On the Ruelle rotation for torus diffeomorphisms

Math. Z. 234, 225–239 (2000)

c Springer-Verlag 2000

On the Ruelle rotation for torus diffeomorphisms Konstantin Athanassopoulos Department of Mathematics, University of Crete, GR-71409 Iraklion, Greece (e-mail: [email protected]) Received March 22, 1999; in final form May 17, 1999

1. Introduction In this note we are concerned with the Ruelle rotation number, which is defined for a C 1 diffeomorphism h of the 2-torus T 2 that is isotopic to the identity. Intuitively, the Ruelle rotation number is the mean value, with respect to an h-invariant Borel probability measure ν, of the asymptotic rate at which the derivative of h rotates the tangent planes. In order to measure it, we need to have fixed in advance a trivialization of the tangent bundle T T 2 of T 2 . The Ruelle rotation number depends on the trivialization, as well as on the isotopy from id to h. As was mentioned by Ruelle in [7], there is an independence on the isotopy if we take the number mod Z, i.e. consider the corresponding point on the unit circle S 1 . In Sect. 4 we clarify the dependence on the trivialization of T T 2 . It turns out that the two points of S 1 taken starting with two different trivializations of T T 2 differ by an element of the group ImRν (h), where Rν (h) : H 1 (T 2 ; Z) → S 1 is the ν-rotation number map of h. This may be seen as the discrete analogue of proposition 3.4 in [4]. It follows from this and the continuity of the Ruelle rotation number with respect to the C 1 topology [7], that the differential of the Ruelle rotation in group cohomology is a real bounded 2-cocycle on the ”universal covering” group Dif^ f 1 (T 2 , ν) 0

of the identity path component of the group of the ν-preserving C 1 diffeomorphisms of T 2 , which depends only on ν and not on the trivialization of T T 2 . So it defines an element of the second bounded cohomology with real coefficients of Dif^ f 1 (T 2 , ν) made discrete, the Ruelle rotation class. This 0

remark is perhaps the core of the present paper.

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The Ruelle rotation class belongs to the kernel of the natural homomorphism H 2 (Dif^ f 1 (T 2 , ν); R) → H 2 (Dif^ f 1 (T 2 , ν); R), which is known b

0

0

to be injective if Dif^ f01 (T 2 , ν) is uniformly perfect [5]. Thus, it can be considered as an obstruction to the uniform perfectness of Dif^ f 1 (T 2 , ν). 0

For instance, it follows from theorem 4.3 in [2] that if ν is the Haar measure, then the Ruelle rotation class is not trivial and so in this case Dif^ f 1 (T 2 , ν) 0

is not uniformly perfect and not amenable. The same is true in the case of a Dirac point measure also. 2. Preliminaries on rotation vectors in homology Let X be a compact metrizable space carrying a continuous flow (φt )t∈R . Let tx denote the translation of the point x ∈ X along its orbit in time t ∈ R. For every continuous function f : X → S 1 there is a continuous function g : R × X → R, called the cocycle of f such that f (tx) = f (x) exp(2πig(t, x)) and g(t + s, x) = g(s, tx) + g(t, x) for every x ∈ X and t, s ∈ R. The Ergodic Theorem of Birkhoff implies that for every φ-invariant Borel probability measure µ on X the limit g(t, x) t R R ∗ exists µ-almost for every x ∈ X and X g dµ = X g(1, .)dµ. This integral describes the µ-average rotation of points moving along their orbits with respect to the projection f . The cocycle property of g implies that g ∗ is an µ-almost everywhere defined measurable flow invariant function, that is g ∗ (tx) = g ∗ (x) for every t ∈ R, whenever g ∗ (x) is defined. If f1 , f2 : X → S 1 are homotopic continuous functions with cocyR R cles g1 , g2 , respectively, then X g1∗ dµ = X g2∗ dµ. Since the first integral ˇ ˇ 1 (X; Z) is isomorphic to the group of homotopy Cech cohomology group H classes of continuous functions of X to S 1 , there is a group homomorphism ˇ 1 (X; Z) → R defined by Aµ : H Z g(1, .)dµ Aµ [f ] = g ∗ (x) = lim

t→+∞

X

where g is the cocycle of f : X → S 1 and [f ] the homotopy class of f .The homomorphism Aµ was defined by S.Schwartzman [8] and is called the µ-asymptotic cycle of the flow. It describes how a µ-average orbit winds around X. The image of Aµ is called the µ-winding numbers group of the flow φ and is denoted by Wµ . An exposition of the basic theory of asymptotic cycles with details is given in [1].

