A BRIEF INTRODUCTION TO ERGODIC THEORY ALEX FURMAN Abstract. These are expanded notes from four introductory lectures on Er- godic Theory, given at the Minerva summer school Flows on homogeneous spaces at the Technion, Haifa, Israel, in September 2012. 1. Dynamics on a compact metrizable space Given a compact metrizable space X, denote by C(X) the space of continuous functions f : X ! C with the uniform norm kfku = max jf(x)j: x2X This is a separable Banach space (see Exs 1.2.(a)). Denote by Prob(X) the space of all regular probability measures on the Borel σ-algebra of X. By Riesz represen- tation theorem, the dual C(X)∗ is the space Meas(X) of all finite signed regular Borel measures on X with the total variation norm, and Prob(X) ⊂ Meas(X) is the subset of Λ 2 C(X)∗ that are positive (Λ(f) ≥ 0 whenever f ≥ 0) and normalized (Λ(1) = 1). Thus Prob(X) is a closed convex subset of the unit ball of C(X)∗, it is compact and metrizable with respect to the weak-* topology defined by weak∗ Z Z µn −! µ iff f dµn−! f dµ (8f 2 C(X)): X X Definition 1.1. Let X be a compact metrizable space and µ 2 Prob(X). 1 1 A sequence fxngn=0 is µ-equidistributed if n (δx0 +δx1 +···+δxn−1 ) weak-* converge to µ, that is if N−1 1 X Z lim f(xn) = f dµ (f 2 C(X)): N!1 N n=0 X Let T : X ! X be a continuous map and µ 2 Prob(X). We say that a point x 2 X n 1 is µ-generic if the sequence fT xgn=0 is µ-equidistributed. Exercise 1.2. Prove that (a) If X is compact metrizable then C(X) 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 2 C(X)" to "for all f from a subset with a dense linear span in C(X)". (c) If x 2 X is µ-generic, then µ is a T -invariant measure. (d) If A ⊂ X is Borel set, and µ(@A) = 0 then for any µ-generic point x 2 X #f0 ≤ n < N j T nx 2 Ag ! µ(A): N 1 2 ALEX FURMAN 1.1. Irrational rotation of the circle. We consider the example of the circle T = R=Z (or the one-dimensional torus). 1 Lemma 1.3 (Weyl). A sequence fxngn=0 of points in T is equidistributed with respect to the Lebesgue measure m on T if and only if for every k 2 Z − f0g N−1 1 X lim e2πikxn = 0: N!1 N n=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 2πikx combinations of ek(x) = e , k 2 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 2 T is irrational, that is kα 6= 0 for all non-zero integers k. Theorem 1.4. Let T (x) = x + α be an irrational rotation on X = R=Z. Then every x 2 T is equidistributed for the Lebesgue measure m on T. Proof. By Lemma 1.3 it suffices to check that N−1 1 X e2πik(x+nα) −! 0 (k 2 n f0g) N Z n=0 Denoting w = e2πikx and z = e2πikα the LHS above is just N−1 1 X w 1 − zN wzn = · ! 0 N N 1 − z n=0 because jzj = 1 and z 6= 1 due to the irrationality of α. 1.2. 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 Pinv(X) = fµ 2 Prob(X) j T∗µ = µg 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 Pinv(X) is non-empty for any continuous map T : X ! X of a compact metrizable space X. In fact, for any sequence xn 2 X and Nn ! 1 every weak-* limit point of the sequence N 1 Xn µ = δ n n N T xn n n=0 of atomic measures is in Pinv(X). A BRIEF INTRODUCTION TO ERGODIC THEORY 3 Proof. Suppose µn ! µ in weak-* topology. Then for every f 2 C(X) Z Z Z f dµ − f dT∗µ = lim (f − f ◦ T ) dµn n!1 Nn−1 1 X n n+1 = lim (f(T xn) − f(T xn)) n!1 N n n=0 1 Nn = lim (f(xn) − f(T xn)) = 0: n!1 Nn Theorem 1.6 (Uniquely ergodicity). Let T : X ! X be a continuous map of a compact metrizable space. TFAE: (a) There is µ 2 Prob(X) so that every x 2 X is µ-generic. (b) There is only one T -invariant measure: Pinv(X) = fµg. (c) For every f 2 C(X) the averages n−1 1 X A f(x) = f(T kx) n n k=0 converge uniformly to a constant, which is R f dµ. Such systems (X; 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 X, useing Proposition 1.9 and Krein-Milman's theorem below. Proof. (a) =) (b). Take any ν 2 Pinv(X) and arbitrary f 2 C(X). The sequence Anf(x) of averages is uniformly bounded by kfku, and for every x 2 X converges to the constant R f dµ. Using T -invariance of ν and Lebesgue's dominated convergence theorem, gives: Z Z Z Z Z f dν = Anf(x) dν(x) −! ( f dµ) dν = f dµ. X X X X X R (b) =) (c). Consider the function f0 = f − f dµ and the linear subspace B = fg − g ◦ T j g 2 C(X)g: Let Λ 2 C(X)∗ be an arbitrary functional vanishing on B. By Riesz' representa- tion theorem, Λ is given by integration of a signed measure, which has a unique representation as λ = (a1µ1 − a2µ2) + i(b1ν1 − b2ν2) with a1; a2; b1; b2 ≥ 0, µ1; µ2; ν1; ν2 2 Prob(X) and µ1 ? µ2, ν1 ? ν2. As Λ vanishes R R on B, we have g dλ = g ◦ T dλ for all g 2 C(X), meaning λ = T∗λ and (a1µ1 − a2µ2) + i(b1ν1 − b2ν2) = (a1T∗µ1 − a2T∗µ2) + i(b1T∗ν1 − b2T∗ν2): Hence µ1; µ2; ν1; ν2 are T -invariant, and therefore equal to µ. Thus λ = cµ for some c 2 C and Z Λ(f0) = c · f0 dµ = 0: 4 ALEX FURMAN We just showed that every functional vanishing on B, vanishes on f0, hence f0 2 B by Hahn-Banach theorem. Given any > 0 there is g 2 C(X) so that kf0 −hku < for h = (g − g ◦ T ). Hence for n > 2kgku/ one has Z 2kgk kA f − f dµk = kA (f )k ≤ kA hk + ≤ u + < 2. n u n 0 u n u n (c) =) (a). Note that if the averages Anf converges to a constant a(f), then for any T -invariant probability measure ν one has Z Z Z f dν = Anf dν −! a(f) dν = a(f): X X X R It follows that Pinv(X) is a singleton fµg, 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 : X ! X. We say that µ is T -ergodic if for every Borel set E ⊂ X one has µ(E 4 T −1E) = 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 2 C is called extremal if it is not an interior point of a segment contained in C, that is if c = tc1 + (1 − t)c2; with 0 < t < 1; c1; c2 2 C =) c1 = c2 = 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 : X ! X be a continuous map of a compact metrizable space X. Then the set ext(Pinv(X)) is precisely the set Perg(X) of T -ergodic measures. −1 Proof. If µ 2 Pinv(X) n Perg(X), then there is E 2 B with µ(E4T E) = 0 and 0 < µ(E) < 1. The probability measures −1 −1 µE = µ(E) · µjE; µXnE = (1 − µ(E)) · µjXnE are T -invariant. Since µ = µ(E) · µE + (1 − µ(E)) · µXnE, it follows that µ is not extremal: µ 62 ext(Pinv(X)). Conversely, if µ is not extremal in Pinv(X), write µ = tµ1 + (1 − t)µ2 with 0 < t < 1 and µ1 6= µ2 2 Pinv(X).