Furstenberg Transformations on Cartesian Products of Infinite-Dimensional Tori

Transcription

1 Potential Anal :43 51 DOI /s y Furstenberg Transformations on Cartesian Products of Infinite-Dimensional Tori P. A. Cecchi 1 Rafael Tiedra de Aldecoa 2 Received: 10 February 2015 / Accepted: 13 July 2015 / Published online: 26 July 2015 Springer Science+Business Media Dordrecht 2015 Abstract We consider in this note Furstenberg transformations on Cartesian products of infinite-dimensional tori. Under some appropriate assumptions, we show that these transformations are uniquely ergodic with respect to the Haar measure and have countable Lebesgue spectrum in a suitable subspace. These results generalise to the infinite-dimensional setting previous results of H. Furstenberg, A. Iwanik, M. Lemańzyk, D. Rudolph and the second author in the one-dimensional setting. Our proofs rely on the use of commutator methods for unitary operators and Bruhat functions on the infinite-dimensional torus. Keywords Furstenberg transformations Infinite-dimensional torus Commutator methods Mathematics Subject Classifications D10 37A30 37C40 58J51 1 Introduction We consider in this note the generalisation of Furstenberg transformations on Cartesian products T d d 2 of one-dimensional tori [8, Section 2] to the case of Cartesian products Supported by the Chilean Fondecyt Grant and by the Iniciativa Cientifica Milenio ICM RC Mathematical Physics from the Chilean Ministry of Economy. Rafael Tiedra de Aldecoa rtiedra@mat.puc.cl P. A. Cecchi paulina.cecchi@usach.cl 1 Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Av. Libertador Bernardo O Higgins 3363, Estación Central, Santiago, Chile 2 Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, Santiago, Chile

2 44 P.A. Cecchi, R. Tiedra de Aldecoa T d of infinite-dimensional tori. Using recent results on commutator methods for unitary operators [14, 15], we show under a C 1 regularity assumption on the perturbations that these transformations are uniquely ergodic with respect to the Haar measure on T d and are strongly mixing in the orthocomplement of functions depending only on variables in the first torus T see Assumption 3.2 and Theorem 3.3. Under a slightly stronger regularity assumption C 1 + Dini continuous derivative, we also show that these transformations have countable Lebesgue spectrum in that orthocomplement see Theorem 3.4. These results generalise to the infinite-dimensional setting previous results of H. Furstenberg, A. Iwanik, M. Lemańzyk,D. Rudolph and the secondauthor [8, 10, 15] in the one-dimensional setting. Apart from commutator methods, our proofs rely on the use of Bruhat test functions [5] which provide a natural analog to the usual C -functions which do not exist in our infinite-dimensional setting see Section 3 for details. We also mention that all the results of this note apply to Furstenberg transformations on Cartesian products T n d of finitedimensional tori T n of any dimension n 1. One just has to consider the particular case where the functions defining the Furstenberg transformations on T d depend only on a finite number of variables. Here is a brief description of the content of the note. First, we recall in Section 2 the needed facts on commutators of unitary operators and regularity classes associated with them. Then, we define in Section 3 the Furstenberg transformations on T d, and prove Theorems 3.3 and 3.4 on the unique ergodicity, strong mixing property and countable Lebesgue spectrum of these transformations. We refer to [9, 11 13] and references therein for other works building on Furstenberg transformations. 2 Commutator Methods for Unitary Operators We recall in this section some facts borrowed from [7, 14] on commutator methods for unitary operators and regularity classes associated with them. Let H be a Hilbert space with scalar product, antilinear in the first argument, denote by BH the set of bounded linear operators on H, and write both for the norm on H and the norm on BH. LetA be a self-adjoint operator in H with domain DA, anake S BH. Foranyk N, we say that S belongs to C k A, with notation S C k A, if the map R t e ita S e ita BH 2.1 is strongly of class C k. In the case k = 1, one has S C 1 A if and only if the quadratic form DA ϕ ϕ,saϕ Aϕ,Sϕ C is continuous for the topology induced by H on DA. We denote by [S,A] the bounded operator associated with the continuous extension of this form, or equivalently i times the strong derivative of the function 2.1att = 0. A condition slightly stronger than the inclusion S C 1 A is provided by the following definition: S belongs to C 1+0 A, with notation S C 1+0 A, ifs C 1 A and if [A, S] satisfies the Dini-type condition 0 t e ita [A, S] e ita [A, S] <.

