A BRIEF INTRODUCTION TO ERGODIC THEORY

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1 A BRIEF INTRODUCTION TO ERGODIC THEORY ALE FURMAN Abstract. These are expanded notes from four introductory lectures on Ergodic Theory, given at the Minerva summer school Flows on homogeneous spaces at the Technion, Haifa, Israel, in September Dynamics on a compact metrizable space Given a compact metrizable space, denote by C() the space of continuous functions f : C with the uniform norm f u = max x f(x). This is a separable Banach space (see Exs 1.2.(a)). Denote by Prob() the space of all regular probability measures on the Borel σ-algebra of. By Riesz representation theorem, the dual C() is the space Meas() of all finite signed regular Borel measures on with the total variation norm, and Prob() Meas() is the subset of Λ C() that are positive (Λ(f) 0 whenever f 0) and normalized (Λ(1) = 1). Thus Prob() is a closed convex subset of the unit ball of C(), it is compact and metrizable with respect to the weak-* topology defined by µ n weak µ iff f dµ n f dµ ( f C()). Definition 1.1. Let be a compact metrizable space and µ Prob(). A sequence {x n } n=0 is µ-equidistributed if 1 n (δ x 0 +δ x1 + +δ xn 1 ) weak-* converge to µ, that is if lim N N 1 1 N n=0 f(x n ) = f dµ (f C()). Let T : be a continuous map and µ Prob(). We say that a point x is µ-generic if the sequence {T n x} n=0 is µ-equidistributed. Exercise 1.2. Prove that (a) If is compact metrizable then C() is separable. (b) In the definitions of weak-* convergence and in the definition of µ-equidistribution, one can reduce the verification of the convergence for all f C() to for all f from a subset with a dense linear span in C(). (c) If x is µ-generic, then µ is a T -invariant measure. (d) If A is Borel set, and µ( A) = 0 then for any µ-generic point x #{0 n < N T n x A} N 1 µ(a).

2 2 ALE FURMAN 1.1. Irrational rotation of the circle. We consider the example of the circle T = R/Z (or the one-dimensional torus). Lemma 1.3 (Weyl). A sequence {x n } n=0 of points in T is equidistributed with respect to the Lebesgue measure m on T if and only if for every k Z {0} lim N N 1 1 N n=0 e 2πikxn = 0. Proof. We only need to prove the if direction, as the only if is obvious. By Stone-Weierstrass theorem, trigonometric polynomials (same as that is finite linear combinations of e k (x) = e 2πikx, k Z), are dense in C(T). So the proof is completed by Exercise 1.2.(b). Next consider the transformation on the circle T (x) = x + α, where α T is irrational, that is kα 0 for all non-zero integers k. Theorem 1.4. Let T (x) = x + α be an irrational rotation on = R/Z. Then every x T is equidistributed for the Lebesgue measure m on T. Proof. By Lemma 1.3 it suffices to check that N 1 1 e 2πik(x+nα) 0 (k Z \ {0}) N n=0 Denoting w = e 2πikx and z = e 2πikα the LHS above is just N 1 1 wz n = w N N 1 zn 1 z n=0 0 because z = 1 and z 1 due to the irrationality of α More on invariant measures. Recall (Exercise 1.2.(c)) that that one can talk about µ-generic points only of T -invariant measures. The set of all T -invariant measures P inv () = {µ Prob() T µ = µ} is a closed convex subset. Hence it is compact in the weak-* topology. Crucially, this set is never empty as the following classical result shows: Theorem 1.5 (Krylov-Bogoloubov, Markov-Kakutani?). The set P inv () is non-empty for any continuous map T : of a compact metrizable space. In fact, for any sequence x n and N n every weak-* limit point of the sequence of atomic measures is in P inv (). µ n = 1 N n N n δ T n x n n=0

3 A BRIEF INTRODUCTION TO ERGODIC THEORY 3 Proof. Suppose µ n µ in weak-* topology. Then for every f C() f dµ f dt µ = lim (f f T ) dµ n n = lim n 1 N n (f(t n x n ) f(t n+1 x n )) N n 1 n=0 1 = lim (f(x n ) f(t Nn x n )) = 0. n N n Theorem 1.6 (Uniquely ergodicity). Let T : be a continuous map of a compact metrizable space. TFAE: (a) There is µ Prob() so that every x is µ-generic. (b) There is only one T -invariant measure: P inv () = {µ}. (c) For every f C() the averages A n f(x) = 1 n 1 f(t k x) n k=0 converge uniformly to a constant, which is f dµ. Such systems (, T ) are called uniquely ergodic. To clarify: such µ is unique, it is T -invariant and ergodic (see Theorem 1.6 and Definition 1.8 below), and the setting can also be characterized by saying that there is only one T -ergodic probability measure on, useing Proposition 1.9 and Krein-Milman s theorem below. Proof. (a) = (b). Take any ν P inv () and arbitrary f C(). The sequence A n f(x) of averages is uniformly bounded by f u, and for every x converges to the constant f dµ. Using T -invariance of ν and Lebesgue s dominated convergence theorem, gives: f dν = A n f(x) dν(x) ( f dµ) dν = f dµ. (b) = (c). Consider the function f 0 = f f dµ and the linear subspace B = {g g T g C()}. Let Λ C() be an arbitrary functional vanishing on B. By Riesz representation theorem, Λ is given by integration of a signed measure, which has a unique representation as λ = (a 1 µ 1 a 2 µ 2 ) + i(b 1 ν 1 b 2 ν 2 ) with a 1, a 2, b 1, b 2 0, µ 1, µ 2, ν 1, ν 2 Prob() and µ 1 µ 2, ν 1 ν 2. As Λ vanishes on B, we have g dλ = g T dλ for all g C(), meaning λ = T λ and (a 1 µ 1 a 2 µ 2 ) + i(b 1 ν 1 b 2 ν 2 ) = (a 1 T µ 1 a 2 T µ 2 ) + i(b 1 T ν 1 b 2 T ν 2 ). Hence µ 1, µ 2, ν 1, ν 2 are T -invariant, and therefore equal to µ. Thus λ = cµ for some c C and Λ(f 0 ) = c f 0 dµ = 0.

4 4 ALE FURMAN We just showed that every functional vanishing on B, vanishes on f 0, hence f 0 B by Hahn-Banach theorem. Given any ɛ > 0 there is g C() so that f 0 h u < ɛ for h = (g g T ). Hence for n > 2 g u /ɛ one has A n f f dµ u = A n (f 0 ) u A n h u + ɛ 2 g u n + ɛ < 2ɛ. (c) = (a). Note that if the averages A n f converges to a constant a(f), then for any T -invariant probability measure ν one has f dν = A n f dν a(f) dν = a(f). It follows that P inv () is a singleton {µ}, and a(f) = f dµ. Exercise 1.7. Prove the implication (b) = (c) using Theorem 1.5. Definition 1.8. Let µ be an invariant probability measure for a transformation T :. We say that µ is T -ergodic if for every Borel set E one has µ(e T 1 E) = 0 = µ(e) = 0 or µ(e) = 1. Recall that if C V is a convex set in some (real) vector space V, a point c C is called extremal if it is not an interior point of a segment contained in C, that is if c = tc 1 + (1 t)c 2, with 0 < t < 1, c 1, c 2 C = c 1 = c 2 = c. The set of extremal points of C is denoted ext(c). By Krein-Milman theorem, any convex compact subset C in a locally convex topological vector space V, is the closure of the convex hull of the extremal points C = conv(ext(c)). In particular, any non-empty convex compact set has extremal points. Proposition 1.9. Let T : be a continuous map of a compact metrizable space. Then the set ext(p inv ()) is precisely the set P erg () of T -ergodic measures. Proof. If µ P inv () \ P erg (), then there is E B with µ(e T 1 E) = 0 and 0 < µ(e) < 1. The probability measures µ E = µ(e) 1 µ E, µ \E = (1 µ(e)) 1 µ \E are T -invariant. Since µ = µ(e) µ E + (1 µ(e)) µ \E, it follows that µ is not extremal: µ ext(p inv ()). Conversely, if µ is not extremal in P inv (), write µ = tµ 1 + (1 t)µ 2 with 0 < t < 1 and µ 1 µ 2 P inv (). Then µ 1 (A) 1/t µ(a) for every Borel set A. Hence µ 1 is absolutely continuous with respect to µ, µ 1 µ. Let φ = dµ 1 dµ be the Radon-Nikodym derivative, which is a positive unit vector in L 1 (, µ), uniquely determined by the relation µ 1 (A) = φ dµ (A B ). A

