Some Classical Ergodic Theorems

Transcription

1 Soe Classical Ergodic Theores Matt Rosenzweig Contents Classical Ergodic Theores. Mean Ergodic Theores Maxial Ergodic Theore Pointwise Ergodic Theore Application to Law of Large ubers Classical Ergodic Theores. Mean Ergodic Theores Proposition. (von euann Ergodic Theore) Let H be a Hilbert space, T : H H be a unitary operator. Then for all f H, the it f : A nf exists in H. In fact, f P T f, where P T is the orthogonal projection onto H T : f H : T f f. Proof. We first reark that H T is evidently closed by the continuity of the T. Consider two cases. If f H T, then A f f for all, so the stateent is obvious. Suppose f / H T. Consider f T g g for soe g H. Then A f ( T + T ) g 0, since T is bounded. Set H diff T g g : g H. I clai that A f 0 for all f H diff. oting that A for all and applying the triangle inequality,for f H diff and a sequence T g n g n f, A f A f A (T g n g n ) + A (T g n g n ) f (T g n g n ) + A (T g n g n ) f (T g n g n ) + 2 g n 0, n, The proof will be coplete if we can show that H H T H diff, for which it suffices to show that H T H diff. Suppose T f f, g H. Then f, T g g T f, T g f, g f, g f, g 0, where we use the fact that T is unitary. By continuity, we obtain f H diff. For the reverse inclusion, if f H diff, then 0 f, T f f f, T f T f, T f T f f, T f Taking the coplex conjugate of the RHS and adding it to f, T f f gives 0 T f f, T f f f H T Definition 2. A easure-preserving syste is (, A, µ, T ) consists of a probability space (, A, µ) and a easurable transforation T : such that µ(t (E)) µ(e) for all E F. We define a function T f : f T.

2 Lea 3. For p, the operator T : L p L p, f T f is an isoetry. Moreover, T : L 2 L 2 is unitary. Proof. Since the siple functions are dense in L p for all p [, ], it suffices to verify the clai on this subspace. The case p is evident, so we assue p <. Let f n a i Ei be a siple function where we ay assue that the E i are pairwise disjoint. The preiages T (E j ) are pairwise disjoint. Since T is easure-preserving, p T f P dµ a i T (E i) dµ a i p µ(t (E j )) a i p µ(e i )dµ f p dµ For the second assertion, we need only verify that T ft gdµ fgdµ for all f, g L2 (, µ). By density, we ay assue that f n a i Ei, g j b j Fj, where the E i, F j are each pairwise disjoint collections. ( n T ft gdµ a i T (E i)) b j T (E j) dµ j j j j j a i b j T (E i) T (F j)dµ a i b j µ(t (E i F j )) a i b j µ(e i F j ) ( n fgdµ a i Ei ) b j Fj dµ j Proposition 4. (Mean Ergodic Theore) Let (, F, µ, T ) be a easure-preserving syste, p <, and f L p. Then A f converges in L p. Proof. Since T is an isoetry L p L p, p <, it follows that A L p L. It is enough to prove p the stateent for a dense subspace of all L p, in particular L. Choose f L. Von euann s ergodic theore tells us that A f f L in L 2 -nor. By Hölder s inequality, A f f in L p -nor, for p 2. For p > 2, observe that A f f p dµ A f f p 2 A f f 2 dµ (2 f L ) p 2 A f f 2 dµ 0, Definition 5. Let (, A, µ) be a probability space and T :. We say that T is ergodic if for all E F, T (E) E a.e. µ(e) 0 or µ(e). Lea 6. T is ergodic if and only if for all f L, T f f a.e. f is constant a.e. Proof. We first show the direction. Let E A satisfy T (E) E a.e.. Then E L and T ( E ) T (E) E a.e. E is constant a.e. For, we assue that f is real-valued (for the coplex-valued case, just decopose f into its real and iaginary parts). Define E a : f a Then E a satisfies T (E a ) E a a.e. µ(e a ) 0, a R. If we take inf a R : µ(e a ), then f a.e. 2

