Dispersion Under Iteration of Strongly Mixing Transformations on Metric Spaces **

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.


Introduction
In this work we shall study actions of the group Z of integers on a probability space X, i.e., we study a transformation T: X  X and its iterates , n  Z. Specifically, we shall investigate some metric properties and dispersive effects of strongly mixing (SM) transformations on general metric spaces endowed with a probability measure; in particular, we shall study connections of SM discrete time dynamical systems with the theory of generalized diameters. We shall generally refer to Billingsley [1], Cornfeld, Fomin and Sinai [2], Fatkić [5], Hille [10], Schweizer and Sklar [14].
Suppose (X, , P) is a probability space. As usual, a transformation T : X X is called: (i) measurable (P -measurable) if, for any A in , the inverse image T -1 (A) is in ; (ii) measure-preserving if T is measurable and P (T -1 (A)) = P (A) for any A in (or, equivalently, measure P is said to be invariant under T ); (iii) ergodic if the only members A of with T -1 (A) = A satisfy P(A) = 0 or P (X \ A) = 0; (iv) weakly mixing (or weak-mixing) (with respect to P) if T is P -measurable and for any two P-measurable subsets A, B of X.
We say that the transformation T : X  X is invertible if T is one-to-one (monic) and such that T(A) is P -measurable whenever A is P-measurable subset of X.
A transformation T on a probability space ( X, ,  ) is said to be In this case we also say that the transformation T preserves P -measurability.
If ( X, , P) is a probability space, and T : X X is a measurepreserving transformation (with respect to P), then we say that  : = ( X, , P,T) is an abstract dynamical system. An abstract dynamical system is often called a dynamical system with discrete time or a measure-theoretic dynamical system or an endomorphism. We shall say that the abstract dynamical system  is: (i) invertible if T is invertible; (ii) ergodic if T is ergodic; (iii) weakly (resp. strongly) mixing if T is weakly (resp. strongly) mixing (see [2, pp. 6 -26]). If T is a strongly mixing transformation of a probability space ( X, , P ), then, as is well-known, T is both measure-preserving and ergodic. Furthermore, if T : X  X, in addition (to being strongly _______________________________________________________________________________________________________________________________________________________________ mixing on X with respect to P), is invertible, then (1.1) is equivalent to (the well-known result): for any P -measurable subsets A, B of X.
In this work we consider a metric space/(extended metric spaces, i.e., if we allow d to take values in [0,∞] (the nonnegative lower reals) instead of just in [0,∞), then we get extended metric spaces) (X, d) on which a probability measure P is defined. The domain of P, a  -algebra of subsets of X, is assumed to include all Borel sets in (X, d); in particular, therefore, all open balls in (X, d) are Pmeasurable. We call P nonsingular if there exist two open balls K 1 , K 2 , and a positive number x 0 such that P(K 1 ) > 0, P(K 2 ) > 0, and d(w 1 ,w 2 ) > 0 for all w 1 in K 1 , w 2 in K 2 . We call P pervasive if P(K) > 0 for all open balls K in X. Note that any pervasive measure is nonsingular if X has more than one point. The (ordinary) diameter of a subset A of X , i.e., the supremum of the set { d(x, y) | x, y ∈ A}, will be denoted by diam (A) . We can define the diameter of the empty set (the case A = ) as 0 or  ∞, as we like. But, like many other authors, we prefer to treat the empty set as a special case, assigning it a diameter equal to 0, i.e., diam () = 0, which corresponds to taking the codomain of d to be the set of all nonnegative real numbers (if the distance function d is viewed here as having codomain R, this implies that diam () =  ∞) (see [5] and [10]). If A is P-measurable, then the (ordinary) essential diameter of A, denoted by ess diam (A), is the infimum of the set of diameters of all P-measurable sets B such that B  A and P (B) = P(A). Let (X, d) be a metric space/(extended metric space), and let A be a subset of X. For any positive integer k ≥ 2 and α > 0, we define the α-harmonic diameter of order k of A (kth αharmonic diameter of A), denoted by ( ) ( ), to be the quantity  Note that 2 (1) ( ) is the (ordinary) diameter (diam (A)) of the set A. The sequence ( ( ) ( )) can be shown to be monotone nonincreasing (see, e.g., [5], [8], [10] and [11]), and therefore has a limit as k tends to infinity. By definition, the α-harmonic If A is -measurable subset of X, then, for any positive integer k (1.5) If A consists of only finitely many points, then ( ) ( ) = 0. The general theory of generalised diameters and transfinite diameters plays an important role in complex analysis. It is related to the logarithmic potential theory with applications to approximation theory and the Čebyšev constant (see, e.g., [4], [10], [13] and [16]). In the classical case, when  = 1, X = R 3 , d the Euclidean metric on R 3 , and A is a compact subset of X, the value ( ) ( ) ( ∈ ( \ {1})), which is now called the harmonic diameter of the order k of the set A, has a simple physical interpretation. In fact, if we look the distribution in A of k point sources of electric charge of size 1/k, then the minimum potential energy of such a system of charges obtained when these charges are in the points ( ) ( = 1, … , ) where the function f , defined by achieves a minimum value. In addition, the value of the potential energy is Any set of k points that attains this minimal energy is called an equilibrium configuration for A.

