K-THEORY FOR CUNTZ-KRIEGER ALGEBRAS ARISING FROM REAL QUADRATIC MAPS

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PII. S0161171203209236 http://ijmms.hindawi.com

K-THEORY FOR CUNTZ-KRIEGER ALGEBRAS ARISING FROM REAL QUADRATIC MAPS

NUNO MARTINS, RICARDO SEVERINO, and J. SOUSA RAMOS Received 20 September 2002

We compute theK-groups for the Cuntz-Krieger algebrasᏻA_᏷(fµ), whereA_᏷_(f_µ₎ is the Markov transition matrix arising from the kneading sequence᏷(fµ)of the one-parameter family of real quadratic mapsfµ.

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

Consider the one-parameter family of real quadratic mapsfµ:[0,1]→[0,1]

defined byfµ(x)=µx(1−x), withµ∈[0,4]. Using Milnor-Thurston kneading theory [14], Guckenheimer [5] has classified, up to topological conjugacy, a certain 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,Idenotes the unit interval[0,1]andcthe unique turning point offµ. Forx∈I, let

εn(x)=











−1, iffµⁿ(x) > c, 0, iff_µⁿ(x)=c, +1, iff_µⁿ(x) < c.

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The sequenceε(x)=(εn(x))^∞_n=0is called the itinerary ofx. The itinerary of fµ(c) is called the kneading sequence of fµ and will be denoted by ᏷(fµ). Observe thatεn(fµ(x))=εn+1(x), that is,ε(fµ(x))=σ ε(x), whereσ 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- pingx→ε(x)because the mapfµreverses orientation on[c,1]. To take this into account, for a sequenceε=(εn)^∞n=0of the symbols−1, 0, and+1, another sequenceθ=(θn)^∞_n=0is deﬁned byθn=_n

i=0εi. Ifε=ε(x)is the itinerary of a pointx∈I, thenθ=θ(x)is called theinvariant coordinate ofx. 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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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 somen∈N. The sequencesσⁱ(᏷(fµ))=εi+1(c)εi+2(c)···, i=0,1,2,..., will then determine a Markov partition ofIinton−1 line intervals{I1,I2,..., In−1}[15], whose deﬁnitions will be given in the proof ofTheorem 1. Thus, we will have a Markov transition matrixA᏷(fµ)deﬁned by

A᏷(fµ):=

aij

withaij=





1, iffµ intIi

⊇intIj,

0, otherwise. (4)

It is easy to see that the matrixA᏷(fµ)is not a permutation matrix and no row or column ofA᏷(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 ᏻA_᏷(fµ). In [2], Cuntz proved that

K0 ᏻA

Z^r/ 1−A^T

Z^r, K1 ᏻA

ker

I−A^t:Z^r →Z^r

, (5)

for anr×r matrix Athat satisfies a certain condition (I) (see [3]), which is readily verified by the matricesA᏷(fµ). In [1], Bowen and Franks introduced the groupBF(A):=Z^r/(1−A)Z^ras an invariant for flow equivalence of topological Markov subshifts determined byA.

We can now state and prove the following theorem.

Theorem1. Let᏷(fµ)=(ε1(c)ε2(c)···εn(c))^∞for somen∈N\{1}. Thus,

K0 ᏻA_᏷(fµ)

Za witha= 1+

n−1

l=1

l i=1

εi(c) , K1 ᏻA_᏷(fµ)



{0}, ifa≠0, Z, ifa=0.

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Proof. Setzi=εi(c)εi+1(c)···fori=1,2,....Letz_i=f_µⁱ(c)be the point on the unit interval[0,1]represented by the sequencezifori=1,2,....We haveσ (zi)=zi+1fori=1,...,n−1 andσ (zn)=z1. Denote byωthen×n matrix representing the shift mapσ. LetC0be the vector space spanned by the formal basis{z1,...,zn}. Now, letρbe the permutation of the set{1,...,n}, which allows us to order the pointsz1,...,z_non the unit interval[0,1], that is, 0< z_ρ(1)< z_ρ(2)<···< z_ρ(n)<1. (7)

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K-THEORY FOR CUNTZ-KRIEGER ALGEBRAS... 2141 Setxi:=z_ρ(i)withi=1,...,nand letπ denote the permutation matrix which takes the formal basis{z1,...,zn}to the formal basis{x1,...,xn}. We will denote byC1the(n−1)-dimensional vector space spanned by the formal basis {xi+1−xi:i=1,...,n−1}. Set

Ii:=

xi,xi+1

fori=1,...,n−1. (8) Thus, we can deﬁne the Markov transition matrixA᏷(fµ)as above. Letϕdenote the incidence matrix that takes the formal basis{x1,...,xn}ofC0to the formal basis{x2−x1,...,xn−xn−1}of C1. Put η:=ϕπ. As in [7, 8], we obtain an endomorphismαofC1, that makes the following diagram commutative:

C0 η

ω

C1 α

C0 η C1.

