MEASURE-PRESERVING DYNAMICAL SYSTEMS AND APPROXIMATION TECHNIQUES

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1 MEASURE-PRESERVING DYNAMICAL SYSTEMS AND APPROIMATION TECHNIQUES JASON LIANG Abstract. I this paper, we demostrate how approximatio structures called sufficiet semirigs ca provide iformatio about measure-preservig dyamical systems. We describe the basic properties of measure-preservig dyamical systems ad illustrate their coectios to sufficiet semirigs by ivestigatig the dyadic addig machie. Cotets. Itroductio. Ivariat Trasformatios 3. Recurrece 3 4. Ergodicity 4 5. The Dyadic Addig Machie ad Approximatio Techiques 8 Ackowledgmets 3 Refereces 3. Itroductio Cosider a measure space (, L, λ), where is a set, L is a σ-rig of measurable subsets of, ad λ is the measure. A dyamical system cosists of the measure space ad a time rule that describes how poits i the space chage over time. Whe the time rule is discrete, the dyamical system is give by (, L, λ, T ), where the time rule T is a trasformatio, or a mappig from a set to itself, o. For a give atural umber, we ca iterpret the iterate T (x) as the positio at time of a object that starts at x ad the orbit of x {T (x)} =0 as the set of all poits visited by the path begiig at x. The first three sectios of the paper-proper cotai a discussio of the geeral properties of measurepreservig dyamical systems. I Sectio, we defie ivariat, or measure-preservig, trasformatios ad prove a theorem relatig ivariat trasformatios to itegrable fuctios. I Sectio 3, we defie recurrece ad prove Poicaré s Recurrece Theorem. I Sectio 4, we state some of the motivatig questios of ergodic theory, demostrate the coectios betwee these questios, ad aswer them by provig ad applyig Birkhoff s Ergodic Theorem. We the demostrate, i Sectio 5, oe method of aalyzig dyamical systems for these properties by costructig ad ivestigatig a trasformatio called the dyadic addig machie. The dyadic addig machie is closely related to the set of dyadic itervals, or itervals whose edpoits are of the form k, o [0, ), ad we defie a approximatio structure called a sufficiet semirig usig the set of dyadic itervals as a example. The mai results of this sectio are Theorem 5., which allows us to exted the measurepreservig property o a sufficiet semirig to the etire space, ad Lemma 5.6, a approximatio result that we ca use to prove ergodicity. The mai prerequisite for this paper is a uderstadig of the defiitios, costructios, ad basic properties of the Lebesgue measure ad the Lebesgue itegral. May results ad defiitios are stated ad proved for geeral measure spaces, but readers should feel free to metally substitute a Lebesgue space. Readers may also choose to begi with the costructio of the dyadic addig machie o pp. 8-9 ad to keep this trasformatio i mid while readig the first three sectios. Date: 9 AUGUST 04; Revised SEPTEMBER 04.

2 JASON LIANG. Ivariat Trasformatios Give a map T from a measure space (, L, λ) to a measure space (Y, S, µ), two importat questios are whether T preserves the measurability of sets, ad, if so, whether it preserves the measure of sets. T is called measurable if A S implies T (A) L. If, i additio, T is ivertible ad T is also measurable, the T is called ivertible-measurable. Give that T is measurable, T is called measure-preservig if A S implies λ(t (A)) = µ(a). Whe T maps from a measure space to itself, it is called a trasformatio, ad if T is also measurepreservig, the it is called ivariat. A measure-preservig map is ot called ivariat uless it is also a trasformatio. This paper maily cocers measure-preservig dyamical systems (, L, λ, T ) o a fiite measure space. I other words, is assumed to be of fiite measure ad T is assumed to be a ivariat trasformatio. The use of preimage, rather tha image, i the defiitios above preserves the σ-rig structure. This is demostrated by the followig propositio, which follows from the defiitios of image ad preimage. Propositio.. Suppose f is a mappig from a set to a set Y. The followig are true. () If, the f ( ) = f ( ) ad f( ) = f( ). () I additio, f ( ) = f ( ) ad f( ) f( ). For the latter case, there exist examples where iclusio is strict; however, if f is ivertible, the the iclusio is always a equality. (3) If,, the f ( \ ) = f ( ) \ f ( ) ad f( \ ) f( ) \ f( ). Oce agai, there exist examples where the latter iclusio is strict, but if f is ivertible, the the iclusio is always a equality. O the other had, the ext propositio shows that takig the image of a set uder a ivertiblemeasurable, ivariat trasformatio also preserves its measure. Propositio.. If T is a ivertible-measurable, ivariat trasformatio o, the A L implies λ(t (A)) = λ(a). Proof. Sice T is ivertible-measurable, T (A) is a measurable set, ad sice T is measure-preservig By