3 Furstenberg Transformations on Infinite Tori 45 As banachisable topological vector spaces, the sets C 2 A, C 1+0 A, C 1 A and C 0 A = BH satisfy the continuous inclusions [2, Section 5.2.4] C 2 A C 1+0 A C 1 A C 0 A. Now, let U C 1 A be a unitary operator with complex spectral measure E U and spectrum σu S 1 := {z C z =1}. If there exist a Borel set S 1, a number a>0 and a compact operator K BH such that E U U [A, U]E U ae U + K, 2.2 then one says that U satisfies a Mourre estimate on and that A is a conjugate operator for U on. Also, one says that U satisfies a strict Mourre estimate on if Eq. 2.2 holds with K = 0. One of the consequences of a Mourre estimate is to imply spectral properties for U on. We recall here these spectral results in the case U C 1+0 A. We also recall a result on the strong mixing property of U in the case U C 1 A see [7, Thm.2.7&Rem.2.8] and [14, Thm. 3.1] for more details. Theorem 2.1 Absolutely continuous spectrum Let U and A be respectively a unitary and a self-ajoint operator in a Hilbert space H, with U C 1+0 A. Suppose there exist an open set S 1, a number a>0 and a compact operator K BH such that E U U [A, U]E U ae U + K. 2.3 Then, U has at most finitely many eigenvalues in, each one of finite multiplicity, and U has no singular continuous spectrum in. Furthermore, if Eq. 2.3 holds with K = 0, then U has only purely absolutely continuous spectrum in no singular spectrum. Theorem 2.2 Strong mixing Let U and A be respectively a unitary and a self-adjoint operator in a Hilbert space H, with U C 1 A. Assume that the strong limit D := s-lim U l [A, U]U 1 U l N N exists, and suppose that ηdda DA for each η C c a lim N ϕ,u N ψ = 0 for each ϕ kerd and ψ H, b U kerd has purely continuous spectrum. R \{0}. Then, 3 Furstenberg Transformations on Cartesian Products of Infinite-Dimensional Tori Let T R /Z be the infinite-dimensional torus with elements x = {x k },and let T be the dual group of T. For each n N,letμ n be the normalised Haar measure on T n,andleth n := L 2 T n,μ n be the corresponding Hilbert space. In analogy to Furstenberg transformations on Cartesian products of one-dimensional tori see [8, Sec. 2.3], we consider Furstenberg transformations on Cartesian products of infinite-dimensional tori T d : T d T d, d 2, given by T d x 1,x 2,...,x d := x 1 + α, x 2 + φ 1 x 1,...,x d + φ d 1 x 1,x 2,...,x d 1,

4 46 P.A. Cecchi, R. Tiedra de Aldecoa with α T such that {nα} n Z is dense in T and φ j C T j ; T for each j {1,...,d 1}.SinceT d is invertible and preserves the measure μ d, the Koopman operator W d : H d H d, ϕ ϕ T d, is a unitary operator. Furthermore, W d is reduced by the orthogonal decompositions H d =H 1 Hj H j 1 =H1 H j,χ, H j,χ := { } η χ η H j 1, j {2,...,d} j {2,...,d} χ T \{1} and the restriction W d Hj,χ is unitarily equivalent to the unitary operator U j,χ η := χ φ j 1 Wj 1 η, η H j 1. In order to define later a conjugate operator for U j,χ, we first define a suitable group of translations on T j 1. For this, we choose {y t } t R an ergodic continuous one-parameter subgroup of T such that each map R t y t 1,...,y t n T n, n N, 3.1 is of class C 1. An example of such a subgroup {y t } t R is given by the formula y t k = y k t mod Z, t R, k N, with y R a transcendental number see [6, Ex ]. Then, we associate to {y t } t R the translation flow F j 1,t x 1,...,x j 1 := x 1,...,x j 1 + y t, t R, x 1,...,x j 1 T j 1, and the translation operators V j 1,t : H j 1 H j 1 given by V j 1,t η := η F j 1,t.Due to the continuity of the map R t y t T and of the group operation, the family {V j 1,t } t R defines a strongly continuous unitary group in H j 1 with self-adjoint generator H j 1 η :=s-lim t 0 it 1 V j 1,t 1 η,η DH j 1 := { η H j 1 lim t 0 t 1 V j 1,t 1 η < When dealing with differential operators on compact manifolds, one typically does the calculations on an appropriate core of the operators such as the set of C -functions. But here, the are no such functions on T j 1,sinceT is not a manifold T does not admit any differentiable structure modeled on locally convex spaces, see [4, Sec. 10.2] for details. So, we use instead the set B j 1 of Bruhat test functions on T j 1, whose definition is the following see [3, Sec. 2.2] or [5] for details. Set T 0 := {0}, and for each n N let π n : T j 1 T n j 1 be the projection given by { 0,...,0 if n = 0 π n x 1,...,x j 1 := x1 1,...,x 1 n,...,x j 1 1,...,x j 1 n if n 1. }. Then, B j 1 := n N { ζ πn ζ C T n j 1}, that is, B j 1 is the set of all functions on T j 1 that are obtained by lifting to T j 1 any C -function on one of the Lie groups T n j 1.ThesetB j 1 is dense in H j 1,itis left invariant by the group {V j 1,t } t R, and satisfies the inclusion B j 1 DH j 1 to show the latter, one has to use the C 1 -assumption 3.1. Therefore, it follows from Nelson s criterion [1, Prop. 5.3] that H j 1 is essentially self-adjoint on B j 1.

