Itroductory Ergodic Theory ad the Birkhoff Ergodic Theorem James Pikerto Jauary 14, 2014 I this expositio we ll cover a itroductio to ergodic theory. Specifically, the Birkhoff Mea Theorem. Ergodic theory is geerally described as the study of dyamical systems that have a ivariat measure. This reaches has relatios to flows (such as the Poicaré s Recurrece Theorem, etc.). Let s begi by discussig what it meas for a dyamical system to be ergodic. We ll start by defiig a probability space. Defiitio 0.1. A probability space is a triple (X, F, µ), where: X is a set of possible outcomes F is a set of evets, where each evet is a set cotaiig outcomes µ is a assigmet of probabilities to each evet i F. Defiitio 0.2. Ergodicity Let T : X X be a measure-preservig trasformatio. T is ergodic with respect to µ if F F s.t. T 1 (F ) = F either µ(f ) = 0 or µ(f ) = 1. Here are some examples: Example 0.3. If X is fiite ad has uiform measure, T : X X is ergodic IFF it s a cycle. Example 0.4. If T is ergodic, the so is T 1. T is ot ecessarily ergodic for all, as we ca see from the previous example (this property total ergodicity). 1
Ituitively, we ca thik of T as a trasformatio w.r.t. time. The, a ergodic fuctio describes a radom process for which the time average of oe sequece equals the total average. Now there is a atural aalogy to measurable flows; I discussed flows i my presetatio earlier i the course. Let T t be a measurable flow o a probability space (X, F, µ). A evet F F is ivariat mod 0 uder T t if t R, X(T t (F ) F ) = 0 ( meas symmetric differece). Now that we uderstad the defiitio of a probability space ad of Ergodicity, we ca begi provig the Birkhoff Theorem. We will first prove a lemma. The best summary of the probability of a evet is the frequecy of its occurrece over some large spa of time. This is captured i ergodic theory by the measure preservig map, T, from the probability space to itself. T is the chage from oe outcome of a radom series of evets to the ext. So suppose we have some measurable fuctio f. The we defie the average of measures over time, the Cesáro average, which is defied as follows: Defiitio 0.5. Cesáro Average A (f, x) = 1 1 f(t i (x)). Defiitio 0.6. σ-fiite measure space A measure µ defied o a σ-algebra Σ of subsets of a set X is called fiite if µ(x) R. It is σ-fiite if X ca be writte as the coutable uio of measurable sets with fiite measure. A set i a measure space is of σ- fiite measure if it ca be writte as the coutable uio of sets with fiite measure. Lemma 0.7. Maximal Ergodic Theorem Let T be a measure-preservig trasformatio o the σ-fiite measure space (X, F, µ). Suppose f L 1 (µ). Set E = {x A j (f, x) > 0 for some j }. The E fdµ 0. 2
Proof. The trick here is to fid a o-decreasig series of fuctios ad the redefie our itegral i terms of differeces i the series (so that we are itegratig over a etirely positive space). j 1 Let F (x) = max(0, f(t i (x)) : j ). This will be our o-decreasig series. This gives us F +1 = max(0, f + F T ). But ote that the secod term is always at least 0 over E +1, meaig we have F +1 = f + F T = f = F +1 F T over E +1. Now take the itegral: fdµ = (F +1 F T )dµ. E +1 E +1 But this is coveiet sice F +1 = 0 everywhere except E +1 ad F T 0 everywhere, givig F +1 F T 0 everywhere but E +1. So we ca chage the boudaries o the itegral to prove our theorem! fdµ = (F +1 F T )dµ (F +1 F T )dµ E +1 E +1 X = (F +1 F )dµ 0. X So we ve just show that if the Cesáro average of a L 1 fuctio if positive for a small eough time frame over a subset, the so is its itegral. Corollary 0.8. Set E = =1 E ad E fdµ 0. Theorem 0.9. Birkhoff Ergodic Theorem Let G be a measure-preservig trasformatio o a σ-fiite measure space (X, F, µ), ad f L 1 (µ). There exists a f with A (f, x) f(x) 3
almost everywhere. Proof. This is the deep theorem which we ve bee aimig for. First let s cosider some ratioal u, v. Ad let s set E u,v = {x lima (f, x) > v > u > lima (f, x) > 0}. Now we ca say: if x E u,v for ay u, v the lim A (f, x) exists. So assumig v > 0, otherwise replace f by f ad u > 0. Assume µ(e u,v ) > 0. From its defiitio, T (x) E u,v, the x is also. I other words, T 1 (E u,v ) E u,v, so we may, w.l.o.g. assume E u,v is the etire measure space. So x X = E u,v, s.t.. 1 1 (f(t i (x)) v) > 0 So, by the maximal ergodic lemma(!), we have (f vχ a )dµ 0. Meaig, X X fdµ vµ(a) because X = E from the first Corollary. Now there are several iterestig corollaries that result from Birkhoff s Theorem. Corollary 0.10. Defiig the map L(f) = f,for f L 1 (µ), L(f) 1 f 1 ad so L(f) is a cotiuous projectio from L 1 (µ) oto the subspace of T -ivariat L 1 -fuctios. Proof. As A (g) 1 = g 1 for g 0, ad sice A (f) coverges poitwise to f, L(f) dµ L( f )dµ lim A ( f )dµ = f dµ. ad L(F ) 1 f 1. 4
So we have: L(f)(T (x) = lim A (f, T (x)) = lim 1 f(t i (x)) i=1 = lim( + 1 A +1(f, x) f(x) ) = L(f)(x). So L(f) is T -ivariat. As L(f g) 1 = L(f) L(g) 1 f g 1, L is a cotiuous projectio oto the T -ivariat L 1 fuctios. Corollary 0.11. If (X, F, µ) has o T -ivariat subsets of fiite measure, the L(f) 0, f L 1 (µ). Proof. If X has o T -ivariat sets of fiite measure, the oly T -ivariat L 1 fuctio is idetically equal to 0. Corollary 0.12. If µ(x) <, the A (f) L(f) 1 0. Proof. Defie A = {f L 1 (µ) : A (f) L(f) 1 0}. As the operator L is a cotractio i L 1, A is L 1 -closed (if f i is Cauchy so is L(f i )). f < = all A (f) have the same boud. By the domiated covergece theorem, A (f) L(f) i L 1. L 1 (µ) is the oly closed subspace of L 1 (µ) that cotais all bouded fuctios. Last, we should uderstad why the Birkhoff Theorem is importat i Ergodic Theory. This lies i the relatioship to ergodic maps. I fact we ca use the theorem to directly characterize ergodic maps! Corollary 0.13. Applicatio of Birkhoff s Theorem If {A i } is a coutable collectio of sets L 1 dese i the collectio of all sets ad 1 1 χ Aj (T i (x)) µ(a j ) i=1 for all i ad almost every x the T is ergodic. 5
Proof. Pick A measure-ivariat ad A j arbitrarily from our collectio. µ(a)µ(a c j) = µ(a c i)dµ ( 1 = lim 1 = lim 1 A A χ A c j (T i (x))dµ ) χ T i(a A c j ) (x)dµ = µ(a A c j). Selectig A j so µ(a A j ) 0, µ(a)µ(a c ) = 0. So µ(a) {0, 1}. Refereces: A Itroductio to Ergodic Theory by Walters Fudametals of Measurable Dyamics by Rudolph 6