Orbit Representations and Circle Maps

Orbit representations and circle maps

C. Correia Ramos, Nuno Martins and Paulo R. Pinto

To the memory of Jose de Sousa Ramos

Abstract. We yield C∗-algebras representations on the orbit spaces from thefamily of interval maps f(x) = βx + α (mod 1) lifted to circle maps, in whichcase β ∈ N.

Each orbit will encode an unitary equivalence class of an irreduciblerepresentation of: a Cuntz algebra Oβ if α = 0 and β > 1; an irrationalrotation algebra Aβ if α /∈ Q and β = 1; and a Cuntz-Krieger OAα,β wheneverβ > 1 and the critical point is periodic, where Aα,β is the underlying Markovtransition matrix of f .

Mathematics Subject Classification (2000). Primary 46L05; Secondary 37B10.

Keywords. Interval maps, irrational rotation algebra, Cuntz-Krieger algebra,irreducible representations.

1. Introduction

The interplay between operator algebras and discrete dynamical systems has beenvery fruitful since their origins. Von Neumann algebras theory has a strong re-lation with ergodic theory, see for example [20] and references therein or [6, 7].After the work of Cuntz and Krieger [9] and Rieffel [19], where C*-algebras deeplyrelated with Markov chains were found, the interplay and interest between discretedynamical systems and C*-algebras has greatly increased.

Besides interest in its own right, applications of representation theory of, e.g.,Cuntz algebras to wavelets, fractals, dynamical systems, see e.g. [3, 4, 13, 16], andquantum field theory in [1] are particularly remarkable.

In the sequel of [8], we will use symbolic dynamics tools to yield furtherrepresentations of Cuntz, Cuntz-Krieger and rotation algebras from interval maps.We consider the family of maps

f(x) = βx + α (mod 1) (1.1)

2 C.C. Ramos, N. Martins and P.R. Pinto

defined on the unit interval I = [0, 1] with real parameters α and β. The mapf can naturally be lifted to a map Φ(e2πix) = e2πif(x) on S1, and the conditionf(0) = f(1) implies that β ∈ N.

The generalized orbit R(x) =⋃

i∈Z f i(x) underlines again the Hilbert spaceattached to every point x upon we construct a representation. However, in contrastwith [8], we produce not only representations of the Cuntz-Krieger algebra butalso representations of the rotation algebra Aα, depending on the parameters αand β. Again the generalized orbit R(x) will encode an unitary equivalence class ofirreducible representations, see Proposition 3.1 in the Cuntz algebra case, Theorem3.4 for the rotation algebra and Theorem 3.5 in the Cuntz-Krieger case. We remarkthat our techniques are different from those employed in [5], where a countablefamily of irreducible representations of the irrational rotation algebra are yielded.

2. Preliminaries

A representation of a ∗-algebraA on a complex Hilbert spaceH is a ∗-homomorphismπ : A → B(H) into the ∗-algebra B(H) of bounded linear operators on H. Usu-ally representations are studied up to unitary equivalence. Two representationsπ : A → B(H) and π : A → B(H) are (unitarily) equivalent if there is a unitaryoperator U : H →H (i.e., U is a surjective isometry) such that

Uπ(a) = π(a)U, for every a ∈ A.

A representation π : A → B(H) of some ∗-algebra is said to be irreducible ifthere is no non-trivial subspace of H invariant with respect to all operators π(a)with a ∈ A. A well known result, see e.g. [17, Proposition 3.13.2], says that π isirreducible if and only if

x ∈ B(H) : xπ(a) = π(a)x, for all a ∈ A =⇒ x = λI, (2.1)

for some complex number λ, where I denotes the identity of A. By the verydefinition of commutant, (2.1) can be restated as follows: π(A)′ = CI. The repre-sentation is called faithful if it is injective.

