Cuntz–Krieger algebras representations from orbits of interval maps

J. Math. Anal. Appl. 341 (2008) 825–833 www.elsevier.com/locate/jmaa

Cuntz–Krieger algebras representations from orbits of interval maps ✩ C. Correia Ramos a , Nuno Martins b , Paulo R. Pinto b,∗ , J. Sousa Ramos b a Centro de Investigação em Matemática e Aplicações, Department of Mathematics, Universidade de Évora, R. Romão Ramalho, 59,

7000-671 Évora, Portugal b Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Department of Mathematics, Instituto Superior Técnico, Av. Rovisco Pais 1,

1049-001 Lisboa, Portugal Received 29 March 2007 Available online 7 November 2007 Submitted by G. Corach

Abstract Let f be an expansive Markov interval map with finite transition matrix Af . Then for every point, we yield an irreducible representation of the Cuntz–Krieger algebra OAf and show that two such representations are unitarily equivalent if and only if the points belong to the same generalized orbit. The restriction of each representation to the gauge part of OAf is decomposed into irreducible representations, according to the decomposition of the orbit. © 2007 Elsevier Inc. All rights reserved. Keywords: Interval maps; Generalized orbits; Cuntz–Krieger algebras; Irreducible representations

1. Introduction A class of representations of Cuntz algebras On are studied in [3,4,8], where the special family of the permutative ones are classified. Besides interest in its own right, applications of Cuntz algebras representations to wavelets, fractals, dynamical systems, see e.g. [3,4,11], and quantum field theory in [1] are remarkable. Some results have been extended to Cuntz–Krieger algebras OA for finite 0–1 matrices A in [12], where Kawamura relates some representations of these algebras to Perron–Frobenius operators of certain measure spaces transformations. In this paper, we study a family of Cuntz–Krieger algebras representations that naturally arise from dynamical systems, by analyzing the orbits of (a large family) interval maps through the study of Cuntz–Krieger algebra OAf representations, where Af is the transition Markov 0–1 matrix of a fixed interval map f : I → I . For that we first ✩ First author acknowledges CIMA-UE for financial support. The other authors were partially supported by the Fundação para a Ciência e a Tecnologia through the Program POCI 2010/FEDER. * Corresponding author. E-mail addresses: [email protected] (C. Correia Ramos), [email protected] (N. Martins), [email protected] (P.R. Pinto), [email protected] (J. Sousa Ramos).

0022-247X/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2007.10.059

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consider the equivalence relation   Rf = (x, y): f n (x) = f m (y) for some n, m ∈ N0 .

(1)

For each point x, the equivalence class R(x), so-called generalized orbit of x, being countable can be endowed with the structure of a Hilbert space on which we can define a representation πx of the Cuntz–Krieger algebra OAf . We prove that two such representations are unitarily equivalent if and only if the points live in the same Rf -equivalence class, and moreover every such representation is indeed irreducible (see Theorem 6). In the particular case where Af is full, i.e. aij = 1, then these classes of irreducible representations reproduce a subfamily of those in [3]. Besides, we will also study the diagonal equivalence relation   (2) Qf = (x, y): f n (x) = f n (y) for some n ∈ N0 . It is clear that Qf ⊆ Rf , so each equivalence class R(x) inherits an equivalence relation from Q, the quotient R(x)/Q as in [10]. Then we decompose the equivalence class R(x) into R(x)/Q-equivalence classes (see Lemma 5). T of O , the representation π breaks up as the direct sum of irreWhen restricted to the gauge subalgebra OA Af x f T , ducible representations of OA f  ρx,y , πx |AF = y∈R(x)/Q

where ρx,y is the restriction of πx into the subspace generated by equivalence class of y in R(x)/Q (see Theorem 7). T . Moreover the irreducible representations {πx } of OAf are multiplicity free when restricted to the subalgebra OA f ∗ Hence we get a C -algebra version of what in group theory is called “Gelfand pairs” (G, K), see [2,3], with K a subgroup of G. This direct sum is finite whenever the forward-orbit of the base point x is finite, nevertheless this set of points is countable and thus has null measure. In turn, this implies that the index of Qf in Rf is infinite as |R(x)/Q| = ∞ a.e. We work with a certain (large) class of interval maps to ensure either the irreducibility and unitarity equivalence of the representations (see Definition 1 or [5,6]). 2. Preliminaries 2.1. Cuntz–Krieger algebras input A representation π of a ∗-algebra A on a complex Hilbert space H is a ∗-homomorphism π : A → B(H) into  are the ∗-algebra B(H) of bounded linear operators on H. Two representations π : A → B(H) and  π : A → B(H)  (unitarily) equivalent if there is a unitary operator U : H →H (i.e., U is a surjective isometry) so that U π(a) =  π (a)U , for every a ∈ A. A representation π : A → B(H) of some ∗-algebra is said to be irreducible if there is no non-trivial subspace of H invariant with respect to all operators π(a) with a ∈ A. According to, e.g., [15, Proposition 3.13.2], π is irreducible if x ∈ B(H): xπ(a) = π(a)x,

for all a ∈ A

⇒

x = λ1,

(3)

for some complex number λ, where 1 denotes the identity of A. Using the definition of commutant, (3) can be restated as: π(A) = C1. The representation is called faithful if it is injective. Here we deal with a special family of ∗-algebras defined as follows: Let A = (aij ) be an n × n 0–1 matrix 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 [7] generated by (non-zero) partial isometries s1 , . . . , sn satisfying   si∗ si = aij sj sj∗ (i = 1, . . . , n), si si∗ = 1, (4) j

i

where 1 denotes the identity. The algebra OA is uniquely determined by the relations (4), see [7, Theorem 2.13]. A special case is the Cuntz algebra On when A is full: aij = 1 for all i and j . For a word α = (α1 , α2 , . . . , αk ) in the alphabet {1, . . . , n} we set sα := sα1 . . . sαk . Hence, the length of α is |α| = k. It is well known that OA is linearly spanned by the set of sα sβ∗ with words α and β in {1, . . . , n} with lengths |α| and |β|, possibly different.

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By the universality properties of OA we have a gauge action ω of the circle T on OA defined by ωz (si ) = zsi with T is an AF-algebra (a C∗ -algebra which is an inductive limit of a sequence of z ∈ T. Then the fixed point algebra OA ∗ finite dimensional C -algebras) linearly spanned by all the monomials of the form sα sβ∗ with |α| = |β|, see e.g. [7]. This AF-algebra is sometimes denoted by FA in the literature. Of interest in the sequel is when A is an irreducible matrix, i.e., for any i and j , there exists a natural number m such that the (i, j )-entry of Am is non-zero, i.e. (Am )ij = 0. A result of Cuntz and Krieger [7] shows that OA is a simple algebra (i.e., it does not contains non-trivial two-sided closed ideal) whenever A is irreducible. Matrices A for which there exists a positive integer m such that all the entries of Am are non-zero are called aperiodic, and obviously form a subclass of irreducible matrices. If A is an aperiodic matrix, then, besides OA , FA is also a simple C∗ -algebra [7]. 2.2. Interval maps Definition 1. A map f is in the class M(I ), with I = [0, 1], if it satisfies the following properties: (P1) There is a partition C = {I1 , . . . , In } of open intervals with Ii ∩ Ij = ∅ for i = j , dom(f ) = im(f ) = I .  (P2) For every i = 1, . . . , n the set f (Ii ) ∩ ( nj=1 Ij ) is a non-empty union of intervals from C. (P3) f |Ij ∈ C 1 (Ij ), monotone and |f |Ij (x)| > b > 1, for every x ∈ Ij , j = 1, . . . , n, and some b.

n

j =1 Ij

⊂ I and

(P4) For every interval Ij with j = 1, . . . , n there is a natural number q such that dom(f ) ⊂ f q (Ij ). Given a map f in M(I ), we call every partition of dom(f ) an f -partition. The minimal f -partition is denoted by Cf . Let f ∈ M(I ) and Cf = {I1 , . . . , In }. Then we set 

n Ij for all k = 0, 1, . . . . Ωf := x ∈ I : f k (x) is in j =1

 Thus Ωf is the set of points that remain in nj=1 Ij under iteration of f , and is sometimes called a cookie-cutter set, see  [9]. Let f ∈ M(I ) and let {1, 2, . . . , n} be the alphabet indexing the elements of Cf . The address map ad : nj=1 Ij → {1, 2, . . . , n} is given as follows ad(x) = i if x ∈ Ii . The itinerary map it : Ωf → {1, 2, . . . , n}N0 is defined as   it(x) = ad(x)ad f (x) ad f 2 (x) · · · . By property (P2) of Definition 1, there is a transition matrix Af = {aij }ni,j =1 , induced by f on Cf and defined by

