Furstenberg transformations on Cartesian products of infinite-dimensional tori

0.13 MB·English

by  P. A. Cecchi

Furstenberg transformations on Cartesian products of
infinite-dimensional tori
5
1 P. A. Cecchi1 and R. Tiedra de Aldecoa2∗
0
2

n
a 1 Departamento de Matem´aticas, Universidad de Santiago de Chile,
J
Av. Alameda Libertador Bernardo O’Higgins 3363, Estaci´on Central, Santiago, Chile

6
2 2 Facultad de Matem´aticas, Pontificia Universidad Cat´olica de Chile,
Av. Vicu˜na Mackenna 4860, Santiago, Chile

]
S E-mails: [email protected],[email protected]
D
Abstract
.
h
WeconsiderinthisnoteFurstenbergtransformationsonCartesianproductsofinfinite-dimensional
t
a tori. Under some appropriateassumptions, we show that these transformations are uniquely ergodic
m
withrespecttotheHaarmeasureandhavecountableLebesguespectruminasuitablesubspace.These
[ resultsgeneralisetotheinfinite-dimensionalsettingpreviousresultsofH.Furstenberg,A.Iwanik,M.

Lema´nzyk,D.Rudolphandthesecondauthorintheone-dimensionalsetting.Ourproofsrelyon the
1
use of commutator methods for unitary operators and Bruhat functions on the infinite-dimensional
v
5 torus.
4
2010 Mathematics Subject Classification: 28D10, 37A30, 37C40, 58J51.
2
6 Keywords: Furstenberg transformations, infinite-dimensional torus, commutator methods.
0
.
1
1 Introduction
0
5
1 We consider in this note the generalisation of Furstenberg transformations on Cartesian products Td
:
v (d ≥2)ofone-dimensionaltori[8,Sec.2]tothecaseofCartesianproducts(T∞)d ofinfinite-dimensional
Xi tori. Using recent results on commutator methods for unitary operators [14, 15], we show under a C1
regularity assumption on the perturbations that these transformations are uniquely ergodic with respect
r
a 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 (C1 + 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. Lema´nzyk, D. Rudolph
and the second author [8, 10, 15] in the one-dimensional setting.
Apartfromcommutatormethods,ourproofsrelyontheuseofBruhattestfunctions[5]whichprovide
a natural analog to the usual C∞-functions which do not exist in our infinite-dimensional setting (see
Section3fordetails).WealsomentionthatalltheresultsofthisnoteapplytoFurstenbergtransformations
onCartesianproducts(Tn)d offinite-dimensionaltoriTn ofanydimensionn ≥1.Onejusthastoconsider
the particular case where the functions defining the Furstenberg transformations on (T∞)d depend only
on a finite number of variables.
∗SupportedbytheChileanFondecytGrant1130168andbytheIniciativaCientificaMilenioICMRC120002“Mathematical
Physics” fromtheChileanMinistryofEconomy.
1
Hereisabriefdescriptionofthecontentofthenote.First,werecallinSection2theneededfactson
commutators of unitary operators and regularity classes associated to 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 propertyand countable Lebesgue spectrum of these transformations.
We refer to [9, 11, 12, 13] and references therein for other works building on Furstenberg transfor-
mations.
Acknowledgements. The authors thank M. M˘antoiu and J. Rivera-Letelier for discussions which partly
motivated this note.
2 Commutator methods for unitary operators
We recallin this section some facts borrowedfrom[7, 14] on commutator methods forunitary operators
and regularity classes associated to them.
Let H be a Hilbertspace with scalar producth·,·i antilinearin the first argument,denote by B(H)
thesetofboundedlinearoperatorsonH,andwritek·kbothforthenormonHandthenormonB(H).
Let A be a self-adjoint operator in H with domain D(A), and take S ∈ B(H). For any k ∈ N, we say
that S belongs to Ck(A), with notation S ∈Ck(A), if the map
R∋t 7→e−itASeitA ∈B(H) (2.1)
is strongly of class Ck. In the case k =1, one has S ∈C1(A) if and only if the quadratic form
D(A)∋ϕ7→ ϕ,SAϕ − Aϕ,Sϕ ∈C
(cid:10) (cid:11) (cid:10) (cid:11)
iscontinuousforthetopologyinducedbyHonD(A).Wedenoteby[S,A]theboundedoperatorassociated
with the continuousextension of this form,or equivalently −i times the strong derivative of the function
(2.1) at t =0.