2.1. Cuntz-Krieger and Rotation algebras

In the sequel we yield ∗-representations of two classes of C∗-algebras. The first oneis that of Cuntz-Krieger algebras. Let A = (aij) be an n × n matrix with entriesequal to 0 or 1 such that each row and column has at least one non-zero entry.The Cuntz-Krieger algebra OA associated to the matrix A is the C∗-algebra [9]generated by (non-zero) partial isometries s1, ..., sn satisfying:

s∗i si =n∑

j=1

aijsjs∗j (i = 1, ..., n),

n∑

i=1

sis∗i = I, (2.2)

where I denotes the identity. Recall that si is a partial isometry if and only ifsis

∗i si = si. The reader acquainted with the work of Cuntz and Krieger will notice

Orbit representations and circle maps 3

that s∗i sj = 0 for i 6= j and sisj = aijsisj . Moreover, the range projectionspi = sis

∗i and support projections qi = s∗i si obey the following:

pipj = δijpi, qiqj = qjqi and qisj = aijsj .

It is well known that the algebra OA is uniquely determined by the relations in(2.2), see [9, Theorem 2.13]. As a special case, we obtain the Cuntz algebra On

when A is full: aij = 1 for all i and j.We now consider the second family of C*-algebras. Given θ ∈ R, as in [19],

the rotation algebra Aθ is the universal C∗-algebra generated by two unitaries uand v obeying

uv = e2πiθvu. (2.3)

The C∗-algebra Aθ is simple if and only if θ is irrational. It is clear thatAθ

∼= An±θ, n ∈ Z. Conversely, if Aθ∼= Aθ′ , then θ = n± θ′ for some n ∈ Z (see

[12] for the case θ rational and [18] for the case θ irrational). We also note thatfor any angle θ and k ∈ Z, we have uvk = exp(2πikθ)vku, and hence Akθ ⊆ Aθ.

If θ = pq is rational with gcd(p, q) = 1, then the irreducible representations of

Aθ are all finite dimensional. Moreover, the dimension is q and the representationson Cq are given by u → zU , v → wV , where V en = en+1, V eq−1 = e0, Uen =ωnen, with ω = e2πip/q, enq−1

n=0 is the canonical orthonormal basis of Cq and z, ware two complex numbers of absolute value 1.

If θ is irrational, the representation on l2(Z) given by V en = en+1 andUen = e2πiθen (where en is an orthonormal basis in l2(Z)) is irreducible, butclassifying all irreducible representations seems to be an impossible task, as for theCuntz and Cuntz-Krieger algebras, see [3, 4, 8] where various families of irreduciblerepresentations are exhibited.

2.2. Piecewise monotone interval maps

Let f : I → I be a piecewise monotone map of the interval I into itself, that is,there is a minimal partition of open sub-intervals of I, I = I1, .., Im such that∪m

j=1Ij = I and f|Ijis continuous monotone, for every j = 1, ..., m. We define

fj := f|Ij. The inverse branches are denoted by f−1

j : f(Ij) → Ij . Let χIi be thecharacteristic function on the interval Ii. The following are naturally satisfied

f f−1i (x) = χf(Ii)(x) x, f−1

i f|Ii(x) = χIi(x) x.

Let 1, 2, · · · ..., n be the alphabet associated to some partition P = I1, ..., Inof open sub-intervals of I so that ∪n

j=1Ij = I, not necessarily I. The address map,is defined by

ad :n⋃

j=1

Ij → 1, 2, ..., n, ad(x) = i if x ∈ Ii.

We defineΩf := x ∈ I : fk(x) ∈ ∪m

j=1Ij for all k = 0, 1, ....


