A MIXING DYNAMICAL SYSTEM ON THE CANTOR SET

. In this paper we give mixing properties (ergodic, weak-nixng and strong-mixing) to a dynamical system on the Cantor set by showtn,, that the one-sided .(! l-shift map is isomorphic to a measure preserving transformation defined on the Cantor set.


INTRODUCTION
A dynamical system is a quadruple (X,.,q, tn, T), where X is a non-empty space, .i s a o-algebra of subsets of X, m s a measure defined on ..q and T is a measure-preserving transformation on X. Ergodic theory may be defined to be the study of transformations or groups of transformations, which are defined on some measure space, which are measurable with respect to the measure structure of that space, and which leave the measure of all measurable subsets of the space. If arbitrary orbit of transformation of a gven dynamical system passes through every point of this system, it is said that this dynamical system satisfies the ergodic hypothesis. The ergodic hypothesis was introduced by L. Boltzman and W. Gibbs to establish the following principle: which says that the time mean off is equal to the space (or phase) mean off. Many contrarguments about this hypothesis came out but in 1912 H. Poincare proved the so-called recurrence theorem and in 1931 G. D. Birkhoff and J. Von Neumann proved the existence of time mean and thus this hypothesis was accepted. This is the historical beginning of mathematical study of ergodic theory. The study of ergodic theory can be categorized into one of four types, that is, (i) Measure theoretic, (ii) Topological, (iii) Mixture of (i) and (ii), and (iv) Smooth. In this paper we are concerned about measure theoretic type and we shall assume that the measure is finite and normalized to have total measure one (the probability measure). We refer [1], [2] as general texts for ergodic theory.
The main purpose in this paper is to give ergodic property to a dynamical system on the Cantor ternary set. In Section 2 we first summarize properties of the generalized Cantor set and the Cantor measure. In Section 3 we introduce mixing properties (i.e. strong-mixing, weak-mixing and ergodic properties) for measure preserving transformations. The one-sided shift map is shown to satisfy these properties. In Section 4 we show that the one-sided shift transformation is isomorphic to a measure preserving transformation defined on the Cantor ternary set so that a dynamical system on the Cantor set has the three mixing properties.

THE GENERALIZED CANTOR SET AND THE CANTOR MEASURE
In this section we summarize some properties of the generalized Cantor set and the Cantor measure. We first define the Cantor set, the Cantor function and the Cantor measure. We describe the Lebesque measure of the Cantor set together with the topological properties of the set and the relation between lhe Cantor measure and the kebesque mea,,ure otlhe et. The Cantor set s knon to have the same cardnalty as the inlerval [0,1]. Fnally n ths see[ion we present the ternary rcpresenlation of Cantor's mddle third DEFINITION 2.1. Le n,n n be a sequence of real numbers such thai < tiC < Ir every k 1,2 We define the generalized Cantor set denoted by C(, [0,1 ], by setting it equal to 1/2 on the first interval removed, 1/2 and 3/2 on the two intervals removed at the second step, 1/2 , 3/2 , 5/3 and 7/2 on the third removal, etc. The values are chosen in the obvious way so that the Ihnction F is monotone non-decreasing on [0,1 (F is extended to [0,1] by the coninuily).
We note that the above r becomes a Borel measure by the following argument: If we let A be the semialgebra consisting of all intervals of the form [a ,b), 0 < a < b _< 1, and set lar([a, b )) F(b F(a ), then I.tr satisfies the following two conditions: 1. l.tF(B)= ',':tl.tF(B,) for any finite disjoint union B(=w','=B,) of B,'s in . . 2. tF(B) <--Y= F(B,) for any countable disjoint union B (= w7= B,) of B,'s in . Then lar admits a unique extension to a measure on the algebra generated by .a nd thus, from Caratheodo extension theorem, r can extended to a -algebra containing Hence we have an extension of to a Borel measure.
In the following we describe the topological stcture of the generalized Cantor set C d the relation between the Cantor measure and the Lebesque measure of the Cantor set. 1. There is a bijection map between the Cantor set C and the inteal [0,1]. 2. The Ctor set C is always nowhere dense. 3. The Cantor set C is a perfect set. 4. The Lebesque measure of the Cantor set C is given by the infinite product -. 5. The Ctor measure g of the Cantor set C is absolutely continuous with respect to the besque measure of the set if 2= 6. The Cantor measure ge is mutually singular with respect to the Lebesque measure if nk If n, is a fixed positive rational number (constant disection ratio) for all k 1,2 then we can ohtaln the concrete rcpre<,cniatllln of the Cailor sot (of [3l). if in particular #ix 3 for all k= !,2 the tlol'lllillOll of the gCllOlall/cd CalllOr sol C, lhcll c (_" iI aild oi11 if h,ls ti IClllary icpl-c,OlllilliOll t)f the l)llll ") ,; whore a,, e {(), I} for all #l " In this case, Ihc called the Carll(li lciilary sol {)i (-'alllOi"s middle third sol and the ('alllOr ftliWIIOll P" on (' I, given hv F _kL 3