Torus diffeomorphisms

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The suspensions of homeomorphisms is a class of flows whose winding numbers groups can be computed in terms of initial data. Let Y be a compact metrizable space and let h : Y → Y be a homeomorphism. On [0, 1]×Y we consider the equivalence relation (1, x) ∼ (0, h(x)), x ∈ Y . The quotient space X = [0, 1] × Y / ∼ is compact metrizable and is called the mapping torus of h. Let [s, x] denote the class of (s, x) ∈ [0, 1] × Y . The flow on X defined by t[s, x] = [t + s − n, hn (x)] if n ≤ t + s < n + 1, is called the suspension of h. If ν is an h-invariant Borel probability measure on Y and λ is the Lebesgue measure on [0, 1], then the product measure λ × ν induces a Borel probability measure on X which is invariant by the suspension of h. Every invariant by the suspension of h Borel probability measure on X is of this form. Let C(Y,Z) be the group of integer valued continuous functions on Y . ˇ 1 (X; Z) be defined by γ(ψ) = [f ], where f : X → S 1 Let γ : C(Y,Z)→ H is the continuous function defined by f [t, x] = exp(2πitψ(x)) and j ∗ : ˇ 1 (Y ; Z) be the homomorphism induced by the inclusion ˇ 1 (X; Z) → H H j : Y → X with j(x) = [0, x]. Then one can easily verify that the following sequence is exact. h∗ −id

j∗

γ

h∗ −id

ˇ 1 (X; Z)−→H ˇ 1 (Y ; Z) −→ H ˇ 1 (Y ; Z) C(Y, Z) −→ C(Y, Z)−→H If ν is an h-invariant Borel probability measure on Y and µ is the corresponding invariant measure on X, then nZ o Wµ = ψdν : ψ ∈ Log(Y, h) Y

where Log(Y, h) is the set of continuous functions ψ : Y → R satisfying f ◦ h = f exp(2πiψ) for some continuous function f : Y → S 1 [1]. Let Y be in addition connected and let f : Y → S 1 be a continuous function such that [f ] ∈ Ker(h∗ − id). There is a continuous function g : Y → R such that f ◦ h = f exp(2πig) and any two such functions differ by an integer. One can easily see that there is a well defined group homomorphism Rν : Ker(h∗ − id) → S 1 with Z Rν [f ] = exp 2πi gdν Y

called the ν-rotation number map of h. The subgroup ImRν of S 1 is called the ν-rotation numbers group of h. We have now the following commutative