Note that Ωf = I. The itinerary map it : Ωf → 1, 2, ..., nN is defined by

it(x) = ad(x))ad(f(x))ad(f2(x))...

and let Σf = it(Ωf ). The space Σf is invariant under the shift map σ : 1, 2, ..., nN →1, 2, ..., nN defined by

σ(i1i2...) = (i2i3...),

and we have it f = σ it. We will use σ meaning in fact σ|Σf. A sequence in

1, 2, ..., nN is called admissible, with respect to f , if it occurs as an itineraryfor some point x in I, that is, if it belongs to Σf . An admissible word is a finitesub-sequence of some admissible sequence. The set of admissible words of size k isdenoted by Wk = Wk(f). Given i1...ik ∈ Wk, we define Ii1...ik

as the set of pointsx in Ωf which satisfy

ad(x) = i1, ..., ad(fk(x)) = ik.

As in [8], we consider the equivalence relation on the set Ωf ,

Rf = (x, y) : fn(x) = fm(y) for some n,m ∈ N0. (2.4)

We write x ∼ y whenever (x, y) ∈ Rf . Consider the equivalence class Rf (x) and setHx the Hilbert space l2(Rf (x)) with canonical orthonormal basis |y〉 : y ∈ Rf (x),in Dirac notation. Note that Hx = Hy (are the same Hilbert spaces) wheneverx ∼ y. The inner product (·, ·) is given by

〈y|x〉 = (|y〉 , |z〉) = δy,z.

For each i = 1, ..., n, let us define an operator Si on Hx, with respect to somepartition P = I1, ..., In of I, as follows:

Si |y〉 = χf(Ii)(y)∣∣f−1

i (y)⟩.

Note that χf(Ii)(x) = 1 if and only if there is a pre-image of x in Ii. We haveS∗i |y〉 = χIi(y) |f(y)〉 . In fact

〈y|Si |z〉〉 =⟨y|f−1

i (z)⟩

= δy,f−1i (z).

On the other hand we have

〈y|S∗i |z〉〉 = χIi(y) 〈f(y)|z〉 = χIi(y)δf(y),z.

Since δy,f−1i (z) = χIi(y)δf(y),z we have shown that the operators Si, S∗i are adjoint

of each other. We further remark that Si is a partial isometry: namely, Si is anisometry on its restriction to span|y〉 : y ∈ f(Ii) ∩Hx and vanishes in theremaining part of Hx.

For µ = µ1...µk ∈ Wk we define Sµ = Sµ1 ...Sµk. Also S∗µ = S∗µk

...S∗µ1. Note

that SµS∗µ |y〉 = χIµ(y).


3. Linear mod 1 interval maps, circle maps and representations

Now, let us consider the family of maps

f(x) = βx + α (mod 1) (3.1)

such that f(0) = f(1), with β ≥ 1 and α ∈ [0, 1[. This family satisfies the con-ditions given in the preliminaries. There is a circle homeomorphism induced byf , as follows: Let φ : [0, 1[→ S1 ⊂ C with φ(x) = e2πix. We set Φ : S1 → S1

with Φ(e2πix) = e2πi ef(x), where f is any extension of f to the interval I, obtainedby choosing either f(cj) = 0 or f(cj) = 1, for each discontinuity point cj . Thecondition f(0) = f(1) implies that β = n ∈ N and the continuity of the map Φ.

The behavior of the dynamical system (I, f) (and consequently of both(Ωf , f) and (S1,Φ)) is characterized by the sequences it(f(c+

j )) and it(f(c−j )), foreach discontinuity point cj , see [14]. Therefore, it suffices to consider the itineraryof 0, since it(f(0)) = it(f(1)), and f(c+

j ) = 1, f(c−j ) = 0. Let ωβ,α = f j(0)j∈N0

be the orbit of 0. Let us consider the partition I = I0, ..., In−1, with

I0 = ]0, (1− α)/β[ , ..., Ij = ](j − α)/β, (j + 1− α)/β[ , ...,

... In−1 = ](n− 1− α)/β, 1[ , (3.2)

which is the minimal partition of monotonicity for f . The set 0, 1, ..., n− 1 willbe the alphabet. Set ξβ,α = (ξi)i∈N = itf (0) and consider the σ-invariant compactsubset Σβ,α = it(I) of 0, 1, ...n − 1N. Depending on the parameters α, β theorbit of 0 can be finite, in which case we obtain a Markov partition, see [14],which is a refinement of the above partition I. We denote the Markov partitionby M =J1, ..., Jm. A characterization of the values of α, for which there is aMarkov partition, is partially given by the following:

Proposition 3.1. If the orbit ωn,α is finite then α ∈ Q. In particular

α =ξl + ξl−1n + ξl−2n

2 + ... + ξ1nl−1

1 + n + n2 + ... + nl−1, with ξβ,α = (ξi)i∈N = itf (0).