5 A BRIEF INTRODUCTION TO ERGODIC THEORY 5 Since both µ 1 and µ are T -invariant, it follows that µ-a.e. φ(t x) = φ(x). Thus for all a > 0 the set E a = {x φ(x) < a} satisfies µ(e a T 1 E a ) = 0. As a function of a [0, ] the measure µ(e a ) is monotonically non-decreasing with µ(e 0 ) = 0 and µ(e ) = 1. Let a 0 = sup{a 0 µ(e a ) = 0}, a 1 = inf{a µ(e a ) = 1}. The situation a 0 = a 1 leads to a contradiction, because it forces φ to be µ-a.e. equal to this constant. Consequently a 0 = a 1 = 1 and µ 1 = µ = µ 2, contrary to the assumption. Hence a 0 < a 1. Let E = E a for some fixed intermediate value a 0 < a < a 1. Then 0 < µ(e) < 1 and µ(e T 1 E) = 0, establishing that µ is not T -ergodic. Remark It follows from Birkhoff s individual ergodic theorem (see Theorem 2.3) that if µ P erg () is T -ergodic, then µ-a.e. point is µ-generic. This implies that distinct T -ergodic measures are mutually singular An interesting example of a uniquely ergodic system. Suppose T : be a continuous map of a compact metrizable space, and φ : T is a continuous function. Consider the system ( ˆ, ˆT ) where (1) ˆ = T, ˆT (x, s) = (T x, s + φ(x)). Let m denote the normalized Lebesgue measure on the torus T = R/Z. Given a T -invariant measure µ P inv () the product measure is ˆT -invariant. ˆµ = µ m Theorem 1.11 (Furstenberg [2]). Suppose µ is the unique T -invariant measure on, and ˆµ = µ m is ˆT -ergodic. Then ˆµ is the unique ˆT -invariant measure. Proof. Let ν 0 be some ˆT -invariant probability measure. For t T consider the translation ν t of ν 0 by the homeomorphism τ t : (x, s) (x, s + t) of ˆ that commutes with ˆT. In other words f dν t = f(x, s + t) dν 0 (x, s) (f C( ˆ)). ˆ ˆ Then ν t P inv ( ˆ). Consider the average ν = ν t dt, defined by f d ν = f dν t dt = f(x, s + t) dν 0 (x, s) dt ˆ T ˆ T ˆ ( ) = f(x, s + t) dt dν 0 (x, s) = f(x, t) dt dν 0 (x, s) ˆ T ˆ T = f dη m ˆ where η Prob() is the push-forward of ν 0 under the projection ˆ. Since such η is necessarily T -invariant, it follows that η = µ and 1 0 ν t dt = ν = µ m.

6 6 ALE FURMAN Since µ m is ergodic, it is an extremal point of P inv ( ˆ). Hence ν t = µ m for (almost) all t. Thus ν 0 = µ m, by applying τ t to some ν t = µ m. As ν 0 was an arbitrary ˆT -invariant probability measure, unique ergodicity of ( ˆ, ˆT, ˆµ = µ m) is proven. Corollary Let α be irrational. The transformation T : (x, y) (x + α, y + x) of the 2-torus T 2 is uniquely ergodic, the invariant measure is Lebesgue. Proof. Applying Theorem 1.11 with irrational rotation x x + α on T as the base, and φ(x) = x, we only need to verify that T is ergodic on T 2 with respect to the Lebesgue measure. Let E T 2 be some Borel set with m 2 (E T 1 E) = 0. Consider the Fourier series decomposition of the characteristic function 1 E L 2 (T 2, m 2 ) 1 E = (k,l) Z 2 c k,l u k,l where u k,l (x, y) = e 2πi(kx+ly). We have (k,l) Z 2 c k,l u k,l = (q,r) Z 2 c q,r u q,r T = Comparing the coefficients we get the identities c k,l = c k l,l e 2πi(k l)α. (q,r) Z 2 c q,r e 2πiqα u q+r,r. In particular, c k,l = c k l,l = c k 2l,l =... Since c k,l should be square summable on Z 2, it follows that c k,l = 0 whenever l 0. For l = 0 we have c k,0 = e 2πikα c k,0. As e 2πikα 1 for k 0, we get vanishing c k,0 = 0 for all k 0, and conclude that c k,l = 0 for all (k, l) (0, 0). Thus 1 E is an essentially constant function on T 2, which means that E is either null or co-null. This proves ergodicity of the Lebesgue measure, and unique ergodicity of the transformation on the torus. Corollary For an irrational α the sequence {n 2 α} n=0 is equidistributed on T = R/Z. Proof. Given an irrational α take β = 2α (still irrational) and consider the iterates of the uniquely ergodic (Corollary 1.12!) transformation T (x, y) = (x + β, y + x) of the torus: T n (x, y) = (x + nβ, y + nx + n2 n β) = (x + 2nα, y + n(x α) + n 2 α). 2 Given f C(T) consider F (x, y) = f(y) and apply the equidistribution of {T n (α, 0)} on T 2 to deduce N 1 1 f(n 2 α) = 1 N 1 1 F (T n (α, 0)) F (x, y) dx dy = f(y) dy. N N T 2 0 n=0 n=0

7 A BRIEF INTRODUCTION TO ERGODIC THEORY 7 Similar equidistribution on the circle can be shown for such sequences as {n 3 α} or {n 3 + 5n 2 πn} by showing unique ergodicity of certain transformations of T 3. This approach allowed Furstenberg [2] to give a dynamical proof for the following theorem Theorem 1.14 (Weyl). Let p(x) = a d x d + + a 1 x + a 0 be a polynomial with at least one irrational nonconstant coefficient. Then {p(n)} n=0 is equidistributed on T Additional remarks and exercises. The central notion of topological dynamics is minimality. Definition Let T : be a continuous map of compact metrizable space. The system (, T ) is minimal if contains no proper closed T -invariant subsets. A system (, T ) is strictly ergodic if it is both uniquely ergodic and minimal. The two examples that we discussed so far irrational rotation x α on T, and the skew-product map (x, y) (x + α, y + x) on T 2 are minimal systems (hence strictly ergodic), because the unique invariant measure has full support (see Exercise 1.16.(a)). Yet there exist minimal systems that are not uniquely ergodic. In the above mentioned beautiful paper [2] Furstenberg studied skew-products (1) and characterized (unique) ergodicity and minimality for such systems in terms of φ : T and (, T ). He then constructed an example an irrational (Liouville) α and a C -smooth φ : T T so that T : (x, y) (x + α, y + φ(x)) is minimal (C -diffeomorphism) that is not uniquely ergodic; in fact not ergodic for the Lebesgue measure. Exercise Prove that: (a) A uniquely ergodic (, µ, T ) system is minimal iff supp(µ) =. (b) Check that the transformation T : x x + 1 of the one point compactification = Z { } of Z is uniquely ergodic but not minimal. (c) Prove that the following are equivalent for a dynamical system (, T ): (1) (, T ) is minimal, (2) Every orbit {(T n x)} is dense in, (3) For every non-empty open V there finite cover = V T 1 V T n V. Let A be a finite set. The infinite product A Z is a Cantor set (a perfect totally disconnected compact metrizable space). The shift T : (..., x 1, x 0, x 1,... ) (..., x 0, x 1, x 2,... ) is a homeomorphism of A Z. Given a sequence x A Z the closure of its orbit = {T n x A Z n Z} is a closed T -invariant subset. Consider the resulting dynamical system (, T ). Exercise Let (, T ) be constructed from some point x A Z as above.

