Conjecture 5.3.5. Let Kv(ω)be an attractor of a 1-variable random graph directed system (Γ, ~π) satisfying the UOSC, all conditions in Definition 3.2.9 and Condi- tion 5.1.1. Then there existsβ, β0∈Rsuch that for all ε >0, almost surely,
Hh1(t,β,ε)(K
v(ω)) = 0 and Hh1(t,β 0,−ε)
(Kv(ω)) =∞.
Since V-variable attractors, see Section 2.2.1, are nothing but a special case of random graph directed systems, this would show thatV-variable sets behave intrin- sically like 1-variable sets. This means that these systems cannot truly interpolate between 1-variable and∞-variable attractors.
5.4
Packing measure for
1-variable constructions
Recall that dimP(F) = dimB(F) ifF is compact and dimBF∩O= dimBF for every open O that intersects F, see Theorem 1.7.8. Similarly, we can prove the following Lemma.
Lemma 5.4.1. Let hbe a doubling gauge function and let Fω be the 1-variable at- tractor of a RIFS (L, µ). Assume that all maps in the IFSs are strict contractions such that there exist0 < cmin≤cmax <1 such thatcmin|x−y| ≤ |fij(x)−f
j i(y)| ≤
cmax|x−y|for all i, j and allx, y∈Rd. Let ω∈Ω, then
Ph
0(Fω) =∞ =⇒ Ph(Fω) =∞ and
Ph
0(Fω) = 0 =⇒ Ph(Fω) = 0.
Note that we did not make any assumption on the contractions and separation conditions in this Lemma.
Proof. The second claim follows by definition ofPhand it remains to prove the first, i.e.we need to show that
inf (∞ X i=1 Ph 0(Ei)|Fω⊆ ∞ [ i=1 Ei ) =∞ ifPh
0(Fω) =∞. Now Fω is compact, and so we can assume the subcover of{Ei}is finite. Thus there existsj andφ∈Ck
ωfor someksuch thatf(φ, Fσkω)⊂Ej and so, for somendependent on the cover,
Ph(F ω) = inf ( n X i=1 Ph 0(Ei) Fω⊆ n [ i=1 Ei ) ≥inf ( Ph 0(Ej) Fω⊆ n [ i=1 Ei ) (j as above) ≥inf ( Ph 0(f(φ, Fσkω)) Fω⊆ n [ i=1 Ei ) ≥inf ( lim δ→0κP h δ(Fσkω) =∞ Fω⊆ n [ i=1 Ei ) (a.s.)
where the infimum is taken over all finite covers and κ is a finite constant arising from the maximal distortion of the mapf(φ, .) (bounded byck
min andckmax) and the
doubling ofh, see Lemma 5.3.1.
Inspired by the recent progress on the packing measure of random recursive at- tractors mentioned above, we would hope that using the gaugeh1(t, β, γ) should give
94 CHAPTER 5. HAUSDORFF AND PACKING MEASURES
similar similar convergence and divergence, depending on the sign ofγ. This can be achieved by considering the natural dual toh1. Letα≥0,γ∈Randβ >0, we set
h∗1(t, β, γ) =tαexp−p
2βlog(1/t) log log(βlog(1/t))
1−γ
.
We remark that, in light of Lemma 5.4.1, we only sketch proofs.
Theorem 5.4.2. LetFωbe the random homogeneous attractor associated to the self- similar RIFS (L, µ) satisfying the UOSC and suppose that ciλ = cλ ∈ [cmin, cmax]
for every i ∈ {1, . . . ,#Iλ} and λ ∈ Λ, where 0 < cmin ≤ cmax < 1. Let ε > 0,
α= ess dimHFω = ess dimPFω andβ0∗ =η0Var(logSαω1) for some η
∗
0 ∈R(arising
in the proof ). ThenPh∗1(t,β0∗,ε)(F
ω) =∞ almost surely. Proof. By Lemma 5.4.1 we only have to analyse limδ→0P
h∗1(t,β0∗,ε)
δ (Fω). Let hXi denote the compact convex hull of X. Since cλ is uniformly bounded away from 0 and 1 and #Iλ is uniformly bounded above there exist l, M < ∞ such that there exists at least oneφch(ω)∈Clω for whichf(φch(ω),hFωi)⊂ hFωi. Thus we get, in a similar fashion to the Hausdorff measure argument,
lim δ→0P h∗1(t,β∗0,ε) δ (Fω) = lim δ→0sup ( ∞ X i=1 h∗1(2ri, β0∗, ε) {B(xi, ri)}is a disjoint
collection of balls with 2ri< δ andxi∈Fω
) ≥lim sup k→∞ X φ∈Ck ω h∗1(|f(φφch(σkω), Fσk+lω)|, β0∗, ε) ≥lim sup k→∞ k Y i=1 Nωi ! h∗1(cω1cω2. . . cωkc l min, β ∗ 0, ε) ≥lim sup k→∞ k Y i=1 Nωi ! κ(cω1cω2. . . cωk) αexp −(1−ε) ·qβ∗
0log(1/(cω1. . . cωk) log log(β
∗ 0log(1/(cω1. . . cωk)) ≥lim sup k→∞ κexp k X i=1
logSαωi−(1−ε)pvklog logvk !
