Generalized Moran sets Generated by Step-wise Adjustable Iterated Function Systems
Tynan Lazarus Qinglan Xia
math.CA math.DS nlin.PS 28A80

	

In this article we provide a systematic way of creating generalized Moran sets using an analogous iterated function system (IFS) procedure.  We use a step-wise adjustable IFS to introduce some variance (such as non-self-similarity) in the fractal limit sets.  The process retains the computational simplicity of a standard IFS procedure.   In our construction of the generalized Moran sets, we relax the second Moran Structure Condition so that the limit set is not necessarily self-similar. We also weaken the fourth Moran Structure Condition so that there are no limitations on the ratios of the diameters of the sets across a generation.
Moreover, we provide upper and lower bounds for the Hausdorff dimension of the fractals created from this generalized process.  Specific examples (Cantor-like sets, Sierpinski-like Triangles, etc) with the calculations of their corresponding dimensions are studied. 

	
	
[2010]28A80
	

	
Bullen's parameter as a seismic observable for spin crossovers in the lower mantle
    Adrian C. Murza
    December 30, 2023
==================================================================================



§ INTRODUCTION



The Moran construction is a typical way to generate self-similar fractals, and has been studied extensively in the literature (e.g. <cit.>,  <cit.>, <cit.>, <cit.>,<cit.>, <cit.>, <cit.>, and references therein).  In this paper, we extend ideas from iterated function systems (IFS) and Moran constructions by describing a new process that allows for the functions to be updated at every iteration while still maintaining the computational simplicity of an IFS.  This process provides more variance in the limit sets (such as non-self-similarity) using an analogous approach to an IFS procedure.  We also give estimates of the Hausdorff dimension of the limit sets created from such a process, and provide concrete examples.



The classic construction of Moran sets was introduced in <cit.>.  We reproduce the definition here with a more current interpretation to introduce notations. 

Let {n_k}_k≥1 be a sequence of positive integers for k ≥ 1.  Here k will represent the generation, and n_k will be the number of children in generation k that each parent set from generation k-1 has.  For any k ∈, define 

    D_k = { (i_1, i_2, ⋯, i_k) : 1 ≤ i_j ≤ n_k, 1 ≤ j ≤ k} and  D = ⋃_k ≥ 0 D_k with  D_0=∅.
 
Let σ = (σ_1, ⋯, σ_k) ∈ D_k and τ = (τ_1, ⋯, τ_m) ∈ D_m, then denote 
    σ * τ = (σ_1, ⋯, σ_k, τ_1, ⋯, τ_m) ∈ D_k+m.

Using this notation, we may express

    D_k ={σ * j |σ∈ D_k-1, 1 ≤ j ≤ n_k}
 to emphasize the process of moving between generations.

Suppose 𝒥 := {J_σ : σ∈ D} is a collection of subsets of ^N.  
Set 

    E_k = ⋃_σ∈ D_k J_σ,  and  F = ⋂_k ≥ 0 E_k.

We call F the limit set associated with the collection 𝒥.


	Suppose that J ⊂^N is a compact set with nonempty interior.  Let {n_k}_k ≥ 1 be a sequence of positive integers, and {Φ_k}_k ≥ 1 be a sequence of positive real vectors with 
	
    Φ_k = (c_k,1, c_k,2, …, c_k,n_k), ∑_1 ≤ j ≤ n_k c_k,j≤ 1, k ∈.
 
	Suppose that := {J_σ : σ∈ D} is a collection of subsets of ^N, where D is given in (<ref>).  We say that the collection  fulfills the Moran Structure provided it satisfies the following Moran Structure Conditions (MSC):
	
		
  MSC(1) J_∅ = J.
		
  MSC(2) For any σ∈ D, J_σ is geometrically similar to J.  That is, there exists a similarity S_σ: ^N →^N such that J_σ = S_σ(J). 
		
  MSC(3) For any k ≥ 0 and σ∈ D_k, J_σ⋆ 1, … , J_σ⋆ n_k are subsets of J_σ, and int(J_σ⋆ i) ∩ int(J_σ⋆ j) = ∅ for i ≠ j.
		
  MSC(4) For any k ≥ 1 and σ∈ D_k-1, 1 ≤ j ≤ n_k, 
		
    diam(J_σ⋆ j)diam(J_σ) = c_k,j.

		
	

	For the collection  fulfilling the MSC, the limit set F given in (<ref>) is a nonempty compact set. This limit set F is called the Moran set associated with the collection .  This Moran set is self-similar, and has been studied extensively by many authors with various approaches (e.g. <cit.>, <cit.>, <cit.>, <cit.>, <cit.>).


The fact that there are four conditions to create a Moran set makes the area ripe for generalizations or restrictions.  Note that in condition MSC(2), the sets in the new generation are geometrically similar, which is a rather strong condition.  In MSC(3) the interiors of the next generation may not overlap, but says nothing else of the spacing between these sets.  Condition MSC(4) requires that the sets in the new generation all have the same pattern of ratios for each iteration.  There is even a hidden condition in MSC(2) that one may want to do away with in that the locations of the sets J_σ *j are completely determined by the similarities used on J_σ. 

 Often times in the literature one may define a self-similar set to be a set satisfying the MSC as well as the fact that the set of similarities {S_σ} has finite cardinality, the ratios described in MSC(4) do not change over generations (i.e. c_k,j = c_j), and that the system is deterministic.  In this case, the dimensions (Hausdorff, Box, Packing etc.) of the Moran set are known to coincide, see <cit.>.  However, the dimensions may differ when we begin to modify the MSC.

Several approaches have been used to relax MSC in order to create more general limit sets.  There are many generalizations for MSC(2).  For a self-similar set, one could change MSC(2) to use conformal maps <cit.> or affine maps <cit.> instead of similarities.  In this setting, however, calculations of the dimension of limit sets can become particularly difficult.  One could also study the limit sets generated by infinitely many similarities, as in <cit.>. In <cit.>, the authors removed MSC(2), but required int(J_σ) = J_σ in their construction, and studied the dimension of the resulting fractals.  In <cit.>, Holland and Zhang studied a construction that replaced similarity maps in MSC(2) with a more general class of functions that are not necessarily contractions.  In <cit.>, Pesin and Weiss removed the requirement for similarities from MSC(2), but also relaxed MSC(3) from non-intersecting basic sets to non-intersecting balls contained in the basic sets.  In particular they pursued sufficient conditions for which the box dimension and Hausdorff dimensions coincide.  For more examples of modifications to the Moran set definition, see <cit.>, <cit.> and the references therein. 

	A special case of Moran sets can be constructed from an iterated function system (IFS).  An iterated function system {S_1, S_2, ⋯, S_m} is a finite family of similarities for a fixed natural number m≥2 (see <cit.> for more details and applications).  In MSC(2), define n_k=m and set S_σ = S_i_k∘ S_i_k-1∘⋯∘ S_i_1 for σ = (i_1, i_2, ⋯, i_k) ∈ D.  Then the resulting Moran set is self-similar and agrees with the attractor of the IFS {S_1, S_2, ⋯, S_m}.  The dimension of the limit set can be quickly calculated from the Moran-Hutchinson formula in <cit.>.  Using iterated function systems is a popular way to construct fractals, and has been used to great effect (e.g. <cit.>, <cit.>, <cit.>, <cit.> ).   
	
	A natural question arises: Can we construct more general fractals (e.g. non-self-similar Moran type sets) using an analogous approach while preserving the computational simplicity of the IFS?  In this paper, we present a method to do so.
	



We first make the following observations about the general construction of a Moran set.  Note that in the construction of a Moran set described in (<ref>),

    J_σ * i = S_i (J_σ),  for all  i=1, ... m,  and σ∈ D.


