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IJMMS 2003:34, 2139–2146PII. S0161171203209236

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K-THEORY FOR CUNTZ-KRIEGER ALGEBRAS ARISINGFROM REAL QUADRATIC MAPS

NUNO MARTINS, RICARDO SEVERINO, and J. SOUSA RAMOS

Received 20 September 2002

We compute the K-groups for the Cuntz-Krieger algebras �A�(fµ), where A�(fµ)

is the Markov transition matrix arising from the kneading sequence �(fµ) of theone-parameter family of real quadratic maps fµ .

2000 Mathematics Subject Classification: 37A55, 37B10, 37E05, 46L80.

Consider the one-parameter family of real quadratic maps fµ : [0,1]→ [0,1]defined by fµ(x)= µx(1−x), with µ ∈ [0,4]. Using Milnor-Thurston kneading

theory [14], Guckenheimer [5] has classified, up to topological conjugacy, a cer-

tain class of maps, which includes the quadratic family. The idea of kneading

theory is to encode information about the orbits of a map in terms of infinite

sequences of symbols and to exploit the natural order of the interval to es-

tablish topological properties of the map. In what follows, I denotes the unit

interval [0,1] and c the unique turning point of fµ . For x ∈ I, let

εn(x)=

−1, if fnµ (x) > c,

0, if fnµ (x)= c,+1, if fnµ (x) < c.

(1)

The sequence ε(x) = (εn(x))∞n=0 is called the itinerary of x. The itinerary of

fµ(c) is called the kneading sequence of fµ and will be denoted by �(fµ).Observe that εn(fµ(x)) = εn+1(x), that is, ε(fµ(x)) = σε(x), where σ is the

shift map. Let∑ = {−1,0,+1} be the alphabet set. The sequences on

∑N are

ordered lexicographically. However, this ordering is not reflected by the map-

ping x → ε(x) because the map fµ reverses orientation on [c,1]. To take this

into account, for a sequence ε = (εn)∞n=0 of the symbols −1, 0, and +1, another

sequence θ = (θn)∞n=0 is defined by θn =∏ni=0 εi. If ε = ε(x) is the itinerary

of a point x ∈ I, then θ = θ(x) is called the invariant coordinate of x. The

fundamental observation of Milnor and Thurston [14] is the monotonicity of

the invariant coordinates:

x <y �⇒ θ(x)≤ θ(y). (2)

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2140 NUNO MARTINS ET AL.

We now consider only those kneading sequences that are periodic, that is,

�(fµ)= ε0

(fµ(c)

)···εn−1(fµ(c)

)ε0(fµ(c)

)···εn−1(fµ(c)

)···= (ε0

(fµ(c)

)···εn−1(fµ(c)

))∞ ≡ (ε1(c)···εn(c))∞ (3)

for some n ∈ N. The sequences σi(�(fµ)) = εi+1(c)εi+2(c)··· , i = 0,1,2, . . . ,will then determine a Markov partition of I into n−1 line intervals {I1, I2, . . . ,In−1} [15], whose definitions will be given in the proof of Theorem 1. Thus, we

will have a Markov transition matrix A�(fµ) defined by

A�(fµ) := (aij) with aij =1, if fµ

(intIi

)⊇ intIj,

0, otherwise.(4)

It is easy to see that the matrix A�(fµ) is not a permutation matrix and no row

or column of A�(fµ) is zero. Thus, for each one of these matrices and following

the work of Cuntz and Krieger [3], one can construct the Cuntz-Krieger algebra

�A�(fµ). In [2], Cuntz proved that

K0(�A)� Zr /(1−AT )Zr , K1

(�A)� ker

(I−At : Zr �→ Zr ), (5)

for an r × r matrix A that satisfies a certain condition (I) (see [3]), which is

readily verified by the matrices A�(fµ). In [1], Bowen and Franks introduced the

group BF(A) := Zr /(1−A)Zr as an invariant for flow equivalence of topological

Markov subshifts determined by A.

We can now state and prove the following theorem.

Theorem 1. Let �(fµ)= (ε1(c)ε2(c)···εn(c))∞ for some n∈N\{1}. Thus,

K0

(�A�(fµ)

)� Za with a=

∣∣∣∣∣1+n−1∑l=1

l∏i=1

εi(c)

∣∣∣∣∣,

K1

(�A�(fµ)

)�{0}, if a≠ 0,

Z, if a= 0.