the ivertibility of T, λ(a) = λ(t (A)). λ(t (T (A))) = λ(t (A)). It is clear that measurable maps ca be composed, whe appropriate, to form ew measurable maps. May questios i dyamical systems pertai to the compositio of a itegrable fuctio, or a measurable real fuctio with a fiite Lebesgue itegral, f with a iterate of a trasformatio T. For physical applicatios, we ca iterpret f(t (x)) as the state of a object at x at time. The ext theorem allows us to characterize ivariat trasformatios i terms of f T. I the proof of this theorem ad the rest of the paper, K A deotes the idicator fuctio for ay set A. Theorem.3. Suppose T is a trasformatio of a fiite measure space (, L, λ). The T is a ivariat trasformatio iff for ay itegrable fuctio f o, f T = f. Proof. First, assume that the equality of itegrals holds for ay itegrable fuctio f. Let A be a measurable set. The K A (T (x)) = iff T (x) A, so K A T = K T (A). Therefore λ(t (A)) = K A T = K A = λ(a), so that T is measure-preservig. Now assume that T is a measure-preservig fuctio. We use the stadard strategy of provig the theorem first for idicator fuctios the geeralizig to itegrable fuctios. If A is a measurable set, the K A T = λ(t (A)) = λ(a) = K A.

3 MEASURE-PRESERVING DYNAMICAL SYSTEMS AND APPROIMATION TECHNIQUES 3 This shows that the theorem holds for idicator fuctios, ad by the liearity of the itegral, the theorem is also true for simple fuctios. Now suppose that f is a o-egative, itegrable fuctio. It is well kow that there exists a sequece {s } of bouded simple fuctios that coverges poitwise to f ad that s = s = f. The f T = s T = s = f. Fially, rewrite ay itegrable fuctio f as f = f + f, where f + := max(f, 0) ad f := max( f, 0), ad apply the argumet for the o-egative case to complete the proof of the theorem. 3. Recurrece Suppose T is a measurable trasformatio o a measure space (, L, λ). Oe questio is whether, give a set A ad a elemet x A, there exists a atural umber such that T (x) A. I other words, does the path startig at x ever retur to A? Is this true of almost every x i A? T is called recurret if for every set A of positive measure ad almost every x A, there exists a atural umber such that T (x) A. The mai theorem, called Poicaré s Recurrece Theorem, of this sectio shows that ivariat trasformatios of a fiite measure space are always recurret. I the proof of this theorem, we use the followig equivalet defiitio of recurrece. T is called coservative if for every set A of positive measure, there exists a atural umber such that λ(t (A) A) > 0. That is, there is a subset B of A such that B is of positive measure ad every object that starts i B lies i A at some specific time. Propositio 3.. Suppose T is a measurable trasformatio o a measure space. T is recurret iff it is coservative. Proof. Suppose that T is ot coservative. The there exists a set A of positive measure such that for ay N, the set of x such that T (x) A has measure zero, ad this meas T is ot recurret. Now suppose that T is ot recurret. The there exist sets A ad B of positive measure such that B A ad for all x B, T (x) B A for ay. Sice x (T (B) B) meas that T (x) B, λ(t (B) B) = 0 for all. This meas that T is ot coservative. If the sets T (A) are of equal positive measure, it seems that if is of fiite measure, the evetually oe of the preimages must overlap with A. The proof of Poicaré s Recurrece Theorem is a formalizatio of this idea. Theorem 3. (Poicaré s Recurrece Theorem). Suppose T is a ivariat trasformatio o a fiite measure space (, L, λ). The T is recurret. Proof. Assume, i cotradictio to the theorem, that T is ot recurret. The there exists a set A of positive measure such that λ(t (A) A) = 0 for all atural umbers. Now suppose that j ad k are atural umbers. The λ(t (j+k) (A) T j (A)) = λ(t j (T k (A) A)) = 0. The last equality holds because T is measure-preservig. If j < k, the λ(t k (A) T j (A)) = 0. This meas that λ( T (A)) = λ(t (A)) = λ(a) =, ad this cotradicts the fact that is of fiite measure. To reach the cotradictio, we eeded to be of fiite measure. The trasformatio T o the Lebesgue space R defied by T (x) := x + 3 provides a example of a ivariat trasformatio that is ot recurret. Oe last questio is whether the orbit of a elemet uder a recurret trasformatio has a ifiite umber of returs. The ext propositio aswers this questio i the affirmative. Propositio 3.3. If T is recurret, the for every set A of positive measure ad almost every x i A, there exists a sequece { xi } such that T x i (x) A. Proof. Let N be the set of all x A such that there does ot exist a atural umber x such that T x (x) A. Sice T is recurret, N has measure zero. Now let N be the set of all x A \ N such that there does ot exist a atural umber x such that T x (x) A \ N, ad defie N 3, N 4, etc. aalogously.