3 Proposition 7. If T is ergodic, then for f L p, p <, then A f : n T n f Lp Proof. Fro the ean ergodic theore, we know that A f Lp f with T f f. By the preceding propostion, f is constant a.e. Since A is bounded L p L p, it follows fro the doinated convergence theore and the fact that T is an isoetry L p L p, f f dµ A fdµ A fdµ fdµ n T n fdµ n fdµ fdµ.2 Maxial Ergodic Theore Lea 8. Suppose (, A, µ) (Z, P(Z), #), and T is the right-shift operator. Then if f L, sup T n f C f L, n L, Proof. Replacing f by f, we ay assue without loss of generality that f 0. We first observe that sup A f is easurable, since each A is easurable, being the linear cobination of easurable functions. We extend f on Z to f on R by f(x) : f(n) [n,n+), n Z For E Z, we define Ẽ R by Ẽ : n E [n, n+). ote that µ(ẽ) #(E), where µ denotes the Lebesgue easure on R, and f(x)dx R n Z f f L (Z). Moreover, L (R) Since + Since + 2, sup f(n + t)dt + 0 f(n + k) k k f(n + k) sup f(n + k) k + f(x + t)dt, x [n, n + ), ( + ( + f(n + t)dt ) + f(x + t)dt, x [n, n + ) + 0 ) + f(x + t)dt 2(Mf)(x), x [n, n + ), + 0 where Mf is the Hardy-Littlewood axial function on R. Applying the (, ) weak-type estiate for M, we see that ( # Z : sup T n f > µ x R : (Mf)(x) > ) 6 f L (R) 6 f L (Z) 2 n Proposition 9. (Maxial Ergodic Theore) Let (, A, µ, T ) be a easure-preserving syste. Then, for f L, sup T n f C f L n L, 3

4 Proof. Without loss of generality, assue that f 0. Furtherore, by the onotone convergence theore, it is enough to show that sup T n f C f L, 0 where C is a constant independent of 0. We define a function F : Z R by Then Since T n (T (x)) T n+ (x), we have F (x, n) A (f)(x) n L, (T n f)(x) n 0 n < (T n f)(x) n A (f)(t (x)) F (x, n) k F (x, n + ) Fix a, a 0, and set b a + 0. We define the function F b : Z R by F (x, n) n < b F b (x, n) 0 otherwise Then which iplies that A (f)(t (x)) k F b (x, n + ), 0, < a, n sup A (f)(t (x)) sup 0 F b (x, n + ) M(F b )(x, ), Set E x : sup 0 A (f)(x) >. Since T is easure-preserving, for any, µ( x : sup A (f)(t x) > ) µ(e ) 0 Hence, n n < a (µ #)( (x, ) Z : sup A (f)(t (x)) >, 0 < a ) aµ(e ) 0 By Fubini s theore and the estiate sup 0 A (f)(t (x)) M(F b )(x, ), n < a, (µ #)( (x, ) Z : sup A (f)(t (x)) >, 0 < a ) # Z : M(F b )(x, ) > dµ 0 6 F b (x, ) L (Z) dµ 6 b f(t n (x))dµ, where we use the weak-type estiate obtained in the preceding lea. Since f(t n (x))dµ f(x)dµ, we conclude that n aµ(e ) 6(a + 0) f L () ( ) a f L µ(e ) () a Letting a shows that µ(e ) 6 f L (). 4