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diagram. ˇ 1 (X; Z) H   j∗y

Aµ

−−−→ R  exp y R

ν Ker(h∗ − id) −−−→ S1

Thus we arrive at the following. 2.1. Proposition. Let Y be a connected, compact, metrizable space and h : Y → Y a homeomorphism. Let ν be an h-invariant Borel probability measure on Y . If j ∗ has a right inverse, then Rν ∈ ImF , where F : Hom(Ker(h∗ − id), R) → Hom(Ker(h∗ − id), S 1 ) is the homomorphism defined by F (α) = exp(2πiα). ˇ 1 (Y ; Z) is free, then Rν If h : Y → Y is homotopic to the identity and H 1 ˇ (Y ; Z), R)/ Hom(H ˇ 1 (Y ; Z), Z), can be considered an element of Hom(H ˇ 1 (Y ; Z), Z), where F is the homomorphism of since Ker F = Hom(H Proposition 2.1. ˇ 1 (Y ; Z), R)/ Hom(H ˇ 1 (Y ; Z), If Y is a closed manifold, then Hom(H Z) is isomorphic to H1 (Y ; R)/H1 (Y ; Z), by the Universal Coefficient Theorems (see [9]). Thus for any h-invariant Borel probability measure ν on Y , the rotation number map Rν can be considered an element of H1 (Y ; R)/H1 (Y ; Z). A representative of Rν in H1 (Y ; R) is Aµ ◦ r, where r is a right inverse of j ∗ . The set ρ(h) = {Rν : ν ∈ Mh (Y )} is the ergodic rotation set of h, where Mh (Y ) denotes the set of h-invariant Borel probability measures on Y and is weakly compact in H1 (Y ; R)/H1 (Y ; Z) [6]. 2.2. Example. Let h : T n → T n be a homeomorphism of the n-torus homotopic to the identity. If pi : T n → S 1 is the projection onto the i-th factor, 1 ≤ i ≤ n, then the homotopy classes [p1 ], ..., [pn ] form a basis of H 1 (T n ; Z). Let ν be a h-invariant Borel probability measure and µ the corresponding measure that is invariant by the suspension of h. Then Rν is represented by the vector (Aµ (r[p1 ]), ..., Aµ (r[pn ])) with respect to the dual basis of H1 (T n ; R). ˜ : Rn → Rn be a lift of h and ψ : T n → Rn be the continuous Let h ˜ function defined by ψ(x) = h(y) − y, where y ∈ p−1 (x) and p : Rn → T n is the canonical covering projection. Then, Z (Aµ (r[p1 ]), ..., Aµ (r[pn ])) = ψdν (mod Zn ). Tn


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That is, Rν is represented by the ν-mean translation vector of h. In case ν is ergodic, the Ergodic Theorem of Birkhoff implies that Z ˜ k (y) − y h ψdν = lim k→+∞ k Tn p−1 ν-almost for every y ∈ [0, 1)n . 2.3. Remark. Let H be a homotopy from id to h. We can extend it to [0, +∞) × Y by setting inductively Ht = Ht−[t] ◦ H[t] . If f : Y → S 1 is a continuous function, there exists a unique continuous function g : [0, +∞)× Y → R such that g(0, x) = 0 and f (Ht (x)) = exp(2πig(t, x)), f (x) for every t ≥ 0 and x ∈ Y . Then, g(t, x) = g(t − [t], h[t] (x)) + g([t], x) and inductively n−1 X g(n, x) = g(1, hk (x)) k=0

for every n ∈ N. It follows from the Ergodic Theorem of Birkhoff that for every ν ∈ Mh (Y ) the limit n−1

g(t, x) 1X = lim g (x) = lim g(1, hk (x)) n→+∞ n t→+∞ t ∗

k=0

exists ν-almost for every x ∈ Y , is ν-integrable and Z Rν [f ] = exp 2πi g ∗ dν . Y

2.4. Remark. Let h1 , h2 be two homeomorphisms of the compact manifold Y , which are homotopic to the identity and preserve the Borel probability measure ν. If Rν (hk ) is the ν-rotation number map of hk , k = 1, 2, then Rν (h1 ◦ h2 ) = Rν (h1 )Rν (h2 ). Indeed, if f : Y → S 1 is a continuous function for which there are continuous functions gk : Y → R such that f ◦ hk = f exp(2πigk ), k = 1, 2, then f ◦ h1 ◦ h2 = f exp(2πi(g1 ◦ h2 + g2 )). Since ν is h2 -invariant, we have Z Rν (h1 ◦ h2 )[f ] = exp(2πi (g1 ◦ h2 + g2 )dν) = Rν (h1 )[f ]Rν (h2 )[f ]. Y

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3. Rotation number maps and SL(2, R) Every A ∈ SL(2, R) has a unique decomposition A = U S, where U ∈ SO(2, R) and S is a symmetric, positive definite matrix. This is the polar decomposition of A. In this way we get a C ∞ diffeomorphism S 1 × (0, +∞) × R ∼ = SL(2, R). It is easily verified [10] that U = | det(A + (At )−1 )|−1/2 (A + (At )−1 ). ^R) → SL(2, R) is the universal covering of SL(2, R), If p : SL(2, then the preceeding diffeomorphism can be lifted to a C ∞ diffeomorphism ^R). The group of the deck transformations is R × (0, +∞) × R ∼ = SL(2, isomorphic to Z. Let d be a generator and D = d(I˜2 ), where I˜2 is the unit ele^R). Since p is a homomorphism, we have P −1 d(P ) ∈ Ker p ment of SL(2, ^R). The continuity of d, and d(P )P −1 ∈ Ker p, for every P ∈ SL(2, ^R) and the discreteness of Ker p imply that the connectedness of SL(2,