Proof. See [14] for full details. ¤

3.1. Case n > 1, α = 0In this situation we have f(Ij) = [0, 1], for every interval Ij , with j = 0, ..., n− 1,in the partition in Eq. (3.2). See Fig. 1 for an example of a graph of f . Therefore,the operators Si, on Hx, for each i = 1, ..., n, take the particular form:

Si |y〉 =∣∣f−1

i (y)⟩.

We have: S∗i |y〉 = χIi(y) |f(y)〉 .These operators Si are partial isometries and besides they satisfy the relations∑n

i=1 SiS∗i = 1 and S∗i Si = 1, for i = 1, ..., n. In fact, consider SiS

∗i acting on a

vector |y〉 of the canonical basis, and let k ∈ 1, ..., n such that y ∈ Ik. Then wehave

SiS∗i |y〉 = χIi(y)Si |f(y)〉 = χIi(y)

∣∣f−1i f(y)

⟩= χIi(y) |y〉 .


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Figure 1. Graph of f with α = 0 and n = 2

Thus∑n

i=1 SiS∗i |y〉 =

∑ni=1 χIi

(y) |y〉 = χIk(y) |y〉 = |y〉. Now, consider S∗i Si

acting on a vector |y〉 of the canonical basis. We have

S∗i Si |y〉 = S∗i∣∣f−1

i (y)⟩

= χIi(f−1i (y))

∣∣f f−1i (y)

⟩= |y〉 .

We can now state the main result of this section.

Proposition 3.2. The map πx : On → B(Hx), given by πx(si) = Si is a faithfulirreducible representation of the Cuntz algebra On, where Hx = l2(Rf (x)). More-over, two such representations πx and πy are unitarily equivalent if and only ifRf (x) = Rf (y).

Proof. The fact that πx is a representation of the Cuntz algebra On follows fromthe computation above. The remaining statements follow from [8, Theorem 7]. ¤

3.2. Case n = 1Let n = 1 and assume α is an irrational number in ]0, 1[. Then the orbit of 0 isinfinite. We consider P = I0, I1, with

I0 =]0, 1− α[, I1 =]1− α, 1[.

Thus, every orbit will be codified in the alphabet 0, 1, via the itinerary map.The map f is invertible and we set f0(x) = x + α, and f1(x) = x− 1 + α. See Fig.2 for an example of a graph of f .

Let S0 and S1 be the operators on Hx defined by Si |y〉 = χf(Ii)(y)∣∣f−1

i (y)⟩,

with i = 0, 1. We obviously have S∗i |y〉 = χIi(y) |f(y)〉 , with i = 0, 1. Let V =Vx = S∗0 + S∗1 . This operator acts on the vector basis |y〉 as

V |y〉 = |f(y)〉 . (3.3)

Since f is invertible, the operator V is unitary with V ∗ |y〉 =∣∣f−1(y)

⟩.


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Figure 2. Graph of f with α = 1−√2 and n = 1

Let U = Ux be the diagonal operator (multiplication operator)

U |y〉 = φ(y) |y〉 = e2πiy |y〉 , (3.4)

which is also an unitary operator, with U∗ |y〉 = φ(y) |y〉 = e−2πiy |y〉 . The vonNeumann algebra generated by S0 and S1 is the same algebra as the algebragenerated by p0 = S0S

∗0 , p1 = S1S

∗1 and V , since Si = piV

∗, with i = 0, 1.Moreover, we obtain the following result.

