VARIATIONAL PRINCIPLE FOR THE ENTROPY

Transcription

1 VARIATIONAL PRINCIPLE FOR THE ENTROPY LUCIAN RADU. Metric entropy Let (X, B, µ a measure space and I a countable family of indices. Definition. We say that ξ = {C i : i I} B is a measurable partition if: µ( i I C i = µ(x and µ(c i > 0 for every i I; µ(c i C j = 0 for every i, j I with i j. We define also ξ η by: ξ η = {C D : C ξ, D η, µ(c D > 0}. Definition 2. The entropy of a measurable partition is given by H µ (ξ = µ(c log µ(c C ξ Let T : X X be measurable, µ-t invariant and ξ a measurable partition. Definition 3. The entropy of T with respect to µ and the partition ξ is given by: h µ (T, ξ = inf n n H µ ( n T k ξ = lim n n H µ and the entropy of T with respect to µ by: ( n T k ξ h µ (T = sup{h µ (T, ξ : ξ a measurable partition with H µ (ξ < } From ( we observe that h µ (T, ξ H µ (ξ. We can see that in order to calculate the entropy of T with respect to µ we have to consider h µ (T, ξ for every measurable partition ξ with H µ (ξ <. Sometimes it is possible to compute the entropy of T with respect to µ using only one special partition described below. Definition 4. If T is invertible (mod 0 let k= T k ξ be the smallest σ-algebra which contains n k= n T k ξ for every n N, then we say that ξ is a (two-sided generator if T k ξ = B (mod 0. In the same way, k= (

2 2 LUCIAN RADU Definition 5. If T is non-invertible (mod 0 let T k ξ be the smallest σ-algebra which contains n T k ξ for every n N, then we say that ξ is a (one-sided generator if T k ξ = B (mod 0. Now we have Proposition 6. If ξ is a generator with H µ (ξ < then h µ (T = h µ (T, ξ Proposition 7. The following proprieties hold:. h µ (id = 0; 2. h µ (T k = kh µ (T for every k N; 3. If an invertible measure preserving transformation possesses a onesided generator then h µ (T = Examples of calculation of metric entropy 2.. Markov measures. We will study the entropy of the two-sided shift, σ : {,..., k} Z {,..., k} Z, with respect to a Markov measure i.e. a stochastic pair (P, p where P is a stochastic matrix, p is a probability vector, k i= p i =, associated to P and:. k j= p ij = for each i =,..., k; 2. k i= p ip ij = p j for each j =,..., k. The measurable partition ξ = {C,..., C k } is in fact a two-sided generator because: n σ k ξ = {C i n...i n : i n,..., i n {,..., k}} k= n and the cylinders generate the Borel σ-algebra. Then and so ( n H µ σ k ξ = µ(c i0...i n log µ(c i0...i n i 0...i n = p i0 p i0 i...p in 2 i n log(p i0 p i0 i...p in 2 i n i 0...i n = p i0 p i0 i...p in 2 i n log p i0 i 0...i n n 2 p i0 p i0 i...p in 2 i n log p ij i j+ i 0...i n j=0 k k = p i log p i (n p i p ij log p ij i= h µ (σ = h µ (σ, ξ = lim n n H µ ( n i,j= σ k ξ = k p i p ij log p ij i,j=

3 VARIATIONAL PRINCIPLE FOR THE ENTROPY Rotations. We will se that any rotation R w : T T, where w = e 2πiτ and R w (z = wz, has zero entropy with respect to the Lebesgue measure. There are two cases: τ Q or {w n : n Z} is not dense in T i.e. w is not a root of unity. Them exists m N such that Rw m (z = z for every z T so h λ (R w = 0; τ Q or {w n : n Z} is dense in T. let ξ = {A, A 2 } the partition of the circle into the upper semi-circle and the lower semi-circle. For every n > 0, Rw n ξ consists of semi-circles beginning at w n and w n. Since {w n : n N} is dense, any semi-circle belong to n=0 R n w ξ. Hence any arc belong to w ξ and so h λ (R w = 0. n=0 R n 3. The entropy map Let (X, d be a compact metric space T : X X continuous. Let M(X, T be the space of all probability measures on (X, B(X that are T-invariant. We know that M(X, T is a non-empty convex set which is compact in the weak*-topology, by the Krylov-Bogolubov theorem. Definition 8. The entropy map is µ h µ (T from M(X, T to [0, ]. Proposition 9. The entropy map is affine, i.e. t [0, ] then if µ, m M(X, T and h tµ+( tm (T = th µ (T + ( th m (T. The entropy map is not in general continuous. We will give a counterexample in the case of the two-sided shift on {0, } Z. Let us consider the measures µ p, for p N, concentrated on the p periodic points, giving to each such a point measure /2 p. We have that µ p M(X, σ and h µp (σ = 0, for every p, because the measure is concentrated on a finite set of points. And let µ be the (/2, /2-Bernoulli measure, which we know, has h µ (σ = log 2. Now the collection of functions that depends only on a finite number of coordinates form a dense subset F (X of C(X by the Stone-Weierstrass theorem. If f F (X then exists N such that X fdµ p = X fdµ if p N. Therefor µ p µ and so the entropy map is not continuous. Sometimes is not even upper semi-continuous, but for a special class of maps we will prove that the entropy map is upper semi-continuous. Definition 0. T : X X is called an expansive homeomorphism if δ with the property that if x y then n Z with d(t n x, T n y > δ. Teorem. If T is an expansive homeomorphism then the entropy map is upper semi-continuous, i.e., if µ M(X, T and ε > 0 then exists a neighborhood U of µ in M(X, T such that m U implies that h m (T < h µ (T + ε. Proof. Let δ be an expansive constant for T,µ M(X, T and ε > 0.