^R). d(P ) = P D = DP for every P ∈ SL(2, The map r : SL(2, R) → SO(2, R) defined by r(A) = U is a C ∞ submersion and a strong deformation retraction that can be lifted to a C ∞ ^R) → R, which has the following properties : submersion Θ : SL(2, (i) If P , Q correspond to elements of {0} × (0, +∞) × R, then |Θ(P Q)| < 1, and ^R). (ii) |Θ(P Q) − Θ(P ) − Θ(Q)| < 1 for every P , Q ∈ SL(2, Of course r is not a group homomorphism. 3.1. Lemma. Under the usual multiplication of matrices r is a homomorphism of H-groups. Proof. Let A, B ∈ SL(2, R) have polar decompositions A = r(A)s(A) and B = r(B)s(B). Since s(A) is symmetric and positive definite, s(A)ρ is defined for every ρ ≥ 0. Note that since AB = r(A)r(B)[r(B)−1 s(A)r(B)s(B)], by the uniqueness of the polar decomposition we have r(B)−1 r(A)−1 r(AB) = r(r(B)−1 s(A)r(B)s(B)). The function F : [0, 1] × SL(2, R) × SL(2, R) → SO(2, R) defined by F (ρ, A, B) = r(A)r(B)r((r(B)−1 s(A)r(B))1−ρ s(B)1−ρ ) is continuous and F (0, A, B) = r(AB), F (1, A, B) = r(A)r(B). Hence r is a homomorphism of H-groups.


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Let Y be a compact metrizable space. The set [Y ; SL(2, R)] of homotopy classes of continuous functions of Y into SL(2, R) becomes a group in the obvious way, using the group structure of SL(2, R). It follows from ˇ 1 (Y ; Z) is a Lemma 3.1 that the induced map r∗ : [Y ; SL(2, R)] → H group homomorphism. Since r is a strong deformation retraction, r∗ is an isomorphism whose inverse is induced by the inclusion SO(2, R) ,→ SL(2, R). Suppose that Y is also connected and h : Y → Y is a homeomorphism homotopic to the identity. Let f : Y → SL(2, R) be a continuous function. If H is a homotopy from id to h, then (f ◦ H)f −1 is a homotopy from I2 to (f ◦ h)f −1 . We can extend H to [0, +∞) × Y inductively, by setting Ht = Ht−[t] ◦ H[t] . By the homotopy lifting property, there is a unique ^R) such that G(0, x) = I˜2 and continuous map G : [0, +∞) × Y → SL(2, p(G(t, x)) = f (Ht (x))f (x)−1 for every t ≥ 0 and x ∈ Y . It follows that f (Ht (x))f (x)−1 = p(G(t − [t], h[t] (x))G([t], x). Thus, there exists a function k : [0, +∞) × Y → Z such that G(t − [t], h[t] (x))G([t], x)G(t, x)−1 = Dk(t,x) . For each x ∈ Y , it is obvious that k(., x) is continuous on [0, +∞) \ N and continuous from the right on [0, +∞). Since lim G(t − [t], h[t] (x))G([t], x)G(t, x)−1 = G(1, x)G(0, x)G(1, x)−1

t%1

= I˜2 , it follows inductively that k(., x) is everywhere continuous and hence identically zero, by connectedness. Thus, G(t, x) = G(t − [t], h[t] (x))G([t], x) for every t ≥ 0 and x ∈ Y , and inductively G(n, x) = G(1, hn−1 (x))...G(1, h(x))G(1, x) for every n ∈ N and x ∈ Y . According to the Ergodic Theorem for ^R) [7], which is a direct consequence of the almost subadditive erSL(2, godic theorem proved in [3], if ν is an h-invariant Borel probability measure on Y , the limit 1 G∗ (x) = lim Θ(G(n, x)) n→+∞ n

