Upload
others
View
3
Download
0
Embed Size (px)
Citation preview
IJMMS 2003:34, 2139–2146PII. S0161171203209236
http://ijmms.hindawi.com© Hindawi Publishing Corp.
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)
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)
2142 NUNO MARTINS ET AL.
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)
K-THEORY FOR CUNTZ-KRIEGER ALGEBRAS . . . 2143
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
2144 NUNO MARTINS ET AL.
(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)
K-THEORY FOR CUNTZ-KRIEGER ALGEBRAS . . . 2145
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.
2146 NUNO MARTINS ET AL.
[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
E-mail address: [email protected]
J. Sousa Ramos: Departamento de Matemática, Instituto Superior Técnico, 1049-001Lisboa, Portugal
E-mail address: [email protected]
Submit your manuscripts athttp://www.hindawi.com
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttp://www.hindawi.com
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
International Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com
Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Stochastic AnalysisInternational Journal of