5 Furstenberg Transformations on Infinite Tori 47 Now, if the multiplication operator valued map R t χ φ j 1 F j 1,t BH j 1 is strongly of class C 1,thenχ φ j 1 C 1 H j 1 since χ φ j 1 F j 1,t = V j 1,t χ φj 1 Vj 1, t = e ith j 1 χ φ j 1 e ith j 1. Thus, the operator g j,χ := [ ] d H j 1,χ φ j 1 χ φj 1 = s- iχ φj 1 F j 1,t χ φj 1 is a bounded self-adjoint multiplication operator in H j 1. = s- d iχ φ j 1 F j 1,t φ j 1 Lemma 3.1 Take j {2,...,d} and χ T \{1}, and assume that the map R t χ φ j 1 F j 1,t BH j 1 is strongly of class C 1. Then, U j,χ C 1 H j 1 with [H j 1,U j,χ ]=g j,χ U j,χ. Proof Take η B j 1. Then, the commutation of V j 1,t and W j 1 and the differentiability assumption imply that Hj 1 η, U j,χ η H η, U j 1 j,χ H j 1 η H j 1 = i d { V j 1,t η, χ φ j 1 Wj 1 η η, χ φ H j 1 j 1 Wj 1 V j 1,t η } H j 1 = i d η, χ φj 1 F j 1,t χ φ j 1 Wj 1 V j 1,t η H j 1 = d { } η, iχ φj 1 F j 1,t χ φj 1 1 Uj,χ V j 1,t η H j 1 = d { } η, iχ φj 1 F j 1,t φ j 1 1 Uj,χ V j 1,t η H j 1 = η, g j,χ U j,χ η H j 1, and thus the claim follows from the boundedness of g j,χ and the density of B j 1 in DH j 1. Assumption 3.2 For each j {2,...,d}, themapφ j 1 C T j 1 ; T satisfies φ j 1 = ξ j 1 + η j 1,where i ξ j 1 C T j 1 ; T is a group homomorphism such that T x j 1 χ ξ j 1 0,...,0 + x j 1 S 1 ii is nontrivial for each χ T \{1}, η j 1 C T j 1 ; T is such that there exists η j 1 C T j 1 ; R with η j 1 x 1,...,x j 1 = η j 1 x 1,...,x j 1 modz for eachx 1,...,x j 1 T j 1, and with R t η j 1,k F j 1,t BH j 1 strongly of class C 1 for each k N. In the next theorem, we use the fact that the map R t χ ξ j 1 0,...,0 + yt S 1

6 48 P.A. Cecchi, R. Tiedra de Aldecoa is a character on R, and thus of class C. We also use the notation ξ χ j 1 := d χ ξj 1 0,...,0 + yt i R. Theorem 3.3 Strong mixing and unique ergodicity Suppose that Assumption 3.2 is satisfied. Then, W d is strongly mixing in H d H 1 and T d is uniquely ergodic with respect to μ d. These results of strong mixing and unique ergodicity are an extension to the case of infinite-dimensional tori of results previously obtained in the case of one-dimensional tori see [8, Thm. 2.1] for the unique ergodicity and [10, Rem. 1] for the strong mixing property. For instance, if the functions η j 1 in Assumption 3.2 were to depend only on a finite number of variables, then the strong C 1 regularity condition on η j 1 in Assumption 3.2ii would reduce to a uniform Lipschitz condition of η j 1 along the flow {F j 1,t } t R, as in the onedimensional case treated by Furstenberg in [8, Thm.2.1]. Proof i Take j {2,...,d}, χ T \{1} and t R. Then, there exist k χ N and n 1,...,n kχ Z such that kχ 2πi χx j 1 = e n kx j 1,k, with x j 1 T and x j 1,k [0, 1 the k-th cyclic component of x j 1. Therefore, we infer from Assumption 3.2 that χ φ j 1 F j 1,t φ j 1 = χ ξj 1 F j 1,t ξ j 1 χ ηj 1 F j 1,t η j 1 = kχ χ ξ j 1 0,...,0+ 2πi yt e n k η j 1 F j 1,t η j 1 k, and Lemma 3.1 implies that U j,χ C 1 H j 1 with [H j 1,U j,χ ]=g j,χ U j,χ and g j,χ =s- d kχ iχ φ j 1 F j 1,t φ j 1 = iξ χ j 1 2π n k s- d η j 1,k F j 1,t kχ = iξ χ j 1 + 2πi n k H j 1 η j 1,k. 3.2 ii We now proceed by induction on d to prove the claims. For the case d = 2, we take χ T \{1} and note that point i implies D 2,χ := s-lim U 2,χ l [H 1,U 2,χ ]U 2,χ 1 U 2,χ l N N = s-lim g 2,χ T 1 l N N = iξ χ 1 + 2πi k χ n k s-lim H 1 η 1,k T 1 l. N N