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exists ν-almost for every x ∈ Y , is h-invariant, ν-integrable and Z Z 1 ∗ G dν = lim Θ(G(n, x))dν. n→+∞ n Y Y Since for every t > 1 we have |Θ(G(t, x)) − Θ(G(t − [t], h[t] (x)) − Θ(G([t], x))| < 1 and by compactness and uniform continuity 1 Θ(G(t − [t], h[t] (x)) = 0, t→+∞ t lim

we conclude that

1 Θ(G(t, x)). t→+∞ t

G∗ (x) = lim

3.2. Lemma. The limit G∗ (mod Z) does not depend on the choice of the homotopy H from id to h. Proof. If φ = (f ◦ h)f −1 , then G(1, .) is a lift of φ and it is sufficient to ^R) is another lift of φ and prove that if ψ : Y → SL(2, G0 (n, x) = ψ(hn−1 (x))...ψ(h(x))ψ(x), then

1 1 Θ(G0 (n, x)) = lim Θ(G(n, x)). n→+∞ n n→+∞ n lim

Indeed, there exists m ∈ Z such that ψ(x) = Dm G(1, x) for every x ∈ Y , ^R), we have G0 (n, x) = and since D commutes with every element of SL(2, mn mn D G(n, x). Note that Θ(D ) = mn, since D ∈ Ker p. Thus, we have |Θ(G0 (n, x)) − Θ(G(n, x)) − mn| < 1 and the conclusion follows. 3.3. Proposition. If Rν is the rotation number map of h with respect to the h-invariant Borel probability measure ν, then using the above notation, Z (Rν ◦ r∗ )[f ] = exp 2πi G∗ dν . Y

Proof. There exists a unique continuous function g : [0, +∞) × Y → R such that g(0, x) = 0 and r(f (Ht (x))) = exp(2πig(t, x)) r(f (x))


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^R) such that for every t ≥ 0 and x ∈ Y . Choose any Q(x) ∈ SL(2, p(Q(x)) = f (x). Then, we have r(f (Ht (x))) 1 = r(p(G(t, x)Q(x))) r(f (x)) r(f (x)) and so there exists k(t, x) ∈ Z such that g(t, x) = Θ(G(t, x)Q(x)) − Θ(Q(x)) + k(t, x). By continuity with respect to t and connectedness, we get k(t, x) = k(0, x) = 0. Hence g ∗ (x) = lim

t→+∞

g(t, x) 1 = lim [Θ(G(t, x)Q(x)) − Θ(Q(x))] t→+∞ t t

ν-almost for every x ∈ Y and by property (ii) of Θ, we get g ∗ (x) = G∗ (x). This proves the proposition.

4. The Ruelle rotation Let h : T 2 → T 2 be a C 1 diffeomorphism isotopic to the identity. Let H be an isotopy from id to h. We need not assume that H is C 1 , but merely that H : [0, 1] × T 2 → T 2 is a continuous map such that H0 = id, H1 = h and Ht : T 2 → T 2 , t ∈ [0, 1], is a continuous path of C 1 diffeomorphisms in the C 1 topology. As usual we extend it to [0, +∞) × T 2 by setting inductively Ht = Ht−[t] ◦ H[t] . Let τ : T T 2 → T 2 × R2 be a trivialization of the tangent bundle of T 2 , compatible with its usual orientation. Let τx : Tx T 2 → {x} × R2 denote the restriction to the fiber. The continuous function F : [0, +∞) × T 2 → SL(2, R) defined by F (t, x) =

τH(t,x) ◦ DHt (x) ◦ τx−1

(det(τH(t,x) ◦ DHt (x) ◦ τx−1 ))1/2

^R) can be lifted to a unique continuous function G : [0, +∞)×T 2 → SL(2, such that G(0, x) = I˜2 and p(G(t, x)) = F (t, x), for every t ≥ 0 and x ∈ T 2 . From the chain rule and a same argument used in an analogous situation of Sect. 3 we have G(t, x) = G(t − [t], h[t] (x))G([t], x). So, inductively, G(n, x) = G(1, hn−1 (x))...G(1, h(x))G(1, x). According to the Ergodic ^R) [7], if ν is an h-invariant Borel probability measure, Theorem for SL(2, the limit G∗ (x) = lim