A condition slightly stronger than the inclusion S ∈ C1(A) is provided by the following definition:
S belongs to C1+0(A), with notation S ∈ C1+0(A), if S ∈ C1(A) and if [A,S] satisfies the Dini-type
condition
1 dt
e−itA[A,S]eitA−[A,S] <∞.
t
Z0
(cid:13) (cid:13)
As banachisable topological vector spac(cid:13)es, the sets C2(A), C1+0(cid:13)(A), C1(A) and C0(A) ≡ B(H) satisfy
the continuous inclusions [2, Sec. 5.2.4]
C2(A)⊂C1+0(A)⊂C1(A)⊂C0(A).
Now, let U ∈ C1(A) be a unitary operator with (complex) spectral measure EU(·) and spectrum
σ(U) ⊂ S1 := {z ∈ C | |z| = 1}. If there exist a Borel set Θ ⊂ S1, a number a > 0 and a compact
operator K ∈B(H) such that
EU(Θ)U∗[A,U]EU(Θ)≥aEU(Θ)+K, (2.2)
then one says that U satisfies a Mourre estimate on Θ and that A is a conjugate operator for U on
Θ. Also, one says that U satisfies a strict Mourre estimate on Θ if (2.2) holds with K = 0. One of
the consequences of a Mourre estimate is to imply spectral properties for U on Θ. We recall here these
spectral results in the case U ∈ C1+0(A). We also recall a result on the strong mixing property of U in
the case U ∈C1(A) (see [7, Thm. 2.7 & Rem. 2.8] and [14, Thm. 3.1] for more details).
2
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 ∈ C1+0(A). Suppose there exist an open set Θ ⊂ S1, a number
a>0 and a compact operator K ∈B(H) such that
EU(Θ)U∗[A,U]EU(Θ)≥aEU(Θ)+K. (2.3)
Then, U has at most finitelymany eigenvaluesin Θ, each one of finite multiplicity,and U has no singular
continuous spectrum in Θ. Furthermore, if (2.3) holds with K = 0, then U has only purely absolutely
continuous spectrum in Θ (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 ∈C1(A). Assume that the strong limit
N−1
1
D :=s-lim Uℓ [A,U]U−1 U−ℓ
N→∞ N
ℓ=0
X (cid:0) (cid:1)
exists, and suppose that η(D)D(A)⊂D(A) for each η ∈C∞(R\{0}). Then,
c
(a) lim ϕ,UNψ =0 for each ϕ∈ker(D)⊥ and ψ ∈H,
N→∞
(b) U| (cid:10)has pure(cid:11)ly continuous spectrum.
ker(D)⊥
3 Furstenberg transformations on Cartesian products of infinite-
dimensional tori
Let T∞ ≃ R∞/Z∞ be the infinite-dimensional torus with elements x ≡ {x }∞ , and let T∞ be the
k k=1
dual group of T∞. For each n ∈ N , let µ be the normalised Haar measure on (T∞)n, and let H :=
≥1 n n
L2 (T∞)n,µ bethecorrespondingHilbertspace.InanalogytoFurstenbergtransformationsondCartesian
n
productsofone-dimensionaltori(see[8,Sec.2.3]),weconsiderFurstenbergtransformationsonCartesian
(cid:0) (cid:1)
products of infinite-dimensional tori T :(T∞)d →(T∞)d, d ∈N , given by
d ≥2
T (x ,x ,...,x ):= x +α,x +φ (x ),...,x +φ (x ,x ,...,x ) ,
d 1 2 d 1 2 1 1 d d−1 1 2 d−1
with α ∈ T∞ such that {nα}n∈Z is(cid:0)dense in T∞ and φj ∈ C (T∞)j;T∞ for each j ∈(cid:1){1,...,d −1}.
