Cálculo de voladuras

As before we let_L={Ii}i∈Λ be a collection of IFSs indexed by Λ ={1, . . . , N}. Let

~π={π1, . . . , π#Λ}be a probability vector such thatπi>0 for alli∈Λ. We writeP for the product probability measure on Ω = ΛN_{induced by~π, see Section 1.6 and 5.1.}

For_Ii ={f_ij}#Ii

j=1 we code the mapf

j

i by the lettere j

i and refer to the arrangement of words encoding the IFS_Ii by Wi =e1i te2i t · · · te

#Ii

i . The 1-variable attractor associated withω∈Ω can then be expressed as

Fω= ∞ \ k=1 f(Ck_ω,∆), whereCk

ω=Wω1 · · · Wωk andf is defined recursively as in Definition 4.3.2.

First we obtain a sure upper bound, i.e. an upper bound which holds for all realisations.

Theorem 6.2.1. Let (_L, ~π) be a RIFS consisting of IFSs of similarities such that 0<#_Ii<∞for alli∈Λ. Assume that(_L, ~π)satisfies the UOSC and letFω denote the associated1-variable attractor. Then, for allω∈Ω,

dimAFω ≤ sup i∈Λ

dimAFi.

The proof of Theorem 6.2.1 will be given in Section 6.6.1.1. Note that for each

i∈Λ, dimAFiis the Assouad dimension of the deterministic self-similar setFi, which may be computed via the Hutchinson-Moran formula since the OSC is satisfied. We will provide an example in Section 6.2.1 showing that this upper bound can fail if we do not assume the UOSC. We are also able to obtain an almost sure lower bound. Theorem 6.2.2. Let (_L, ~π) be a RIFS consisting of IFSs of similarities such that 0<#Ii<∞ for alli∈Λ. Let Fω denote the associated 1-variable attractor. Then, for almost allω∈Ω, we have

dimAFω ≥ sup i∈Λ

dimAFi. (6.2.1)

The proof of Theorem 6.2.2 will be given in Section 6.6.1.2. Note that the The- orem 6.2.2 requires no separation conditions, whereas Theorem 6.2.1 requires the UOSC. Combining the upper and lower estimates immediately yields our main result on random self-similar sets.

Theorem 6.2.3. Let (L, ~π) be a RIFS consisting of IFSs of similarities such that 0<#Ii<∞for alli∈Λ. Assume that(L, ~π)satisfies the UOSC and letFω denote the associated1-variable attractor. Then

dimAFω = sup i∈Λ

dimAFi= max

i∈Λ dimAFi,

for almost allω∈Ω.

The results above are in stark contrast to the analogous almost sure formulae for the Hausdorff, packing and box-counting dimension which are some form of weighted average of the deterministic values. As can be deduced from Corollary 3.2.24 the Hausdorff dimension of a random 1-variable self-similar set satisfying the UOSC is almost surely given by the unique zero of the weighted average of the logarithm of the Hutchinson-Moran formulae for the individual IFSs. A neat consequence of this is that the Assouad dimension and the Hausdorff dimension can be almost surely

6.2. THE SELF-SIMILAR SETTING 99

distinct, no matter which separation condition you assume. Recall that in the de- terministic setting the WSP is sufficient to guarantee equality, and in the random setting the Hausdorff and box-counting dimensions almost surely coincide, even if there are overlaps. In fact the only way the Assouad and Hausdorff dimensions can almost surely coincide in the UOSC case is if all of the deterministic IFSs had the same similarity dimension. Also, apart from this special situation, our result shows that random self-similar sets are almost surely not Ahlfors regular, as for Ahlfors reg- ular sets the Hausdorff and Assouad dimensions coincide. Finally we obtain precise information on the size of the exceptional set of Theorem 6.2.3.

Theorem 6.2.4. Let (_L, ~π) be a RIFS consisting of IFSs of similarities such that 0<#_Ii<∞ for alli∈Λ. Assume that(_L, ~π)satisfies the UOSC and letFω denote the associated1-variable attractor. Assume further that dimAFi is not the same for alli∈Λ,i.e.the similarity dimensions of the deterministic attractors are not all the same. Then the exceptional set

E =

ω∈Ω|dimAFω<max

i∈Λ dimAFi

is a set of full Hausdorff dimension, despite being aP-null set,i.e. dimHE= dimHΩ.