. DYNAMICAL SYSTEM AND THE ONE-SIDED StlIFT TRANSFORMATION
In this section we define a nicasurc preserving translbrmation and a dynamical system in terms of lhs transformation. We introduc tho three kinds of mtxino properti (t e. erodic, wak-mixn and strong-mixing) of a measure prsrvin transformalion. Th one-sided shift map is shown to b a transformation which has all of these thre proprtos. Th lllowing thorom is well known in masur theory [4]. It provides th sufficient condition lr the map T to be measure-prsrvin in trms of gnoratin algebra. Sinc wo wish to study the iteration T" of the transformation T, w shall dal with the identical case (X,t,m)= (X,,m). A probability spac (X,,m) together with a measure-preserving transformation T dfins a dynamical sstem (X, m, T). Th above Theorem 3. shows that whthr a given transformation is masur-prserving or not is determined only by the knowldge of an algebra generating the -algebra In th following we defin the thre mixing properties for a dynamical system. In the theory of dynamical systems, mixing is the property of indecomposibility of a dynamical system into nontrivial invariant subsets. "strong-mixing" implies "weak-mixing" and "weak-mixing" implies "ergodic." .I.H. KIN Like measure-preserving property, mixing property .,, determined by the knowledge of generating algebra. From ths we can prove that the one-sided shift transformation defined by the following is strong-mixing and .so v,'eak-mxng and ergodc.

ISOMORPHISM OF MEASURE-PRESERVING TRANSFORMATIONS
The content of this section is our main result. In previous section, we show that the one-sided (p, p p,,)-shift transformation is ergodic, weak-mixing and strong-mixing. In this section, we define a measure-preserving transformation on the Cantor measure space, i.e., the Cantor ternary set, and then ( )-shift transformation defined on the show that this transformation is isomorphic to the one-sided 5, product space X (see Definition 3.3). The Cantor measure space together with the one-sided shift transformation, therefore, becomes a dynamical system having the three mixing properties. THEOREM 4.1. Suppose that (C, Y,, lad is a probability space, where C is the Cantor ternary set, Y-is a '-algebra generated by finite disjoint unions of sets of the form [a, b) C, 0 < a < b < 1, and 1.F is the Cantor measure. Then the function T C--> C defined by T 2,,Z, =2,Y_.,, --, a,, {0,1}  which is a disjoint union because b < a + 2. Then obviously the right-hand side of (4.3) belongs to I/" Then a,, < x,, < b,, up to n no sothat T becomes measurable. Now lctx e [a,b]Cwithx 2Y,: e'" '"-.__! where xo is either 0 or and thus T-(x) belongs to the right side for some "o and T-' (x) " Y,7= Then y =0 and of (4.3) For the reverse direction of subset, let y e .7,7 C with y 2 Y., v," [,,+2 ,,+2] a,,_<y,,<b,,_t up to no for some n0 and thus a<T(y)<b. Similarly, the set ----,-q-oC can be shown to be a subset of T-([a,b ]cn C). Therefore the equality  PROOF. To define an appropriate map @, we first define the following notation: Let (x,x2 x, be an arbitrary element in X, i.e., x,, 0 or for any n  where we actually take the element of the set, not ct tsclf, for the right hand sde of (4.7). We observe that K,,(x,x2 x,,) is the closed nterval whose left end point s 2 : 7" x,,, ,, and whose right end point s 2 7 7' r,,.