Suppose that there is a tuning parameter in the expression of the function S_i (e.g. the coefficients a_i, b_i in a linear function S_i(x) = a_i x+ b_i).  One can tune the values of the parameter to get a comparable function.  When J_σ is given, applying the comparable function to J_σ, as in equation (<ref>), will not significantly change the computational complexity of constructing J_σ * i. The advantage of doing this at each iteration is that we introduce some variance into the limit set.  Another observation is about which space the functions are defined. In classical IFS constructions, the functions are usually defined on all of the ambient space ^n (as in <cit.>, the functions are C^1+α diffeomorphisms on ^n).  For our construction, we wish to relax the condition MSC(2) as well.  Instead of restricting our attention to functions of higher regularity defined on the whole ambient space ^n, we use maps from a collection of subsets to itself.  



This article is organized as follows. In Section <ref> we find bounds for the Hausdorff dimension of the limit sets in a general metric space setting of a collection of bounded sets, not necessarily satisfying the MSC conditions. In particular, we introduce the concept uniform  covering  condition in Definition <ref> for the purpose of studying the lower bound of the Hausdorff dimension of the limit set. Then in Section <ref> we formulate the general setup for the construction of Moran-type limit sets using the ideas from a modified IFS procedure, as discussed in the previous paragraph.  In our construction we relax MSC(2) so that the limit set is not necessarily self-similar.  More importantly, we drop MSC(4) from the construction process so that there are no limitations on the ratios of the diameters of the sets.  Specifically, the ratio diam(J_σ * j)diam(J_σ) in (<ref>) is not limited to depend on just k and j, but varies with σ.  This change allows us to produce a mosaic of possible fractals.  An important observation is that the computational complexity of generating these fractals is the same as using an analogous, standard IFS.  In Section <ref> we give estimates of the Hausdorff dimension of the limit sets created from the general construction.  In Section <ref> we apply the results to specific examples, including modifications of the Cantor set, the Sierpinski triangle, and the Menger sponge.  We also give a remark to discuss similarities and differences of this construction with V-variable fractals created by Barnsley, Hutchinson, and Stenflo in <cit.>, <cit.>. In section <ref>, we explore the sufficient conditions needed for a fractal to satisfy the uniform covering condition, which plays a vital role in computing a lower estimate for the Hausdorff dimension of a fractal.



§ HAUSDORFF DIMENSION OF THE LIMIT SETS

In this section we investigate the Hausdorff dimension _H(F) of the fractals F defined in (<ref>), which do not necessarily satisfy all the MSC conditions.  To start, we determine an upper bound for the dimension of the limit set F by considering the step-wise relative ratios between the diameters of sets.


	Suppose 𝒥 := {J_σ : σ∈ D} is a collection of bounded subsets of a metric space (X,d), and s>0. 
	Let	E_k = ⋃_σ∈ D_k J_σ,  and  F = ⋂_k ≥ 0 E_k be defined as in (<ref>).  If there exists a sequence of positive numbers {c_k}_k=1^∞ such that
	
    lim inf_k→∞∏_i=1^k c_i =0

	and
	
    ∑_j=1^n_k ((J_σ*j))^s ≤ c_k ( (J_σ))^s,

	for all σ∈ D_k-1 and all k=1,2,⋯, then dim_H(F) ≤ s.



	We prove by using mathematical induction that for k=1,2,⋯,
	
    ∑_σ∈ D_k((J_σ))^s ≤(∏_i=1^k c_i ) (diam(J_∅))^s.

	When k=1, (<ref>) follows from (<ref>). Now assume (<ref>) is true for some k≥ 1.  Then by (<ref>), (<ref>), and (<ref>),
	
	
    ∑_σ∈ D_k+1 ((J_σ))^s    =   ∑_σ∈ D_k(∑_j=1^n_k+1 ((J_σ*j))^s)
       ≤    c_k+1∑_σ∈ D_k((J_σ))^s 
       ≤   (∏_i=1^k+1 c_i )((J_∅))^s

	
	as desired. By the induction principle, (<ref>) holds for all k=1,2,⋯.
	For each k, set
	
    δ_k=max{(J_σ): σ∈ D_k}>0.

	Then, by (<ref>),  δ_k≤(∏_i=1^k c_i )^1/s(J_∅).  Moreover, by (<ref>)
	
    ℋ_δ_k^s(F) ≤ℋ_δ_k^s(E_k)≤∑_σ∈ D_kα(s)((J_σ)/2)^s ≤(∏_i=1^k c_i )α(s)((J_∅)/2)^s.

	Since lim inf_k→∞∏_i=1^k c_i =0, there exists a sequence {k_t}_t=1^∞ such that
	
    lim_t→∞∏_i=1^k_t c_i =0.
 Thus, lim_t→∞δ_k_t= 0,
	ℋ^s(F) =lim_t→∞ℋ_δ_k_t^s(F)=0, and hence _H(F) ≤ s.


Conversely, a lower bound on the Hausdorff dimension of the limit set F can also be obtained as follows.



		Let 𝒥 := {J_σ : σ∈ D} be a collection of compact subsets of a metric space (X, d), and F be the limit set of 𝒥 as given in (<ref>).  𝒥 is said to satisfy the uniform covering condition if there exists a real number γ>0 and a natural number N such that for all closed ball B in X, there exists a subset D_B⊂ D with cardinality of D_B at most N,
	
    B∩ F ⊆⋃_σ∈ D_B J_σ and   diam(B)≥γ∑_σ∈ D_Bdiam(J_σ).

 


Let 𝒥 := {J_σ : σ∈ D} be a collection of compact subsets of a metric space (X, d) with diam(J_∅)>0, and F be the limit set of 𝒥 as given in (<ref>).  If 𝒥 satisfies the uniform covering condition, and if for some s>0,
	
    ∑_j=1^n_k(J_σ*j)^s
    	≥(J_σ)^s

	for all σ∈ D_k-1 and all k=1,2,⋯, then dim_H(F)≥ s.



	
	We first show that under condition (<ref>), there exists a probability measure μ on X concentrated on F such that for each σ∈ D,
	
    μ(J_σ)≤((J_σ)/(J_∅))^s.
	
	Let μ(J_∅)=1, and for each σ∈ D_k for k>0 and i=1, ⋯, n_k, we inductively set
	
    μ(J_σ *i) =(J_σ*i)^s/∑_j=1^n_k(J_σ *j)^sμ(J_σ).

	For any Borel set A in X, define
	
    μ(A) = inf{∑_i=1^∞μ(J_σ_i) : A ∩ F ⊂⋃_i=1^∞ J_σ_i and  J_σ_i∈𝒥}.
	
	One can check that μ defines a probability measure on X, concentrated on F.
	
	To prove (<ref>) for J_σ, ∀σ∈ D, we proceed by using induction on k when σ∈ D_k. It is clear for k=0.  Now assume that (<ref>) holds for each σ∈ D_k for some k.  Then by induction assumption and (<ref>), for each i=1,⋯, n_k+1,
	
    μ(J_σ*i)    =(J_σ*i)^s/∑_j=1^n_k(J_σ*j)^sμ(J_σ) 
       ≤(J_σ*i)^s/∑_j=1^n_k(J_σ*j)^s((J_σ)/(J_∅))^s
       ≤((J_σ*i)/(J_∅))^s.

	This proves inequality (<ref>).
	
	Now, for any δ>0, let {B_i} be any collection of closed balls with (B_i)≤δ and F⊆∪_i B_i. For each i, let D_B_i be the subset of D corresponding to B_i as given in equation (<ref>). Note that
	
    F⊆⋃_i B_i∩ F ⊆⋃_i  ⋃_σ∈ D_B_iJ_σ=⋃_σ∈D̃J_σ,
		
	where D̃:=∪_i=1^∞ D_B_i⊆ D. 
	