Since T is invertible and preserves the measure µ , the Koopman operator
d d
(cid:0) (cid:1)
W :H →H , ϕ7→ϕ◦T ,
d d d d
is a unitary operator. Furthermore, W is reduced by the orthogonal decompositions
d
H =H H ⊖H =H H , H := η⊗χ|η ∈H ,
d 1 j j−1 1 j,χ j,χ j−1
j∈{M2,...,d}(cid:0) (cid:1) j∈{2,...,dM},χ∈Tc∞\{1} (cid:8) (cid:9)
and the restriction W | is unitarily equivalent to the unitary operator
d Hj,χ
U η := χ◦φ W η, η ∈H .
j,χ j−1 j−1 j−1
In order to define later a conjugate o(cid:0)perator fo(cid:1)r U , we first define a suitable group of translations
j,χ
on(T∞)j−1.Forthis,wechoose{yt}t∈R anergodiccontinuousone-parametersubgroupofT∞ suchthat
each map
R∋t 7→ (y ) ,...,(y ) ∈Tn, n ∈N , (3.1)
t 1 t n ≥1
is of class C1. An example of such a su(cid:0)bgroup {yt}t∈R i(cid:1)s for instance given by the formula
(y ) =ykt mod Z, t ∈R, k ∈N ,
t k ≥1
3
with y ∈ R a transcendental number (see [6, Ex. 4.1.1]). Then, we associate to {yt}t∈R the translation
flow
F (x ,...,x ):=(x ,...,x +y ), t ∈R, (x ,...,x )∈(T∞)j−1,
j−1,t 1 j−1 1 j−1 t 1 j−1
and the translation operators V :H →H given by V η := η◦F . Due to the continuity
j−1,t j−1 j−1 j−1,t j−1,t
of the map R ∋ t 7→ yt ∈ T∞ and of the group operation, the family {Vj−1,t}t∈R defines a strongly
continuous unitary group in H with self-adjoint generator
j−1
H η :=s-limit−1 V −1 η, η ∈D(H ):= η ∈H | lim|t|−1 V −1 η <∞ .
j−1 j−1,t j−1 j−1 j−1,t
t→0 t→0
(cid:0) (cid:1) n (cid:13)(cid:0) (cid:1) (cid:13) o
When dealing with differential operatorson compact manifolds,one typically d(cid:13)oes the calcul(cid:13)ationson an
appropriatecoreofthe operatorssuchasthesetofC∞-functions.Buthere,the arenosuchfunctionson
(T∞)j−1, since T∞ 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 of Bruhat test functions
j−1
on (T∞)j−1, whose definition is the following(see [3, Sec. 2.2] or [5] for details). Set T0 :={0},and for
each n ∈N let π :(T∞)j−1 →(Tn)j−1 be the projection given by
n
(0,...,0) if n =0
π (x ,...,x ):=
n 1 j−1
( (x1)1,...,(x1)n,...,(xj−1)1,...,(xj−1)n if n ≥1.
(cid:0) (cid:1)
Then,
B := ζ◦π |ζ ∈C∞ (Tn)j−1 ,
j−1 n
n∈N
[ (cid:8) (cid:0) (cid:1)(cid:9)
thatis,B isthesetofallfunctionson(T∞)j−1thatareobtainedbyliftingto(T∞)j−1anyC∞-function
j−1
ononeoftheLiegroups(Tn)j−1.ThesetBj−1isdenseinHj−1,itisleftinvariantbythegroup{Vj−1,t}t∈R,
and satisfies the inclusionB ⊂D(H ) (to showthe latter, one hasto use the C1-assumption (3.1)).
j−1 j−1
Therefore,it follows from Nelson’s theorem [1, Prop. 5.3] that H is essentially self-adjoint on B .
j−1 j−1
Now, if the (multiplicationoperatorvalued) map R∋t 7→χ◦φ ◦F ∈B(H ) is strongly of
j−1 j−1,t j−1
class C1, then χ◦φ ∈C1(H ) since
j−1 j−1
χ◦φj−1◦Fj−1,t =Vj−1,t χ◦φj−1 Vj−1,−t =e−itHj−1 χ◦φj−1 eitHj−1.