4 4 JASON LIANG Sice T is recurret, each N i has measure zero. For all x i A \ N i, there exists a sequece { xi } such that T x i (x) A, ad we also have λ( N i ) = 0. i= i= 4. Ergodicity Amog the most promiet questios i dyamical systems is the oe that asks whether the time average of a fuctio is equal to its space average. Today, this questio is typically posed i the followig fashio. Suppose (, L, λ, T ) is a measure-preservig dyamical system o a fiite measure space, ad let f be a arbitrary itegrable fuctio. The log-term time average of f over a particular orbit is give by f(t i (x)) while the space average of f is λ() assume that is a probability space. The questio is whether the equatio f(t i (x)) = f. Sice λ ca be scaled so that λ() =, we is true at least for almost every x. The followig specific case of this questio for idicator fuctios asks whether, o average, T distributes elemets proportioally throughout the space. Suppose is a set of locatios such that each locatio cotais exactly oe object. If A ad B are measurable sets, is the proportio of objects that start i B that are i A at time equal to, o average, the proportio of the space that A takes up? I other words, is λ(t i (A) B) = λ(a) λ(b) = λ(a)λ(b)? λ() While these questios are of a physical ature, we will show that they are directly related to the algebraic questio of reducibility. It is with this property that the sectio begis. It is atural to ask whether T ca be decomposed ito smaller trasformatios. Give a ivariat trasformatio T o a probability space (, L, λ), a subset A of is called T -ivariat if x A iff T (x) A. If there exists a T-ivariat set A of positive but ot full measure, the T ca be decomposed ito two trasformatios o two disjoit sets of positive measure. Namely, T restricted to A ad T restricted to A c is oe such decompositio. T is called ergodic if the oly sets that are T -ivariat are those that have zero measure or whose complemets have zero measure. Sice sets which differ by a set of zero measure are typically idetified, the ergodic trasformatios are those that are irreducible i a o-trivial sese. I the ext theorem, we preset several equivalet characterizatios of ergodicity that follow immediately from this defiitio. The first is a relaxatio of the T -ivariat coditio, while the last two provide more geometric iterpretatios of ergodicity. I both the statemet of this theorem ad the rest of this paper, deotes the symmetric differece operatio. Theorem 4.. For a ivariat trasformatio T o a probability space (, L, λ), the followig statemets are equivalet. () T is ergodic. () If A is measurable ad λ(a T (A)) = 0, the either A or A c has zero measure. (3) If A is a set of positive measure, the for almost every x, there exists a atural umber such that T (x) A. I other words, every area of positive measure is reached by the orbit of almost every poit i the space. (4) If A ad B are both sets of positive measure, the there exists a atural umber such that λ(t (A) B) > 0. That is, there is a regio of positive area withi B such that every object that starts i that area lies i A at some particular time. Proof. Assume that T is ergodic. We will show that ay set A satisfyig λ(a T (A)) = 0 ca be approximated up to a differece of zero measure by a T -ivariat set A. Suppose A is such a set, ad let f

5 A := for all N, N=0 =N+ MEASURE-PRESERVING DYNAMICAL SYSTEMS AND APPROIMATION TECHNIQUES 5 T (A). The T (A ) = λ(a \ A ) = λ( ( =N+ N=0 =N+ N= =N+ A \ T (A)) ( T (A) = A, so A is T -ivariat. Sice =N+ A \ T (A)) = N λ( A \ T (A)) =N+ A \ T (A)) = 0. We ca apply a similar argumet to show that λ(a \ A) = 0, so λ(a A ) = 0. Sice T is ergodic, either A or (A ) c has zero measure, ad thus the same is true of A. This meas that () implies (). Now assume that () is true, ad suppose that A is a set of positive measure. Defie A := T (A). The T (A ) = = T (A) A, ad λ(t (A )) = λ(a ) sice T is a ivariat trasformatio. Thus λ(a T (A )) = 0, ad sice T is ergodic, λ(a ) =. Therefore () implies (3). Next