	Let
	
    C(s)   :=   max{∑_i=1^N (x_i)^s: (x_1,x_2,⋯, x_N)∈ [0,1]^N  with ∑_i=1^N x_i=1}
       =   
    			N^1-s,     if 0<s<1 
    
    			1,     if s≥ 1.

	and c(s)=α(s)/C(s)(γ(J_∅)/2)^s>0. Then, by (<ref>) and (<ref>),
	
	
    ∑_i  α(s)((B_i)/2)^s 
       ≥   ∑_i  α(s)/2^s(γ∑_σ∈ D_B_i(J_σ))^s 
       ≥   ∑_i  α(s)/2^s C(s)γ^s∑_σ∈ D_B_i((J_σ))^s 
       ≥   α(s)/2^s C(s)γ^s∑_σ∈D̃((J_σ))^s 
       ≥   α(s)/2^s C(s)γ^s((J_∅))^s∑_σ∈D̃μ(J_σ) 
       ≥    c(s)μ(∑_σ∈D̃J_σ) ≥ c(s)μ(F) = c(s).

	
	Thus, ℋ^s(F) =lim_δ→ 0ℋ_δ^s(F)≥ c(s)>0, and hence _H(F) ≥ s.

 
 In section <ref> we will explore sufficient conditions for 𝒥 to satisfy the uniform covering condition.



§ GENERAL SETUP OF -LIMIT SETS


 
We now formalize the ideas stated in the Introduction to give a description of the construction of generalized fractals.  We concentrate on the maps in order to take advantage of the computational nature of an IFS, but allow for the maps to be updated and changed at each iteration.

Let 𝒳 be a collection of nonempty compact subsets of a metric space.


	A mapping f: 𝒳→𝒳 is called a compression on 𝒳 if f(E) ⊆ E for each E ∈𝒳. 


For each natural number m, let 
    𝒞_m(𝒳) = { (f^(1), f^(2), …, f^(m)): f_i  is a compression on 𝒳, i=1, …, m}.



	Let ℳ be a nonempty set.  A mapping 	
	
    ℱ:    ℳ→𝒞_m(𝒳)  
          k → f_k = (f_k^(1), f_k^(2), ⋯ , f_k^(m))
 is called a marking of 𝒞_m(𝒳) by ℳ.
	Each element k ∈ℳ is called the marker of f_k.
	


Given a marking  and an initial set E_0 ∈𝒳, we will construct a generalized Moran set from any sequence of markers in ℳ.  Note that any sequence {k_ℓ}_ℓ = 0^∞ in ℳ can be represented as a mapping from the ordered set D to ℳ.



	
	Let ℱ be a marking of 𝒞_m(𝒳) by ℳ, let E_0 be any element in 𝒳, and D be as in (<ref>).  Suppose k⃗ : D →ℳ is a map sending σ to k_σ. 
	For each σ∈ D and 1 ≤ j ≤ m, we recursively define J_∅ = E_0 and
	
    J_σ * j = f__̨σ^(j) (J_σ),

	where f_k_σ is given by ℱ as in (<ref>). The limit set 
	
    F = ⋂_k≥ 1⋃_σ∈ D_k J_σ

 associated with 𝒥(k⃗)={J_σ: σ∈ D} is called the ℱ-limit set generated by k⃗ with the initial set E_0.  
	
	


We now make two observations relating the concepts of an -limit set with the attractor of an IFS.

First we observe that the attractor of an IFS { S_1, S_2, …, S_m} on a closed subset Δ of ^n can be viewed as an -limit set as follows.
Let 

    𝒳 = { E : E  is a non-empty compact subset of Δ,  S_i(E) ⊆ E,  for all  i}.
  Since each S_i is a contraction on Δ, the set E_r := Δ∩B(0,r) is a non-empty compact subset of Δ, and S_i(E_r) ⊆ E_r for each i when r is sufficiently large.  In other words, E_r ∈𝒳 for sufficiently large r.  Also, each contraction map S_i acting on Δ naturally determines a map f^(i) :𝒳→𝒳 given by 

    f^(i)(E) = S_i(E) := { S_i(x) | x ∈ E ⊆Δ}
 
for each E ∈𝒳.  Since f^(i)(E) = S_i(E) ⊆ E, f^(i) is a compression for each i.  Set 
    f=(f^(1), f^(2), … , f^(m) ).
  For any non-empty set ℳ, define the marking  of 𝒞_m(𝒳) to be the constant function (k)=f for all k ∈ℳ.  
Thus, for each σ∈ D_k and i=1, … ,m, we have that J_σ * i = S_i(J_σ) from (<ref>). As a result, for any map k⃗: D →ℳ, the collection 𝒥(k⃗) = { J_σ : σ∈ D} is independent of the choice of k⃗.  Thus, the associated ℱ-limit set F = ⋂_k≥ 1⋃_σ∈ D_k J_σ  agrees with the attractor of the given IFS { S_1, S_2, …, S_m}. Conversely, let ℱ be a marking of 𝒞_m(𝒳) by ℳ where 𝒳 is a collection of non-empty compact subsets of Δ.  Suppose there is a mapping k⃗ : D →ℳ such that the sequence { f_k_σ}_σ∈ D is constant in 𝒞_m(𝒳) (i.e. there exists an f ∈𝒞_m(𝒳) such that f_k_σ =f for all σ∈ D)  and for each i=1,2,…,m, there exists a contraction S_i on Δ such that equation (<ref>) holds for each E ∈𝒳. Then the ℱ-limit set F generated by k⃗ is the attractor of the IFS {S_1, S_2, …, S_m }. Therefore, choosing k⃗: D →ℳ to be a constant map will result in a limit set F that is the attractor of an IFS.
In the above sense, our approach is a generalization of the standard IFS construction.

An important observation is that replacing {k_σ}_σ∈ D by another sequence  {k̃_σ}_σ∈ D in (<ref>) will not change the computational complexity of the construction of 𝒥(k⃗).  Thus, generating the limit set F will have a similar computational complexity as generating the attractor of a comparable IFS.

In the following section we will compute the Hausdorff dimension of the constructed -limit sets.  In section 5 we will provide examples along with their dimensions.



§ HAUSDORFF DIMENSIONS OF -LIMIT SETS



As indicated in Propositions <ref> and <ref>, the relative ratio between the diameters of the sets plays an important role in the calculation of the dimension of the limit set.  Therefore, we introduce the following definition.


	For any compression g: 𝒳→𝒳, define
	
    U(g)=sup_E∈𝒳(g(E))/(E),  and  L(g)=inf_E∈𝒳(g(E))/(E).


Note that, for each E∈𝒳,

    L(g)·(E)≤(g(E)) ≤ U(g) ·(E).


For any 𝐤∈ℳ and f_𝐤 = (f_^(1), f_^(2), ⋯ , f_^(m)) ∈𝒞_m(𝒳), define

    𝐔_𝐤=( U(f_𝐤^(1)), ⋯,  U(f_𝐤^(m))) ∈ℝ^m,

and 

    𝐋_𝐤=( L(f_𝐤^(1)), ⋯,  L(f_𝐤^(m))) ∈ℝ^m.


Also, for each x=(x_1, ⋯, x_m)∈ℝ^m and s>0, denote

    ||x||_s=(∑_i=1^m |x_i|^s)^1/s.


These notations, Proposition <ref> and Proposition <ref> motivate our main theorem.
 
	Let F be the -limit set generated by a sequence {_̨σ}_σ∈ D with initial set J_∅, and s> 0. 
	
		
  (a) If F satisfies the uniform covering condition (<ref>) and
		
    inf_σ∈ D{||𝐋__̨σ||_s}≥ 1,

		then _H(F) ≥ s.
		