(cid:0) (cid:1) (cid:0) (cid:1)
Thus, the operator
d
g := H ,(χ◦φ χ◦φ ≡s- iχ◦φ ◦F χ◦φ
j,χ j−1 j−1 j−1 dt j−1 j−1,t j−1 t=0
(cid:2) (cid:1)(cid:3)(cid:0) (cid:1)≡s-ddt (cid:0)iχ◦ φj−1◦Fj−1,t(cid:1)−(cid:0) φj−1 t=(cid:1)0(cid:12)(cid:12)(cid:12)
(cid:12)
(cid:0) (cid:1)(cid:12)
is a bounded self-adjoint multiplication operator in Hj−1. (cid:12)
Lemma3.1. Takej ∈{2,...,d}andχ∈T∞\{1},andassumethatthemapR∋t 7→χ◦φ ◦F ∈
j−1 j−1,t
B(H ) is strongly of class C1. Then, U ∈C1(H ) with [H ,U ]=g U .
j−1 j,χ j−1 j−1 j,χ j,χ j,χ
d
Proof. Take η ∈ B . Then, the commutation of V and W and the differentiability assumption
j−1 j−1,t j−1
4
imply that
H η,U η − η,U H η
j−1 j,χ Hj−1 j,χ j−1 Hj−1
(cid:10) d (cid:11) (cid:10) (cid:11)
=i − V η, χ◦φ W η − η, χ◦φ W V η
dt j−1,t j−1 j−1 Hj−1 j−1 j−1 j−1,t Hj−1 t=0
=i ddt nη,(cid:10)χ◦φj−1(cid:0)◦Fj−1,t −(cid:1)χ◦φj−(cid:11)1 Wj−1(cid:10)Vj−(cid:0)1,tη Hj−1 (cid:1)t=0 (cid:11) o(cid:12)(cid:12)(cid:12)
= ddt (cid:10)η, (cid:0)iχ◦φj−1◦Fj−1,t χ◦φj−1(cid:1) −1 Uj,χVj(cid:11)−1,tη(cid:12)(cid:12)(cid:12)Hj−1 t=0
= d (cid:10)η,(cid:8)i(cid:0)χ◦ φ ◦F (cid:1)−(cid:0)φ −(cid:1)1 U(cid:9) V η (cid:11) (cid:12)(cid:12)
dt j−1 j−1,t j−1 j,χ j−1,t Hj−1 t=0(cid:12)
(cid:12)
= η,(cid:10)gj,χ(cid:8)Uj,χη(cid:0)Hj−1, (cid:1) (cid:9) (cid:11) (cid:12)(cid:12)
and thus the cl(cid:10)aim follows f(cid:11)rom the boundedness of g and the density of B in D(H ).
j,χ j−1 j−1
Assumption3.2. Foreachj ∈{2,...,d},themap φ ∈C (T∞)j−1;T∞ satisfies φ =ξ +η ,
j−1 j−1 j−1 j−1
where
(cid:0) (cid:1)
(i) ξ ∈C (T∞)j−1;T∞ is a group homomorphism such that
j−1
(cid:0) (cid:1) T∞ ∋x 7→(χ◦ξ )(0,...,0+x )∈S1
j−1 j−1 j−1
is nontrivial for each χ∈T∞\{1},
(ii) η ∈C (T∞)j−1;T∞ is such that there exists η ∈C (T∞)j−1;R∞ with
j−1 d j−1
η (cid:0)(x ,...,x )(cid:1)= η (x ,...,x ) (mod Z∞) (cid:0)for each (x ,(cid:1)...,x )∈(T∞)j−1,
j−1 1 j−1 j−1 1 j−1 e 1 j−1
and with R∋t 7→η ◦(cid:0)F ∈B(H ) strongly of(cid:1)class C1 for each k ∈N .
j−1,k ej−1,t j−1 ≥1
In the next theorem, we use the fact that the map
e
R∋t 7→ χ◦ξ (0,...,0+y )∈S1
j−1 t
is a character on R, and thus of class C∞.(cid:0)We also(cid:1)use the notation
d
ξ(χ) := χ◦ξ (0,...,0+y ) ∈iR.
j−1 dt j−1 t t=0
(cid:12)
(cid:0) (cid:1) (cid:12)
Theorem 3.3 (Strong mixing and unique ergodicity). Suppose t(cid:12)hat Assumption 3.2 is satisfied. Then,
W is strongly mixing in H ⊖H and T is uniquely ergodic with respect to µ .
d d 1 d 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 in Assumption 3.2wereto dependonlyona finitenumberofvariables,then the strong C1 regularity
j−1
condition on η in Assumption 3.2(ii) would reduce to a uniform Lipschitz condition of η along the
j−1 j−1
flow {Fj−1,t}t∈R, as in the one-dimensional case treated by Furstenberg in [8, Thm. 2.1].