7 Furstenberg Transformations on Infinite Tori 49 Since T 1 is ergodic and H 1 η 1,k L T, it follows by Birkhoff s pointwise ergodic theorem and Lebesgue dominated convergence theorem that k χ D 2,χ =iξ χ 1 +2πi n k k χ dμ 1 H 1 η 1,k =iξ χ T 1 +2πi n k dμ 1 1,H1 η 1,k T =iξ χ H 1 1. Now, since the character T x j 1 χ ξ 1 0,...,0 + x j 1 S 1 is nontrivial and the subgroup {y t } t R ergodic, we have ξ χ 1 = 0see[6, Thm ]. Thus, D 2,χ = 0, and we deduce from Theorem 2.2a that U 2,χ is strongly mixing. Since this is true for each χ T \{1}, and since U 2,χ is unitarily equivalent to W 2 H2,χ, we infer that W 2 is strongly mixing in χ T \{1} H 2,χ = H 2 H 1. To show that T 2 is uniquely ergodic with respect to μ 2, we take an eigenvector of W 2 with eigenvalue 1, that is, a vector ϕ H 2 such that W 2 ϕ = ϕ.sincew 2 is strongly mixing in H 2 H 1, W 2 has purely continuous spectrum in H 2 H 1 see Theorem 2.2b, and thus ϕ = η 1forsomeη H 1.So, W 2 ϕ = ϕ W 2 η 1 = η 1 W 1 η = η, and thus η is an eigenvector of W 1 with eigenvalue 1. It follows that η is constant μ 1 -almost everywhere due to the ergodicity of T 1. Therefore, ϕ is constant μ 2 -almost everywhere, and T 2 is ergodic. This implies that T 2 is uniquely ergodic because ergodicity implies unique ergodicity for transformations such as T 2 see [6, Thm.4.2.1]. Now, assume the claims are true for d 1 1. Then, W d 1 is strongly mixing in H d 1 H 1. Furthermore, a calculation as in the case d = 2showsthatW d is strongly mixing in χ T \{1} H d,χ = H d H d 1. This implies that W d is strongly mixing in Hd 1 H 1 Hd H d 1 = Hd H 1. This, together with the unique ergodicity of T d 1, allows us to show that T d is uniquely ergodic as in the case d = 2. We know from the proof of Theorem 3.3 that if Assumption 3.2 is satisfied, then U j,χ C 1 H j 1 with [H j 1,U j,χ ]=g j,χ U j,χ. So, it follows from [7, Sec. 4] that the operator A N j,χ η := 1 U j,χ l H j 1 U j,χ l η, N N,η D A N j,χ := DHj 1, N is self-adjoint, and that U j,χ C 1 A N j,χ with [ A N j,χ,u ] j,χ = g N j,χ U j,χ and g N j,χ := 1 g j,χ T j 1 l. N Theorem 3.4 Countable Lebesgue spectrum Suppose that Assumption 3.2 is satisfied, and assume for each j {2,...,d} and k N that H j 1 η j 1,k C T j 1 and that 0 t H j 1 η j 1,k F j 1,t H j 1 η j 1,k L T j 1 <. 3.3 Then, W d has countable Lebesgue spectrum in H d H 1.