  (b) If
		
    sup_σ∈ D{||𝐔__̨σ||_s}<1,

		then _H(F)≤ s.
	



	(a)  By (<ref>) and (<ref>), for all σ∈ D, 
	
    ∑_j=1^m(J_σ * j)^s =∑_j=1^m(f__̨σ^(j)(J_σ))^s 
    		≥∑_j=1^m(L(f__̨σ^(j)))^s (J_σ)^s ≥(J_σ)^s.

	Thus, by Proposition <ref>, _H(F)≥ s.
	
	(b) Similarly, for all σ∈ D, 
	
    ∑_j=1^m(J_σ*j)^s ≤∑_j=1^m(U(f__̨σ^(j)))^s (J_σ)^s ≤ c ·(J_σ)^s,

	
	where
	
    c:=sup_σ{(||𝐔__̨σ||_s)^s}<1.

	By Proposition <ref>, _H(F)≤ s.





	
	For practical reasons, we find that it is more convenient to represent the mapping k⃗ : D →ℳ by a sequence {k_ℓ}_ℓ=0^∞⊆ℳ.  For each σ=(i_1, i_2, …, i_k) ∈ D_k, let 
	
    ℓ(σ) = ∑_p=0^k-1 m^p i_k-p

	be the ordering of σ in the ordered set D.  Using this notation, we can rewrite Definition <ref> as follows. 



	Let  be a marking of 𝒞_m(𝒳) by ℳ, let {_̨ℓ}_ℓ=0^∞ be a sequence in ℳ, and E_0 ∈𝒳 be a starting set.  For each ℓ =0,1, 2,⋯ and j=1,2,⋯, m, we iteratively denote the set
	
    E_mℓ+j=f__̨ℓ^(j)(E_ℓ) ∈𝒳,

	where f__̨ℓ is given by ℱ as in (<ref>).
	Let (0)=0 and for n ≥ 1,
	
    (n)=m+m^2+⋯+m^n=m^n+1-m/m-1
  
	denote the number of sets in the n^th generation, i.e. the cardinality of D_n.
	The limit set 
    F=⋂_n=1^∞⋃_ℓ=(n-1)+1^(n) E_ℓ

	is called the -limit set generated by the triple (, {_̨ℓ}_ℓ =0^∞ , E_0). 


	 

	In the following, we will use the notation from Definition <ref> to describe the construction of the -limit sets.  Clearly, using this notation, Theorem <ref> simply says that if F satisfies the uniform covering condition (<ref>) and 	inf_ℓ{||𝐋__̨ℓ||_s}≥ 1, then _H(F) ≥ s, and if sup_ℓ{||𝐔__̨ℓ||_s}<1,	then _H(F)≤ s.	


When both {||𝐋__̨ℓ ||_s}_ℓ=0^∞ and {||𝐔__̨ℓ ||_s}_ℓ =0^∞ are convergent sequences, the following corollary enables us to quickly estimate the dimension of F.


	Let F be the limit set generated by the triple (, {_̨ℓ}_ℓ =0^∞ , E_0).
	
		
		
  (a) Let s_* := sup{s: lim inf_ℓ→∞{||𝐋__̨ℓ ||_s}>1 }.  Then
		
    _H(F) ≥s_*,

		provided F satisfies the uniform covering condition (<ref>).
		
		
  (b) Let s^* := inf{s: lim sup_ℓ→∞{||𝐔__̨ℓ||_s}<1 }.  Then	
    _H(F)≤s^*.

		
		

	


	For any 0< s< s_*, by the definition of s_*,
	
    lim inf_ℓ→∞{||𝐋__̨ℓ ||_s}>1.

	Thus, when ℓ_*∈ℕ is large enough,
	
    inf_ℓ≥ℓ_*{||𝐋__̨ℓ||_s }≥ 1,     i.e. inf_ℓ≥ 0{||𝐋__̨ℓ_*+ℓ||_s}≥ 1.

	Since F∩ E_ℓ_* is the set generated by
	the triple (, {_̨ℓ_*+ℓ}_ℓ =0^∞ , E_ℓ_*),  by Theorem <ref>, it follows that _H(F∩ E_ℓ_*) ≥ s for any ℓ_* large enough. This implies that _H(F) ≥ s for any s<s_* and hence _H(F) ≥s_*. Similarly, we also have _H(F)≤s^*.


In the following corollaries, we will see that bounds of the dimension of F can also be obtained from corresponding bounds on 𝐋__̨ℓ and 𝐔__̨ℓ. 

Notation. For any two points x=(x_1,⋯, x_m) and y=(y_1,⋯, y_m) in ℝ^m, we say x ≤ y if x_i ≤ y_i for each i=1,⋯, m.

 
	Let 𝐭=(t_1,⋯, t_m)  and 𝐫=(r_1,⋯, r_m) be two points in (0,1)^m ⊂ℝ^m. Let s_* and s^* be the solutions to ||𝐭||_s_*=1, and ||𝐫||_s^*=1 respectively,  i.e. 
	
    t_1^s_*+t_2^s_*+⋯+t_m^s_*=1,   and  r_1^s^*+r_2^s^*+⋯+r_m^s^*=1.

	 
		
  (a) If 𝐋__̨ℓ≥𝐭 for all ℓ and F satisfies the uniform covering condition (<ref>), then _H(F)≥ s_*.
		
		
  (b)If 𝐔__̨ℓ≤𝐫 for all ℓ, then
		_H(F) ≤ s^*. 
		
		
  (c) If 𝐋__̨ℓ=𝐫 =𝐔__̨ℓ for all ℓ and F satisfies the uniform covering condition (<ref>), then _H(F)=s^*.
	


	(a) Let 0< s<s_*. Then,
	
    inf_ℓ{||𝐋__̨ℓ||_s}≥ ||𝐭||_s ≥ ||𝐭||_s_*= 1.

	Thus, by Theorem <ref>, _H(F)≥ s for any s<s_*, and hence _H(F)≥ s_*.
	
	(b) Similarly,  let 0<s^*<s. Then,
	
    sup_ℓ{||𝐔__̨ℓ||_s}≤ ||𝐫||_s < ||𝐫||_s^*= 1.

	Thus, by Theorem <ref>,  _H(F)≤ s for any s>s^*, and hence _H(F)≤ s^*.
	
	(c) follows from (a) and (b).


A special case of Corollary <ref> gives the following explicit formulas for the bounds on the dimension of F.

	Let F be the limit set generated by the triple (, {_̨ℓ}_ℓ =0^∞ , E_0).  Let
	
    𝐭=(t, ⋯, t)  and 𝐫=(r, ⋯, r),

	for some 0<t, r<1. 
	
		
  (a) If 𝐋__̨ℓ≥𝐭 for all ℓ and F satisfies the uniform covering condition (<ref>), then _H(F)≥logm/-log t.
		
		
  (b)If 𝐔__̨ℓ≤𝐫 for all ℓ, then
		_H(F) ≤logm/-log r. 
		
		
  (c) If 𝐋__̨ℓ=𝐫 =𝐔__̨ℓ for all ℓ and F satisfies the uniform covering condition (<ref>), then _H(F)=logm/-log r.
	


Other types of bounds on 𝐋__̨ℓ and 𝐔__̨ℓ can also be used to provide bounds on _H(F), as indicated by the following result.

	Let F be the limit set generated by the triple (, {_̨ℓ}_ℓ =0^∞ , E_0). 
	
		
  (a) If F satisfies the uniform covering condition (<ref>) and 
		
    w:=inf_ℓ{||𝐋__̨ℓ||_1}≥ 1,

		then _H(F) ≥log(m)/log(m)-log(w).
		