=∞,
writingv = Var(Sαω1) and having used he law of the iterated logarithm in the last step.
Finally, we also obtain an upper bound.
Theorem 5.4.3. LetFωbe the random homogeneous attractor associated to the self- similar RIFS (L, µ) satisfying the UOSC and suppose that ciλ = cλ ∈ [cmin, cmax]
for every i ∈ {1, . . . ,#Iλ} and λ ∈ Λ, where 0 < cmin ≤ cmax < 1. Let ε > 0,
α= ess dimHFω= ess dimPFω andβ∗=ηVar(logSαω1)for some η
∗∈
R(arising in the proof ), thenPh∗1(t,β
∗,ε)
(Fω) = 0 holds almost surely. Proof. By the homogeneity of the construction
sup ( ∞ X i=1 h∗1(2ri, β0∗, ε)
{B(xi, ri)}are disjoint balls with 2ri < δ andxi∈Fω )
5.4. PACKING MEASURE FOR1-VARIABLE CONSTRUCTIONS 95 ≤κ sup n≥k(δ) ( n Y i=1 Nωi ! h∗1(cω1. . . cωn, β ∗, ε) )
for someκ > 0 depending on the diameter of Fω and the doubling properties of h1
only. So, for an appropriately chosen η, we obtain the desired conclusion from the law of the iterated logarithm in a similar fashion to results above.
Clearly limδ→0(h∗1(t, β, γ))/(h1(t, β, γ)) = 0 and so h∗1 and h1 are not equivalent
gauge functions. This, however, means that while Hausdorff and packing dimensions coincide, in this simple setting they also require different but related gauge functions for finite and infinite measure.
This of course means that while both the 1-variable as well as the ∞-variable constructions are very natural and the Hausdorff and packing dimensions coincide, their precise asymptotic behaviour measured by the gauge functions differ immensely. This means that any implicit theorem that was to capture these fine details must take into account the random process defining them.
Chapter
6
The Assouad dimension of randomly
generated sets
6.1
Introduction
In this last chapter, we study the generic Assouad dimension for a variety of different models for generating random fractal sets. We start by considering the 1-variable random iterated function systems model. Recall that we already established some results regarding the Assouad dimension in the self-similar setting in Chapter 3. In Section 6.2 we revisit these results and give more precise results in the 1-variable RIFS case, in particular we establish a sharp bound using the uniform open set condition as opposed to the uniform strong separation condition,cf.Theorem 3.2.27. We compute the Hausdorff dimension of the exceptional set where this value is not attained.
We then consider the setting of self-affine carpets in Section 6.3 and establish the almost sure Assouad dimension of 1-variable Bedford-McMullen carpets. In par- ticular, these sections seek the generic dimension from a measure theoretic point of view. In Section 6.4 we consider 1-variable attractors from a topological point of view and compute the Assouad dimension for a residual subset of the sample space. This approach was initiated by Fraser [Fr3] and we compute the generic dimension for a finite collection of IFSs with bi-Lipschitz contractions.
In Section 6.5 we will return to Mandelbrot percolation and compute, conditioned on non-extinction, the almost sure Assouad dimension of fractal percolation as well as the almost sure Assouad dimension of all orthogonal projections of the percolation simultaneously. While the first conclusion follows directly from Theorem 3.3.6 we prove this specific example here on its own. A somewhat surprising corollary of our results is that, conditioned on non-extinction, almost surely the fractal percolation cannot be embedded in any lower dimensional Euclidean space, no matter how small the almost sure Hausdorff dimension is.
The key common theme throughout this chapter is that the Assouad dimension is always generically as large as possible. In the measure theoretic setting this be- haviour is completely different from that observed by other important notions of dimension, such as Hausdorff, packing or box-counting, where these dimensions are generically an intermediate value, which take the form of an appropriately weighted average of deterministic values,cf.Theorem 3.2.23, Corollary 3.2.24, Theorem 3.4.2, Corollary 4.3.7, and Corollary 4.4.6.
In the topological setting, the generic dimensions of random fractals were shown to be ‘extremal’ in [Fr3]: some are generically as small as possible and others are generically as large as possible. Interestingly, the Assouad dimension of random attractors agree in both the measure theoretic and topological framework. This is also in stark contrast with what is ‘usually’ the case. A classical example being that
The content of this chapter is based onThe Assouad dimension of randomly generated fractals
in collaboration with Jonathan M. Fraser and Jun-Jie Miao, and to appear in Ergodic Theory and Dynamical Systems, see [FMT].
98 CHAPTER 6. ASSOUAD DIMENSION OF RANDOM SETS
Lebesgue almost all real numbers are normal, but a residual set of real numbers are as far away from being normal as possible [HLOPS, S].