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To describe the cylinders and points in the attractor of iterated function systems and graph directed systems, one uses a natural coding. In this section we give a more abstract way of manipulating words that will become useful in describing the construction in random systems. We introduce two binary operationst andthat take over the rˆoles of set union and concatenation, respectively, to manipulate strings in a meaningful way.

32 CHAPTER 3. RANDOM GRAPH DIRECTED SYSTEMS

Definition 3.2.2. Let GE _{be a finite alphabet, which in this chapter is the set of} letters identifying the edges of the graphsΓi,i.e.GE_{={e|e∈E(i)andi∈Λ}. We} define the prime arrangements G to be the set of symbolsG={∅, ε0} ∪ GE.

Define i to be the free monoid with generators GE and identity (empty word)

ε0, and defineit to be the free commutative monoid with generatorsiand identity ∅. We define to be left and right multiplicative over t, and ∅to annihilate with respect to . That is, given an element e of _i, we get e∅=∅e= ∅. We define_i∗ be the set of all finite combinations of elements ofG and operations tand

. Using distributivity _i = (_i∗,t,) is the non-commutative free semi-ring with ‘addition’tand ‘multiplication’and generator GE _{and we will call}

ithe semiring of arrangements of words and refer to elements of _i∗ as (finite) arrangements of words.

We adopt the convention to ‘multiply out’ arrangements of words and write them as elements of _i. Furthermore we omit brackets, where appropriate, replaceby concatenation to simplify notation, and for arrangements of wordsφwriteϕ∈φto refer to the maximal subarrangementsϕthat do not containtand are thus elements ofϕ∈i.

Example 3.2.3. LetGE_{={0,1}. The set of prime arrangements is then{}

∅, ε0,0,1}

and the elements of the semiringi∗ are all possible concatenationsand unions t, e.g.

10t1 = 10t1, (110t101tε0)1 = 1101t1011t1, ∅(10t101) =∅, . . . The usefulness of the description above is thati∗is ring isomorphic to the set of all cylinders with set union and concatenation as the binary operations and we can useand tto describe collections of cylinders. For example the set containing all cylinders of lengthkcan be identified with the arrangement of words (0t1)k.

We can now use the algebraic structure above to give descriptions of 1-variable RIFS.

Example 3.2.4. Consider the simple setting of just two Iterated Functions Systems L={I1,I2} that are picked at random according to probability vector ~π ={π1, π2},

πi>0. Letφi=a1it · · · tain, whereaij are the letters in the alphabet associated with IFS_Ii. The arrangement of words describing the cylinders of lengthkwith realisation

ω is then simply

φω1φω2 · · · φωk.

An arrangement of words is nothing more than a formalisation of the standard alphabet one uses to describe words, where is concatenation of letters and t is the union of several letters. Before we can apply this construction to our RGDS we need to extend this concept to the natural analogue of matrix multiplication×and addition, which we also refer to ast.

Definition 3.2.5. LetM andN be squaren×nmatrices andv={v1, . . . , vn}be a

n-vector with entries being arrangements of words. We define matrix multiplication in the natural way,

(M×N)i,j = n

G

k=1

(Mi,kNk,j), (MtN)i,j =Mi,jtNi,j,

(v×M)i= n

G

k=1

(vkMk,i).

We extend this to multiplication of countable (finite or infinite) square matrices with matrix entries.

3.2. NOTATION AND PRELIMINARIES FOR1-VARIABLE RGDS 33

v1 e1 v2 e4

e3

e2

Figure 3.1: Graph used in Example 3.2.7

Definition 3.2.6. Let M∗ and N∗ be elements of Mk,k(Mn,n((i∗)) and v∗ ∈ (Mn,n((i∗))k, wherek∈N∪{N}. We define multiplication and addition by

(M∗×N∗)i,j= k

G

l=1

(M∗i,l×N∗l,j), (M∗tN∗)i,j=M∗i,jtN∗i,j, and (v∗×M∗)i= k G l=1 (v∗_l ×M∗_l,i).

For graph directed attractors we can now describe codes as arrangements of words in matrix form. Recall that a graph directed attractor is a collection of sets which is invariant under maps between them, see (2.1.3). The aim of codings in this setting is to describe all paths in the graph and every point in the attractor corresponds to an infinite such path. We apply arrangements of words to succinctly write and modify such paths.

Example 3.2.7. Let Γ0 be the graph in Figure 3.1. We define

M0= ∅ e1te3 e2 e4 .

All paths of length, sayk= 2, are the arrangements contained in(M0)2=M0×M0,

that is (M0)2= e1e2te3e2 e1e4te3e4 e4e2 e2e1te2e3te4e4 ,

where e.g.e2e1te2e3te4e4 represents nothing but the set of paths starting at vertex

v2 and ending atv2 of length2.

Slightly more abstractly, we can now take multiple graphs and consider paths that traverse edges of graphi at stepi as the following example shows.

Example 3.2.8. Let Γ={Γi}i∈Λ be a finite collection of graphs sharing vertex set

V. Define the matrixM(i)over arrangements of words by

(M(i))u,v=

G

e∈_uE_v(i)

e, u, v∈V.

Let ω ∈ ΛN_{. The arrangement of words describing all paths of length k starting at} v∈V, traversing through graph Γωi at stepi, is then simply

v(v)M(ω1)M(ω2). . .M(ωk), where(v(v))i=

(

ε0 if i=v,

∅ otherwise.

Thus the arrangement of words encode paths that in turn will be associated with sets. The limits of these sets as we multiply more and more matrices are the ob- ject under investigation and we will expand on them after increasing the level of abstractivication one more level.

34 CHAPTER 3. RANDOM GRAPH DIRECTED SYSTEMS