assume that (3) is true, ad suppose that A ad B are sets of positive measure. Sice is the set of all x B such that the orbit of x itersects A, (3) implies λ( T (A) B T (A) B) = λ(b) > 0. Therefore there must exist a m such that λ(t m (A) B) > 0, so (3) implies (4). Lastly, assume that (4) is true, ad suppose that A is a T -ivariat set of positive measure. If, i cotradictio to the theorem, A does ot have full measure, the A c also has positive measure ad λ(t m (A) A c ) > 0 for some m. At the same time, λ(t (A) A c ) = 0 for all sice A is T -ivariat. This leads to a cotradictio, so T must be ergodic. Corollary 4.. If T is a ivertible-measurable, ivariat trasformatio ad T is ergodic, the T is also ergodic. Proof. Suppose A ad B are both sets of positive measure. By Theorem 4., there exists a atural umber such that λ(t (A) B) > 0. Sice, by Propositio., T is measure-preservig, λ(a T (B)) = λ(t (T (A) B)) = λ(t (A) B) > 0. We are ow i a positio to aswer the opeig questio of this sectio. The ext theorem says that whe T is a ergodic trasformatio, the time average of ay itegrable fuctio over almost every orbit is equal to its space average. Theorem 4.3 (Birkhoff s Ergodic Theorem). Suppose T is a ergodic trasformatio o a probability space (, L, λ). If T is ergodic, the for ay itegrable fuctio f o, f(t i (x)) = f for almost every x. Before begiig the proof, we itroduce some otatio to keep track of importat expressios. The otatio ad fudametal argumet of our proof closely follow those i Sectio 5. of Silva s book [3, pp ]. A more geeral form of the theorem ad a proof of that geeral form are also foud there. Notatio 4.4. Defie the followig: () for a give atural umber, let f (x) := f(t i (x)), () let f (x) := sup (3) let f (x) := if f (x), f (x),

6 6 JASON LIANG (4) give a real umber r, let C r := {x : f (x) < r}, ad (5) give a real umber r ad a atural umber p, let E r p := {x : f (x) r for all p}. The outlie of the proof is as follows. If we ca show that f (x) f is true for almost every x, the f (x) f by applicatio to f, ad this proves the theorem. Assumig the first iequality does ot hold for almost every x meas that there is some ratioal r < f such that C r = {x : f (x) < r} has positive measure. We will show that λ(c r T (C r )) = 0, so λ( Ep) r λ() λ(c r ) = = 0. Our mai goal i the ext series of lemmas is to prove a iequality relatig Ep r to f which shows that this caot be the case. Lemma 4.5. The followig statemets are true. p= () For ay x, f (T (x)) = f (x) ad f (T (x)) = f (x). () For ay real r, λ(c r T (C r )) = 0. Proof. For ay atural umber, f (T (x)) = f(t (T i (x))) = f(t i (x)) = [f + f(x)] i= = + + f + f(x). + Sice =, we ca take the appropriate its usig the last form of the equatio to get ad f (T (x)) = f (x) f (T (x)) = f (x). From the last equatio, we get that if x C r, the T (x) C r. This meas that C r T (C r ). Sice T is measure-preservig, λ(c r T (C r )) = 0. Lemma 4.6. Suppose g is a real fuctio o a set, p is a atural umber, ad τ : {,,..., p} is a τ(x) fuctio such that for every x i, g τ(x) (x) = g(t i (x)) 0. The for ay p, for all x. g (x) i= p Proof. Let x ad p be give. Sice g τ(x) (x) 0, g(t i (x)) τ(x) g (x) = g(t i (x)) g(t i (x)) g(t i (x)) = g(t i (x)) g(t i (x)). We ca raise the lower boud o the last sum by removig the ext τ(τ(x)) umbers, ad this is permissible provided there are more tha τ(τ(x)) terms remaiig. We ca keep raisig the lower boud util there are ot eough terms remaiig, ad sice τ(x) < p, there are at most p terms remaiig whe we have to stop. Therefore g (x) g(t i (x)). i= p Lemma 4.7. Suppose f is a itegrable fuctio ad p is a atural umber. The followig statemets are are true. τ(x) τ(x)

