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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(fµ) 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 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,Idenotes the unit interval[0,1]andcthe unique turning point offµ. Forx∈I, let
εn(x)=
−1, iffn µ(x) > c, 0, iffn
µ(x)=c,
+1, iffµn(x) < c.
(1)
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 onNare 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=ni=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:
We now consider only those kneading sequences that are periodic, that is,
fµ=ε0
fµ(c)···εn−1fµ(c)ε0
fµ(c)···εn−1fµ(c)···
=ε0
fµ(c)···εn−1fµ(c)∞≡ε1(c)···εn(c)∞
(3)
for somen∈N. The sequencesσi((f
µ))=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, iffµintIi⊇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
K0ᏻAZr/1−ATZr, K1ᏻAkerI−At:Zr →Zr, (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):=Zr/(1−A)Zras 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.
(6)
Proof. Setzi=εi(c)εi+1(c)···fori=1,2,....Letzi=fµi(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,...,znon the unit interval[0,1], that is,
K-THEORY FOR CUNTZ-KRIEGER ALGEBRAS... 2141
Setxi:=zρ(i)withi=1,...,nand letπ denote the permutation matrix which takes the formal basis{z1,...,zn}to the formal basis{x1,...,xn}. We will de-note 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.
(9)
We haveα=ηωηT(ηηT)−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:=
1nL 0 0 −1nR
, (10)
where 1nLand 1nRare 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.
(12)
The matrixγ(fµ)makes the diagram
C0
η
γ(fµ)
C1
β
C0 η C1
commutative. Finally, setθ(fµ):=θ(fµ)ω. Then, the diagram
C0
η
θ(fµ)
C1
A(fµ)
C0 η C1
(14)
is also commutative. Now, notice that the transpose ofη has the following factorization:
ηT=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 ηT by removing the nth row of ηT. Thus, from the commutative diagram
C1
ηT
AT(fµ)
C0
θT(fµ)
C1
ηT C0,
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,
(19)
where the mappis represented by the 1×nmatrix[0···0 1]and
A =XAT
(fµ)X−1, θ =Y−1θ(fT µ)Y , (20)
that is,A is similar toAT
(fµ)overZandθ is similar toθT(fµ)overZ. Hence, for example, by [10] we obtain, respectively,
Zn−1/1−AZn−1Zn−1/1−A
(fµ)
Zn−1, Zn/1−θZnZn/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−AZn−1Zn/1−θZn,
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)
(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−AT (fµ)
Zn−1Z
a, (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 −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
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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Nuno Martins: Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal
E-mail address:nmartins@math.ist.utl.pt
Ricardo Severino: Departamento de Matemática, Universidade do Minho, 4710-057 Braga, Portugal
E-mail address:ricardo@math.uminho.pt
J. Sousa Ramos: Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal
