IJMMS 2003:34, 2139–2146
PII. S0161171203209236
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
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 the K-groups for the Cuntz-Krieger algebras ᏻA᏷(fµ ) , where A᏷(fµ )
is the Markov transition matrix arising from the kneading sequence ᏷(fµ ) of the
one-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 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 establish 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


−1, if fµn (x) > c,



εn (x) = 0,
if fµn (x) = c,



+1, if f n (x) < c.
µ
(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)) = σ ε(x), where σ is the
N
are
shift map. Let
= {−1, 0, +1} be the alphabet set. The sequences on
ordered lexicographically. However, this ordering is not reflected by the mapping 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
n
sequence θ = (θn )∞
i=0 εi . If ε = ε(x) is the itinerary
n=0 is defined by θn =
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)
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) · · ·
∞ ∞
≡ ε1 (c) · · · εn (c)
= ε0 fµ (c) · · · εn−1 fµ (c)
(3)
for some n ∈ N. The sequences σ i (᏷(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

1, if fµ int Ii ⊇ int Ij ,
A᏷(fµ ) := aij
with aij =
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
ᏻA᏷(fµ ) . In [2], Cuntz proved that
K0 ᏻA Zr / 1 − AT Zr ,
K1 ᏻA 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
l
n−1
ᏻA᏷(fµ ) Za with a = εi (c),
1+
l=1 i=1

{0}, if a ≠ 0,
K1 ᏻA᏷(fµ ) Z,
if a = 0.
(6)
Proof. Set zi = εi (c)εi+1 (c) · · · for i = 1, 2, . . . . Let zi = fµi (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 × n
matrix representing the shift map σ . Let C0 be the vector space spanned by the
}. Now, let ρ be the permutation of the set {1, . . . , n},
formal basis {z1 , . . . , zn
on the unit interval [0, 1], that is,
which allows us to order the points z1 , . . . , zn
0 < zρ(1)
< zρ(2)
< · · · < zρ(n)
< 1.
(7)
K-THEORY FOR CUNTZ-KRIEGER ALGEBRAS . . .
2141
with i = 1, . . . , n and let π denote the permutation matrix which
Set xi := zρ(i)
} to the formal basis {x1 , . . . , xn }. We will detakes the formal basis {z1 , . . . , zn
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 matrix A᏷(fµ ) as above. Let ϕ 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:
η
C0
C1
ω
(9)
α
C0
C1 .
η
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 :=
1 nL
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:


i = 1, . . . , n,
γ = εi (c),


 ii
with γin = −εi (c), i = 1, . . . , n,
γ᏷(fµ ) := γij



γ = 0,
otherwise.
ij
(12)
The matrix γ᏷(fµ ) makes the diagram
C0
η
γ᏷(fµ )
C0
C1
β
η
C1
(13)
2142
NUNO MARTINS ET AL.
commutative. Finally, set θ᏷(fµ ) := θ᏷(fµ ) ω. Then, the diagram
C0
η
C1
(14)
A᏷(fµ )
θ᏷(fµ )
C0
η
C1
is also commutative. Now, notice that the transpose of η has the following
factorization:
ηT = Y iX,
(15)
where Y is an invertible (over Z) n × n integer matrix given by

1

0


 .
 ..

Y := 




0
−1
0
1
0
..
.
0
−1
···
0
..
.
···
..
.
···
···
0
1
−1
0

0

0

.. 
.

,




0
1
(16)
i is the inclusion C1 C0 given by

1


0


i :=  ..
.



0
0
..
.
..
.
..

0
.. 

.


,
0

1

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
C1
ηT
AT
᏷(fµ )
C1
C0
T
θ᏷
(fµ )
ηT
C0 ,
(18)
K-THEORY FOR CUNTZ-KRIEGER ALGEBRAS . . .
2143
we will have the following commutative diagram with short exact rows:
C1
0
i
C0
C1
C0 /C1
θ
A
0
p
C0
i
0
p
C0 /C1
0,
where the map p is represented by the 1 × n matrix [ 0 ···
A = XAT᏷(fµ ) X −1 ,
(19)
0
0 1]
and
T
θ = Y −1 θ᏷
(fµ ) Y ,
(20)
T
that is, A is similar to AT᏷(fµ ) over Z and θ is similar to θ᏷
(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],
A
θ =
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

−ε1 (c)

..