1 if f (Ii ) ⊃ Ij , aij = (5) 0 otherwise.  Let Σf be the subspace of {1, 2, . . . , n}N0 given by it( nj=1 Ij ), which is invariant under the shift map σ : {1, 2, . . . , n}N0 → {1, 2, . . . , n}N0 defined as σ (i1 , i2 , . . .) = (i2 , i3 , . . .). We obviously have it ◦ f = σ ◦ it. We will use σ meaning in fact σ |Σf . A sequence in {1, 2, . . . , n}N0 is called admissible, with respect to f if it occurs as an itinerary of some point x in Ωf , i.e. if it belongs to Σf . An admissible word is a finite sub-sequence of some admissible sequence. The set of admissible words of size k is denoted by Wk = Wk (f ). Let fi : Ii → f (Ii ) be the restriction of f to the subinterval Ii , i = 1, . . . , n. Let fi−1 be the inverse branch of f , that is, fi−1 : f (Ii ) → Ii and for x ∈ f (Ii ) and y ∈ Ii we obviously have f ◦ fi−1 (x) = x,

fi−1 ◦ f |Ii (y) = y.

(6)

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Given i1 . . . ik ∈ Wk , we define Ii1 ...ik as being the set of points x in dom(f ) satisfying  ad(x) = i1 , ..., ad f k (x) = ik . A characterization of the admissible words is obtained as follows. Proposition 2. For k  2, a word i1 . . . ik ∈ {1, 2, . . . , n}k is admissible if and only if fi−1 ◦ · · · ◦ fi−1 ◦ fi−1 (Iik ) = ∅. 1 k−2 k−1 ◦ · · · ◦ fi−1 ◦ fi−1 (Iik ). As a consequence f (Ii1 ...ik ) = Ii2 ...ik . Moreover In that case Ii1 ...ik = fi−1 1 k−2 k−1 Ii1 ...ik = Ii1 ...ik j . j : i1 ...ik j ∈Wk+1

Proof. Since dom(fi ) = Ii , we have that fi−1 (Ij ) ⊂ Ii is the set of points x ∈ Ii so that f (x) ∈ Ij . In this case ad(x) = i and ad(f (x)) = j . Moreover fi−1 (Ij ) = ∅ if and only if Ij ∩ f (Ii ) = ∅ if and only if there is no transition from Ii to Ij under f . A similar argument is applied for every admissible word. 2 Remark 3. The expansiveness condition (P3) guarantees the uniqueness of the itinerary, i.e. it(x) = it(y), then x = y. 3. Representations of type I associated to an orbit of f In the sequel we will consider the equivalence relations Rf and Qf defined by Eqs. (1) and (2) restricted to Ωf . Moreover Rf and Qf are countable equivalence relations in the sense that the equivalence classes Rf (x) and Qf (x) of x ∈ Ωf , are countable sets. We denote x ∼R y whenever (x, y) ∈ Rf and x ∼Q y whenever (x, y) ∈ Qf . Each class Rf (x) has a natural tree structure, considering the points in Rf (x) as the vertices of the tree and a directed edge connecting z to y, with y, z ∈ Rf (x), whenever y = f (z). Let Hx be the Hilbert space l 2 (Rf (x)) with the canonical orthonormal basis {|y: y ∈ Rf (x)} in Dirac notation. Whenever x ∼R y, Hx = Hy (are the same Hilbert spaces). The inner product (·,·) is given by (|y, |z) = δy,z . Likewise, let Kx be the Hilbert space l 2 (Qf (x)). Hence Kx is a Hilbert subspace of Hx and Kx = Ky (are the same Hilbert spaces) whenever x ∼Q y. If x ∈ Ωf we can consider the partition of the set Rf (x) into equivalence classes with respect to Qf , i.e., the quotient Rf (x)/Qf . Note that [y] = {z ∈ Rf (x): y ∼Q z}, for each [y] ∈ Rf (x)/Qf . Lemma 4. Let x ∈ Ωf . Then Qf (y) = Rf (x). [y]∈Rf (x)/Qf

Proof. Let y ∈ Rf (x) which means that f n (y) = f m (x) for some n, m ∈ N0 . Suppose n < m. Then we have f n (y) = ◦ · · · ◦ fμ−1 (x)), for f n (f m−n (x)), and y ∈ Qf (f j (x)) with j = m − n. Suppose m < n. Since f m (x) = f n (fμ−1 n−m 1 −1 −1 some μ1 μ2 . . . μn−m ∈ Wn−m , we have y ∈ Qf (fμ1 ◦ · · · ◦ fμn−m (x)). Let z ∈ Qf (w) for some [w] ∈ Rf (x)/Qf .  Obviously z ∈ Rf (x), since Qf (w) ⊂ Rf (x) if and only if w ∈ Rf (x). Therefore, we have kj =1 Qf (f j (x)) = Rf (x). 2 The partition of Hx can be written in the form  Ky . Hx = [y]∈Rf (x)/Qf

Note that Ky does not depend on the choice of the element y in the class [y], since Ky = Kz (the same Hilbert spaces) whenever z ∼Q y.

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If x is periodic, we say that (x1 , . . . , xk ) is a cycle of minimal period k if xj +1 = f (xj ), x1 = f (xk ) and x1 = f (xj ) for j < k − 1. Hence |Rf (x)/Qf | = k and f j1 (x), f j2 (x) are in distinct classes of Qf whenever j1 = j2 mod k. Thus in the periodic case and thanks to Lemma 4 we have the following decomposition of Hx : Hx = Kx ⊕ Kf (x) ⊕ · · · ⊕ Kf k−1 (x) . For each i = 1, . . . , n, let us define an operator Si on Hx as follows:   Si |y = χf (Ii ) (y)fi−1 (y) , for all y ∈ Rf (x)

(7)

where χB denotes the characteristic function on a set B. Note that χf (Ii ) (x) = 1 if and only if there is a pre-image of x in Ii . Then every Si is extended by linearity and continuity to all of Hx . We have Si∗ |y = χIi (y)|f (y), again extended by linearity and continuity to all Hx . In fact       y Si |z = y fi−1 (z) = δy,f −1 (z) . i

On the other hand, we have     ∗   y Si |z = χIi (y) f (y)z = χIi (y)δf (y),z . Since δy,f −1 (z) = χIi (y)δf (y),z we have shown that the operators Si , Si∗ are adjoint of each other. i

We further remark that |fi (x) or |fi−1 (x) are identically equal to the zero vector whenever x ∈ / Ii or x ∈ / f (Ii ), respectively. Moreover, Si is a partial isometry: namely, Si is an isometry on its restriction to span{|y: y ∈ f (Ii )}∩Hx and vanishes on the remaining part of Hx . Lemma 5. The operators Si satisfy the relations transition matrix defined in Eq. (5).

n

∗ i=1 Si Si

= 1 and Si∗ Si =

n

∗ j =1 aij Sj Sj ,

where Af = (aij ) is the

Proof. Consider Si Si∗ acting on a vector |y of the canonical basis, and let k ∈ {1, . . . , n} so that y ∈ Ik . Then      Si Si∗ |y = χIi (y)Si f (y) = χIi (y)χf (Ii ) f (y) fi−1 ◦ f (y) = χIi (y)|y, since χIi (y)χf (Ii ) (f (y)) = χf (Ii ) (f (y)). Thus n 

Si Si∗ |y =

i=1

n 

χIi (y)|y = χIk (y)|y = |y.

i=1

Now, let us consider Si∗ Si acting on a vector |y of the canonical basis    Si∗ Si |y = χf (Ii ) (y)Si∗ fi−1 (y) = χf (Ii ) (y)χIi fi−1 (y) |y. Note that χIi (fi−1 (y)) = 1 and the condition χf (Ii ) (y) = 1 is equivalent to the existence of a pre-image of y in Ii . Since y ∈ Ik this means that aik = 1. On the other hand, n 

aij Sj Sj∗ |y =

j =1

Thus Si∗ Si =

n

n 

aij χIj (y)|y = aik |y.

j =1

∗ j =1 ai,j Sj Sj .