Proof. (i)Take j ∈{2,...,d},χ∈T∞\{1}and t ∈R.Then,there existk ∈N andn ,...,n ∈Z
χ ≥1 1 kχ
such that
dχ(xj−1)=e2πiPkkχ=1nkxj−1,k,
5
withx ∈T∞andx ∈[0,1)thek-thcycliccomponentofx .Therefore,weinferfromAssumption
j−1 j−1,k j−1
3.2 that
χ◦ φ ◦F −φ =χ◦ ξ ◦F −ξ ·χ◦ η ◦F −η
j−1 j−1,t j−1 j−1 j−1,t j−1 j−1 j−1,t j−1
(cid:0) (cid:1)= χ◦(cid:0)ξj−1 (0,...,0+yt)(cid:1)·e2πi(cid:0)Pkkχ=1nk(ηej−1◦Fj−1,t−ηej−1)(cid:1)k,
and Lemma 3.1 implies that U ∈C1(H(cid:0) ) with(cid:1)[H ,U ]=g U and
j,χ j−1 j−1 j,χ j,χ j,χ
d kχ d
g =s- iχ◦(φ ◦F −φ ) =iξ(χ) −2π n s- η ◦F
j,χ dt j−1 j−1,t j−1 t=0 j−1 k dt j−1,k j−1,t t=0
(cid:12) Xk=1 (cid:18) (cid:12) (cid:19)
(cid:12) (cid:12)
(cid:12) kχ e (cid:12)
=iξ(χ) +2πi n (H η ). (3.2)
j−1 k j−1 j−1,k
k=1
X
(ii)Wenowproceedbyinductionond toprovetheclaims.Forthecasede=2,wetakeχ∈T∞\{1}
and note that point (i) implies
d
N−1
1
D :=s-lim (U )ℓ [H ,U ](U )−1 (U )−ℓ
2,χ N→∞ N 2,χ 1 2,χ 2,χ 2,χ
ℓ=0
X (cid:0) (cid:1)
N−1
1
=s-lim g ◦(T )ℓ
N→∞ N 2,χ 1
ℓ=0
X
kχ 1 N−1
=iξ(χ)+2πi n s-lim (H η )◦(T )ℓ .
1 k N→∞ N 1 1,k 1 !
k=1 ℓ=0
X X
SinceT isergodicandH η ∈L∞(T∞),itfollowsbyBirkhoff’spoeintwiseergodictheoremandLebesgue
1 1 1,k
dominated convergence theorem that
keχ kχ
D =iξ(χ)+2πi n dµ (H η )=iξ(χ)+2πi n dµ 1,H η =iξ(χ).
2,χ 1 k=1 kZT∞ 1 1 1,k 1 k=1 kZT∞ 1 1 1,k H1 1
X X (cid:10) (cid:11)
Now, since the character T∞ ∋ x 7→ (χe ◦ξ )(0,...,0+x ) ∈ S1 is nontrivieal and the subgroup
j−1 1 j−1
{yt}t∈R ergodic,we have ξ1(χ) 6=0 (see [6, Thm. 4.1.1’]). Thus, D2,χ 6=0, and we deduce fromTheorem
2.2(a) that U is strongly mixing. Since this is true for each χ ∈ T∞ \{1}, and since U is unitarily
2,χ 2,χ
equivalent to W | , we infer that W is strongly mixing in ⊕ H =H ⊖H .
2 H2,χ 2 χ∈T∞\{1} 2,χ 2 1
ToshowthatT isuniquelyergodicwithrespecttoµ ,wetakceadneigenvectorofW witheigenvalue
2 2 2
1, that is, a vector ϕ∈H such that W ϕ=ϕ. Since W is strongly mixing in H ⊖H , W has purely
2 2 2 2 1 2
continuous spectrum in H ⊖H (see Theorem 2.2(b)), and thus ϕ=η⊗1 for some η ∈H . So,
2 1 1
W ϕ=ϕ ⇐⇒ W (η⊗1)=η⊗1 ⇐⇒ W η =η,
2 2 1
and thus η is an eigenvector of W with eigenvalue1. It followsthat η is constant µ -almost everywhere
1 1
duetotheergodicityofT .Therefore,ϕisconstantµ -almosteverywhere,andT isergodic.Thisimplies
1 2 2
that T is uniquely ergodic because ergodicity implies unique ergodicity for transformations such as T
2 2
(see [6, Thm. 4.2.1]).