.


θ᏷(fµ ) := 


−ε
 n−1 (c)
0
ε1 (c)
0
..
.
0
..
.
..
.
0
···
···
..
.
0
..
.






.

0

εn−1 (c)

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
2144
NUNO MARTINS ET AL.
(over Z) matrices U1 and U2 with integer entries such that


l
n−1
1 +
εi (c)


l=1 i=1



1 − θ᏷(fµ ) = U1 











 U2 .





1
..
.
(25)
1
Thus, we obtain
K0 ᏻA᏷(fµ ) Zn−1 / 1 − AT᏷(fµ ) Zn−1 Za ,
(26)
where
l
n−1
εi (c),
a = 1 +
n ∈ N\{1}.
(27)
l=1 i=1
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 :

0

0

A᏷(fµ ) = 
0

0
1

0

0

0

ω=
0

0

1
1
0
0
0
0
0


1
0
0
0
1
1
0
0
1
0
0
1
0
1
0
0

1

1
,

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

0

0

0

0

π =
0

0

1
−1
0
0
0
0
0
1

−1

−1

θ᏷(fµ ) = 
 1

 1

0




ϕ=



1
0
0
0
0
0
0
1
0
0
0
0
−1
0
0
0
1
0
0
1
−1
0
0
0
0
1
0

0

0

1

.
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

0

1
−1
1
−1
1
−1



,
 (29)


1
K-THEORY FOR CUNTZ-KRIEGER ALGEBRAS . . .
2145
We have
K0 ᏻA᏷(fµ ) Z2 ,
K1 ᏻA᏷(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 calculated with respect to another kind of maps on the interval.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
R. Bowen and J. Franks, Homology for zero-dimensional nonwandering sets, Ann.
of Math. (2) 106 (1977), no. 1, 73–92.
J. Cuntz, A class of C ∗ -algebras and topological Markov chains. II. Reducible
chains and the Ext-functor for C ∗ -algebras, Invent. Math. 63 (1981), no. 1,
25–40.
J. Cuntz and W. Krieger, A class of C ∗ -algebras and topological Markov chains,
Invent. Math. 56 (1980), no. 3, 251–268.
B. Derrida, A. Gervois, and Y. Pomeau, Iteration of endomorphisms on the real
axis and representation of numbers, Ann. Inst. H. Poincaré Sect. A (N.S.) 29
(1978), no. 3, 305–356.
J. Guckenheimer, Sensitive dependence to initial conditions for one-dimensional
maps, Comm. Math. Phys. 70 (1979), no. 2, 133–160.
Y. Katayama, K. Matsumoto, and Y. Watatani, Simple C ∗ -algebras arising from
β-expansion of real numbers, Ergodic Theory Dynam. Systems 18 (1998),
no. 4, 937–962.
J. P. Lampreia and J. Sousa Ramos, Trimodal maps, Internat. J. Bifur. Chaos Appl.
Sci. Engrg. 3 (1993), no. 6, 1607–1617.
, Symbolic dynamics of bimodal maps, Portugal. Math. 54 (1997), no. 1,
1–18.
S. Lang, Algebra, Addison-Wesley, Massachusetts, 1965.
D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.
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.
N. Martins and J. Sousa Ramos, Cuntz-Krieger algebras arising from linear mod
one transformations, Differential Equations and Dynamical Systems (Lisbon, 2000), Fields Inst. Commun., vol. 31, American Mathematical Society,
Rhode Island, 2002, pp. 265–273.
, Bowen-Franks groups associated with linear mod one transformations, to
appear in Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2003.
2146
[14]
[15]
NUNO MARTINS ET AL.
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.
P. Štefan, A theorem of Šarkovskii on the existence of periodic orbits of continuous
endomorphisms of the real line, Comm. Math. Phys. 54 (1977), no. 3, 237–
248.
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
E-mail address: sramos@math.ist.utl.pt