2

Theorem 6. The map πx : OAf → B(Hx ) given by πx (si ) = Si is a faithful irreducible representation of the Cuntz–Krieger algebra OAf . Moreover, two such representations πx and πy are unitarily equivalent if and only if Rf (x) = Rf (y). Proof. By definition the operators Si , with i = 1, . . . , n, are all distinct and by Lemma 5 we conclude that πx (si ) = Si yields a ∗-representation of OAf on B(Hx ). Thanks to [7, Theorem 2.14], OAf is a simple algebra as Af is an irreducible matrix. Since ker(πx ) is an ideal we conclude that πx is faithful.

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We prove now that every non-zero vector in Hx is cyclic proving the irreducibility of the representation, see Pedersen’s textbook [15, Theorem 3.13.2]. Let |z, |y ∈ Hx . Since z ∼R y there are natural numbers n, m with f n (z) = f m (y). Then, there are admissible words μ ∈ Wn and ν ∈ Wm so that z = fμ−1 ◦ · · · ◦ fμ−1 ◦ fνm ◦ · · · ◦ fν1 (y) n 1 which is equivalent to |z = Sν Sμ∗ |y. Therefore every vector |z, z ∈ Rf (x), is cyclic. Let ξ ∈ Hx be a non-zero vector. Then we aim to prove that ξ is cyclic, and for that it is enough to find a sequence tn ∈ OA such that π(OA )ξ  Tn ξ → |x0  for some x0 ∈ Rf (x), since |x0  is a cyclic vector and Tn := π(tn ). Choose x0 ∈ Rf (x) such that ξ |x0  = 0. Let it(x0 ) = α1 α2 · · · be the itinerary of the point x0 and α(k) := (α1 , . . . , αk ) ∈ Wk its kth prefix. For every n ∈ N, let Tn := π(sα(n) sα∗(n) ) and denote by Hx0 (n) the range of the projection Tn . Then (Tn ) is a decreasing sequence ∞ of orthogonal projections, thus it strongly converges to T , the orthogonal projection onto ∞ n=1 Hx0 (n). Since f is expansive, x0 is the unique point in Ωf such that it(x) = α1 α2 · · · , see Remark 3, hence n=1 Hx0 (n) = C|x0 . Since |x0  is a norm one vector, we conclude that T ξ = ξ |x0 |x0 . We prove the second part of the theorem. Let x, y ∈ Ωf so that Rf (x) = Rf (y). In this case Hx = Hy and the operators πx (si ) and πy (si ) are equal. Thus πx and πy are the same. Let x, y ∈ Ωf so that Rf (x) = Rf (y). Assume that πx and πy are equivalent, then there is an unitary operator U ∈ B(Hx , Hy ) so that U πx (a) = πy (a)U

for every a ∈ OA .

This means that U |xcan be arbitrarily approximated by a linear  combination of vectors |z ∈ Hy . Thus let us assume that U |x = z∈R(y) cz |z with cz ∈ C. Note that  z∈R(y) cz |zHy = U |xHy = |xHx = 1. Let it(x) = α1 α2 · · ·. Then    cz πy sα(n) sα∗(n) |z. πy sα(n) sα∗(n) U |x = z∈R(y)

However we have also   πy sα(n) sα∗(n) U |x = U πx sα(n) sα∗(n) |x = U |x. Since x R y, for every |z ∈ Hy there is a natural number k so that πy (sα(k) sα∗(k) )|z = 0, then       ∗  lim  cz πy sα(k) sα(k) |z  = 0, k→∞

Hy

z∈R(y)

and we have a contradiction. Therefore, πx and πy are not unitarily equivalent.