5 .3 Pointwise Ergodic Theore Theore 0. Suppose f L (). Then for alost every x, the averages A (f) n f(t n (x)) converge to a it. Proof. Since µ(), L 2 (, A, µ) L (, A, µ) with f L 2 f L. Furtherore, by consider the n th truncates f n : f f n, we see that L 2 is dense in L. Let f L and ɛ > 0 be given. By the preceding rearks, we can write f f + f 2, where f L 2 and f 2 L < ɛ 2. Since the operator f T f is unitary, the proof of von euann s ergodic theore shows that the subspaces H T f L 2 : T f f, H diff T g g : g L 2 are orthogonal copleents. Hence, we can write f f,t + (T g g ) + h, where h L 2 < ɛ 2. Hence, there is a function f F + (T G G) L 2, where T F F such that f (F + (T G G)) L 2 < ɛ. Observe that A ( f) A (F + (T G G)) F + (T + G T G) I clai that (T + G G) 0 a.e. Indeed, since T is easure-preserving, n n 2 (G(T n (x)) 2 )dµ(x) G 2 L 2 (G(T n (x))) 2 so by the onotone convergence theore, n n 2 0, n by the continuity of x x. We conclude that A ( f)(x) F (x) n n 2 <, converges a.e., which iplies that G(T n (x)) n We now prove the convergence stateent for A (f). By the copleteness of R, it suffices to show that (A (f)(x)) is a Cauchy sequence for alost every x. Set E : x : sup A (f)(x) A M (f)(x) > 0,M 0 Since f f + h, where f L 2 and therefore A ( f)(x) 0 a.e., we have that ( ) µ x : sup A (h)(x) A M (h)(x) > \ x : sup A (f)(x) A M (f)(x) > 0, 0,M 0 0,M 0 hence it suffices to estiate the easure of E : x : 0 sup,m 0 A (h)(x) A M (h)(x) >. ote that E x : 2 sup A (h)(x) >, so by Proposition 9, µ (E ) µ(e ) µ a.e. ( ) x : 2 sup A (h)(x) > 2 6 h L 2 h L 2 2ɛ Since ɛ > 0 was arbitrary, we conclude that µ(e ) 0, which shows that (A (f)(x)) is Cauchy for alost every x. Corollary. (Pointwise Ergodic Theore) If f L (, A, µ) and T is ergodic, then A (f)(x) fdµ a.e. Proof. Since T is ergodic, Proposition 4 shows that A f f L 0,, where f fdµ. It is a consequence of the Borel-Cantelli Lea that we can choose a subsequence (A k f) such that A k (f)(x) f (x) for alost every x. By the preceding proposition and the uniqueness of its, we conclude that A (f)(x) fdµ a.e. 5

6 .4 Application to Law of Large ubers Lea 2. Let (Ω i, A i, µ i ) be a collection of identical probability spaces. Set Ω Ω i, let A be the cylinder σ-algebra, and let µ be the product easure. Let T : Ω Ω be the shift Then T is an ergodic transforation. T ((x, x 2, )) (x 2, x 3, ) Proof. We first verify that T is easure-preserving. Since the collection of sets E for which µ(t (E)) µ(e) fors a σ-algebra, it suffices to show that µ(t (E)) µ(e), where E A is a cylinder set. Fix a cylinder set E, and let the axial index such that π (E) Ω. Then E E i i+ Ω, where E i A i. It is evident that ( ) T (E) Ω E i Ω, µ(t (E)) µ E i µ(e) i+ By the sae arguent, it suffices to show that T (E) E µ(e) 0, for a cylinder set E A. Suppose T (E) E and µ(e) 0. Then E i i+ Ω E T (E) Ω E i i+ which iplies that E Ω, E i+ E i, i E Ω, which has easure. Proposition 3. Let ( n ) n be a sequence of i.i.d. rando variables on a probability space (Ω, F, P) such that E[ ] <. Then Ω, i E[ ], alost surely Proof. Consider the probability space (R, B(R ), µ), where µ is the product easure on B(R ) induced by the laws of the i. Define f : R R by f(x, x 2, ) f(x ) Let T : R R be the shift operator defined above. So (R, B(R ), µ, T ) is a easure-preserving syste. I clai that f L. Indeed, By the pointwise ergodic theore, x i By definition of the easure µ, it follows f(t ((x i ) )) fdµ E[ ] R alost surely i f(t (( i ) )) E[ ] alost surely References [] Stein, Elias, and Rai Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces. Princeton University Press,