Investigations in ([3], [5]
- [9], [12] and [14]) have shown, however, that many important consequences of (1.2) persist in the absence of invertibility and/or the strongly mixing property.   [7]). It is therefore interesting to investigate how the conclusions of Theorem 1.2 must be modified when the the ordinary diameter is replaced by the α-harmonic diameter of any finite order.
A strongly mixing transformation T that is one to one cannot exhibit such extreme behavior, for in this case −1 is also measure preserving. Nevertheless, as was shown in [3] and [12], under iteration, all strongly mixing transformations tend to spread sets (of positive measure) out in ordinary diameter. Our aim in this work is to show that the known R. E. Rice's Theorem 1.2 [12, Theorem 2] (motivated by some physical phenomena and offer some clarifications of these phenomena) can be generalized in such a way that (1.7) remains valid when "(ordinary) diameter" is replaced by " α-harmonic diameter of any finite order" and that (1.8) remains valid when "(ordinary) essential diameter" is replaced by "essential α-harmonic diameter of any finite order". These results also complement the previous results of N. Faried and M. Fathey [4], H. Fatkić ([5] and [7]), Huse Fatkić, S. Sekulović, and Hana Fatkić ([8] and [9]), E. B. Saff [13] and V. Zakharyuta [16]. We first show that (1.7) holds, not only for the diameter, but also for the α-harmonic diameter ( ) ( ) of order k which is defined for any positive integer k ≥ 2 and any A  X by (1.3). Next we show that essential diameter in (1.8) can be replaced by the essential α-harmonic diameter ess ( ) ( ) of order k which is defined for any positive integer k ≥ 2 and any A  X by (1.5). But here the requirement that transformation T be measurability -preserving is necessary because the essential α-harmonic diameter is defined only for measurable sets. All of these results have some importance, not only in Mathematics, but also in Statistical Mechanics. Note that measurability-preserving strongly mixing transformations on a finite measure space are generalizations of the invertible strongly mixing transformations (they are not necessarily one-to-one).  Proof. Throughout this proof we will denote 2 ( ) ( ), the αdiameter of X, by D, and Let k be an arbitrary positive integer greater than 1. Since ( ) is a monotone nondecreasing set function it is clear that (2.4) holds when ( ) ( ) = 0.
Suppose next that 0 < ( ) ( ) < +∞, whence also 0 < D < +, and let   0 be given. Then, by (1.3), there exist points 1 , 2 , … , in X such that Theorem 2.1 shows that any set A of positive measure necessarily spreads out, not only in (ordinary) diameter, but also in αharmonic diameter of any finite order. Thus, even though A may not spread out in "volume" (measure), there is a very definite sense in which A does not remain small.

Conclusion
There is considerable evidence (see the proof of Theorem 2.1 and 2.2) to support a conjecture that our results (for strongly mixing dynamical systems with discrete time) which are contained in Theorems 2.1 and 2.2 can be extended to strongly mixing dynamical systems with continuous time (see [2, pp. 6-26]). 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 generalized diameters (see [14, pp. 181-190 and (in Dover Edition) 295-297] and also [5], [8], [10], [11] - [13] and [16]). B. Schweizer and A. Sklar have been concluded that comparison of (11. 6.1) with (11.3.5) in [14] and the proof of the Rice's Theorem 1.2 quickly leads to the following conjecture [14, Problem 11.6.5]: Problem 3.1. (Schweizer and Sklar, 1983) Does Theorem 1.2 remain valid when "diameter" is replaced by "(geometric) transfinite diameter"? Note that our Theorems 2.1 and 2.2 in this paper is a first step toward the resolution of the above Problem 3.1. Also note that this problem could be formulated in a more general setting, i.e., not only for the compact set, but also for any set (with positive measure ) in a general metric space. The transfinite diameter of a compact subset A of a metric space is closely related, and often equal, to the capacity, and also to the Chebyshev constant. By the Fundamental Theorem of Classical Potential Theory, we have that, for any compact set A in the complex plane, cap (A) = τ (A) = čeb (A) (see, e. g., [13]). Thus we may also pose the following conjecture (open problem): Problem 3.2. Does the Rice's Theorem 1.2 remain valid when "diameter" is replaced by "α-harmonic transfinite diameter"?