t→+∞

1 1 Θ(G(t, x)) = lim Θ(G(n, x)) n→+∞ n t

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exists ν-almost for every x ∈ T 2 , is h-invariant, ν-integrable and Z Z 1 G∗ dν = lim Θ(G(n, x))dν. n→+∞ n T 2 T2 The method of proof of Lemma 3.2 applies here also to show that G∗ (mod Z) is independent on the isotopy H. It does depend however on the trivialization τ of the tangent bundle of T 2 . Let Z ρτν (H) = G∗ dν, T2

and Pντ (h) = exp(2πiρτν (H)). Let now τ1 and τ2 be two trivializations of T T 2 compatible with the usual orientation of T 2 and f : T 2 → SL(2, R) be the continuous function defined by −1 τ1x ◦ τ2x f (x) = −1 1/2 . (det(τ1x ◦ τ2x )) 4.1. Proposition. If Rν (h) is the ν-rotation number map of h, then Rν (h)(r∗ [f ]) = Pντ1 (h)Pντ2 (h)−1 . τ

Proof. Let Fj , Gj be the functions used in the definition of ρνj as above, ^R) be the unique continuous map j = 1, 2 and G : [0, +∞) × T 2 → SL(2, ˜ such that G(0, x) = I2 and p(G(t, x)) = f (Ht (x))f (x)−1 for every t ≥ 0 and x ∈ T 2 . It is easy to see that since F1 (t, x) = (f (Ht (x))f (x)−1 )(f (x)F2 (t, x)f (x)−1 ), we have G1 (t, x) = G(t, x)Q(x)G2 (t, x)Q(x)−1 , where p(Q(x)) = f (x). This imlpies that G∗1 = G∗ + G∗2 , ν-almost everywhere and the conclusion follows from Proposition 3.3. The preceeding Proposition shows that ρν (h) = Pντ (h)ImRν (h) is a well defined element of the quotient group S 1 /ImRν (h), independent of the trivialization τ , and is called the Ruelle rotation of h with respect to the invariant measure ν. 4.2. Remark. For every x ∈ T 2 and n ∈ N we have n−1

1X 1 | Θ(G(n, x)) − Θ(G(1, hk (x)))| < 1. n n k=0


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From the Ergodic Theorem of Birkhoff the limit n−1

X ˜ x) = lim 1 G(t, Θ(G(1, hk (x))) n→+∞ n k=0

exists ν-almost for every x ∈ T 2 , is h-invariant, ν-integrable and Z T2

˜ x)dν = G(1,

Z T2

Θ(G(1, x))dν.

Thus, taking limits and integrating, we get |ρτν (H)

Z −

T2

Θ(G(1, x))dν| ≤ 1.

Let now ν be a Borel probability measure on T 2 and Dif f01 (T 2 , ν) be the path component of the identity in the C 1 topology of the group of C 1 diffeomorphisms of T 2 which preserve ν. In the space of continuous paths in Dif f01 (T 2 , ν) with initial point id endowed with the compactopen topology the relation of homotopy with fixed endpoints is an equivalence relation. Let Dif^ f 1 (T 2 , ν) be the corresponding quotient space and 0

q : Dif^ f01 (T 2 , ν) → Dif f01 (T 2 , ν) be the continuous map q([H]) = H1 . If Dif f01 (T 2 , ν) is locally contractible, then q is its universal covering projection. The space Dif^ f 1 (T 2 , ν) has a topological group structure, if we 0

define the composition of paths by (H ◦ H 0 )t = Ht ◦ Ht0 , t ∈ [0, 1], and then q becomes a group homomorphism. 4.3. Lemma. If H and H 0 are two homotopic paths with fixed endpoints the initial point being id in Dif f01 (T 2 , ν), then ρτν (H) = ρτν (H 0 ) for every trivialization τ of T T 2 . Proof. Let (H s )s∈[0,1] be a homotopy with fixed endpoints such that H 0 = H and H 1 = H 0 . Let F s and Gs be the functions as above in the definition of the Ruelle rotation corresponding to H s . Then, F s (n, x) = F 0 (n, x), for every x ∈ T 2 and n ∈ N, and Gs (1, .) is a lift of F 0 (1, .) for every s ∈ [0, 1]. Thus, there exist m(s) ∈ Z such that Gs (1, .) = G0 (1, x)Dm(s) , for every x ∈ T 2 and s ∈ [0, 1]. By continuity and connectedness, we must have m(s) = m(0) = 0. So, Gs (1, .) = G0 (1, .) for every s ∈ [0, 1] and the conclusion follows.