(6)

Proof. Set zi = εi(c)εi+1(c)··· for i = 1,2, . . . . Let z′i = f iµ(c) be the point

on the unit interval [0,1] represented by the sequence zi for i = 1,2, . . . . We

have σ(zi) = zi+1 for i = 1, . . . ,n−1 and σ(zn) = z1. Denote by ω the n×nmatrix representing the shift map σ . Let C0 be the vector space spanned by the

formal basis {z′1, . . . ,z′n}. Now, let ρ be the permutation of the set {1, . . . ,n},which allows us to order the points z′1, . . . ,z′n on the unit interval [0,1], that is,

0< z′ρ(1) < z′ρ(2) < ···< z′ρ(n) < 1. (7)

K-THEORY FOR CUNTZ-KRIEGER ALGEBRAS . . . 2141

Set xi := z′ρ(i) with i= 1, . . . ,n and let π denote the permutation matrix which

takes the formal basis {z′1, . . . ,z′n} to the formal basis {x1, . . . ,xn}. We will de-

note by C1 the (n−1)-dimensional vector space spanned by the formal basis

{xi+1−xi : i= 1, . . . ,n−1}. Set

Ii := [xi,xi+1]

for i= 1, . . . ,n−1. (8)

Thus, we can define the Markov transition matrixA�(fµ) as above. Letϕ denote

the incidence matrix that takes the formal basis {x1, . . . ,xn} of C0 to the formal

basis {x2 −x1, . . . ,xn −xn−1} of C1. Put η := ϕπ . As in [7, 8], we obtain an

endomorphism α of C1, that makes the following diagram commutative:

C0η

ω

C1

α

C0 η C1.

(9)

We have α = ηωηT(ηηT )−1. Remark that if we neglect the negative signs on

the matrix α, then we will obtain precisely the Markov transition matrix A�(fµ).

In fact, consider the (n−1)×(n−1) matrix

β :=[

1nL 0

0 −1nR

], (10)

where 1nL and 1nR are the identity matrices of ranks nL and nR , respectively,

with nL (nR) being the number of intervals Ii of the Markov partition placed

on the left- (right-) hand side of the turning point of fµ . Therefore, we have

A�(fµ) = βα. (11)

Now, consider the following matrix:

γ�(fµ) := (γij) with

γii = εi(c), i= 1, . . . ,n,

γin =−εi(c), i= 1, . . . ,n,

γij = 0, otherwise.

(12)

The matrix γ�(fµ) makes the diagram

C0η

γ�(fµ)

C1

β

C0 η C1

(13)


commutative. Finally, set θ�(fµ) := θ�(fµ)ω. Then, the diagram

C0η

θ�(fµ)

C1

A�(fµ)

C0 η C1

(14)

is also commutative. Now, notice that the transpose of η has the following

factorization:

ηT = YiX, (15)

where Y is an invertible (over Z) n×n integer matrix given by

Y :=

1 0 ··· 0

0 1 0 ··· 0... 0

. . .. . .

...... 0

0 0 ··· 0 1 0

−1 −1 ··· −1 1

, (16)

i is the inclusion C1 ↩ C0 given by

i :=

1 0 0

0. . .

. . ....

.... . . 0

1

0 ··· 0

, (17)

and X is an invertible (over Z) (n−1)× (n−1) integer matrix obtained from

the (n− 1)×n matrix ηT by removing the nth row of ηT . Thus, from the

commutative diagram

C1ηT

AT�(fµ)

C0

θT�(fµ)

C1ηT

C0,

(18)


we will have the following commutative diagram with short exact rows:

0 C1i

A′

C0p

θ′

C0/C1

0

0

0 C1i

C0 p C0/C1 0,

(19)

where the map p is represented by the 1×n matrix [0 ··· 0 1] and

A′ =XAT�(fµ)X−1, θ′ = Y−1θT�(fµ)Y , (20)

that is, A′ is similar to AT�(fµ) over Z and θ′ is similar to θT�(fµ) over Z. Hence,

for example, by [10] we obtain, respectively,

Zn−1/(1−A′)Zn−1 � Zn−1/

(1−A�(fµ)

)Zn−1,

Zn/(1−θ′)Zn � Zn/(1−θ�(fµ)

)Zn.

(21)

Now, from the last diagram we have, for example, by [9],

θ′ =[A′ ∗0 0

]. (22)

Therefore,

Zn−1/(1−A′)Zn−1 � Zn/(1−θ′)Zn,

Zn−1/(1−A�(fµ)

)Zn−1 � Zn/(1−θ�(fµ)

)Zn.

(23)

Next, we will compute Zn/(1−θ�(fµ))Zn. From the previous discussions and

notations, the n×n matrix θ�(fµ) is explicitly given by

θ�(fµ) :=

−ε1(c) ε1(c) 0 ··· 0... 0

. . .. . .

......

. . . 0

−εn−1(c) εn−1(c)0 0 ··· 0

. (24)

Notice that the matrix θ�(fµ) completely describes the dynamics of fµ . Finally,

using row and column elementary operations over Z, we can find invertible


(over Z) matrices U1 and U2 with integer entries such that

1−θ�(fµ) =U1

1+n−1∑l=1

l∏i=1

εi(c)

1. . .