  (b)
		If 
		
    u:=sup_ℓ{||𝐔__̨ℓ||_1} < 1,

		then _H(F) ≤log(m)/log(m)-log(u).
	


	(a). In this case, for s = log(m)/log(m)-log(w)≥ 1, we have
	
    ∑_j=1^m(L(f__̨ℓ^(j)) )^s/m≥(∑_j=1^mL(f__̨ℓ^(j))/m)^s ≥(w/m)^s

	for each ℓ. Thus,
	
    inf_ℓ{||𝐋__̨ℓ||_s}≥ m^1/sw/m = 1,

	then by Theorem <ref>, _H(F) ≥ s.
	
	(b).  In this case, for any 1 ≥ s > log(m)/log(m)-log(u), we have
	
    ∑_j=1^m(U(f__̨ℓ^(j)))^s/m≤(∑_j=1^mU(f__̨ℓ^(j))/m)^s ≤(u/m)^s

	for each ℓ. Thus,
	
    sup_ℓ{||𝐔__̨ℓ||_s}≤ m^1/su/m <1.

	By Theorem <ref>, _H(F) ≤ s. Hence, _H(F) ≤log(m)/log(m)-log(u).


Note that this corollary generally provides better bounds on _H(F) than those obtained from directly applying Theorem <ref>.





§ EXAMPLES OF -LIMIT SETS


In this section we describe the construction of both classical fractals and generalized Moran sets in the language of Section <ref>, and calculate the dimension using the results from Section <ref>.  

Cantor-Like Sets
	
	We first consider Cantor-like sets.  Let 
    𝒳 = { [a,b] : a,b ∈}
 be the collection of closed intervals, m=2, and let ℳ = [0,1]^2 ⊆. For each 𝐤=(k^(1),k^(2)) ∈ℳ, we consider the following two maps, 
	
    f^(1)_𝐤 :    𝒳   →𝒳
       [a,b]      ↦  [a, k^(1)(b-a)+a]

	
    f^(2)_𝐤 :    𝒳   →𝒳
        [a,b]      ↦ [k^(2)(a-b)+b, b].

	
	
	
	Note that both f^(1)_$̨ andf^(2)_are compression maps for any∈̨ℳ.  Thus, this defines a marking
	
    :    ℳ   →𝒞_2(𝒳) 
       ↦  f_= (f^(1)_, f^(2)_).

	Here, for each𝐤=(k^(1), k^(2)) ∈ℳ, one can clearly see that 
    (f^(i)_𝐤([a,b]))=k^(i)·([a,b]).
  Thus,L (f^(i)_𝐤 )=k^(i)=U(f^(i)_𝐤), and hence 
    𝐋_𝐤=𝐤=𝐔_𝐤.

	
	
	
LetE_0 = [0,1] ∈𝒳be fixed. For any sequence{_̨ℓ}_ℓ=0^∞ ∈ℳ, we define the following:
	
	
    E^(0)   = E_0
       E^(1)   = f^(1)_𝐤_0( E_0 ) ∪  f^(2)_𝐤_0( E_0 ) =: E_1∪ E_2
       E^(2)   = f^(1)_𝐤_1( E_1) ∪  f^(2)_𝐤_1( E_1) ∪  f^(1)_𝐤_2( E_2) ∪  f^(2)_𝐤_2( E_2) 
          :=     E_3    ∪    E_4  ∪    E_5  ∪    E_6
       ⋮
       E^(n)   = ⋃_i=2^n-1-1^2^n-2(f__̨i^(1)(E_i) ∪ f__̨i^(2)(E_i) ):=⋃_i=2^n-1-1^2^n-2(E_2i+1∪ E_2i+2)=⋃_ℓ= 2^n-1^2(2^n-1) E_ℓ.

	
	Note that when𝐤_ℓ=(1/3, 1/3)for allℓ,E^(n) is then^th-generation of the Cantor setandF =lim_n →∞ E^(n) = ⋂_n E^(n) =. 
	
	
	Observe that the process of constructing the sequence{E^(n)}_n=0^∞is independent of the values of{_̨ℓ}_ℓ=0^∞. To allow for more general outcomes, we can update the linear functionsf^(1)_$̨ and f^(2)_$̨ simply by changing the value of$̨ at each stage of the construction, which does not change the computational complexity of the process.  Using this idea, we now construct some examples of Cantor-like sets by choosing suitable sequences {_̨ℓ}_ℓ=0^∞.
	
	
	
	
		Let 𝐤_ℓ = (ℓ+1/4ℓ+6, 2ℓ +5/8ℓ+16) for ℓ≥ 0, and let F be the -limit set generated by the triple (, {_̨ℓ}_ℓ=0^∞, E_0).  In Figure <ref> we plot the usual Cantor set  (in blue) below the set F (in red) to illustrate the comparison. We can see that the set F has the same basic shape as the Cantor set , but is no longer strictly self-similar.
		In order to compute the Hausdorff dimension of the new Cantor-like set F, we apply Corollary <ref>.  Note that by equation (<ref>),
		
    lim_ℓ→∞||𝐋_𝐤_ℓ||_s=lim_ℓ→∞||𝐤_ℓ ||_s=2^1/s/4.

		So,  
		
    s_* =sup_s{lim inf_ℓ→∞||𝐋_𝐤_ℓ||_s>1} = sup_s {2^1/s/4>1}=1/2.

		Similarly, we also have s^*=1/2. By Corollary <ref>, _H(F)=1/2.  Here, F satisfies the uniform covering condition (<ref>) since 
    sup{k_ℓ^(1)+k_ℓ^(2): ℓ=0,1,2, ⋯}=1/2<1,
 according to Proposition <ref>.
		
	
	
	In the next example, we will construct a random Cantor-like set as follows.
	
	 
		For each ℓ≥ 0, we take 𝐤_ℓ=(q_ℓ, 1/2-q_ℓ) where q_ℓ is a random number between 1/8 and 3/8. Let F be the corresponding -limit set generated by the triple (, {_̨ℓ}_ℓ=0^∞, E_0).  We plot the first few generations in Figure <ref>.  In this example, the total length of the n^th generation E^(n) is chosen to be (1/2)^n, while the scaling factors of the left subintervals at each stage are randomly chosen.
		
		
		
		We now estimate the dimension of F.  By (<ref>),
		
    (1/8,1/8)≤𝐋_𝐤_ℓ =𝐤_ℓ=𝐔_𝐤_ℓ≤(3/8,3/8).

		By Corollary <ref>, 
		
    log(2)/-log(1/8)≤_H(F) ≤log(2)/-log(3/8).

		That is,
		
    1/3≤_H(F) ≤log(2)/log(8/3)≈ 0.7067.

		Note that due to Proposition <ref>, F satisfies the uniform covering condition (<ref>) since q_ℓ + 1/2-q_ℓ = 1/2 <1 for each ℓ≥ 0.
		
	
	
	
	 
		In this example, we create a sequence {_̨ℓ}_ℓ=0^∞ that results in a limit set with a given measure, e.g. 1/3.   Of course, the classic example of such a limiting set is the fat Cantor set.  For a different approach, let ∑_n=0^∞ a_n be any convergent series of positive terms with limit L.  We consider a sequence {_̨ℓ}_ℓ=0^∞ defined in the following way.  
		
		Let n≥1 be the generation of the construction and for each ℓ with 2^n-1-1 ≤ℓ≤ 2^n-2, define 𝐤_ℓ=(b_n, b_n) where 
    b_1:= 3/2L-a_02(3/2L)   and    b_n := 3/2L-∑_i=0^n-1 a_i2(3/2L-∑_i=0^n-2a_i) for  n≥ 2.
 