7 MEASURE-PRESERVING DYNAMICAL SYSTEMS AND APPROIMATION TECHNIQUES 7 () For ay p, () For ay real r, f (x) f(t i (x))k E 0 p (x) + f E r p i= p f + r( λ(e r p)). f(t i (x)). Proof. Defie the fuctios h(x) := f(x)k E 0 p (x) ad g(x) := f(x) h(x). Sice x E 0 p implies f(x) 0, h (x) 0 for all. Suppose we could fid a fuctio τ correspodig to g ad p ad satisfyig the hypothesis of Lemma 4.6. The for ay p, g (x) = f(t i (x)) f(t i (x))k E 0 p (T i x) i= p f(t i (x)) f(t i (x))k E 0 p (T i x) i= p f(t i (x)). Defie a fuctio τ : {,,...p} by the rule τ(x) = for x i E 0 p ad τ(x) = mi {k f k (x) < 0} for all other x. The for x i E 0 p, g τ(x) (x) = g(x) = f(x) f(x)k E 0 p (x) = 0 ad for x ot i E 0 p, g τ(x) (x) = f τ(x) (x) h τ(x) (x) 0 h τ(x) (x) 0. Thus τ satisfies the hypothesis of Lemma 4.6, ad this proves the first part of Lemma 4.7. To prove the secod part of the lemma, we first apply () to the fuctio (f r). The for ay p, (f r) (x) (f r)(t i (x))k E r p (x) + i= i= p (f r)(t i (x)). Sice T is measure-preservig, we ca itegrate both sides the divide by to get f f + r( λ(ep)) r + p f r, ad sice this holds for arbitrary, the lemma is true. E r p Readers who are strugglig with the last iequality i the proof above may wat to recall that we assumed that λ() =. Proof of Theorem 4.3 Assume, i cotradictio to the theorem, that the iequality f f (x) fails o a set A of positive measure. Let P := {q Q < f}. Sice A = C q, there exists a ratioal r < f such that λ(c r ) > 0. We apply Lemma 4.5 i combiatio with the ergodicity of T to get that λ(c r ) =, so λ( Ep) r = p λ(ep) r = 0 as discussed earlier. Now we apply () from Lemma 4.7 to get f r, p= ad this cotradicts the choice of r. We ca use Birkhoff s Ergodic Theorem to show that the trasformatios that distribute elemets proportioally i the sese described at the begiig of this sectio are exactly those that are ergodic. Corollary 4.8. Suppose T is a ivariat trasformatio o a probability space (, L, λ). The T is ergodic iff λ(t i (A) B) = λ(a)λ(b) wheever A ad B are measurable. q P

8 8 JASON LIANG Proof. First, assume that T is ergodic. By Birkhoff s Ergodic Theorem, for each atural umber N there is a set Y N of full measure such that for all y Y N, K A (T +i (y))k B (T i (y)) = λ(t (A) B). Sice each Y N has full measure, we ca fix x such that x is i Y N for every N. The m m m k=0 λ(t k (A) B) = m m m k=0 K A (T k+i (x))k B (T i (x)). We apply Birkhoff s Ergodic Theorem to covert the ier it ito a itegral, ad the equatio becomes m m m k=0 λ(t k (A) B) = m m m k=0 K A (T k (x))k B (x). Sice the itegrad has a absolute value of at most, it satisfies the hypothesis of Lebesgue s Domiated Covergece Theorem. This allows us to move the it iside the itegral to yield the equatio m m m k=0 λ(t k (A) B) = B m m m k=0 K A (T k (x)) = λ(a)λ(b). Next, assume that the equatio i the corollary holds for all measurable sets A ad B. If A is T -ivariat, the λ(t (A) A c ) = λ(a A c ) = 0 for all. This meas that λ(a)λ(a c ) = 0, so either A or A c has measure zero. 5. The Dyadic Addig Machie ad Approximatio Techiques There are several methods for provig the ergodicity of a ivariat trasformatio. Here we itroduce a method based o approximatio structures, while Walke s lecture otes [4, p ] provide a example of a alterate techique that uses Fourier series. We begi by defiig a trasformatio kow as the dyadic addig machie, ad we will develop the approximatio techiques metioed by usig this trasformatio as a example. The costructio of the dyadic addig machie uses the idea of replicatio, which is helpful for geeratig examples ad couterexamples i ergodic theory. Friedma s article [] ad the sectios of Silva s book [3, pp ad 8-6] that deal with the Hajia-Kakutai trasformatio ad Chacó s trasformatio demostrate this latter poit. Suppose I = [a, a + b) ad I = [(a + c), (a + c) + b) are two disjoit itervals of equal legth. If the map T : I I is said to be represeted by Figure, the T is give by the traslatio T (x) = x + c. I I Figure. A represetatio of T Similarly, if the iterval I 3 is of the same legth ad type (left-closed, right-ope) as I ad I ad disjoit from both, the the map T : I I I I 3 represeted by Figure is defied by traslatig I to I ad I to I 3, ad maps ivolvig ay fiite umber of disjoit itervals of equal legth ad idetical type may be represeted ad defied aalogously.

