondary: 37A05, 47A35
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