8 50 P.A. Cecchi, R. Tiedra de Aldecoa Proof Take j {2,...,d} and χ T \{1}. Then, we know from Lemma 3.1 that U j,χ C 1 H j 1 with [H j 1,U j,χ ]=g j,χ U j,χ. Furthermore, Eq. 3.2 and Eq. 3.3 imply that 0 t = 0 t k χ 2π n k <. e ith j 1 g j,χ e ith j 1 g j,χ BHj 1 gj,χ F j 1,t g j,χ L T j 1 0 t Hj 1 η j 1,k F j 1,t H j 1 η j 1,k L T j 1 So, we obtain that U j,χ C 1+0 H j 1 with [H j 1,U j,χ ] = g j,χ U j,χ, and thus deduce from [7, Sec. 4] that U j,χ C 1+0 A N [ j,χ with A N j,χ,u ] j,χ = g N j,χ U j,χ. Now, H j 1 η j 1,k C T j 1 and T j 1 is uniquely ergodic with respect to μ j 1 due to Theorem 3.3. Thus, we infer from Eq. 3.2 that lim N gn j,χ = iξχ j 1 + 2πi kχ kχ = iξ χ j 1 + 2πi n k = iξ χ j 1 n k H j 1 η j 1,k T j 1 l N T j 1 dμ j 1 1,Hj 1 η j 1,k H j 1 uniformly on T j 1.Sinceξ χ j 1 = 0 by the proof of Theorem 3.3, one has g N j,χ > 0if N is large enough. So, N g j,χ a with a := infx T g N j 1 j,χ x > 0, and Uj,χ satisfies the following strict Mourre estimate on S 1 : U j,χ [ sgn g N j,χ A N j,χ,u ] j,χ = Uj,χ N g j,χ Uj,χ a. Therefore, all the assumptions of Theorem 2.1 are satisfied, and thus U j,χ has purely absolutely continuous spectrum. Since this occurs for each j {2,...,d} and χ T \{1}, and since U j,χ is unitarily equivalent to W d Hj,χ, one infers that W d has purely absolutely continuous spectrum in j {2,...,d},χ T \{1} H j,χ = H d H 1. Finally, the fact that W d has has countable Lebesgue spectrum in H d H 1 can be shown in a similar way as in [10, Lemmas 3 & 4]. The result of Theorem 3.4 is an extension to the case of infinite-dimensional tori of results previously obtained in the case of one-dimensional tori see [10, Cor.3]or [15, Thm.5.3]. Acknowledgments motivated this note. The authors thank M. Măntoiu and J. Rivera-Letelier for discussions which partly

9 Furstenberg Transformations on Infinite Tori 51 References 1. Amrein, W.O.: Hilbert Space Methods in Quantum Mechanics. Fundamental Sciences. EPFL Press, Lausanne Amrein, W.O., de Monvel Boutet, A., Georgescu, V.: C 0 -Groups, Commutator Methods and Spectral Theory of N-Body Hamiltonians, Volume 135 of Progress in Mathematics. Basel, Birkhäuser Verlag Bendikov, A., Saloff-Coste, L.: On the sample paths of Brownian motions on compact infinite dimensional groups. Ann. Probab. 313, Bogachev, V.I.: Differentiable measures and the Malliavin calculus. J. Math. Sci. New York 874, Analysis, 9 5. Bruhat, F.: Distributions sur un groupe localement compact et applications àl étude des représentations des groupes -adiques. Bull. Soc. Math. France 89, Cornfeld, I.P., Fomin, S.V., Sinaĭ, Y.G.: Ergodic Theory, Volume 245 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, New York Translated from the Russian by A. B. Sosinskiĭ 7. Fernández, C., Richard, S., de Aldecoa Tiedra, R.: Commutator methods for unitary operators. J. Spectr. Theory 33, Furstenberg, H.: Strict ergodicity and transformation of the torus. Amer. J. Math. 83, Greschonig, G.: Nilpotent extensions of Furstenberg transformations. Israel J. Math. 183, Iwanik, A., Lemańczyk, M., Rudolph, D.: Absolutely continuous cocycles over irrational rotations. Israel J. Math , Ji, R.: On the crossed product C*-algebras associated with furstenberg transformations on tori. ProQuest LLC, Ann Arbor, MI. Thesis Ph.D. State University of New York at Stony Brook Osaka, H., Phillips, N.C.: Furstenberg transformations on irrational rotation algebras. Ergodic Theory Dynam. Systems 265, Reihani, K.: K-theory of Furstenberg transformation group C -algebras. Canad. J. Math. 656, Tiedra de Aldecoa, R.: Commutator criteria for strong mixing. to appear in Ergodic Theory Dynam. Systems, preprint on Tiedra de Aldecoa, R.: Commutator methods for the spectral analysis of uniquely ergodic dynamical systems. Ergodic Theory Dynam. Syst. 353,