8 8 ALE FURMAN (a) Prove that consists of those y A Z for which every word in y appears in x; more precisely, for every n Z and l N there is k Z so that (y n+1, y n+2,..., y n+l ) = (x k+1, x k+2,..., x k+l ). (b) Prove that (, T ) is minimal iff every finite word w = (x k,..., x m ) in x reappears in x with bounded gaps, namely there exists n = n(w) so that for any i Z there is j {i,..., i + n(w)} with (x j, x j+1,..., x j+m k ) = w. (c) Prove that (, T ) is uniquely ergodic iff every finite word w = (x k,..., x m ) in x reappears in x with fixed frequency, namely the following limits exist 1 a(w) = lim N 2N #{ N j < N (x j, x j+1,..., x j+m k ) = w}. (d) Construct minimal non-uniquely ergodic system as above. Theorem 1.6 shows that in uniquely ergodic systems averages A n f of any continuous functions uniformly converge to a constant. A closer examination of the proof of that theorem shows that such a conclusion can be obtained for a specific function (or functions) on systems that are not necessarily uniquely ergodic. Exercise Let T : be a continuous map of a compact metrizable space. For real valued continuous function f : R define I (f) = inf f dµ, I (f) = sup f dµ. µ P inv() µ P inv() and A n f(x) = 1 n 1 n k=0 f(t k x) as before. (a) Show that in the above definition inf and sup can be replaced by min and max, and P inv () by P erg (). (b) Prove that for any f C(, R) the following inequalities hold for every x and uniformly on : I (f) lim inf n A nf(x), lim sup A n f(x) I (f). n (c) Assuming (, T ) is minimal, for any f C(, R), µ P erg () prove that { } x f dµ is a limit point of A n f(x) is a dense G δ -set in. (Hint: this result relies on Birhoff s theorem 2.3). (d) Deduce, that if (, T ) is minimal then for f C(, R) the set { } x I (f) = lim inf A nf(x), lim sup A n f(x) = I (f) n n is a dense G δ -set in. Given a topological dynamical system (, T ) let U(, T ) C() be the collection of uniformly averaged functions f C(), namely those for which A n f converges uniformly to a constant c(f). Let V (, T ) = {f C() I (f) = I (f)}. These are closed linear subspaces of C(). By part (a) we have V (, T ) U(, T ) and, if (, T ) is minimal, part (d) implies the equality V (, T ) = U(, T ).

9 A BRIEF INTRODUCTION TO ERGODIC THEORY 9 2. Ergodic theorems in the measurable context Dynamics can be studied in a purely measure-theoretical context, where (, B, µ) is a standard probability space 1, T : is a measurable map, preserving the measure µ. The latter means that Equivalently f dt µ = T µ(e) = µ(t 1 E) = µ(e) f(t x) dµ(x) = (E B). f(x) dµ(x) (f L 1 (, µ)). Definition 2.1. A measure-preserving map T of (, B, µ) is ergodic if every essentially invariant set (that is a set E B with µ(e T 1 E) = 0) is µ-null (µ(e) = 0) or µ-co-null (µ( \ E) = 0). Here are some other characterizations of ergodicity Exercise 2.2. Let T be a measure-preserving transformation of (, B, µ). Show that the following conditions are equivalent: (a) T is ergodic. (b) Any measurable a.e. T -invariant function f : C (that is one for which µ-a.e. f(t x) = f(x)), is µ-a.e. equal to a constant. (c) The only T -invariant vectors f in the Hilbert space L 2 (, µ) are constants Birkhoff s Individual Ergodic Theorem. The following central result is known as the individual ergodic theorem or pointwise ergodic theorem; it was proved after the mean ergodic theorem (Theorem 2.9). Theorem 2.3 (Birkhoff). Let (, B, µ, T ) be an ergodic probability measure preserving system and f L 1 (, µ). Then for µ-a.e. x and in L 1 -norm the ergodic averages converge to the integral: n 1 1 f(t k x) n k=0 f dµ If (, B, µ, T ) is a probability measure preserving that is not necessarily ergodic, the ergodic averages converge µ-a.e. and in L 1 to the conditional expectation n 1 1 f(t k x) E(f B inv ) n k=0 to the sub-σ-algebra of T -invariant sets B inv = {E B µ(e T 1 E) = 0}. We bring here the short elegant proof by Katznelson and Weiss [5]. 1 Hereafter we shall assume taht B is complete with respect to µ. 2 Recall that vectors in L 2 are equivalence classes of measurable functions

10 10 ALE FURMAN Proof. We think of f : R as a fixed Borel function. For n 1 denote the ergodic sums and ergodic averages by S n (x) = f(x) + f(t x) + + f(t n 1 x), A n (x) = 1 n S n(x). Note the cocycle relation S n+m (x) = S n (x) + S m (T n x). Define measurable functions f : R { } and f : R {+ } by f (x) = lim inf n A n(x), Since S n+1 (x) = f(x) + S n (T x) we have f (x) = lim sup A n (x). n A n+1 (x) = 1 n + 1 f(x) + n n + 1 A n(t x), and it follows that f (x) and f (x) are µ-a.e. T -invariant functions. For the sake of clarity of exposition, we focus on the ergodic case. Then f (x) and f (x) equal µ-a.e. to some constants a <, < a +, respectively. We want to show that a = f dµ = a and due to symmetry, it suffices to show a = f dµ. Fix arbitrary ɛ > 0 and a < a (if < a then take a = a + ɛ, otherwise take a = R with R 1 large). Define n(x) = min {n N { } A n (x) < a}. Then n(x) is a Borel function with n(x) < for µ-a.e. x. For k N let E k = {x n(x) > k}. Then E 1 E 2... with µ(e k ) 0 as k. Choose k large enough so that (2) ( f(x) a ) dµ(x) < ɛ E k and consider n large enough so that k (3) n ( f(x) a ) dµ(x) < ɛ. Define a measurable function ñ : {1, 2,..., k} by { n(x) if x \ Ek ñ(x) = 1 if x E k. Given x define a sequence of points x n by x 0 = x and inductively x i+1 = ñ(x n ). Given n k as in (3), let r N be such that ñ(x 0 ) + ñ(x 1 ) + + ñ(x r 1 ) n < ñ(x 0 ) + ñ(x 1 ) + + ñ(x r ) and set m = n ñ(x 0 ) + + ñ(x r 1 ). So 0 m < ñ(x r ) k. We are interested in a good upper estimate for (the integral of) the average A n (x) = 1 n S n(x) = 1 n (S ñ(x 0)(x 0 ) + + Sñ(xr 1)(x r 1 ) + S m (x r )).