2

Theorem 7. For x, y ∈ Ωf with x ∼R y, the map ρx,y = πx |FAf : FAf → B(Ky ) is a faithful irreducible representation of FAf . Let x, y1 , y2 ∈ Ωf be such that x ∼R y1 ∼R y2 , and let ρx,y1 , ρx,y2 be two representations on Ky1 , Ky2 , respectively, of the algebra FAf . Then ρx,y1 and ρx,y2 are unitarily equivalent if and only if y1 ∼Q y2 . Thus πx |FAf  decomposes into irreducibles [y]∈Rf (x)/Qf ρx,y with simple multiplicities. Proof. The algebra FAf is generated by elements of the form sμ sν∗ with μ, ν ∈ Wn for some n ∈ N. Then, we easily see that ρx,y is well defined, i.e. ρx,y (Ky ) ⊆ Ky . Since Af is aperiodic the algebra FAf is simple, see [7, Theorem 2.14], thus the representation ρx,y is faithful. The rest of the proof closely follows that of Theorem 6, since the projections sα(n) sα∗(n) belong to FAf . We only detail the last part. Let us assume that y1 Q y2 . We also assume that ρx,y1 and ρx,y2 are equivalent. Then there is an unitary operator U ∈ B(Ky1 , Ky2 ) so that Uρx,y1 (a) = ρx,y2 (a)U

for every a ∈ FAf .

This means that U |y 1  can be arbitrarily approximated by a linear combination of vectors |z ∈ Ky2 . Thus let us assume that U |y1  = z∈Q(y2 ) cz |z with cz ∈ C. Note that  z∈Q(y2 ) cz |zHy2 = U |y1 Hy2 = |y1 Hy1 = 1. Let it(y1 ) = α1 α2 · · · . Then    ρx,y2 sα(n) sα∗(n) U |y1  = cz ρx,y2 sα(n) sα∗(n) |z. z∈Q(y2 )

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However we have also   ρx,y2 sα(n) sα∗(n) U |y1  = Uρx,y1 sα(n) sα∗(n) |y1  = U |y1 . Since y1 Q y2 , for every |z ∈ Ky2 there is a natural number k so that πy (sα(k) sα∗(k) )|z = 0, and as above in the Theorem 6 we have a contradiction with respect to the sequence of the values πy (sα(n) sα∗(n) )U |y1 Hy2 , n = 1, 2, . . . . Therefore, ρx,y1 and ρx,y2 are not unitarily equivalent. 2 Let Nf,x be the von Neumann algebra generated by πx (OAf ), that is to say: Nf,x = πx (OAf ) . Similarly, let us denote by Nf,x,y the von Neumann algebra generated by ρx,y (FA ). Thanks to Theorems 6 and 7, πx and ρx,y are irreducible representations, hence πx (OAf ) = B(Hx ) and ρx,y (FA ) = B(Ky ). In particular we obtain the following result. Corollary 8. The von Neumann algebras Nf,x and Nf,x,y are all (the unique AF) factors of type I∞ . Remark 9. Let f ∈ M(I ) for some interval I with f -partition Cf = {I1 , . . . , In }. Assume further that f satisfies n N0 is admissible and the transition matrix A is the full f j =1 Ij ⊂ f (Ik ) for all k. Thus every sequence in {1, 2, . . . , n} matrix. In this case we obtain representations of the Cuntz algebra On , see first example, in Section 4. Following [3], a representation of the Cuntz algebra On , on a Hilbert space H , is permutative if there is a basis {el }l∈N of H such that Si el = em for some m ∈ N, with i = 1, . . . , n. Since Rf (x) is countable we can give a one-to-one correspondence y ∈ Rf (x) ↔ l(y) ∈ N. We thus obtain a one-to-one correspondence |y ∈ Hx ↔ el(y) ∈ l 2 (N). Therefore Si |y = |fi−1 (y) which is equivalent to Si el(y) = el(f −1 (y)) , and therefore the representation πx is permutative. i