		With this sequence {_̨ℓ}_ℓ =0^∞, one can find that the length of each interval in the n^th generation is 
    b_1 b_2 ⋯ b_n=3/2L-∑_i=0^n-1 a_i2^n ·3/2L.
  Thus, the total length of the n^th generation is 
    3/2L-∑_i=0^n-1 a_i3/2L=1-2/3L∑_i=0^n-1 a_i
 which converges to 1/3 as desired.  As an example, we take the convergent series ∑_n=0^∞1n! = e and use it to create the -limit set F with measure 1/3.  The first few generations are shown in Figure <ref>.
		
	


Sierpinski Triangle

	The Sierpinski triangle is another well known fractal. Following the general setup in Section <ref>, we take 
    𝒳 = { (A,B,C) | A,B,C ∈^2}
 representing the collection of all triangles Δ ABC in ^2, m=3, and ℳ = [0,1]^6 ⊆^6.  For each 𝐤=(k^(1), k^(2), k^(3), k^(4), k^(5), k^(6)) ∈ℳ and i=1,2,3 we can define affine transformations f_^(i) : 𝒳→𝒳 as 
	
    f^(1)_ (A,B,C)    =    (A,A+k^(1)(B-A), A+k^(2)(C-A))
    
    		f^(2)_ (A,B,C)    =    (B+k^(4)(A-B),B, B+k^(3)(C-B))
    
    		f^(3)_ (A,B,C)    =    (C+k^(5)(A-C),C+k^(6)(B-C), C)

	for every (A,B,C) ∈𝒳.
	
	
Note that each f^(i)_$̨ is a compression map fori=1,2,3and  any∈̨ℳ.  Thus, this defines a marking
	
    :    ℳ   →𝒞_3(𝒳) 
       ↦  f_= (f^(1)_, f^(2)_, f^(3)_).

	
	Of course, to prevent overlaps we can require thatk^(1) + k^(4) ≤1, k^(2) +k^(5) ≤1, k^(3)+k^(6) ≤1. When each of the inequalities are strict, the images off^(i)_are three disconnected triangles, as illustrated in Figure <ref>. When all equalities hold, the images are connected, as illustrated in Figure <ref>. 
	
	
	
	In the case of the connected sets, the values of𝐤=(k^(1), k^(2), k^(3), k^(4), k^(5), k^(6)) are determined byk^(1), k^(2), k^(3)sincek^(4) = 1-k^(1),k^(5)=1-k^(2),k^(6)=1-k^(3).  In this case, we may also view=̨(k^(1), k^(2), k^(3))as a vector in[0,1]^3 ⊆^3.  
	
	
	
	To create the normal Sierpinski triangle, we choose
	
    E_0 = [   -1/2    1/2      0;      0      0 √(3)/2 ],

	the equilateral triangle of unit side length, and_̨ℓ ∈ℳto be the constant sequence_̨ℓ ==̨(1/2, 1/2, 1/2, 1/2, 1/2, 1/2)so that each iteration maps a triangle to three triangles of half the side length with the desired translation.  In this case the-limit set generated by(, {_̨ℓ}_ℓ=0^∞, E_0)corresponds to the standard Sierpinski Triangle as seen in Figure <ref>.  
	
	
	To generate Sierpinski-like fractals, we now adjust the values of the marking parameters{_̨ℓ}_ℓ=0^∞.  For each𝐤=(k^(1), k^(2),⋯, k^(6))∈ℳand1≤i ≤3, 
	
    U(f_𝐤^(i))=sup_(A,B,C)∈𝒳diam (f_𝐤^(i)(A,B,C))/diam((A,B,C)) =max{k^(2i-1), k^(2i)},

	and
	
    L(f_𝐤^(i))=inf_(A,B,C)∈𝒳diam (f_𝐤^(i)(A,B,C))/diam((A,B,C)) =min{k^(2i-1), k^(2i)}.

	
	When$̨ is bounded, i.e. if λ≤ k^(j)≤Λ<1 for all j=1,⋯ ,6, then
	
    𝐔_𝐤≤𝐫:=(r,⋯,r)  and 𝐋_𝐤≥𝐬:=(s,⋯,s),

	where r=max{1-λ, Λ} and s=min{1-λ, Λ}.
	
	Following our general process, we construct some random Sierpinski-like sets by introducing randomness into the choice of the sequence {_̨ℓ}_ℓ=0^∞.
	
		Let {_̨ℓ}_ℓ=0^∞ ={ (k_ℓ^(1), k_ℓ^(2), k_ℓ^(3) )}_ℓ=0^∞ be a sequence in [0,1]^3 with each k_ℓ^(i) a random number between given numbers λ and Λ for each i=1,2,3.  Let F be the -limit set generated by (, {_̨ℓ}_ℓ=0^∞, E_0).  Then the 6^th generation of the construction results in images like Figure <ref>. Here, in Figure <ref>, λ=1/4 and Λ=3/4;  while in Figure <ref>, λ=0.45 and Λ=0.55. Note that the sets are no longer self-similar.
	
	
	
	
	In Figure <ref>, we pick λ=0.45 and Λ=0.55. By Corollary <ref>,
	
    log(m)/-log(s)≤_H(F) ≤log(m)/-log(r),

	where m=3, r=0.55 and s=0.45. That is,
	
    1.3758 ≤_H(F)≤ 1.8377,
 provided F satisfies the uniform covering condition (<ref>).
	
	
	
		
		As in Example <ref>, but replacing E_0 with  Ẽ_̃0̃ = [ 0 1 0; 0 0 1 ], the 7^th generation of the construction results in an image like Figure <ref>, when λ=1/4 and Λ=3/4.
		
	
		
	
	  
		For each ℓ=0,1, ⋯, let 𝐤_ℓ=(k_ℓ^(1), k_ℓ^(2), ⋯, k_ℓ^(6)) where
		
    k_ℓ^(1)   =   1/2+a_ℓ/√(ℓ+1),    k_ℓ^(2)=1-k_ℓ^(1),
    
    			k_ℓ^(3)   =   1/2+b_ℓ/√(ℓ+1),    k_ℓ^(4)=1-k_ℓ^(3),
    
    			k_ℓ^(5)   =   1/2+c_ℓ/ℓ+1,    k_ℓ^(6)=1-k_ℓ^(5).

		for random numbers a_ℓ, b_ℓ, c_ℓ∈ [-1/3, 1/3].  Let F be the -limit set F generated by (, {_̨ℓ}_ℓ=0^∞, E_0).
		Then the seventh generation of the construction of F results in an image like Figure <ref>. 
		
		
		
		In this case, we can calculate the exact value of the Hausdorff dimension of F.  Indeed, by Corollary <ref>,
		
    lim_ℓ→∞ (||𝐔_𝐤_ℓ||_s)^s=3/2^s=lim_ℓ→∞ (||𝐋_𝐤_ℓ||_s)^s.

		Thus, _H(F)=log(3)/log(2), provided F satisfies the uniform covering condition (<ref>).
	
	


Menger Sponge
	
	
	Let 
    𝒳 = { (O,A,B,C) | O,A,B,C ∈^3 }
 representing the collection of all rectangular prisms (OABC) in ^3, m=20, and
	
    ℳ = {(k^(1), k^(2), k^(3), k^(4), k^(5), k^(6)) ∈ [0,1]^6 :  k^(1)≤ k^(2),  k^(3)≤ k^(4), k^(5)≤ k^(6)}.

	
			

	For each ∈̨ℳ and i =1, 2, … , 20, we can define affine transformations f^(i)_ : 𝒳→𝒳 as follows.  For any =̨(k^(1), k^(2), k^(3), k^(4), k^(5), k^(6)) ∈ℳ, define  
	
    T = [     0 k^(1) k^(2)     1 ],    R= [     0 k^(3) k^(4)     1 ],    S= [     0 k^(5) k^(6)     1 ].

