Cohomology-free diffeomorphisms on tori

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1 Cohomology-free diffeomorphisms on tori arxiv: v2 [math.ds] 21 May 2014 Nathan M. dos Santos 1 Instituto de Matemática, Universidade Federal Fluminense, , Niterói, RJ, Brazil santosnathanbr@gmail.com Abstract. A C diffeomorphism ϕ: M M on a smooth closed orientablemanifoldm iscohomology-free(c.f.) ifforeachc function f: M R there exists a constant f 0 and a C function h: M R such that h h ϕ = f f 0. In this article we show that (c.f.) diffeomorphisms on tori are conjugate to Diophantine translations. This is part of a conjecture of A. Katok [H, Problem 17]. 1. Introduction We first show that a (c.f.) diffeomorphism of a torus T n = T p T q may be given on the covering by ϕ(x,y) = (x+α,cx+by +G(x,y)+β) (1) where x T p, y T q, G: R n R q is a C function and α R p is a Diophantine vector [AS2] i.e. k,α C, C,δ > 0 (2) Kp+δ for all k Z p 0 and x = inf{ x l,l Z p } defines a metric on T p where x = sup x j, 1 j p and k,α = k 1 α 1 + +k p α p. A (c.f.) diffeomorphism leaves invariant volume form and it is uniquely ergodic and minimal. 1 Correspondence address: Rua Lopes Quintas, 255 ap. 401-A, Jardim Botânico, Rio de Janeiro, CEP , Brazil. 1

2 Proposition 1. There exists an invariant volume form for a cohomologyfree diffeomorphism. Proof: Let Ω 0 be a volume form on M. Thus ϕ Ω 0 = detdϕω 0 (1) since ϕ: T n T n is (c.f.) then there exist a constant c and a C function h: M R such that log detϕ = h h ϕ+c. (2) We will show that c = 0. Consider the volume form ω = exphω 0. (3) Now from (2) and (3) we have ϕ Ω = (exph ϕ)ϕ Ω 0 = exph ϕ(detdϕ)ω 0 = exp(h ϕ+log detdϕ )Ω 0 = exp(h+c)ω 0 = exp(c)exphω 0 (4) thus M ϕ Ω = deg(ϕ) = expc M Ω (5) from (2) we may assume that Ω = 1 by choosing a convenient M function h: M R. Now from (5) we get deg(ϕ) = exp(c) = 1 (6) thus c = 0 and from (3) and (4) we have ϕ Ω = deg(ϕ)ω 2

3 2. The topological entropy of a (c.f.) diffeomorphism is zero Proposition 2. Let ϕ: M M be a C diffeomorphism and Ω be a ϕ-invariant volume form and ϕ be uniquely ergodic. Then the topological entropy h(ϕ) is zero. Proof: ϕ Ω = deg(ϕ)ω = detdϕω. (1) Thus deg(ϕ) = detdϕ and detdϕ = 1. Now 1 h µ (ϕ) = lim log D x ϕ n dµ(x) by [M; 10.2b)] n n M where µ is a ϕ-invariant measure and Dϕ n = detdϕ n. So the topological entropy of ϕ is given by 1 h(ϕ) = lim log detdϕ n dµ (2) n n since ϕ is uniquely ergodic. Now M (ϕ n ) Ω = (degϕ) n Ω = detdϕ n Ω. Thus and detdϕ n = degϕ n detdϕ n = 1 log detdϕ n = 0 (3) and from (2) and (3) we see that h(ϕ) = 0. Proposition 3. The (c.f.) diffeomorphisms ϕ: T n T n are quasiunipotent on homology and 1 is an eigenvalue of ϕ : H 1 (T n,r) H 1 (T n,r). Proof: By [Mann] we have the following inequality logspϕ h(ϕ) = 0 (1) where Spϕ is the spectral radius of ϕ : H 1 (T n,r) H 1 (T n,r). By the above inequality we see that Spϕ = 1. Thus by a well-known 3

4 theorem in algebra all eigenvalues of ϕ are roots of the unity and 1 is an eigenvalue of ϕ by the Lefschetz fixed point theorem since ϕ has no fixed points. Proposition 4. Let ϕ: T n T n be a diffeomorphism such that ϕ w j = w j, w j = dx j +dh j, 1 j p, then there exists a diffeomorphism ψ: T n T n homotopic to the identity such that ψ dx j = w j, 1 j p and ψ dy j = dy j, 1 j q. So if ϕ 0 = ψ ϕ ψ 1 then ϕ 0 dx j = dx j, 1 j = p Proof: Let ϕ: T p T q T p T q, n = p+q Z + be given on the covering R n = R p +R q by ϕ(x,y) = (x+f +α,cx+by +G(x,y)+β) (1) where x R p, y R q, F: R n R p and G: R n R q are C functions. Now ϕ w j = w j where w j = dx j + dh j, 1 j p iff the cohomological equation H H ϕ = F (2) has a solution H: R n R p, H = (h 1,...,...,h p ) is a C function. For ϕ w = w, dx + dh = w and ϕ w = dx + df + d(h ϕ) = w = (dx + dh) iff dh d(h ϕ) = df or H H ϕ = F. Let ψ: T n T n, ψ(x,y) = (x + H,y), then ψ is a C diffeomorphism and ψ dx = dx+dh = w, ψ dy = dy. Thus if ϕ 0 = ψ ϕ ψ 1 then ϕ 0 (x,y) = (x+α,cx+by +G(x,y)). (3) 3. Canonical form of a (c.f.) diffeomorphism Proposition 5. If ϕ: T n T n, n Z + is a (c.f.) diffeomorphism then there exists a C diffeomorphism ζ: T n T n homotopic to the identity conjugating ϕ to a skew-product ϕ 0 (x,y) = (x+α,cx+by +S(x)) (1) 4