11 A BRIEF INTRODUCTION TO ERGODIC THEORY 11 Let I = {0 i < r x n / E k } and J = {0,..., r 1} \ I. Then the terms Sñ(xn)(x n ), 0 i < r, can be estimated as follows: for i I while for i J Sñ(xn)(x n ) = S n(xn)(x n ) < n(x n ) a = ñ(x n ) a, Sñ(xn)(x n ) = f(x n ) f ( x n ) a + a = (1 Ek f a )(x n ) + ñ(x n ) a. For the last summand we have S m (x r ) = m 1 i=0 f(t l x r ) n 1 l=n k f(t l x) a + ma. Therefore A n (x) = 1 ( Sñ(x0)(x 0 ) + + Sñ(xr 1)(x r 1 ) + S m (x r ) ) n 1 r 1 ñ(x n ) a + ma + (1 Ek f a )(x j ) + n i=0 j J a + 1 n 1 n (1 Ek f a )(T j x) + 1 n j=0 n 1 l=n k n 1 l=n k f(t l x) a. f(t l x) a Integrating this estimates with respect to x we obtain, using (2) and (3), f dµ = A n (x) dµ a + f a + k E k n f a dµ < a + 2ɛ. As a > a and ɛ > 0 were arbitrary, it follows that a f dµ. Similarly (by replacing f(x) by f(x)) we deduce that a f dµ, and a = a = f dµ follows, using the obvious a a. This proves the µ-a.e. convergence n 1 1 lim f(t i x) = n n i=0 f dµ. The above estimate also gives max(a n (x) a, 0) dµ < 2ɛ, that gives the L 1 - convergence when combined with the similar estimate for f(x). Exercise 2.4. Let T : be a continuous map of a compact metrizable space. (a) Let µ P erg () be a T -ergodic measure. Show that µ-a.e. x is µ-generic. (b) Deduce that any two distinct T -ergodic measures µ ν P erg () are mutually singular: µ ν. Definition 2.5. A measure-preserving system (, B, µ, T ) is mixing if A, B B : lim µ(a T n B) = µ(a) µ(b). n Exercise 2.6. (a) Prove that any mixing system is ergodic. (b) Show that an irrational rotation is not mixing (hence the converse to (a) is false).

12 12 ALE FURMAN (c) Say that a collection C B is dense if for every ɛ > 0 and A B there is C C with µ(a C) < ɛ. Prove that if µ(c T n D) µ(c)µ(d) for all sets C, D from a dense collection C, then the transformation is mixing. Exercise 2.7 (Bernoulli shift). Let (Z, ζ) be a probability space, and (, µ) = (Z, ζ) N the infinite product of probability spaces. Prove that the shift T on T (z 1, z 2, z 3,... ) = (z 2, z 3, z 4,... ) is a measure preserving transformation, which is mixing (hence also ergodic). Hint: use Exercise 2.6.(c) and the collection C of so called cylinder sets: C = E 1 E 2 E k Z Z where k N and E 1,..., E k are measurable subsets of Z. Remark 2.8. Given a probability space (Z, ζ) and some function φ L 1 (Z, ζ), one can apply Birkhoff s ergodic theorem to the Bernoulli shift on (, µ) = (Z, ζ) N and the function f : R given by f(z 1, z 2,... ) = φ(z 1 ) to deduce the Strong Law of Large Numbers in Probability Theory Von Neumann s Mean Ergodic Theorem. Birkhoff s individual ergodic theorem was predated by von Neumann s mean ergodic theorem, asserting that for an ergodic (, B, µ, T ) for every f L 2 (, µ) there is an L 2 -convergence: 1 n 1 f(t k x) f dµ 2 0. n k=0 If the system (, B, µ, T ) is not ergodic, the constant f dµ should be replaced by the conditional expectation E(f B inv ), which is the orthogonal projection of f to the closed subspace H inv = L 2 (, B inv, µ Binv ) H = L 2 (, B, µ). In fact, the beauty of von Neumann s proof is that it is a purely Hilbert space argument, that applies to general isometry U of a Hilbert space, and not only to ones coming from a transformations of (an) underlying measure space. A linear map U : H H is an isometry if Ux, Uy = x, y for all x, y H. Note that in an infinite dimensional Hilbert space, not every isometry is invertible. Theorem 2.9 (von Neumann). Let U be a linear isometry of a Hilbert space H. Then lim 1 n 1 U k x P inv x = 0 n n k=0 (x H), where P inv is the orthogonal projection to the subspace H inv = {y H Uy = y}. Proof. We claim that H inv is the orthogonal complement H0 of the linear subspace H 0 = {Uz z z H} of H. Indeed y H0 iff for every z H 0 = Uz z, y = Uz, y z, y = z, U y z, y = z, U y y

13 A BRIEF INTRODUCTION TO ERGODIC THEORY 13 that is equivalent to U y = y and to y 2 = U y, y = y, Uy y Uy = y 2, that is equivalent, by Cauchy-Schwartz, to Uy = y. Hence H0 = H inv, and therefore H = H inv H 0. Given x H and ɛ > 0, there is z H so that x P inv x (Uz z) < ɛ. Applying any power U k of the isometry U, we have and taking averages we obtain and as soon as n > 2 z /ɛ. U k x P inv x (U k+1 z U k z) < ɛ 1 n 1 U k x P inv x 1 n n (U n z z) < ɛ 1 n k=0 n 1 U k x P inv x < 2 z + ɛ < 2ɛ n k= Kingman s Subadditive Ergodic Theorem. Ergodic theorems are concerned with ergodic averages of a function along the orbit. There are several important generalizations of these results concerning compositions of non-commuting transformations along an orbit of a measure preserving transformation, such as Oseledets theorem (see [8], [4], [3]). The following result of Kingman, known as the subadditive ergodic theorem, is an essential tool in these considerations. Definition Let (, B, µ, T ) be a probability measure-preserving system. A subadditive cocycle is a sequence h n : R, n N, of measurable functions, satisfying µ-a.e. h n+m (x) h n (x) + h m (T n x) (n, m N). A particular case of subadditive cocycles, are additive cocycles sequences of functions a n : R, satisfying equalities: a n+m (x) = a n (x) + a m (T n x) Given any function f : R, the ergodic sums (n, m N). S n f(x) = f(x) + f(t x) + + f(t n 1 x) form an additive cocycle. Conversely, any additive cocycle {a n (x)} is given by the ergodic sums a n = S n f of the function f(x) = a 1 (x). Subadditive cocycles appear, for example, as h n (x) = log A(T n 1 A(T x)a(x) where A : G is some measurable function taking values in a group (e.g. G = GL d (R)), and : G R + is a sub-multiplicative norm. We formulate the ergodic case of Kingman s subadditive ergodic theorem.

14 14 ALE FURMAN Theorem 2.11 (Kingman). Let h n : R, n N, be a subadditive cocycle on an ergodic system (, B, µ, T ) with h + 1 (x) = max(h 1(x), 0) L 1 (, µ). Then there is µ-a.e. and L 1 (, µ) convergence 1 lim n n h n(x) = L where the constant L [, ) is given by lim and inf of the integrals 1 L = lim n n 1 h n dµ = inf n 1 n h n dµ. In the general measure preserving case (not necessarily ergodic), the limit is a T -invariant function L(x) given by lim and inf of conditional expectations E( 1 n h n B inv ). We refer the reader to Katznelson-Weiss [5] for a short proof of this theorem, along the same lines as in their proof of the Birkhoff s ergodic theorem, presented above.