4 4 LUCIAN RADU Let now ξ = {C,..., C k } be a finite partition with diam(c j < δ. Then h µ (T = h µ (T, ξ. Let N so that N H µ ( N T k ξ < h µ (T + ε Fix ε > 0 to be chosen later. As µ is regular we choose compact sets: K i0...i N N ( N with µ T k C ik \ K i0...i N < ε. Then N i k =j T k C ik T k K i0,...i N C j The sets L j := N i k =j T k K i0,...i N are compact and disjoint so there is a partition η = {D,..., D k } with diam(d j < δ and L j int(d j. We have K i0...i N int ( N T k D ik By Uryson s lemma we can choose f i0...i N C(X such that: 0 f i0...i N equals on K i0...i N ( N vanishes on X \ int T k D ik Let now U i0...i N := { m M(X, T : f i0...i N dm } f i0...i N dµ < ε The set U i0...i N is open is M(X, T and if m U i0...i N then ( N m T k D ik f i0...i N dm > f i0...i N dµ ε µ(k i0...i N ε and µ ( N Now if U := i 0...i N U i0...i N µ ( N T k C ik m T k C ik m ( N and m U then: ( N T k D ik < 2ε T k D ik < 2ε k N because if m i= a i = = m i= b i and also exists c > 0 with a i b i < c foe every i then a i b i < cm i, as b i a i = j i (a j b j < cm. So if m U and ε small enough the continuity of x log x gives: N H m ( N T k η < N H µ ( N T k ξ + ε 2

5 VARIATIONAL PRINCIPLE FOR THE ENTROPY 5 Hence, m U and ε small enough then: h m (T = h m (T, η N H m < N H µ ( N ( N T k η T k ξ + ε 2 < h µ(t + ε. 4. Topological entropy We can define topological entropy in the same way we defined the metric entropy, with open covers, α, taking the place of the measurable partitions and H(α = log N(α, where N(α is the minimal cardinality of a subcover of α. But in order to prove the variational principle we will use also Bowen s definition. Let (X, d be a compact metric space, T : X X a continuous map: h top (T = lim ε 0 n n log N d(t, ε, n Where N d (T, ε, n is the maximum number of point in X with pairwise d T n -distances at least ε, with d T n := max 0 i n d(t i x, T i y. We will call such a set of points (n, ε-separated. 5. Variational principle In this section we prove the relationship between topological entropy and the metric entropy for a continuous map T of a compact metric space i.e. : h top (T = sup{h µ (T : µ M(X, T }. In 968 L.W.Goodwyn proved the inequality: sup{h µ (T : µ M(X, T } h top (T. In 970 E.I.Dinaburg proved equality when X has finite covering dimension. In 970 T.N.T.Goodman proved equality in the general case. The elegant proof we will give is due to M.Misiurewicz. Lemma. Let X be a compact metric space, µ a Borel probability measure on X.. For x X, δ > 0 there exists δ (0, δ such that µ( B(x, δ = For δ > 0 there exists a finite measurable partition ξ = {C,..., C k } with diam(c i < δ for all i and µ( ξ = 0. Proof.. δ (0,δ B(x, δ is an uncountable disjoint union with finite measure.