1

U2. (25)

Thus, we obtain

K0

(�A�(fµ)

)� Zn−1/

(1−AT�(fµ)

)Zn−1 � Za, (26)

where

a=∣∣∣∣∣1+

n−1∑l=1

l∏i=1

εi(c)

∣∣∣∣∣, n∈N\{1}. (27)

Example 2. Set

�(fµ)= (RLLRRC)∞, (28)

where R = −1, L = +1, and C = 0. Thus, we can construct the 5×5 Markov

transition matrix A�(fµ) and the matrices θ�(fµ), ω, ϕ, and π :

A�(fµ) =

0 1 1 0 0

0 0 0 1 1

0 0 0 0 1

0 0 1 1 0

1 1 0 0 0

, θ�(fµ) =

1 −1 0 0 0 0

−1 0 1 0 0 0

−1 0 0 1 0 0

1 0 0 0 −1 0

1 0 0 0 0 −1

0 0 0 0 0 0

,

ω=

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

1 0 0 0 0 0

, ϕ =

−1 1

−1 1

−1 1

−1 1

−1 1

,

π =

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 0 0 1

0 0 0 1 0 0

0 0 0 0 1 0

1 0 0 0 0 0

.

(29)


We have

K0

(�A�(fµ)

)� Z2, K1

(�A�(fµ)

)� {0}. (30)

Remark 3. In the statement of Theorem 1 the case a = 0 may occur. This

happens when we have a star product factorizable kneading sequence [4]. In

this case the correspondent Markov transition matrix is reducible.

Remark 4. In [6], Katayama et al. have constructed a class of C∗-algebras

from the β-expansions of real numbers. In fact, considering a semiconjugacy

from the real quadratic map to the tent map [14], we can also obtain Theorem 1

using [6] and the λ-expansions of real numbers introduced in [4].

Remark 5. In [13] (see also [12]) and [11], the BF-groups are explicitly cal-

culated with respect to another kind of maps on the interval.

References

[1] R. Bowen and J. Franks, Homology for zero-dimensional nonwandering sets, Ann.of Math. (2) 106 (1977), no. 1, 73–92.

[2] J. Cuntz, A class of C∗-algebras and topological Markov chains. II. Reduciblechains and the Ext-functor for C∗-algebras, Invent. Math. 63 (1981), no. 1,25–40.

[3] J. Cuntz and W. Krieger, A class of C∗-algebras and topological Markov chains,Invent. Math. 56 (1980), no. 3, 251–268.

[4] B. Derrida, A. Gervois, and Y. Pomeau, Iteration of endomorphisms on the realaxis and representation of numbers, Ann. Inst. H. Poincaré Sect. A (N.S.) 29(1978), no. 3, 305–356.

[5] J. Guckenheimer, Sensitive dependence to initial conditions for one-dimensionalmaps, Comm. Math. Phys. 70 (1979), no. 2, 133–160.

[6] Y. Katayama, K. Matsumoto, and Y. Watatani, Simple C∗-algebras arising fromβ-expansion of real numbers, Ergodic Theory Dynam. Systems 18 (1998),no. 4, 937–962.

[7] J. P. Lampreia and J. Sousa Ramos, Trimodal maps, Internat. J. Bifur. Chaos Appl.Sci. Engrg. 3 (1993), no. 6, 1607–1617.

[8] , Symbolic dynamics of bimodal maps, Portugal. Math. 54 (1997), no. 1,1–18.

[9] S. Lang, Algebra, Addison-Wesley, Massachusetts, 1965.[10] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cam-

bridge University Press, Cambridge, 1995.[11] N. Martins, R. Severino, and J. Sousa Ramos, Bowen-Franks groups for bimodal

matrices, J. Differ. Equations Appl. 9 (2003), no. 3-4, 423–433.[12] N. Martins and J. Sousa Ramos, Cuntz-Krieger algebras arising from linear mod

one transformations, Differential Equations and Dynamical Systems (Lis-bon, 2000), Fields Inst. Commun., vol. 31, American Mathematical Society,Rhode Island, 2002, pp. 265–273.

[13] , Bowen-Franks groups associated with linear mod one transformations, toappear in Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2003.


[14] J. Milnor and W. Thurston, On iterated maps of the interval, Dynamical Systems(College Park, Md, 1986–87), Lecture Notes in Math., vol. 1342, Springer,Berlin, 1988, pp. 465–563.

[15] P. Štefan, A theorem of Šarkovskii on the existence of periodic orbits of continuousendomorphisms of the real line, Comm. Math. Phys. 54 (1977), no. 3, 237–248.

Nuno Martins: Departamento de Matemática, Instituto Superior Técnico, 1049-001Lisboa, Portugal

E-mail address: [email protected]

Ricardo Severino: Departamento de Matemática, Universidade do Minho, 4710-057Braga, Portugal


J. Sousa Ramos: Departamento de Matemática, Instituto Superior Técnico, 1049-001Lisboa, Portugal


mailto:[email protected]



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