Proposition 3.3. The operator U belongs to the von Neumann algebra generated bythe operators S0 and S1.

Proof. For each µ ∈ Wk, let m(µ) be some point in Iµ ∩ Rf (x). Note that if wehave it(y) = (αj)∞j=1, for some point y ∈ Rf (x) then limj→∞m(α1...αj) = y, andthe limit is independent on the choice of m(α1...αj) ∈ Iα1...αj , since for each j ∈ Nthere is r > j so that Iα1...αj ⊂ Iα1...αr . Let Mk =

∑µ∈Wk

e2πim(µ)SµS∗µ. We cansee that limk→∞Mk = U in the strong topology, since limk→∞ ‖Mkv − Uv‖ = 0,for every v ∈ Hx. Therefore, U is in the von Neumann algebra generated by theoperators S0, S1. ¤

Now, since f is invertible, the equivalence class Rf (x) is given by a singleorbit (the orbit of x), that is, Rf (x) = f j(x) : j ∈ Z, and Hx = l2(Z).

Theorem 3.4. Let x ∈ I. The map ρx : Aθ → B(Hx), given by ρx(u) = Ux andρx(v) = Vx is a faithful representation of the irrational rotation algebra Aθ =C∗(u, v : uv = e2πiθvu), with θ = α. Moreover, two such representations ρx andρy are unitarily equivalent if and only if Rf (x) = Rf (y).

Proof. The definition of U = Ux and V = Vx gives UV |y〉 = e2πi(α+y)|f(y)〉 andV U |y〉 = e2πiy|f(y)〉. Therefore UV = e2πiαV U and ρx defines a (necessarily


faithful) representation of the irrational rotation algebra Aα. Moreover, ρx is irre-ducible. Indeed, let T ∈ B(Hx) commuting with both U and V . Since U |f j(x)〉 =e2πi(x+jα)|f j(x)〉, j ∈ Z, f j(x)j∈Z is an orthonormal basis in Hx and the eigen-values e2πi(x+jα) are all distincts, it follows that T |f j(x)〉 = λj |f j(x)〉, j ∈ Z. ButV |f j(x)〉 = |f j+1(x)〉, so V T = TV gives λj = λj+1, j ∈ Z. Therefore T is ascalar and thus ρx is irreducible.

It is clear that Rf (x) = Rf (y), then ρx and ρy are unitarily equivalent. Noticethat Ux and Uy have the same eigenvalues if and only if x ∼ y. Hence ρx and ρy

can be unitarily equivalente only when x ∼ y. ¤

3.3. Case n > 1, α ∈]0, 1[We have two distinct situations to consider: the orbit of 0, ωn,α, is either finite orinfinite. We will only treat the finite case, i.e. we will assume that n and α ∈]0, 1[are such that ωn,α is a finite set. See Fig. 3 for an example of a graph of f . Thishappens when 0 is a periodic point or a pre-image of a periodic point. In both caseswe obtain a Markov partition M = J1, ..., Jm and a transition matrix denotedby Aβ,α, see [14]. For each i = 1, ...,m, let us consider the operator Si on Hx, with

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Figure 3. Graph of f with α = 1−√2 and n = 2

respect to the Markov partition M:

Si |y〉 = χf(Ji)(y)∣∣f−1

i (y)⟩.

Note that χf(Ji)(x) = 1 if and only if there is a pre-image of x in Ji (also note thatthe operators Si are defined, in this case, with respect to the Markov partitionand not with the partition I). We have S∗i |y〉 = χJi(y) |f(y)〉 .