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So we have a well defined function ρτν : Dif^ f01 (T 2 , ν) → R and the following diagramm commutes. ρτ

ν Dif^ f01 (T 2 , ν) −−−→   qy

R  exp y

Pτ

ν Dif f01 (T 2 , ν) −−−→ S1

^R) implies also that The continuous case of the Ergodic Theorem for SL(2, τ 1 ρν is continuous with respect to the C topology in Dif f01 (T 2 , ν) (see [7]). It is immediate from the definition that ρτν (H m ) = mρτν (H) for every m ∈ Z. If H and H 0 are two paths in Dif f01 (T 2 , ν) with initial point id, then one can easily verify using Remark 4.2 that |ρτν (H ◦ H 0 ) − ρτν (H) − ρτν (H 0 )| ≤ 4. Thus, ρτν is a homogeneous quasi-morphism. Using this one can show that Pντ is constant on conjugancy classes in Dif f01 (T 2 , ν) (see the remark following proposition 2.8 in [4]). We recall now the notion of bounded cohomology with real coefficients of a discrete group G. Let B n (G) be the Banach space of all bounded real functions on Gn , n ≥ 1. If dn : B n (G) → B n+1 (G) is the continuous operator defined by (dn f )(g0 , ..., gn ) = f (g1 , ..., gn ) +

n X

(−1)k f (g0 , ..., gk−1 gk , ..., gn )

k=1

+(−1)n+1 f (g0 , ..., gn−1 ), then dn+1 ◦ dn = 0. The bounded cohomology of G with real coefficients Hb∗ (G; R) is the cohomology of the cochain complex of Banach spaces (B ∗ (G), d∗ ). It is always true that Hb1 (G; R) = 0. It follows from the above that dρτ : Dif^ f 1 (T 2 , ν) × Dif^ f 1 (T 2 , ν) → ν

0

0

R defined by (dρτν )(H, H 0 ) = ρτν (H ◦ H 0 ) − ρτν (H) − ρτν (H 0 ) is a bounded 2-cocycle. If τ1 , τ2 are two trivializations of T T 2 , it follows from Remark 2.4 and Proposition 4.1 that dρτν1 − dρτν2 takes on integer values. Since Dif^ f 1 (T 2 , ν) is connected, and ρτk , k = 1, 2, are continuous, both taking ν

0

the value zero at id, we conclude that dρτν1 = dρτν2 . Thus, in the sequel we shall simply write dρν . We shall call the element [dρν ]b of the second bounded cohomology with real coefficients of the group Dif^ f 1 (T 2 , ν) 0

made discrete, the Ruelle rotation class. It belongs to the kernel of the natural f 1 (T 2 , ν); R) → H 2 (Dif^ f 1 (T 2 , ν); R). homomorphism H 2 (Dif^ b

0

0


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4.4. Lemma. If ν is a Borel probability measure on T 2 , then [dρν ]b = 0 if and only if ρτ : Dif^ f 1 (T 2 , ν) → R is a group homomorphism for any ν

trivialization τ of T T 2 .