5 where T n = T p T q, x T p, y T q and S: T p R q is C, α R p is a Diophantine vector and B is a quasi-unipotent q q integer matrice. Proof: By Proposition 3, ϕ is quasi-unipotent on homology thus by [SL; Remark 1.4] ϕ is given on the covering R n = R p R q by ϕ(x,y) = (Ax+F(x,y),Cx+By +G(x,y) (2) ( ) A 0 where L =, L GL(n,Z) and A = I +N, N nilpotent and C B C is q p and B is q q are integer matrices and w j = dx j +dh j, 1 j p (3) where the twisted cohomological equation AH (H ϕ)+α = F (4) is solved below: since ϕ: T n T n is (c.f) then we have h 1 h 1 ϕ = F 1 α 1 h 2 h 2 ϕ = F 2 A 21 h 1 α 2... p 1 h p h p ϕ = F p A pj h j α j j=1 (5) and α j = F j dµ, h j dµ = 0, 1 j p T n T n and µ is the measure given by the ϕ-invariant volume form Ω. Thus by (4) we have in matrice notation where w 1 dx 1 w =., dx =. ϕ w = t Adx+d(H ϕ)+df = t Aw (6), H = h 1., and F = F 1.. w p dx p h p F p 5

6 To show that {w 1,...,w p } is linearly independent on T n consider the p-form W given by W = w 1 w p (7) which by (6) is ϕ-invariant. Now {w 1,...,w p } is linearly dependent at a point z T n iff W(z) = 0 and as W is ϕ-invariant and ϕ is minimal then W vanishes on T n giving the contradiction dx = dθ. So {w 1,...,w p } is linearly independent. We claim: the affine mapping a: T p T p, a(x) = Ax+α (8) is a factor of ϕ. For consider the submersion p: T n T p given by p(x,y) = x+h. Thus (p ϕ)(x,y) = p(ax+f,cx+by +G) = Ax+F +H ϕ = A(x+H)+α = (a p)(x,y) (9) by (4). Now by [SL; Corollary 1.11] the affine diffeomorphism a: T p T p is (c.f.) since ϕ: T n T n is (c.f.) and by [SL; Theorema 1.5] a(x) = x+α is a Diophantine translation. So by [SL; Remark 1.3] and Proposition 4 there exists a diffeomorphism ψ: T n T n homotopic to the identity such that ψ dx j = w j, 1 j pandby[sl;corollary6.2]wemayassumethatψ (dx dy) = Ω. So up to a conjugation by ψ we may assume that ϕ(x,y) = (x+α,cx+by +G(x,y)) (10) preserves the canonical volume Ω 0 = dx dy and ϕ 0 dx j = dx j, 1 j p. We claim: there exists a C diffeomorphism ζ: T n T n homotopic to the identity conjugating (10) to the skew-product ϕ 0 (x,y) = (x+α,cx+by +S(x)). (11) For, consider the C diffeomorphisms ϕ x : T q T q given by ϕ x (y) = By +G x (y) (12) 6

7 where G x (y) = G(x,y), x T p, y T q. From (12) we get ϕ x dy = t Bdy +dg x. (13) Now let H x = t B 1 G x and from (13) we have ϕ x(y) = t B(dy +dh x ) (14) since ϕ is (c.f.) we have a C solution k x : T q R q to the cohomological equation and from (15) we have For each x T p consider the 1-form and from (14), (16) and (17) we get k x k x ϕ = H x (15) dk x d(k x ϕ) = dh x. (16) w x = dy +dk x (17) ϕ xw x = t B(dy +dh x +d(k x ϕ)) = t B(dy +dk x ) = t Bw x (18) now from (18) and [SL; Lemma 1.6] there exists a C diffeomorphism ψ x : T q T q homotopic to the identity conjugating ϕ x to the affine map ψ x ϕ x ψ 1 x = By +S(x) = a x (19) notice that S: T p R q is a C. Consider the C diffeomorphism given by ζ: T p T q T p T q ζ(x,y) = (x,ψ x (y)), x T p and y T q. (20) Thus ϕ 0 = ζ ϕ ζ 1 7

8 where ϕ 0 (x,y) = (x+α,cx+by +S(x)) and ϕ 0 : T n T n is a skew-product, α R p is a diophantine vector and B is quasi-unipotente, proving the Proposition 5. Theorem. The cohomology-free diffeomorphisms of the tori T n, n Z + are conjugate to Diophantine translations and the conjugating diffeomorphisms are homotopic to the identity. Proof: This theorem follows from [SU; Theorem 3], Proposition 5 above and [SL; Theorem 1.5]. REFERENCES [AS1] J.L. Arraut and N.M. dos Santos, Differentiable conjugation of actions of R p. Bol. Soc. Brasil. Math. 21 (1988), [AS2] J.L. Arraut and N.M. dos Santos, Linear foliation of T n. Bol. Soc. Brasil. Math. 21 (1991), [H] S. Hurder, Problems on rigidity of group actions and cocycles. Ergod. Th. & Dynam. Sys. 5 (1985), [SL] R.U. Luz and N.M. dos Santos, Cohomology-free diffeomorphisms of low-dimension tori. Ergod. Th. & Dynam. Sys. 18 (1998), [Kat] Combinatorial Constructions in Ergodic Theory and Dynamics. A. Katok, University Lectures Series, volume 30. [SU] N.M. dos Santos and R. Urzúa-Luz, Minimal homeomorphisms of low-dimensional tori. Ergod. Th. & Dynam. Sys. (2009), 29, [M] R. Mañé, Ergodic Theory and Differentiable Dynamics. Springer, New York, [Mann] A.K. Manning, Topological entropy and first homology group. Lectures Notes in Mathematics, 468, (1975),