The operators Si satisfy the relationsm∑

i=1

SiS∗i = 1, S∗i Si =

m∑

j=1

aijSjS∗j , for i = 1, ..., m,

where Aβ,α = (aij) is the transition matrix defined above. In fact, consider SiS∗i

acting on a vector |y〉 of the canonical basis, and let k ∈ 1, ..., m such thaty ∈ Jk. Then we have

SiS∗i |y〉 = χJi(y)Si |f(y)〉 = χJi(y)χf(Ji)(f(y))

∣∣f−1i f(y)

⟩= χJi(y) |y〉 ,

since χIi(y)χf(Ji)(f(y)) = χf(Ji)(f(y)). Thus

m∑

i=1

SiS∗i |y〉 =

m∑

i=1

χJi(y) |y〉 = χJk

(y) |y〉 = |y〉 .

Now, consider S∗i Si acting on a vector |y〉 of the canonical basis by

S∗i Si |y〉 = χf(Ji)(y)S∗i∣∣f−1

i (y)⟩

= χf(Ji)(y)χJi(f−1i (y)) |y〉 .

Note that χIi(f−1i (y)) = 1 and the condition χf(Ji)(y) = 1 is equivalent to the

existence of a pre-image of y in Ji. Since y ∈ Jk this means that aik = 1. On theother hand

m∑

j=1

aijSjS∗j |y〉 =

m∑

j=1

aijχJj (y) |y〉 = aik |y〉 .

Thus S∗i Si =∑m

j=1 ai,jSjS∗j . Therefore, we have the following result:

Theorem 3.5. Let x ∈ I. Let α,n be such that the orbit of 0 is finite. The mapπx : OAα,β

→ B(Hx), given by πx(si) = Si is a faithful irreducible representationof the Cuntz-Krieger algebra OAα,β

. Moreover, two such representations πx andπy are unitarily equivalent if and only if Rf (x) = Rf (y).

Proof. The fact that πx is a representation of the Cuntz-Krieger algebra OAα,β

follows from the computation above. The remaining statements follow from [8,Theorem 7]. ¤

As in the n = 1 case, see proof of Proposition 3.3 above, the multiplicationoperator U |y〉 = φ(y) |y〉 = e2πiy |y〉 is in the von Neumann algebra generated bythe operators S1, ..., Sm.

Proposition 3.6. The operators U, V satisfy the commutation relation

UV = e2πiαV Un.

Proof. We have

UV |y〉 = e2πif(y) |f(y)〉 = e2πiαe2πiny |f(y)〉 .On the other hand

V Un |y〉 = e2πiny |f(y)〉 .Therefore, UV = e2πiαV Un. ¤


It seems interesting to look into our proofs and to analyze the C*-algebragenerated by two unitaries u and v satisfying the relation

uvn = zvu (3.5)

with z ∈ S1. For z=1 (the “commutative” case) we get uvu−1 = vn, which isthe defining relation1 of a Baumslag-Solitar group B(1, n), see [2]. One non-trivialrepresentation in 2× 2 matrices is given by

u →(

n 00 1

), v →

(1 10 1

).

Therefore the C*-algebra generated by u and v obeying Eq. (3.5) should be re-garded as a “twisted” group algebra of the untwisted z = 1 one.

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Acknowledgment

First author acknowledges CIMA-UE for financial support. The other authors werepartially supported by the Fundacao para a Ciencia e a Tecnologia through theProgram POCI 2010/FEDER.

C. Correia RamosDepartment of Mathematics, Universidade de Evora,

R. Romao Ramalho, 59, 7000-671 Evora, Portugale-mail: [email protected]

Nuno MartinsCentro de Analise Matematica, Geometria e Sistemas Dinamicos,Department of Mathematics, Instituto Superior Tecnico,Av. Rovisco Pais 1, 1049-001 Lisboa, Portugale-mail: [email protected]

Paulo R. PintoCentro de Analise Matematica, Geometria e Sistemas Dinamicos,Department of Mathematics, Instituto Superior Tecnico,Av. Rovisco Pais 1, 1049-001 Lisboa, Portugale-mail: [email protected]

Orbit Representations and Circle Maps

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Transcript of Orbit Representations and Circle Maps