	Let
	
    I   =   { (a,b,c) | 1 ≤ a,b,c≤3  with  a,b,c ∈, and no two of a,b,c equal to 2}.

	For each (a,b,c) ∈ I and ∈̨ℳ, define 
	
    M_(a,b,c)=[   1-( T(a) + R(b) +S(c))                     T(a)                     R(b)                     S(c); 1-( T(a+1) + R(b) +S(c))                   T(a+1)                     R(b)                     S(c); 1-( T(a) + R(b+1) +S(c))                     T(a)                   R(b+1)                     S(c); 1-( T(a) + R(b) +S(c+1))                     T(a)                     R(b)                   S(c+1);                          ].

	Note that the set I contains 20 elements, so we can express it as 
    I = { (a_i, b_i, c_i) | 1 ≤ i ≤ 20}.

	For each ∈̨ℳ and 1 ≤ i ≤ 20, we consider the affine transformation f^(i)_ : 𝒳→𝒳 given by
	
    f^(i)_ (O,A,B,C)  = M_ (a_i,b_i,c_i)  [ O; A; B; C ]

		for every (O, A, B, C) ∈𝒳.  
		Note that for i=1, …, 20 and ∈̨ℳ, f^(i)_ is a compression.   Thus, we can define a marking : ℳ→𝒞_20(𝒳) by sending ↦̨f_= (f^(1)_, … , f^(20)_).
		Using this, for any starting rectangular prism E_0 = (O,A,B,C) ∈𝒳, we can generate a sequence of sets that follows a similar construction to the Menger Sponge.
	
	
		
		Let
		
		
    E_0 = [ 0 1 0 0; 0 0 1 0; 0 0 0 1 ]

		be the cube of unit side length and choose _̨ℓ∈ℳ to be the constant sequence _̨ℓ ==̨(1/3, 2/3, 1/3, 2/3, 1/3, 2/3).  Then the -limit set F generated by the triple (, {_̨ℓ}_ℓ =0^∞, E_0 ) is the classical Menger sponge, as seen in Figure <ref>. 
		


		
		
	
	Now we consider variations of Menger Sponge. For each 𝐤=(k^(1), k^(2), ⋯, k^(6))∈ℳ and 1≤ i ≤ 20, 
	
    U(f_𝐤^(i))   =   sup_(O,A,B,C)∈𝒳diam (f_𝐤^(i)(O,A,B,C))/diam((O,A,B,C))
       =   sup_(O,A,B,C)∈𝒳diam (M_𝐤(a_i, b_i, c_i)[O,A,B,C]')/diam((O,A,B,C))
       =   max{T(a_i+1)-T(a_i), R(b_i+1)-R(b_i), S(c_i+1)-S(c_i)}.

	Similarly,
	
    L(f_𝐤^(i))=min{T(a_i+1)-T(a_i), R(b_i+1)-R(b_i), S(c_i+1)-S(c_i)}.

	When k^(2j)=1-k^(2j-1) for each j=1, 2, 3, it is easy to check that
	
    ∑_i=1^20U(f_𝐤^(i))^s   =   ∑_i=1^20max{T(a_i+1)-T(a_i), R(b_i+1)-R(b_i), S(c_i+1)-S(c_i)}^s
       =   8max{k^(1), k^(3), k^(5)}^s+4max{1-2k^(1), k^(3), k^(5)}^s  
           +4max{k^(1), 1-2k^(3), k^(5)}^s +4max{k^(1), k^(3), 1-2k^(5)}^s.

	
	
	
		Let 
    Ẽ_̃0̃ = [ 0 3 0 0; 0 0 1 0; 0 0 0 2 ].
 Let (k^(1), k^(2), k^(3), k^(4), k^(5), k^(6)) ∈ℳ where each k^(i) is a random number in [0,1], but still satisfying the condition k^(1)≤ k^(2), k^(3)≤ k^(4), k^(5)≤ k^(6).  Then the first generation E^(1) of the construction results in a set like Figure <ref>.  
		
	
	
	
		Let _̨ℓ =(k_ℓ^(1), k_ℓ^(2), k_ℓ^(3), k_ℓ^(4), k_ℓ^(5), k_ℓ^(6)) ∈ℳ with each k_ℓ^(2j-1) a random number between given parameters λ and Λ and k_ℓ^(2j)=1-k_ℓ^(2j-1) for each j=1,2,3.  Let F be the -limit set generated by (, {_̨ℓ}_ℓ =0^∞, E_0). Then the third iteration of the construction of F results in images like Figure <ref>. Here, in Figure <ref> the parameters λ=0 and Λ=1/2, while in Figure <ref> the parameters λ=0.32 and Λ=0.35.
		
		
		
	
	
	We now calculate the dimension of the limit fractal F illustrated by Figure <ref> in Example <ref>.
	Note that in general, when λ≤ k^(2j-1)≤Λ for each j=1, 2, 3, it follows that
	
    (||𝐔_𝐤||_s)^s=∑_i=1^20U(f_𝐤^(i))^s ≤ 8Λ^s +12max{1-2λ, Λ}^s.

	Similarly,
	
    (||𝐋_𝐤||_s)^s ≥ 8λ^s +12min{1-2Λ, λ}^s.

	In particular, when λ=0.32 and Λ=0.35,  for any s>2.901,
	
    (||𝐔_𝐤||_s)^s    ≤    8Λ^s+12max{1-2λ, Λ}^s ≤ 8*0.35^s+12*0.36^s 
       <    8*0.35^2.901+12*0.36^2.901≈ 1.000.

	By Theorem <ref>, _H(F)≤ 2.901. Similarly, for any s ≤ 2.546,
	
    (||𝐋_𝐤||_s)^s
    		   ≥    8λ^s+12min{1-2Λ, λ}^s 
       ≥    8*0.32^s+12*0.3^s ≥ 8*0.32^2.546+12*0.3^2.546≈ 1.000.

	By Theorem <ref> again, _H(F)≥ 2.546, provided F satisfies the uniform covering condition (<ref>). As a result,
	
    2.546 ≤_H(F)≤ 2.901.

	
	
	
		
		For each ℓ≥ 0, let 𝐤_ℓ=(k_ℓ^(1), k_ℓ^(2), ⋯, k_ℓ^(6)) where
		
    k_ℓ^(1)   =   1/3+(-1)^ℓ/12(ℓ+1)^2,    k_ℓ^(2)=1-k_ℓ^(1),
    
    			k_ℓ^(3)   =   1/3-(-1)^ℓ/6(ℓ+1)^2,    k_ℓ^(4)=1-k_ℓ^(3),
    
    			k_ℓ^(5)   =   1/3+(-1)^ℓ/18(ℓ+1)^2,    k_ℓ^(6)=1-k_ℓ^(5).

		Let F be the -limit set generated by (, {_̨ℓ}_ℓ =0^∞, E_0).  Then the third generation of the construction of F leads to an image like Figure <ref>.
		
		
	
	
	In this case, we can still calculate the exact Hasudorff dimension of F. By direct computation, 
	
    lim_ℓ→∞ (||𝐔_𝐤_ℓ||_s)^s=20/3^s=lim_ℓ→∞ (||𝐋_𝐤_ℓ||_s)^s.

	Thus, by Corollary <ref>, _H(F)=log(20)/log(3)≈ 2.7268, since F satisfies the uniform covering condition according to Example <ref>.


  Here, we discuss similarities and differences of this construction with V-variable fractals created by Barnsley, Hutchinson, and Stenflo in <cit.>, <cit.>. These authors have described a similar approach to creating more generalized fractals that can take on a prescribed amount of randomness.  In <cit.> and <cit.>, they describe a generating process for some fractals along with calculations of their dimensions.  In essence, a V-variable fractal set has at most V ∈ number of distinct patterns in each generation of the construction.  This is done through the following process.