0

f01 (T 2 , ν)) such that d1 σ = Proof. Suppose that there exists σ ∈ B 1 (Dif^ τ τ dρν . Then, σ(id) = ρν (id) = 0. Moreover, it follows inductively that σ is homogeneous, since ρτν is. But then we must have σ = 0, because σ is bounded. Hence ρτν is a group homomorphism. The converse is trivial. 4.5. Theorem. The Ruelle rotation class with respect to the Haar measure on T 2 is not trivial. Proof. Let D2 be the closed unit disc, λ be the normalized Lebesgue measure on D2 and K be a collar closed neighbourhood of ∂D2 . The group Dif f ∞ (D2 , K, λ) of λ-preserving diffeomorphisms of D2 , which are the identity on K, is contractible in the C ∞ topology. There is a continuous monomorphism l : Dif f ∞ (D2 , K, λ) → Dif f0∞ (T 2 , ν), where ν is the Haar measure on T 2 . The latter is a locally contractible group in the C ∞ topology. By contractibility, l is lifted to a continuous homomorphism 2 ∞ 2 ˜l : Dif f ∞ (D2 , K, λ) → Dif f^ 0 (T , ν). Let τ be the trivialization of T T 2 2 defined by the derivative of the universal covering projection p : R → T . According to theorem 4.3 in [2], ρτν ◦ ˜l is not a group homomorphism. Hence ρτν is not a group homomorphism and [dρν ]b is not trivial, by Lemma 4.4. From corollary 2.11 in [5], we have : 4.6. Corollary. The group Dif^ f01 (T 2 , ν) is not uniformly perfect for the Haar measure ν. Finally, we shall be concerned with the Ruelle rotation class of a Dirac measure ν at a point x ∈ T 2 . In this case Dif f01 (T 2 , ν) is the path component of the identity in the C 1 topology of the group of C 1 diffeomorphisms that fix x. If h : T 2 → T 2 is a C 1 diffeomorphism isotopic to the identity that fixes x, then Rν (h) = 1 and ρ(x, h) = Pντ (h) is independent of the trivialization τ of T T 2 , by Proposition 4.1. ρ(x, h) is the infinitesimal rotation number of the fixed point x of h. It measures the rate at which the fixed point x is rotating infinitesimally. If H is an isotopy from id to h and F , G as above, then F (n, x) =

(τx ◦ Dh(x) ◦ τx−1 )n (det(τx ◦ Dh(x) ◦ τx−1 ))n/2

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K. Athanassopoulos

for every n ∈ N and ρτν (H) = lim

n→+∞

1 Θ(G(n, x)). n

^R). Applying the Ergodic Theorem for SL(2, ^R) for Let P ∈ SL(2, the trivial one point probability space, we conclude that the limit Ω(P ) = lim

n→+∞

1 Θ(P n ) n

^R) → R is a continuous funcexists. Since Θ is continuous, Ω : SL(2, tion, which is also constant on conjugancy classes, by property (ii) of Θ. Moreover, we have |Θ((Dk P )n ) − Θ(P n ) − kn| < 1, from which follows that Ω(Dk P ) = Ω(P ) + k. Thus, Ω descends to a continuous function ω : SL(2, R) → S 1 defined by ω(A) = exp(2πiΩ(P )), where P is any lift of A, which is constant on conjugancy classes. If A is hyperbolic with positive eigenvalues, then ω(A) = 1. If A has negative eigenvalues, then ω(A) = −1. If A is parabolic, then ω(A) = 0. If A is elliptic with eigenvalues cos 2πφ ± i sin 2πφ, then ω(A) = e2πiφ . Let A be a rotation by an angle 2πφ, with 0 < φ < 1/4 and B be diagonal hyperbolic with eigenvalues −4 and −1/4. Then AB is hyperbolic with negative eigenvalues and according to the above we have ω(AB) = ω(B) = −1 and ω(A) = e2πiφ . Hence ω is not a group homomorphism. 4.7. Theorem. The Ruelle rotation class with respect to a Dirac point measure is not trivial. Proof. With the above notation we have ρ(x, h) = ω(Dh(x)/(det Dh(x))1/2 ), for every h ∈ Dif f01 (T 2 , ν), where ν is the Dirac measure at x. The above remarks imply that ρ(x, .) : Dif f01 (T 2 , ν) → S 1 is not a group homomorphism and the result follows.

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5. Matsumoto, S., Morita, S.: Bounded cohomology of certain groups of homeomorphisms. Proc. Amer. Math. Soc. 94, 539-544 (1985). 6. Pollicott, M.: Rotation sets for homeomorphisms and homology. Trans. Amer. Math. Soc. 331, 881-894 (1992). 7. Ruelle, D.: Rotation numbers for diffeomorphisms and flows. Ann. Inst. H. Poincar´e, Physique Theorique 42, 109-115 (1985). 8. Schwartzman, S.: Asymptotic cycles. Ann. of Math. 66, 270-284 (1957). 9. Spanier, E.H.: Algebraic Topology. New York: McGraw-Hill 1966. 10. Uhlig, F.: Explicit polar decomposition and a near-characteristic polynomial: The 2 × 2 case. Linear Algebra Appl. 38, 239-249 (1981).