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We haveα=ηωη^T(ηη^T)⁻¹. Remark that if we neglect the negative signs on the matrixα, then we will obtain precisely the Markov transition matrixA᏷(fµ). In fact, consider the(n−1)×(n−1)matrix

β:=

1_n_L 0 0 −1_n_R

, (10)

where 1_n_Land 1_n_Rare the identity matrices of ranksnLandnR, respectively, withnL(nR) being the number of intervalsIiof the Markov partition placed on the left- (right-) hand side of the turning point offµ. 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.

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The matrixγ᏷(fµ)makes the diagram C0

η γ_᏷(fµ)

C1 β

C0 η C1

(13)

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commutative. Finally, setθ᏷(fµ):=θ᏷(fµ)ω. Then, the diagram

C0 η

θ_᏷(fµ)

C1 A_᏷(fµ)

C0 η C1

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is also commutative. Now, notice that the transpose ofη has the following factorization:

η^T=Y iX, (15)

whereY is an invertible (overZ)n×ninteger matrix given by

Y:=







1 0 ··· 0

0 1 0 ··· 0

... 0 . .. . .. ...

... 0

0 0 ··· 0 1 0

−1 −1 ··· −1 1







, (16)

iis the inclusionC1C0given by

i:=







1 0 0

0 . .. . .. ... ... . .. 0 1

0 ··· 0







, (17)

andXis an invertible (overZ)(n−1)×(n−1)integer matrix obtained from the (n−1)×nmatrix η^T by removing the nth row of η^T. Thus, from the commutative diagram

C1 η^T

A^T_᏷(fµ)

C0 θ^T_᏷(fµ)

C1

η^T C0,

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K-THEORY FOR CUNTZ-KRIEGER ALGEBRAS... 2143 we will have the following commutative diagram with short exact rows:

0 C1 i

A

C0 p θ

C0/C1 0

0

0 C1

i C0 p C0/C1 0,

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where the mappis represented by the 1×nmatrix[⁰^···^{0 1}]and

A =XA^T_᏷_(f_µ₎X⁻¹, θ =Y⁻¹θ_᏷^T_(f_µ₎Y , (20)

that is,A is similar toA^T_᏷_(f_µ₎overZandθ is similar toθ^T_᏷_(f_µ₎overZ. Hence, for example, by [10] we obtain, respectively,

Zⁿ⁻¹/ 1−A

Zⁿ⁻¹Zⁿ⁻¹/

1−A᏷(fµ) Zⁿ⁻¹, Zⁿ/

1−θ

ZⁿZⁿ/

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⁻¹Zⁿ/ 1−θ

Zⁿ, Zⁿ⁻¹/

1−A᏷(fµ)

Zⁿ⁻¹Zⁿ/

1−θ᏷(fµ)

Zⁿ. (23)

Next, we will computeZⁿ/(1−θ᏷(fµ))Zⁿ. From the previous discussions and notations, then×nmatrixθ᏷(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 matrixθ᏷(fµ)completely describes the dynamics offµ. Finally, using row and column elementary operations over Z, we can ﬁnd invertible

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(overZ) matricesU1andU2with 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 ᏻA_᏷(fµ)

Zⁿ⁻¹/ 1−A^T_᏷_(f_µ₎

Zⁿ⁻¹Za, (26) where

a= 1+

n−1

l=1

l i=1

εi(c)

, n∈N\{1}. (27)

Example2. Set

᏷fµ

=(RLLRRC)^∞, (28)

whereR= −1, L= +1, andC=0. Thus, we can construct the 5×5 Markov transition matrixA᏷(fµ)and the matricesθ᏷(fµ),ω,ϕ, andπ:

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





 ,

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





 .

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K-THEORY FOR CUNTZ-KRIEGER ALGEBRAS... 2145 We have

K0 ᏻA_᏷(fµ)

Z2, K1 ᏻA_᏷(fµ)

{0}. (30)

Remark3. In the statement ofTheorem 1the casea=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.

Remark4. In [6], Katayama et al. have constructed a class ofC^∗-algebras from theβ-expansions of real numbers. In fact, considering a semiconjugacy from the real quadratic map to the tent map [14], we can also obtainTheorem 1 using [6] and theλ-expansions of real numbers introduced in [4].

Remark5. In [13] (see also [12]) and [11], the BF-groups are explicitly cal- culated with respect to another kind of maps on the interval.

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of Math. (2)106(1977), no. 1, 73–92.

[2] J. Cuntz,A class ofC^∗-algebras and topological Markov chains. II. Reducible chains and the Ext-functor forC^∗-algebras, Invent. Math.63(1981), no. 1, 25–40.

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

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[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.

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

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[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.

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248.

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

E-mail address:[email protected]

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

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