15 A BRIEF INTRODUCTION TO ERGODIC THEORY Howe-Moore theorem on vanishing of matrix coefficients Translations on homogeneous spaces of Lie groups provide very interesting examples of dynamical systems with many applications in different areas. More specifically, consider the following examples: Example 3.1. Let G be a connected Lie group, Γ < G a discrete subgroup so that G/Γ has a G-invariant probability measure m G/Γ, with the transformation T α : gγ αgγ where α G is some fixed element. Circle rotations correspond to G = R with Γ = Z. Highly non-commutative settings, e.g. G = SL d (R) with Γ = SL d (Z), provide more interesting examples (see next section). The following very general result about unitary representations proves ergodicity (in fact, a stronger property of mixing) for many of such examples (Corollary 3.4). Let G be a semi-simple Lie group with finite center and no non-trivial compact factors. For example G = SL d (R), SO(n, 1), etc. A unitary representation of G is a continuous group homomorphism π : G U(H) into the unitary group of some Hilbert space H, where U(H) is taken with the strong operator topology. In other words, the continuity requirement is that whenever g n g in G for every x H one has π(g n )x π(g)x 0. Theorem 3.2 (Howe-Moore). Let G = k i=1 G i be a connected, semi-simple Lie group with finite center and no non-trivial compact factors, let π : G U(H) be a unitary representation, and assume that each factor G i acts with no non-trivial invariant vectors: H π(gi) = {0}. Then for every u, v H one has π(g)u, v 0 as g in G. For u, v H the function f u,v (g) = π(g)u, v is called matrix coefficient of π. Indeed if H is finite dimensional and {u 1,..., u d } is an orthonormal basis, then f un,u j (g) is the ij-entry of the unitary matrix representing π(g). Matrix coefficients are bounded ( f u,v (g) u v ) continuous functions on G. Howe-Moore theorem states that matrix coefficients vanish at infinity on G: f u,v C 0 (G) (u, v H) provided H π(gi) = {0} for all factors G i of G. This strengthens and generalizes a prior result of Moore: Theorem 3.3 (Moore). Let G = k i=1 G i and π : G U(H) with H π(gi) = {0} be as above. Then H π(h) = {0} for every non-precompact subgroup H < G. Hereafter we shall focus on simple Lie groups G, such as SL d (R), SO(d, 1). Corollary 3.4. Let G be a connected simple Lie group and α G does not lie in a compact subgroup. Then the system (, µ, T ) = (G/Γ, m G/Γ, T α ) is ergodic, and furthermore mixing.

16 16 ALE FURMAN Before proving Theorem 3.2, we recall a few facts about connected semi-simple group G. Every such G contains a maximal compact subgroup K, a maximal connected Abelian diagonalizable subgroup A (called Cartan subgroup); such subgroups are unique up to conjugation. For the key case of G = SL d (R) one can take K = SO(d), A = { diag(e t1,..., e t d ) t t d = 0 }. Any semi-simple group admits so called Cartan decomposition: G = K A K. In fact, the A component can be assumed to belong to the positive Weyl chamber A + A, so G = K A + K. In the case of G = SL d (R) A + = { diag(e t1,..., e t d ) t t d = 0, t 1 t 2 t d }, and the Cartan decomposition, stating that every g G can be written (not always uniquely) as a product g = kar with k, r SO(d) and a A +, is just the polar decomposition of a matrix. Proof of Howe-Moore s theorem 3.2. Let π : G U(H) be a unitary representation, and assume that for some u, v H the matrix coefficient f u,v (g) does not vanish at infinity, namely f u,v (g n ) = π(g n )u, v ɛ 0 > 0 for some fixed ɛ 0 > 0 and a sequence g n G leaving compact subsets of G. Step 1: non-vanishing along the Cartan. Using Cartan decomposition G = K A + K we write g n = kn 1 a n r n where a n A +, k n, r n K. Since K is a compact group, upon passing to a subsequence, we may assume that k n k and r n r in K. Continuity of π implies convergence of the vectors We have x n = π(r n )u x = π(r)u, y n = π(k n )v y = π(r)v. f xn,y n (a n ) = π(a n )x n, y n = π(a n r n )u, π(k n )v = π(k 1 n a n r n )u, v = f u,v (g n ) and can estimate f xn,y n (a n ) f x,y (a n ) f xn,y n (a n ) f x,yn (a n ) + f x,yn (a n ) f x,y (a n ) because y n = y is constant. Thus = π(a n )(x n x), y n + π(a n )x, (y n y) x n x y n + x y n y 0 (4) lim n f x,y(a n ) = lim n π(a n)x n, y n ɛ 0 > 0. We observe that since g n = k 1 n a n r n one has a n. Step 2: use of Mautner s Lemma. In a Hilbert space any closed ball is weakly compact (as is any closed bounded convex set). In particular, upon passing to a subsequence we may assume that the sequence {π(a n )x} i=1 of equal length vectors converges weakly to some vector z: π(a n )x w z, where weak convergence means that w H : π(a n )x, w z, w.

17 A BRIEF INTRODUCTION TO ERGODIC THEORY 17 A crucial point here is that z 0, because by (4) z, y = lim i π(a n )x, y ɛ 0 > 0. Lemma 3.5 (Generalized Mautner Lemma). Let π : G U(H) be a unitary representation of some topological group, and elements {a n } and h in G satisfy a 1 n ha n e in G. If vectors y, z H are such that π(a n )x z w then π(h)z = z. In particular, if π(a n )z = z then π(h)z = z. Proof. (Strong) continuity of π gives π(ha n )x π(a n )x = π(a 1 n ha n )x x 0. At the same time π(a n )x w z and π(ha n )x w π(h)z. Hence π(h)z = z. Step 3: proof for G = SL 2 (R). We can now prove Theorem 3.2 in the case of G = SL 2 (R). Let π : SL 2 (R) U(H) be a unitary representation with some matrix coefficient not vanishing at infinity Applying Step 1 we get a sequence ( ) e t n 0 a tn = A + 0 e tn with t n, and non zero vectors x, z H with π(a tn )x z. w horocyclic subgroup { ( ) } H = h s 1 s = s R 0 1 It is normalized by the diagonal subgroup A, and for every h s H one has a tn h s a tn = h e 2tn s e Consider the as i. Mautner s Lemma 3.5 yields that the non-zero vector z is π(h)-invariant. The matrix coefficient f z,z (g) = π(g)z, z is a continuous function on G, which is bi-h-invariant: (a) f z,z (gh) = π(g)π(h)z, z = π(g)z, z = f z,z (g) (b) f z,z (hg) = π(g)z, π(h 1 )z = π(g)z, z = f z,z (g) for all g G and h H. By (a) f z,z : G C descends to a continuous function φ : G/H C, that is H-invariant for the left H-action on G/H. Observe that G/H = R 2 {(0, 0)} with the H-action given by h s (x, y) = (x + sy, y) it follows that φ is a constant c y on each horizontal line l y = {(x, y) x R}, for y 0. By continuity, φ is a constant c 0 on the punctured x-axis as well Since φ(1, 0) = f z,z (e) = z 2, we have {(x, 0) x 0}. z 2 = φ(e 2t, 0) = π(a t )z, z (t R) where a t = diag(e t, e t ) A. By the strict case of the Cauchy-Schwartz inequality it follows that z is π(a)-invariant.