4. Examples 4.1. Quadratic maps Let us consider the one-parameter family of quadratic maps given by fb (x) = 4bx(1 − x), see Fig. 1(a). If b  1 we have a countable set of values for which the critical orbit of fb is finite. For a fixed b in that set, we can associate a Cuntz–Krieger algebra as in [14]. There is an uncountable set of values that the parameter b can take for which the critical orbit of fb is not finite and we do not obtain Cuntz–Krieger algebras. We address this case in a future paper. Now, let us consider b > 1. In this case the set Ωf is a Cantor set with Hausdorff dimension less than 1, depending on the parameter b. Moreover, for every b > 1 the Cuntz–Krieger algebra OAf coincides with the Cuntz algebra O2 , since the transition matrix is always given by   1 1 , Af = 1 1 with respect to the partition Cf = {I0 , I1 } with I0 = ]0, x1 [, I1 = ]x2 , 1[, where x1 , x2 are the solutions of fb (x) = 1. Every sequence in {0, 1}N corresponds to a number in Ωf and Wk = {0, 1}k . Let x be the non-zero fixed point. Let us analyze the tree structure of Rf (x): every word in Wk = {0, 1}k corresponds to a unique pre-image of x and consequently a unique element in Rf (x). There is only one class in Rf (x)/Qf , and Rf (x) = Qf (x) = Rf (x)/Qf . Every point in the class [x] ∈ Rf (x)/Qf has an itinerary of the type i1 . . . ik (1)∞ , for some word i1 . . . ik ∈ Wk . Let y be the point of period 2. In this case there are two classes in Rf (y)/Qf . One class corresponds to [y] ∈ Rf (y)/Qf , where it(y) = (01)∞ , and the other corresponds to [f (y)] ∈ Rf (y)/Qf , where it(f (y)) = (10)∞ . The points in [y] have the itinerary of the type i1 . . . ik (01)∞ with k even, or i1 . . . ik (10)∞ with k odd, for some word i1 . . . ik ∈ Wk . The points in [f (y)] have the itinerary of the type i1 . . . ik (01)∞ with k odd, or i1 . . . ik (10)∞ with k even, for some word i1 . . . ik ∈ Wk . 2 Let us consider now another family of quadratic maps: gα (x) = λx 2 − (λ + α)x + α, with λ = 1−α+α α(α−1) and 0 < α < 1, see Fig. 1(b). This condition on λ guarantees that gα (α) = 1. The domain of gα is the set [0, α] ∪ [c, 1], where c > α is the point verifying gα (c) = 1. Therefore gα belongs to M([0, 1]). Since gα (0) = α, gα (α) = 1, gα (c) = 1

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(a)

(b)

Fig. 1. (a) Graph of the map fb with b = 5/4. (b) Graph of the map gα with α = 0.56984 . . . .

and gα (1) = 0 the transition matrix for gα is   1 1 Agα = . 1 0 The set Ωgα is a Cantor set with Hausdorff dimension less than 1, depending on the parameter α. Every sequence in Σgα = {(ij )j ∈N : ij ij +1 = 11} ⊂ {0, 1}N corresponds to an element in Ωgα . Let y be the point of period 2 with itinerary it(y) = (01)∞ . As previously, to each word in Wk it corresponds a unique pre-image of y and consequently a unique element in Rgα (y). There are two classes in Rgα (y)/Qgα . One class corresponds to [y] ∈ Rgα (y)/Qgα , where it(y) = (01)∞ , the other corresponds to [gα (y)] ∈ Rgα (y)/Qgα , where it(gα (y)) = (10)∞ . The points in [y] have the itinerary of the type i1 . . . ik (01)∞ with k even, or i1 . . . ik (10)∞ with k odd, for some word i1 . . . ik ∈ Wk . The points in [gα (y)] have the itinerary of the type i1 . . . ik (01)∞ with k odd, or i1 . . . ik (10)∞ with k even, for some word i1 . . . ik ∈ Wk . 4.2. Beta transformations Let τβ (x) = βx mod 1, the beta transformation. There is a countable set of values β > 1 for which the map τβ (x) belongs to M([0, 1]). For each value of the parameter β we have a different topological structure. The transition matrix √ can be obtained, see e.g. [13], from the orbit of 1. Let β = 1+2 5 . In this case the partition for τβ is Cτβ = {I0 , I1 } with I0 = ]0, β −1 [, I1 = ]β −1 , 1[. Every sequence in Στβ = {(ij )j ∈N : ij ij +1 = 11} ⊂ {0, 1}N corresponds to an element in Ωτβ . The transition matrix of τβ with respect to Cτβ is given by   1 1 Aτβ = . 1 0 Let x be the point of period 2. The tree associated with Rτβ (x) is isomorphic to the tree associated with Rgα (y), in the example above, although the sets Rτβ (x) and Rgα (y) are distinct. Acknowledgment We would like to thank the anonymous referee for helpful comments, particularly his assessment to arguments of the proofs of Theorems 6 and 7.

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