9 MEASURE-PRESERVING DYNAMICAL SYSTEMS AND APPROIMATION TECHNIQUES 9 I 3 I I Figure. A represetatio of T A map represeted i the form above is called a tower, its represetatio is called a stack, ad a iterval i the stack is called a level. For example, the image i Figure is a stack, the map T (x) = x+c is the tower it represets, ad I ad I are the oly levels of the stack. We costruct the dyadic addig machie by iductively defiig for each iteger > 0 a tower T ad usig these towers to defie a ifiitely piecewise fuctio. Let I be the iterval [0, ). To defie the stack S, divide I i half ito [0, ) ad [, ) ad place the latter iterval above the former. Let T be the tower represeted by S. To defie S, divide S i half vertically to form two stacks ad place the right stack above the left stack. Let T be the tower represeted by S. As before, the itervals are left-closed ad right-ope. This process is illustrated i Figure S 3 4 S 0 I Figure 3. Costructig the dyadic addig machie If S has bee costructed for some N ad T is the tower correspodig to S, the S + is the stack formed by dividig S i half ad stackig the right stack above the left stack i the maer described above, ad T + is defied to be the tower correspodig to S +. Readers may fid it helpful to explicitly costruct a few more stacks. It is apparet from this process that these towers are closely related to the dyadic umbers, or umbers of the form k where both k ad are itegers. For the rest of this paper, let D refer to the set of left-closed, right-ope dyadic itervals, or itervals whose edpoits are dyadic umbers, i [0, ). For a give, let D refer to the subset of D cosistig of itervals whose edpoits are of the form k. Several key facts about the towers T follow immediately from the costructio above, ad we summarize these i the ext propositio. Propositio 5.. Suppose N, ad let S ad T be as above. The followig statemets are true. () Each level of S is i D.

10 0 JASON LIANG () T maps from D \ [, ) oto D \ [0, ). D D D D (3) For ay atural umber k less tha, T = T k o the domai of the latter. (4) Let m be the Lebesgue measure. If D D \ [0, ), the m(t (D)) = m(d). (5) If D D \ [, ), the m(t (D)) = m(d). The dyadic addig machie is the trasformatio K o [0, ) defied by the followig piecewise fuctio. T (x) : x [0, ) K(x) := T (x) : x [, 3 4 ) T 3 (x) : x [ 3 4, 7 8 )... We see from this defiitio that if N, the K agrees with T o the domai of the latter. Take care to ote that the itervals i D do ot appear i S i order. Re-examie Figure 3 to see this. It is easy to see from the piecewise defiitio of K that K is a ivariat trasformatio of [0, ). We will re-prove this by itroducig some approximatio techiques. This provides greater geerality ad at the same time gives us tools to prove the ergodicity of K. We ca see from Propositio 5. that K preserves the measure of dyadic itervals i [0, ), ad most of the ext two pages are aimed at uderstadig ad provig the followig theorem, which allows us to exted the measure-preservig property of K o D to all of [0, ). Theorem 5.. Suppose that (, L, λ) is a fiite measure space, S is a sufficiet semirig o, ad T is a trasformatio o such that λ(t (S)) = λ(s) for ay S S. The T is a ivariat trasformatio o. Note that T is ot eve assumed to be measurable. I our proof of this theorem, we show that the measurability of T ca be derived form the fact that T is measure-preservig o S. The set D has the followig structure. If D, D D, the () D D D, ad () there exists a fiite umber of disjoit elemets C,.., C i D such that D \ D = C i. Ay o-empty collectio of sets satisfyig the above properties is called a semirig. Note that ay semirig cotais. If the elemets of a semirig S are subsets of some set, the S is called a semirig o. Suppose that (, L, λ) is a measure space ad S is a semirig cosistig of subsets of of fiite measure. S is called a sufficiet semirig o if A L implies that λ(a) = if A S λ(s ), where S S. These defiitios do ot appear to be widely used ad follow those i Silva s book [3, pp. 39-4]. Let m deote the Lebesgue measure. We show that D is a sufficiet semirig o [0, ) by demostratig that ay ope iterval i [0, ) ca be approximated arbitrarily closely by a fiite uio of dyadic itervals. Suppose that I = (a, b) is a iterval cotaied i [0, ) ad that ɛ > 0 is give. Choose N so that < ɛ k. Sice the umbers of the form where k = 0,,,..., partitio [0, ) ito itervals of legth, there exists a o-egative iteger m such that m is to the left of a (or equal to a if a = 0) ad a m l. Similarly, there exists a o-egative iteger l such that is to the right of b (or equal to b if b = ) ad l b. Let D I := D [ m l, ). The I D D I D i=

11 MEASURE-PRESERVING DYNAMICAL SYSTEMS AND APPROIMATION TECHNIQUES ad m(i) m(d) < m(i) + ɛ. D D I Suppose that A is a measurable set. The there exists a collectio of ope itervals {I } i [0, ) such that m(a) m(i ) < m(a) + ɛ. Sice ɛ is arbitrary, for each I, there exists D := D I such that The Now let m(i ) D D m(d) < m(i ) + (m(a) + ɛ m (A) := D D m(d) < m(a) + ɛ. if A D m(d ), m= m(i m )). where D D. The precedig iequality shows that m (A) < m(a) + ɛ, ad sice ɛ is arbitrary, m (A) m(a). Sice, by defiitio, m(a) m (A), this proves that D is a sufficiet semirig o [0, ). Carathéodory s coditio states that give a outer measure λ, a set A is measurable iff for every set S, λ (S) = λ (S A)+λ (S \A). Therefore ay ull set, or set of outer measure zero, is measurable, ad we will prove Theorem 5. by writig A as the differece of a ull set N ad a set H A costructed usig members of the sufficiet semirig. I the ext lemma, we costruct H A ad highlight its essetial properties. Lemma 5.3. Suppose that (, L, λ) is a fiite measure space ad S is a sufficiet semirig o. If A is measurable, the there exists a set H A that satisfies the followig: () H A = H, where each H is a coutable uio of elemets of S, () H H H 3... H... H A A, ad (3) λ(h A \ A) = 0. Proof. By defiitio, give ɛ > 0 there exists a set H ɛ = where Si ɛ S such that A Hɛ ad λ(h ɛ \ A) < ɛ. For ay atural umber, defie H := H H...H = S i m. S ɛ m= i= Sice S is closed uder fiite itersectios, H is a coutable uio of elemets of S. Now let H A be defied as i (). The () holds by the defiitio of H, ad for ay atural umber, λ(h \ A) λ(h \ A) λ(h \ A) <. Thus H A also satisfies (3). Proof of Theorem 5. Suppose A is a measurable set, ad let H A ad H = m= S,m be as i the proof of Lemma 5.3. Sice takig preimages preserves, i the sese of Propositio., uios ad itersectios, the sets H ad the set H A are measurable. Sice T preserves the measure of elemets of S, λ(t (H A )) = λ( T (H )) = λ( T (S,m )) = λ( (S,m )) = λ(h A ). m= m=