Now, assume the claims are true for d − 1 ≥ 1. Then, W is strongly mixing in H ⊖H .
d−1 d−1 1
Furthermore, a calculation as in the case d = 2 shows that W is strongly mixing in ⊕ H =
d χ∈T∞\{1} d,χ
H ⊖H . This implies that W is strongly mixing in c
d d−1 d
H ⊖H ⊕ H ⊖H =H ⊖H .
d−1 1 d d−1 d 1
This, together with the unique(cid:0)ergodicityof(cid:1)T (cid:0) , allowsus(cid:1)to showthat T is uniquelyergodicas in the
d−1 d
case d =2.
6
We know from the proof of Theorem 3.3 that if Assumption 3.2 is satisfied, then U ∈ C1(H )
j,χ j−1
with [H ,U ]=g U . So, it follows from [7, Sec. 4] that the operator
j−1 j,χ j,χ j,χ
N−1
1
A(N)η := (U )ℓH (U )−ℓη, N ∈N , η ∈D A(N) :=D(H ),
j,χ N j,χ j−1 j,χ ≥1 j,χ j−1
ℓ=0
X (cid:0) (cid:1)
is self-adjoint, and that U ∈C1 A(N) with
j,χ j,χ
(cid:0) (cid:1) N−1
1
A(N),U =g(N)U and g(N) := g ◦(T )ℓ.
j,χ j,χ j,χ j,χ j,χ N j,χ j−1
ℓ=0
(cid:2) (cid:3) X
Theorem 3.4 (CountableLebesguespectrum). SupposethatAssumption3.2issatisfied,andassume for
each j ∈{2,...,d} and k ∈N that H η ∈C (T∞)j−1 and that
≥1 j−1 j−1,k
1 dt (cid:0) (cid:1)
(H η )◦eF −(H η ) <∞. (3.3)
t j−1 j−1,k j−1,t j−1 j−1,k L∞((T∞)j−1)
Z0
(cid:13) (cid:13)
Then, Wd has countable Leb(cid:13)esgue sepectrum in Hd ⊖H1. e (cid:13)
Proof. Take j ∈ {2,...,d} and χ ∈ T∞ \{1}. Then, we know from Lemma 3.1 that U ∈ C1(H )
j,χ j−1
with [H ,U ]=g U . Furthermore, (3.2) and (3.3) imply that
j−1 j,χ j,χ j,χ
d
1 dt
Z0 t e−itHj−1gj,χeitHj−1−gj,χ B(Hj−1)
1 d(cid:13)t (cid:13)
(cid:13) (cid:13)
= g ◦F −g
t j,χ j−1,t j,χ L∞((T∞)j−1)
Z0
kχ (cid:13)(cid:13) 1 dt (cid:13)(cid:13)
≤2π |n | (H η )◦F −(H η )
k t j−1 j−1,k j−1,t j−1 j−1,k L∞((T∞)j−1)
k=1 Z0
X (cid:13) (cid:13)
<∞. (cid:13) e e (cid:13)
So, we obtain that U ∈C1+0(H ) with [H ,U ]=g U , and thus deduce from[7, Sec. 4] that
j,χ j−1 j−1 j,χ j,χ j,χ
U ∈ C1+0 A(N) with A(N),U = g(N)U . Now, H η ∈ C (T∞)j−1 and T is uniquely
j,χ j,χ j,χ j,χ j,χ j,χ j−1 j−1,k j−1
ergodic with respect to µ due to Theorem 3.3. Thus, we infer from (3.2) that
(cid:0) (cid:1) j(cid:2)−1 (cid:3) (cid:0) (cid:1)
e
kχ 1 N−1
lim g(N) =iξ(χ) +2πi n (H η )◦(T )ℓ
N→∞ j,χ j−1 k N j−1 j−1,k j−1
k=1 ℓ=0
X X
kχ e
=iξ(χ) +2πi n dµ 1,H η
j−1 k=1 kZ(T∞)j−1 j−1 j−1 j−1,k Hj−1
X (cid:10) (cid:11)
=iξ(χ) e
j−1
uniformly on (T∞)j−1. Since ξ(χ) 6= 0 by the proof of Theorem 3.3, one has g(N) > 0 if N is large
j−1 j,χ
enough. So, gj(,Nχ) ≥a with a:=infx∈(T∞)j−1 gj(,Nχ)(x) >0, and Uj,χ satisfies the(cid:12)follo(cid:12)wing strict Mourre
estimate on S1: (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (U )∗ sgn g(N) A((cid:12)N),U (cid:12)=(U )∗ g(N) U ≥a.
