Capítulo II: Caracterización del sistema de costo y del sistema de
2.1 Caracterización de la industria del mueble
The checker graphon WC represents a large graph formed by disjoint complete graphs on the 1/2,1/4,1/8, . . . fractions of its vertices. We now present a fam- ily of iterated checker graphons. Informally speaking, we start with the checker graphon WC and at each iteration, we paste a scaled copy of WC on each clique of the current graphon. The formal definition is as follows. Fix k ∈N0. If k = 0,
Figure 2.7: The iterated checker graphons WC0,WC1 and WC2.
defineIj0,j0 ∈N0, to be the interval
Ij0 = 1−2−j0,1−2−j0−1. Ifk >0, we defineIj0,...,jk for (j0, . . . , jk)∈N k 0 as Ij0,...,jk = supIj0,...,jk−1 −2 −jk|I j0,...,jk−1|,supIj0,...,jk−1 −2 −jk−1|I j0,...,jk−1| .
Thek-iterated checker graphon WCkis then defined as follows: WCk(x, y) is equal to 1 if there exists a (k+ 1)-tuple (j0, . . . , jk)∈Nk0 such that bothxand ybelong to the
intervalIj0,...,jk, and it is equal to 0 otherwise. The iterated checker graphons W
0 C,
W1
C and WC2 are depicted in Figure 2.7. Note thatWC0 =WC and the definition of
Ij0 coincides with that given in Subsection 2.2.2. We will also refer to an interval
Ij0,...,jk as to a k-iterated binary interval.
ForX∈ {B, C} and Y ∈ {X, . . . , E}, we set
W0(ηX(x), ηY(y)) =
(
WC1(x, y) ifX=B, and
WC2(x, y) ifX=C
for allx, y∈[0,1)2. We also set the tile D×Dto be such that
W0(ηD(x), ηD(y)) =WC3(x, y)
for allx, y ∈[0,1)2. This also defines the values of W0 in the symmetric tiles, i.e.,
the values for the tileX×Y determine the values for the tile Y ×X.
Consider the decorated constraints depicted in Figures 2.8 and 2.9. We first analyze the structure of the tile B×B, then all the tilesB ×Y,Y ∈ {B, . . . , E}, then the tileC×C, then all the tiles C×Y,Y ∈ {C, . . . , E}, before finishing with the tile D×D. Fix (X, Y) to be one of the pairs (A, B),(B, C) or (C, D). We assume thatW(x, y) =W0(g(x), g(y)) for almost every (x, y)∈X×X and almost
Y Y Y
= 0
Y Y Y P P= 0
X Y Y= 0
Y P Y Y P Y Y Y X X=
X X Y Y X X=
13Figure 2.8: The decorated constraints forcing the structure of the tilesB2,C2, and
D2, where (X, Y)∈ {(A, B),(B, C),(C, D)}. X Z Y X
= 0
Z Z Z Y P P= 0
Y Z Y Y=
Z Z Z Z Z P Z P Y Y X X=
Figure 2.9: The decorated constraints forcing the structure of the iterated checker graphons on the non-diagonal tiles, where (X, Y) ∈ {(A, B),(B, C)} and Z ∈ {C, D, E, F} ifX=Aand Z ∈ {D, E, F} ifX=B.
almost every (x, y)∈Y ×Y.
The first two constraints on the first line in Figure 2.8 imply that there exists a collection J0
Y of disjoint open intervals such that the following holds for almost every (x, y)∈Y2: W(x, y) is equal to 1 if and only iffY(x) andfY(y) belong to the same interval J0 ∈ JY0, and it is equal to 0 otherwise. The third constraint on the first line in Figure 2.8 yields that each interval inJ0
Y is a subinterval of an interval inJX0 .
The first constraint on the second line in Figure 2.8 yields that the following holds for almost every triple (x, y, y0)∈X×Y ×Y such thatfY(y) and fY(y0) are from the same intervalJY0 ∈ JY0 and fX(x) is from the interval JX0 ∈ J
0
X that is a superinterval of JY0 : the measure of JY0 (which is equal to the left hand side of the equality) is the same as the measure of the set of ally00 such that fY(y00)∈JX0 and
fY(y00)>supJY0 (which is equal to the right hand side). It follows that
JY0 = (supJX0 −2γ,supJX0 −γ)
for some γ ∈ (0,|J0
X|/2]. The very last constraint in Figure 2.8 yields for every
JX0 ∈ JX0 that X JY0 ∈J0 Y,J 0 Y⊆J 0 X |JY0 |2 = 1 3|J 0 X|2.
However, this is only possible if the setJ0
Y contains all intervals of the form (supJX0 − 2γ,supJX0 −γ) for every JX0 ∈ JX0 and every γ =|JX0 | ·2−i, i∈N. It follows that
W(x, y) =W0(g(x), g(y)) for almost every (x, y)∈Y ×Y.
We continue to fix a pair (X, Y) ∈ {(A, B),(B, C)}, but in addition we now fix Z ∈ {Y, . . . , E} \ {Y} where Y ∈ {B, C}. Our next goal is to show that
W(y, z) =W0(g(y), g(z)) for almost every (y, z)∈Y×Z, which is achieved using the
decorated constrains given in Figure 2.9. The first constraint in Figure 2.9 implies that it holds for almost every y∈Y thatfZ(NWZ(y))vJX0 whereJX0 is the unique interval of J0
X containing fY(y). The second constraint in Figure 2.9 yields that for almost every y ∈ Y, there exists an interval Jy such that NWZ(y) and f
−1 Z (Jy) differ on a null set,W(y, z) = 1 for almost every z∈fZ−1(Jy), and W(y, z) = 0 for almost everyz∈Z\fZ−1(Jy). The third constraint yields that degYW(y) = degZW(y) for almost every y ∈ Y, i.e., the measure of Jy is the same as the measure of the interval inJY containing fY(y).
Finally, the last constraint in Figure 2.9 implies that almost every quadruple
x∈ X,y ∈Y, z, z0 ∈Z such that fZ(z) < fZ(z0), fZ(z) and fZ(z0) belong to the intervalJy, which is a subinterval ofJX0 ∈ JX0 with fX(x) ∈JX0 , satisfies that the
measure ofNWZ(y) (note thatNWZ(y) is a subset offZ−1(JX0 )) and the measure of all
z0 ∈fZ−1(JX0 )\NZ
W(y) withfZ(z
0)>supJ
y are equal. In particular, the intervalJy is of the form (supJX0 −2γ,supJX0 −γ) for almost everyy ∈Y, whereJX0 is the unique interval ofJX0 containingfY(y). Hence, the intervalJy is equal to the interval inJY0 containing fY(y) for almost every y ∈Y. It follows that W(y, z) = W0(g(y), g(z))
for almost every (y, z)∈Y ×Z.