12 JASON LIANG Defie the set N := H A \ A; N is a ull set by (3) of Lemma 5.3. We ca apply the argumet i the precedig paragraph to N to show that both H N ad T (H N ) are ull sets, so 0 λ (T (N)) λ (T (H N )) = 0. T (N) is a ull set ad is measurable by Carathéodory s coditio. Sice T (A) = T (H A ) \ T (N), T (A) is measurable ad λ(t (A)) = λ(a) for ay measurable set A. Sice K is a ivariat trasformatio, we ca ask whether K is ergodic. To prove that K is ergodic, we eed to derive some further results about sufficiet semirigs. At this poit, readers are ecouraged to draw a K-ivariat set o some of the stacks S. This will provide some ituitio for the discussio that follows. Propositio 5.4. The dyadic addig machie K is a ergodic trasformatio o the Lebesgue space [0, ). If A is a K-ivariat set of positive measure ad D ad D are dyadic itervals i [0, ), the m(a) = m(a [0, )) m([0, )) = m(a D) m(d) = m(a D ). m(d ) I other words, the measure of A is equal to the proportio of ay dyadic iterval that A occupies. If, for ay give 0 < δ <, we ca fid a dyadic iterval D δ that satisfies the iequality m(d δ ) m(a D δ ) > ( δ)m(d δ ), the m(a) =. I the the ext two lemmas, we show that D δ exists for ay δ. Lemma 5.5. Suppose that (, L, λ) is a fiite measure space ad S is a sufficiet semirig o. For ay measurable set A ad ay ɛ > 0, there exists a set G A ɛ that is a fiite uio of elemets of S such that λ(a G A ɛ ) < ɛ. Proof. Let δ := ɛ, ad let Hδ = S i be as i the proof of Lemma 5.3. Sice N S i M S i for M > N, i= λ( S i ) = λ( S i ). Thus there exists a such that λ(h δ \ S i ) < ɛ. Let G A ɛ := S i. The i= i= i= i= λ(a G A ɛ ) = λ(a G A ɛ ) + λ(g A ɛ A) λ(h δ G A ɛ ) + λ(h δ A) < ɛ. Lemma 5.6. Suppose that (, L, λ) is a fiite measure space ad S is a sufficiet semirig o. For ay set A of positive measure ad ay 0 < δ <, there exists a set S δ S such that λ(a S δ ) > ( δ)λ(s δ ). Proof. Choose ɛ so that 0 < ɛ < δ δ, ad defie β := ɛλ(a). Let G := GA β = N i= i= i= S i be as i the statemet of Lemma 5.5. We apply λ to set iclusio G = [(G A) (G \ A)] [A (A \ G) (G \ A)] to get λ(g \ A) λ(a [A G]) < ( + ɛ)λ(a). Similarly, we apply λ to the set iclusio A = [(A G) (A \ G)] [(A G) (A G)] to get or λ(a) < λ(a G) + ɛλ(a), λ(a G) > ( ɛ)λ(a). Suppose, i cotradictio to the lemma, that there is o set S δ S that satisfies the stated iequality. I particular, oe of the sets S i satisfy the iequality. The λ(a G) ( δ)λ(g) < ( δ)( + ɛ)λ(a), so O the other had, ( + ɛ) < δ ( δ)( + ɛ) > ( ɛ). ad ( ɛ) > δ δ This is a cotradictio, so the lemma is true. = ( δ) δ ( ɛ) > ( δ). ( + ɛ) by the choice of ɛ, so

13 MEASURE-PRESERVING DYNAMICAL SYSTEMS AND APPROIMATION TECHNIQUES 3 Proof of Propositio 5.4 Suppose A is a K-ivariat set of positive measure, ad let 0 < δ < be give. Sice D is a sufficiet semirig o the Lebesgue space [0, ), there exists a dyadic iterval D δ such that By the remarks above, Sice δ is arbitrary, m(a) =. m(a D δ ) m(d δ ) > ( δ). m(a) > ( δ). Silva s book [3, pp ad 07-08] cotais some more examples of usig sufficiet semirigs to prove ergodicity. The latter group of pages cited presets a proof that K is ergodic that is differet from the oe i this paper. Ackowledgmets. I would like to thak my metor Gog Che for aswerig my may questios ad readig my drafts. Refereces [] N.A. Friedma, Replicatio ad Stackig i Ergodic Theory, Amer. Math. Mothly 99 (99), o., 3 4. [] P.R. Halmos, Measurable Trasformatios, Bull. Amer. Math. Soc. 55 (949), o., [3] C.E. Silva, Ivitatio to Ergodic Theory, Stud. Math. Libr., Vol. 4, Amer. Math. Soc., Providece, RI, 007. [4] C. Walke, MATH4/6 Ergodic Theory, Course otes, retrieved July 4, 04 from ac.uk/~cwalkde/ergodic-theory/ergodic_theory.pdf