18 18 ALE FURMAN Consequently z is fixed by the (connected) upper triangular subgroup of G {( ) } e t s AH = 0 e t t, s R and therefore f z,z and φ further descend to a continuous function ψ on G/AH which can be identified with the projective line P(R 2 ) = R { } with G = SL 2 (R) acting by fractional linear transformations. In particular, H acts on P(R 2 ) = R { } by translation: h s : x x + s. Since ψ is invariant under this action, while the H-orbit of x = 1 is dense, it follows that ψ is constant. Hence so are φ : G/H C and f z,z : G C. Therefore π(g)z, z = f z,z (g) = f z,z (e) = z 2 (g G) implying, by Cauchy-Schwartz, that 0 z H π(g). Step 4: proof for G = SL d (R). One of the key facts about semi-simple Lie groups is that they can be built out of copies of SL 2 (R). In the case of G = SL d (R) for each 1 k < l d one considers the image G k,l of the isomorphic embedding τ k,l : SL 2 (R) SL d (R), which maps ( ) α β γ to the d d matrix g which looks like the identity except for the entries g k,k = α, g k,l = β, g l,k = γ, g l,l = δ. For i j in {1,..., d} let H i,j denote the matrices g SL d (R) with 1-s on the diagonal and 0-s in all off diagonal entries, except possibly for the (i, j)-entry. So for 1 k < l d, H k,l < G k,l is the image of the horocycle (α = δ = 1, γ = 0, β R). Let π : G = SL d (R) U(H) be a unitary representation, that has some matrix coefficient f u,v (g) that does not vanish at infinity. By Step 1, there is a sequence a n A +, a n, and vectors x, z 0 so that π(a n )x w z. Write a n = diag(e t1,n,..., e t d,n ), t 1,n + + t d,n = 0, t 1,n t d,n. As a n, one can assume, after passing to a subsequence, that (still denoted by subscript n) so that t k,n t l,n + as n +. for some fixed pair of indices 1 k < l d. Then for every h H k,l one has a 1 n ha n e and by Lemma 3.5, z is fixed by π(h k,l ). Considering the restriction of π to G k,l = SL2 (R) we deduce from (the proof of) Step 3, that z is fixed by all of G k,l. In particular, z is fixed by a n k,l = τ k,l (diag(e n, e n )) = diag(1,..., e n, , e n,..., 1) SL d (R). Another application of Lemma 3.5 shows that z is fixed by π(h k,j ) with any j k, because for every h H k,j a n k,l h an k,l e as n if k < j, and n if 1 j < k. Applying similar arguments to H k,j we deduce that z is fixed by π(h i,j ) for all i j. Since these groups generated G = SL d (R), it follows that H π(g) {0}. δ

19 A BRIEF INTRODUCTION TO ERGODIC THEORY 19 Proof of Corollary 3.4. The probability measure space (, µ) = (G/Γ, m G/Γ ) has a transitive measurepreserving action of the simple Lie group G by translations: g : hγ ghγ. Consider the orthogonal complement in L 2 (, µ) to constant functions: H = L 2 0(, µ) = {f L 2 (, µ) f dµ = 0}. Then π : G U(H), given by (π(g)f)(hγ) = f(g 1 hγ), is a unitary representation without non-trivial invariant vectors (exercise!). By Howe-Moore (Theorem 3.2), for any f 1, f 2 H lim n π(αn )f 1, f 2 = 0 We claim that this corresponds to the property of mixing for T on (, µ) that means, by definition, that for any measurable subsets A, B one has asymptotic independence of T n B from A: lim µ(a T n B) = µ(a) µ(b). n Indeed, given a measurable subset E the projection f E of the characteristic function 1 E to H = L 2 0(, µ) is f E (x) = 1 E (x) µ(e) = (1 µ(e)) 1 E (x) + ( µ(e)) 1 \E (x). One calculates f A, f B = µ(a B) µ(a)µ(b) and π(α n )f A, f B = f A, f T n B = µ(a T n B) µ(a)µ(b). Finally, mixing implies ergodicity, because any set E with µ(e T 1 E) = 0 would have µ(e) = µ(e T n E) µ(e) 2 which is possible only if µ(e) = 0 or µ(e) = 1.

20 20 ALE FURMAN 4. A glimpse of entropy An important class of examples of dynamical systems are homogeneous spaces G/Γ where G is a Lie group (e.g. SL 2 (R)) and Γ < G is a lattice, that is a discrete subgroup of finite covolume in G. Here we shall focus on uniform lattices discrete subgroups Γ < G with G/Γ being compact Classification of a class of systems. Consider the family of examples of dynamical systems T α : G/Γ G/Γ, T α : gγ αgγ where G = SL 2 (R) and Γ < G is a uniform lattice lattice and g G is some fixed element. To construct such Γ < G take a surface Σ of genus 2, and choose a Riemannian metric ρ of constant curvature 1 on it. This metric ρ lifts to the universal cover the hyperbolic plane H 2 and the fundamental group of Σ acts by isometries on H 2, giving rise to an imbedding j ρ : π 1 (Σ, ) Isom(H 2 ) = PSL 2 (R) whose image is a discrete cocompact subgroup Γ ρ, < PSL 2 (R). Changing the base point Σ is equivalent to conjugating Γ. But varying ρ on Σ (there are 6 ginus(σ) 6 many dimensional space of such choices) gives a multi-dimensional family of mutually non-conjugate uniform lattices Γ ρ. One may also think of them as lifted to the double cover SL 2 (R) PSL 2 (R). Note that if α is conjugate to β in G, then T α and T β are indistinguishable as transformations of = G/Γ; indeed, if α = γβγ 1 then T γ : intertwines T β with T α. Note also, that if Λ = b 1 Γb then the map G/Γ G/Λ, gγ gbλ, intertwines the translation by any α on G/Γ and on G/Λ. Therefore, studying translations T α on G/Γ, it suffices to focus on representatives of conjugacy classes of elements α and on representatives of conjugacy classes of lattices Γ. Exercise 4.1. Prove that the following three families of matrices are representative classes of all non-trivial conjugacy classes in G = SL 2 (R): ( ) ( ) ( ) cos θ sin θ 1 1 r θ =, h =, a t e t/2 0 = sin θ cos θ e t/2 with θ (0, 2π), t (0, + ) (the normalization t/2 is a convention). Elements of the compact group SO(2) = {r θ } do not have interesting dynamics on G/Γ. So we can restrict our attention to just two classes of examples: (G/Γ, T h ) where h is the unipotent matrix, (G/Γ, T a t) where a t is a diagonal matrix, known as the discretized horocyclic flow, and discretized geodesic flow, according to their geometric meaning when PSL 2 (R)/Γ ρ is identified with the unit tangent bundle to (Σ, ρ). Let us now list some known facts about this class of dynamical systems. (P1) (G/Γ, T h ) is uniquely ergodic (Furstenberg).