6 6 LUCIAN RADU 2. By (and X compact there is a finite open cover {B,..., B k } by balls of radius less then δ/2 with µ( B j = 0 for all j. Let C = B, C i = B i \ i j= B j for i > and ξ = {C,..., C k }. Then ξ is as desired as ξ k i= B i. ( Note that if µ( ξ = 0 then µ n T k ξ = 0. Now we will describe a method of constructing measures of large entropy. Lemma 2. Let (X, d be a compact metric space, T : X X a homeomorphism, E n X an (n, ε-separated set, ν n := (/card(e n x E n δ x the uniform δ-measure on E n, and µ n := (/n n T ν i n. Then there is an accumulation point µ of {µ n } n N (in the weak* topology that is T -invariant and satisfies n n log card(e n h µ (T. Proof. Since M(X is compact we can choose a subsequence {n k } such that: lim k n k log card(e nk = n n log card(e n µ nk converges in M(X to some µ M(X In fact µ M(X, T because T µ n µ n = (T n ν n ν n /n (T cont. as in the proof of K-B theorem and ν n are probability measures. We choose now a partition ξ with diameter less then ε and µ( ξ = 0 as in Lemma. Then ( n H νn T k ξ = log card(e n Since each C n T k ξ contains at most one x E n so there are card(e n elements of ν n -measure /card(e n in n T k ξ. Now suppose 0 < q < n and for every 0 k < q define a(k := [(n k/q]. If we fix 0 k < q then {0,,..., n } = {k + rq + i : 0 r a(k, 0 < i q} S where S = {0,,..., k, k + a(kq +,..., n } and card(s 2q since k + a(kq k + [ n k q ]q n q. Then we have n T k ξ = ( a(k r=0 T q (rq+k( ( T i ξ j S T j ξ

7 and VARIATIONAL PRINCIPLE FOR THE ENTROPY 7 ( n log card(e n = H νn a(k r=0 a(k r=0 Then q q log card(e n = Thus n and that implies h=0 H νn ( H T rq+k ( n H νn q T h ξ h=0 a(k r=0 ( q nh µn n log card(e n lim k n q T (rq+k ( q ν n T h ξ H T rq+k ν n T i ξ T i ξ + H νn (T j ξ j S T i ξ + 2q log card(ξ. ( q T i ξ + 2q 2 log card(ξ ( q H µnk T i ξ q + 2q log card(ξ =! H µ n log card(e n h µ (T, ξ h µ (T. ( q T i ξ Teorem 2. Let (X, d be a compact metric space, T : X X a homeomorphism, then h top (T = sup{h µ (T : µ M(X, T }. Proof. ( Let µ M(X, T we will show that h µ (T h top (T. Let ξ = {C,..., C k } be a finite measurable partition and 0 < ε < k log k. Since µ is regular there exist compact sets B i C i with µ(c i \ B i < ε. Let η = {B 0, B,..., B k } the partition with B 0 = X \ k i= B i. We have µ(b 0 < kε and k k ( µ(bi A j H µ (ξ η < µ(b i φ µ(b i So j= = µ(b 0 k ( µ(b0 A j φ µ(b 0 j= µ(b 0 log k < kε log k < h µ (T, ξ h µ (T, η + H µ (ξ η h µ (T, η + q

8 8 LUCIAN RADU The reason we can bring in the topological entropy is that α := {B 0 B,..., B 0 B k } is an open cover of X. We have ( n ( n ( ( n H µ T k η log card T k η log 2 n card T k α If δ 0 is the Lebesgue number of α i.e., the supremum of δ > 0 such that every δ-ball is contained in an element of α, then δ 0 is the Lebesgue number of n T k α with respect to the d T n. Since n T k α is a minimal cover every C n point x C not in any other element of n T k α. The x C set form a (δ 0, n-separated set. Then h µ (T, η h top (T + log 2 and h µ (T, ξ h µ (T, η + h top (T + + log 2, as this is also true for any iterate of T we have T k α contain a h µ (T h µ(t n h top(t n + + log 2 = h top (T + + log 2 n n n for all n N and hence h µ (T h top (T (2 On the other hand applying Lemma 2 to maximal (n, ε-separated sets in X we have n n log N d(t, ε, n h µ (T (2 for a corresponding accumulation point µ M(X, T. Thus n n log N d(t, ε, n sup h µ (T. µ and letting ε 0 we get h top (T sup{h µ (T : µ M(X, T }. A measure µ M(X, T is called a measure of maximal entropy for T if h µ (T = h top (T. We observe that for expansive homeomorphisms the existence of such a measure is assured by the upper semi-continuity of the entropy map and the compactness of M(X, T. But the same is true because in (2 the left hand side is equal to the topological entropy if ε is less then half its expansive constant, and so by Lemma 2 we can construct a measure of maximal entropy for such an ε. Departamento de Matemática, Instituto Superior Técnico, Lisboa, Portugal address: lradu@math.ist.utl.pt