The proof of Theorem 6.2.4 can be found in Section 6.6.1.3. The following two figures depict some examples of random self-similar sets. The RIFS is made up of three deterministic IFSs, which are shown in Figure 6.1. Dotted squares indicate the (homothetic) similarities used. In Figure 6.2, three different random realisations are shown, which will (almost surely) all have the same Assouad dimension as the maximum of the three deterministic values.

Figure 6.1: Deterministic self-similar attractors F1,F2andF3.

Figure 6.2: Random self-similar attractors Fα, Fβ and Fγ for different realisations

α= (1,2,3,1,2,1,3,3, . . .), β= (2,1,2,1,1,1,1,3, . . .), γ = (2,3,3,2,1,1,1,3, . . .)∈ Ω.

100 CHAPTER 6. ASSOUAD DIMENSION OF RANDOM SETS

We finish this section by mentioning that Li et al. [LLMX] studied the Assouad dimension of Moran setsE generated by two sequences

{nk∈N}∞k=1 and {φk ∈Rnk}∞k=1,

where nk indicates the number of contractions, andξk = (ck,1,· · ·, ck,nk) gives the contraction ratios at thekth level. They show that

dimAE= lim

m→∞sup_k sk,k+m,

wheresk,k+mis the unique solution to the equation k+m Y i=k+1 ni X j=1 (ci,j)s= 1.

By choosingξk= (ck,1,· · ·, ck,nk) from a fixed number of patterns, such a Moran set may be regarded as a particular realisation of our random self-similar sets. Therefore this result gives information about specific realisations, whereas our results study the generic situation.

6.2.1

An example with overlaps

Here we provide an example showing that the assumption of some separation condition in Theorem 6.2.1 is necessary. Let the RIFSLbe the system consisting of two IFSs of similarities,I1andI2. LetI1be the IFS consisting of the three mapsS1,1, S1,2and

S1,3 andI2 consist of the three mapsS2,1, S2,2 andS2,3, whereSi,j:R→Rand

S1,1= 1₂x, S1,2= ₄1x, S1,3= ₁₆1x+15₁₆,

S2,1= 1₃x, S2,2= ₉1x, S2,3= ₈₁1x+80₈₁.

AsSi,1 andSi,2 have the same fixed point for i= 1,2, both I1 and I2 fail the OSC.

Note that

Id = (Si,1◦Si,1)−1◦Si,2∈ E

where E is the set of composition maps, see Definition 2.1.3. We conclude that the two IFSs fail the OSC and so_Lfails to satisfy the UOSC, see the discussion following Definition 2.1.3. Letcj_i be the contraction rate ofSi,j. If one considers the individual IFSs, since (logc1

i)/(logc2i)∈Qfori= 1,2, one can show directly from the definition that the WSP is satisfied. Therefore, for both systems the Assouad and Hausdorff dimensions coincide and are therefore no greater than their similarity dimensions, see [FHOR]. That is dimAFi ≤ si for i = 1,2, where si is given implicitly by the Hutchinson-Moran formula P3

j=1(c

j i)

si _{= 1. Solving numerically we find that}

s1 ≈ 0.81137 and s2 ≈ 0.511918 and so maxi∈Λ dimAFi < 1. Consider however

ω = (1,2,1,2,1, . . .). This is equivalent to the deterministic IFS consisting of the 9 possible compositions of a map from_I1 with a map from I2. Consider just the two

maps T1=S1,1◦S2,2= 1 18x, T2=S1,2◦S2,1= 1 12x.

One can check that log 18/log 12 ∈/ Q and therefore using an argument similar to the one in [Fr2, Section 3.1] one can show that dimAF(1,2,1,...)= 1, which is strictly

greater than the maximum given by the deterministic IFS, showing that if the UOSC is not satisfied, then the Assouad dimension of particular realisations can exceed the maximum of the deterministic values.