21 A BRIEF INTRODUCTION TO ERGODIC THEORY 21 (P2) (G/Γ, T a t) for any t > 0, has uncountably many different ergodic measures, and minimal sets of arbitrary Hausdorff dimension in [0, 3]. (P3) (G/Γ, T a t) has unique measure of maximal entropy (see below), which is precisely the G-invariant measure m G/Γ. Properties (P2), (P3) follow from properties of Anosov diffeomorphisms and Anosov flows studied in hyperbolic dynamics. Two topological systems ( i, T i ) are considered topologically conjugate, here denoted by ( 1, T 1 ) = top ( 2, T 2 ), if there exists a homeomorphism φ : 1 = 2 so that φ T 1 = T 2 φ. In this context one has the following topological dynamics rigidity results (T1) (G/Γ, T h ) =top (G/Λ, T a t) for any t > 0. (T2) There exists φ : (G/Γ, T h ) = top (G/Λ, T h ) if and only if b G : Γ = bλb 1, φ(gγ) = gbλ. (T3) There exists φ : (G/Γ, T a t) = top (G/Λ, T a s) if and only if t = s, and b G : Γ = bλb 1, φ(gγ) = gbλ. In the context of (ergodic) measure preserving systems the classification is up to measurable isomorphism, where two ergodic probability measure-preserving systems ( i, µ i, T i ), (i = 1, 2), are measurably isomorphic if there exists measure space isomorphism θ : ( 1, µ 1 ) = ( 2, µ 2 ) s. t. µ 1 {x θ T 1 (x) T 2 θ(x)} = 0. Note, that a priori measurable isomorphism is a rather subtle relation as everything is defined up to null sets. Consider the spaces G/Γ with the G-invariant probability measure m G/Γ, and recall that T h and every T a t (any t > 0) is ergodic (Corollary 3.4). Then (E1) All the above systems (G/Γ, m G/Γ, T h ) and (G/Λ, m G/Λ, T a t), t > 0, are unitarily equivalent, meaning that there are isometric isomorphisms between the Hilbert spaces intertwining the unitary operators induced by the underlying measure-space transformations. (E2) (G/Γ, m G/Γ, T h ) = (G/Λ, mg/λ, T a t) for any t > 0. (E3) There exists φ : (G/Γ, m G/Γ, T h ) = (G/Λ, m G/Λ, T h ) iff b G so that Γ = bλb 1, (E4) There exists φ : (G/Γ, m G/Γ, T a t) = top t = s. φ(gγ) = gbλ. (G/Λ, m G/Λ, T a s) if and only if Howe-Moore s theorem implies that the unitary operators are mixing; in fact, it can be shown that the operators have countable Lebesgue spectrum, which implies (E1). Fact (E2) and the only if direction of (E4) can be shown using the notion of entropy (see Corollary 4.15 below), which is a numeric measurable-isomorphism invariant of general systems (, µ, T ). A brief introduction to entropy is given below. The if direction of (E4), is a consequence of the fact that (G/Γ, m G/Γ, T a t) are Bernoulli shifts (Ornstein-Weiss [7]), and the fact that entropy is the complete isomorphism invariant for Bernoulli shifts (Ornstein [6]). Fact (E3) is a result of

22 22 ALE FURMAN Ratner, that predated her proof of Ragunathan conjecture and related results [9], that in turn, easily implies (E3), (T2) and (P1) Introduction to Entropy. Let A be a finite set, Σ = A N the infinite product with the Tychonoff topology (so Σ is a Cantor set, unless A is singleton), and let θ : Σ Σ denote the shift θ : (y 1, y 2, y 3,... ) (y 2, y 3, y 4,... ) Given a word w = (w 1, w 2..., w n ) of length n in the alphabet A (so w A n ), the cylinder set C w is C w = {y Σ y 1 = w 1,..., y n = w n }. For y Σ let [y] n denote the finite word [y] n = (y 1, y 2,..., y n ). There are many θ-invariant probability measures on Σ. Given ν P inv (Σ) we shall refer to (Σ, ν, θ) as a shift space, or a measure-preserving shift transformation. Given a shift space (Σ, ν, θ) define the functions I n : Σ [0, ] by I n (y) = log ν(c [y]n ) = log ν{z Σ z 1 = y 1, z 2 = y 2,..., z n = y n } and denote by H n their averages H n = I n (y) dν(y) = ν(c w ) log ν(c w ). Σ w A n Exercise 4.2. Prove that H n+m H n +H m for any n, m N, using the following: (a) Given words u A n, w A m, denote by uw A n+m their concatenation. Use shift invariance of ν to show ν(c u ) = ν(c uw ), ν(c w ) = ν(c uw ) w A m u A n (b) Use convexity of the function x x log x to prove the claim. Subadditivity of the sequence {H n } enables one to define the (Shannon) entropy of the process (Σ, ν, θ) to be ent(σ, ν, θ) = lim n 1 n H n = inf n 1 1 n H n. Definition 4.3 (Kolmogorov-Sinai). The entropy of a general ergodic system (, µ, T ) is defined to be Ent(, µ, T ) = sup{ent(σ, ν, θ) (, µ, T ) p (Σ, ν, θ)} where the sup is taken over all possible shift systems as quotients. Remark 4.4. Ent(, µ, T ) is a measurable isomorphism invariant. Remark 4.5. To define a shift space as a quotient (, µ, T ) (Σ, ν, θ) of a measurable system is the same as choosing a measurable partition = E 1 E a. Indeed, given such a partition take Σ = {1,..., a} Z, define p(x) = y with y i = k if T i x E k, and take ν = p µ. Note that p T = θ p by construction. Conversely, given a quotient map as above, construct a partition of by = E 1 E a, E k = p 1 (C k ).

23 A BRIEF INTRODUCTION TO ERGODIC THEORY 23 We state the following key result without proof. Theorem 4.6 (Shannon-McMillan-Breiman). Assume ν P erg (Σ) is θ-ergodic. Then 1 n I n(y) ent(σ, ν, θ) where the convergence is in measure, in L 1 (Σ, ν), and ν-a.e. So ent(σ, ν, θ) is the constant describing the exponential rate of shrinkage on ν-size of points with similar n-step trajectory. The weakest form of convergence above, the convergence in measure, allows to characterize the entropy as the growth rate of the number of n-words needed to capture significant size of the space. Corollary 4.7. Let (Σ, ν, θ) be a shift system and h = ent(σ, ν, θ). Then given ɛ > 0 there is N so that for every n N: There is a collection Ω A n of words of length n so that Ω < e (h+ɛ)n and ν( C w ) > 1 ɛ w Ω For any collection W A n of words of length n: W < e (h ɛ)n = ν( C w ) < ɛ. w W Proof. Convergence in measure n 1 I n (y) h (that follows from the L 1 -convergence) implies that there is N so that for n N the set G = {y Σ I n (y) = log ν(c [y]n ) < (h + ɛ)n} has ν(g) > 1 ɛ. This set consists of n-cylinders, so the collection Ω = {[y] n y G} satisfies Σ n = C w. For w Ω one has ν(c w ) > e (h+ɛ)n. Hence Ω < e (h+ɛ)n. The second statement is left to the reader. Exercise 4.8. Prove using Stirling s formula that given a > 1 and ɛ > 0 there is δ > 0 and N N so that for all n N one has ( ) n (5) a δn < a ɛn. δn Proposition 4.9. Given a N and ɛ > 0 there is δ > 0 with the following property: for any ergodic system (, µ, T ) with quotient maps to the shift on Σ = {1,..., a} N w Ω p : (, µ, T ) (Σ, ν, θ), p : (, µ, T ) (Σ, ν, θ) satisfying ν(c i ) ν (C i ) < δ for each i {1,..., a}, one has ent(σ, ν, θ) ent(σ, ν, θ) < ɛ.