j,χ j,χ j,χ j,χ j,χ j,χ j,χ
Therefore,alltheassumptionsof(cid:2)The(cid:0)orem(cid:1)2.1aresat(cid:3)isfied,andt(cid:12)husU(cid:12)j,χhaspurelyabsolutelycontinuous
(cid:12) (cid:12)
spectrum.Sincethisoccursforeachj ∈{2,...,d}andχ∈T∞\{1},andsinceU isunitarilyequivalent
j,χ
to W | , one infers that W has purely absolutely continuous spectrum in H =
d Hj,χ d j∈{2,...,d},χ∈T∞\{1} j,χ
H ⊖H . Finally, the fact that W has has countable Lebesdgue spectrum in H ⊖H can bce shown in a
d 1 d d 1
L
similar way as in [10, Lemmas 3 & 4].
7
TheresultofTheorem3.4isanextensiontothecaseofinfinite-dimensionaltoriofresultspreviously
obtained in the case of one-dimensional tori (see [10, Cor. 3] or [15, Thm. 5.3]).
References
[1] W. O. Amrein. Hilbert space methods in quantum mechanics. Fundamental Sciences. EPFL Press,
Lausanne, 2009.
[2] W. O. Amrein, A. Boutet de Monvel, and V. Georgescu. C -groups, commutator methods and
0
spectraltheoryofN-bodyHamiltonians,volume135ofProgressinMathematics. Birkh¨auserVerlag,
Basel, 1996.
[3] A. Bendikov and L. Saloff-Coste. On the sample paths of Brownian motions on compact infinite
dimensional groups. Ann. Probab., 31(3):1464–1493,2003.
[4] V. I. Bogachev. Differentiable measures and the Malliavin calculus. J. Math. Sci. (New York),
87(4):3577–3731,1997. Analysis, 9.
[5] F. Bruhat. Distributions sur un groupe localement compact et applications `a l’´etude des
repr´esentationsdes groupes ℘-adiques. Bull. Soc. Math. France, 89:43–75,1961.
[6] I. P. Cornfeld,S. V. Fomin, and Y. G. Sina˘ı. Ergodic theory, volume 245 of Grundlehren der Math-
ematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag,
New York, 1982. Translated from the Russian by A. B. Sosinski˘ı.
[7] C. Fern´andez, S. Richard, and R. Tiedra de Aldecoa. Commutator methods for unitary operators.
J. Spectr. Theory, 3(3):271–292,2013.
[8] H. Furstenberg. Strict ergodicity and transformation of the torus. Amer. J. Math., 83:573–601,
1961.
[9] G. Greschonig. Nilpotent extensions of Furstenberg transformations. Israel J. Math., 183:381–397,
2011.
[10] A.Iwanik,M. Lema´nczyk,and D.Rudolph. Absolutelycontinuouscocyclesoverirrationalrotations.
Israel J. Math., 83(1-2):73–95,1993.
[11] R. Ji. ON THE CROSSED PRODUCT C*-ALGEBRAS ASSOCIATED WITH FURSTENBERG
TRANSFORMATIONS ON TORI. ProQuest LLC, Ann Arbor, MI, 1986. Thesis (Ph.D.)–State
University of New York at Stony Brook.
[12] H. Osaka and N. C. Phillips. Furstenberg transformations on irrational rotation algebras. Ergodic
Theory Dynam. Systems, 26(5):1623–1651,2006.
[13] K. Reihani. K-theory of Furstenberg transformation group C∗-algebras. Canad. J. Math.,
65(6):1287–1319,2013.
[14] R. Tiedra de Aldecoa. Commutator criteriafor strong mixing. to appear in Ergodic TheoryDynam.
Systems, preprint on http://arxiv.org/abs/1406.5777.
[15] R. Tiedra de Aldecoa. Commutator methods for the spectral analysis of uniquely ergodicdynamical
systems. Ergodic Theory and Dynamical Systems, FirstView:1–24, 10 2014.
8