24 24 ALE FURMAN Proof. Given a = A and ɛ > 0 use Exercise 4.8 to choose δ > 0 so that ( ) n a 2δn < a 1 2 ɛn. 2δn Assume that the measurable quotient maps p, p : (, µ, T ) (Σ, S) satisfy E = {x p(x) 1 p (x) 1 }, has µ(e) < δ. As a consequence of the ergodic theorem (this follows already from the mean ergodic theorem 2.9) for large n most points x visit E with frequency < 2δ. More precisely A n = { x #{1 k n T k E} < 2δn } has µ(a n ) > 0.9 for all n > N 1. From Corollary 4.7, for large enough N 2 for each n > N 2 there is a set Ω n A n of words so that and the set Ω n < e (ent(σ,ν,θ)+ 1 2 ɛ)n B n = {x [p(x)] n Ω n } has µ(b n ) > 0.9. For every n > max(n 1, N 2 ) the set C n = A n B n has µ(c n ) > 0.8. Let W n A n be the collection of all the words that are obtained as follows: for each w Ω n look at all subsets J {1,..., n} of size J = 2δn, and include in W n all n-long words that agree with w for i / J. Then ( ) n W n a 2δn Ω n < e 1 2 ɛn e (ent(σ,ν,θ)+ 1 2 ɛ)n = e (ent(σ,ν,θ)+ɛ)n 2δn while for x C n one has [p (x)] n W n. Thus by Corollary 4.7 ent(σ, ν, θ) ent(σ, ν, θ) + ɛ and a symmetric argument completes the proof. Exercise Prove (a) For a shift system (Σ, ν, θ) and k N one has ent(σ, ν, θ k ) = k ent(σ, ν, θ). (b) For a general system (, µ, T ) and k N one has Ent(, µ, T k ) = k Ent(, µ, T ). Let A and B be finite sets, c : A k B be a map (k-to-1 encoding), Σ = A N and Z = B N. Consider the shift equivariant map p : Σ Z where p(y) = z means that z i = c(y i, y i+1, y i+2,..., y i+k 1 ) (i N). Exercise Let c : A k B and p : Σ Z be as above, ν P inv (Σ) be shift-invariant measure and η = p ν P inv (Z). Show that ent(σ, ν, θ) ent(z, η, θ) and that injectivity of c : A k B is sufficient for equality. Theorem 4.12 (Kolmogorov-Sinai). For an ergodic measure ν on a shift space Σ = A N one has Ent(Σ, ν, θ) = ent(σ, ν, θ).

25 A BRIEF INTRODUCTION TO ERGODIC THEORY 25 Proof. The inequality Ent(Σ, ν, θ) ent(σ, ν, θ) is clear from the definition of the Kolmogorov-Sinai entropy (4.3). To establish the reverse inequality one needs to show that Shannon entropy of an arbitrary measurable quotient process p : (Σ, ν, θ) (Ω, η, σ) cannot be larger than that of the original, namely ent(ω, η, σ) ent(σ, ν, θ). Here Ω = B N Note that here Σ 1 = A N 1 is a shift space on potentially larger alphabet A 1 and the quotient map p is just measurable. Let ɛ > 0 be arbitrary. Using Proposition 4.9 one can find δ > 0 so that if (Σ 1, ν 2, θ 2 ) is some shift space be the associated to A 1 and ɛ > 0 as in. For every measurable subset E Σ there is k N and collection W A k so that ν(e C w ) < δ. w W Working with individual letters a A 1 we can approximate p 1 (C a ) by Exercise Let ζ be a probability measure on a finite set Z = {1,..., k}, say ζ({i}) = p i, where p 1 + p p k = 1. Let T be the Bernoulli shift on (, µ) = (Z, ζ) N as in Exercise 2.7. Prove: Ent(, µ, T ) = p i log p i Entropy of some homogeneous systems. Let us compute the entropy for translations on the homogeneous space of G = SL 2 (R). These special cases illustrate some general phenomena. Theorem Let G = SL 2 (R) and Γ < G a cocompact lattice. Then (a) Ent(G/Γ, m G/Γ, T a t) = t. (b) Ent(G/Γ, m G/Γ, T h ) = 0. Corollary Let Γ, Λ < G = SL 2 (R) be cocompact lattices, and t, s > 0. Then: (a) (G/Γ, m G/Γ, T h ) = (G/Λ, mg/λ, T a s). (b) If (G/Γ, m G/Γ, T a t) = (G/Λ, m G/Λ, T a s) then t = s. Proof of Theorem Let Γ < G = SL 2 (R) be a uniform lattice. The compact space = G/Γ is equipped with the G-invariant probability measure µ = m G/Γ, and transformation T α for α = h or α = a t for some fixed t > 0. The fact that Ent(, µ, T h ) = 0 can be shown using ideas similar to the upper estimate part of the T a t case below. However, assuming one knows that this entropy is finite, it is immediate that it must be zero, because T h is conjugate (by a measure-space isomorphism) to its square (T h ) 2 = T h 2 = T at ha t = (T a t)t h(t a t) 1 where t = log 2

26 26 ALE FURMAN so using Exercise 4.10 one obtains 2 Ent(, µ, T h ) = Ent(, µ, T 2 h) = Ent(, µ, T h ). So we focus on the case T a t for some fixed t > 0. We shall show the inequalities t Ent(, µ, T a t), Ent(, µ, T a t) t. The Lower estimate. For each x there is r x > 0 so that the map φ x : R 3 given by ( ) ( ) ( ) 1 0 e v/2 0 1 w (6) φ x (u, v, w) = u 1 0 e v/2.x 0 1 is a diffeomorphism from a cube ( r x, r x ) 3 R 3 onto an open neighborhood V x of x. Since {V x } x form an open cover of a compact space, there is an r > 0 (Lebesgue number) so that for every x we have a diffeomorphism φ x : ( r, r) 3 = W x from a fixed cube onto some open neighborhood W x of x. By continuity of T = T a t there exists r > δ > 0 small enough so that the φ x -image U x of the smaller cube ( δ, δ) 3 satisfies T (U x ) W T.x = φ T.x (( r, r) 3 ). In the (u, v, w)-coordinates the map T : is given by (7) θ x = φ 1 T.x T φ x, θ x (u, v, w) = (e t u, v, e t w). In particular, the w-direction is expanded by a factor of e t. Consider an intersection of consecutive preimages of sets U x U x1 T 1 U x2 T (n 1) U xn for some x 1,..., x n. Such intersection is contained in the φ x -image of a very narrow box ( δ, δ) ( δ, δ) ( e nt δ, e nt δ). The measure µ has some (smooth) density with respect to the du dv dw-measure (pushed forward by φ x1 ), and since the space is compact we get an estimate µ(u x1 T 1 U x2 T (n 1) U xn ) < Ce nt with some constant C independent of n and x 1,..., x n. Let = E 1 E a be a measurable partition into sets that are small enough to ensure E i U zi for some points z 1,... z a. Define the equivariant quotient p : Σ = {1,..., a} N as in Remark 4.5, and let ν = p µ to obtain the quotient map: p : (, µ, T ) (Σ, ν, S). Then for every y Σ and n one has (for some x 1,..., x n {z 1,..., z a }) I n (y) = log ν(c [y]n ) = log µ(e y1 T 1 E y2 T (n 1) E yn ) log µ(u x1 T 1 U x2 T (n 1) U xn ) log C + nt. Dividing by n, integrating dµ(x), and passing to the limit, gives (8) Ent(, µ, T ) ent(σ, ν, S) = t.