Let (X,d) be a metric space, Λ an index set, F^λ = {f^λ_1, f^λ_2, …, f^λ_m} an IFS for each λ∈Λ, and P a probability distribution on some σ-algebra of subsets of Λ.  Then denote 𝐅 = {(X,d), F^λ, λ∈Λ, P} to be a family of IFSs (with at least two functions in each IFS) defined on (X,d).  Assume that the IFSs F^λ are uniformly contractive and uniformly bounded, that is, for some 0<r<1,

    sup_λmax_m d ( f^λ_m (x), f^λ_m(y) ) ≤ r d(x,y),
          sup_λmax_m d( f^λ_m(a),a )< ∞
 for all x,y ∈ X and some a ∈ X.

A tree code is a map ω from the set of all finite sequences {1, … ,m } to Λ.  A tree code is V-variable if for each positive integer k, there are at most V distinct tree codes in the tree truncated at the k^th generation.  For example, consider the Sierpinski triangle.  We let F be the IFS that maps the triangle to three copies of 1/2 the size, as usual.  Let G be the IFS that maps the initial triangle to three triangles that are 1/3 the size, with the vertices shared with the initial set being the fixed points of the maps.  See Figure <ref> for the image of the initial step of each.  Thus, 𝐅 = { (^2, d), {F,G}, P=(1/2, 1/2)} is the family {F,G} with probability function uniformly choosing 1/2 for each IFS.  Using these IFSs, three V-variable pre-fractals are given in Figure <ref>, being 1-variable, 2-variable, and 3-variable respectively.





Now, we express V-variable fractals in terms of -limit sets.  Let 𝒳, ℳ,ℱ and E_0 be as in Section <ref>.  Let F = ( 1/2, 1/2, 1/2, 1/2,1/2,1/2) ∈ℳ and G=( 1/3, 1/3, 1/3, 1/3, 1/3, 1/3) ∈ℳ.  We will use F and G to denote terms in the sequence { k_ℓ}_ℓ =1^∞.  Consider the third generation examples in Figure <ref>.

Then from left to right we have the following:


    [                                                                V=1      { k_ℓ} _ℓ =1^13  = { F, F, F, F, G, G, G, G, G, G, G, G, G};                                                                 V=2     { k_ℓ} _ℓ =1^13  = { F, G, F, G, F, G, F, F, F, G, F, G, F};                                                                V=3      { k_ℓ} _ℓ =1^13 = { F, F, F, G, F, F, G, F, G, G, G, G, G}. ]

From these examples, we can see that if we want to create a V-variable fractal, for each generation we should choose at most V distinct triples from the set { (A,B,C) | A,B,C ∈{F,G}} and repeat those triples in any order.  

When V < ∞, there are at most V distinct tree codes in the address of point in the set.  We can create such a situation from our construction described earlier in section <ref> by choosing blocks of {k_ℓ} that repeat across generations.  In the case that V = ∞, the fractal is based off of a probability distribution for applying specific IFSs.  In our construction we also can use a probability distribution to determine the contraction ratios within a generation (as in Examples <ref>, <ref> and <ref>), but we do not require such a choice. We allow for deterministic sequences that also do not repeat any blocks, thus not falling into the category of V-variable.





§ UNIFORM COVERING CONDITION


In previous sections, we have seen that the uniform covering condition (<ref>) plays a vital role in computing a lower estimate for the Hausdorff dimension of a fractal.  In this section we explore the sufficient conditions needed for a fractal to satisfy the uniform covering condition.
 

	Let (X, d) be a metric space with the following property: For any ϵ>0, there exists a natural number N_ϵ such that for any ρ>0, any closed ball in X of diameter ρ contains at most N_ϵ many disjoint balls of diameter ϵρ. Clearly, any Euclidean space satisfies this property.
	
	Let 𝒥 := {J_σ : σ∈ D} be a collection of compact subsets of (X, d), and F be the limit set of 𝒥 as given in (<ref>).  Suppose that 𝒥 satisfies the following conditions: 
	
		
  * there exists a number r∈(0,1] such that for any k∈ℕ and for each σ∈ D_k,
		
    rc_k≤ diam(J_σ)≤c_k/r

		where c_k:=min{	diam(J_σ̅): σ̅∈ D_k-1}.
		
  * there exists a number τ∈ (0, 1] such that for each σ∈ D, the convex hull of J_σ contains a closed ball W_σ
		such that 
		
    diam(W_σ)≥τ· diam(J_σ)

		and for each k∈ℕ, the collection {W_σ: σ∈ D_k} are pairwise disjoint.
	
 Then F satisfies the uniform covering condition (<ref>).

	
	  For any closed ball B in X, let k be the number such that
	
    c_k+1≤ diam(B) <c_k

	where by convention, we set c_0=∞. Let
	
    D_B:={σ∈ D_k: B∩ F∩ J_σ≠∅}.
	
	Note that
	
    B∩ F = B ∩ F ∩⋃_σ∈ D_k J_σ⊆⋃_σ∈ D_B J_σ.
	Also for any σ∈ D_B, since diam(J_σ)≤c_k/r and B∩ J_σ≠∅, it follows that J_σ⊆B̅(x_0, r+2/2r c_k), where x_0∈ X is the center of the ball B.  Thus, W_σ⊆B̅(x_0, r+2/2r c_k). Let ρ=r+2/rc_k and ϵ =r^2 /r+2τ, then
	
    diam(W_σ)≥τ· diam(J_σ) ≥τ r c_k = ϵρ.

	Since {W_σ: σ∈ D_B} are pairwise disjoint, the cardinality of D_B is at most N:=N_ϵ.  
	On the other hand, for γ =r^2/N,	it holds that
	
    diam(B)≥ c_k+1≥ rc_k= γ N c_k/r≥γ∑_σ∈ D_Bc_k/r≥γ∑_σ∈ D_Bdiam(J_σ).

	As a result, 𝒥 satisfies the condition (<ref>) as desired.
		


We now discuss some specific sufficient conditions concerning the types of examples provided in section 3.4.  To start, let's first consider Cantor-like constructions.  Let X be the family of closed intervals described in (<ref>), m=2, and ℳ = [0,1]^2 ⊆. 


	
	Let {_̨ℓ}_ℓ=0^∞ be a sequence in ℳ with
	
    sup{k_ℓ^(1)+k_ℓ^(2): ℓ=0,1,2, ⋯}<1,

	and F be the -limit set generated by the triple (, {_̨ℓ}_ℓ=0^∞, E_0).
	Then F satisfies the uniform covering condition (<ref>).
	

	Let N=1 and 
	
    γ=inf_ℓ{1-k_ℓ^(1)-k_ℓ^(2)}∈ (0,1] .

	For any closed interval B in ℝ with B∩ F ≠∅, consider the set
	
    ℒ:={ℓ(σ): B∩ F ⊆ J_σ, σ∈ D},

	where ℓ(σ) is given in (<ref>).
	Note that ℒ is nonempty because B∩ F ⊆ J_0 implies that ℓ(0)∈ℒ.
	If ℒ is an infinite set, then since diam(J_σ)→ 0 as ℓ(σ)→∞, there exists σ^*∈ D such that ℓ(σ^*)∈ℒ and diam(B)≥ diam(J_σ^*)≥γ· diam(J_σ^*).
	If ℒ is finite, let ℓ(σ^*) be the maximum number in ℒ for some σ^*∈ D.
	Then,  ℓ(σ^*)∈ℒ but ℓ(σ^* * j)∉ℒ for each j=1,2. This implies that  B∩ J_σ^* *j ≠∅ for both j=1, 2 because J_σ^* = J_σ^* *1 ∪  J_σ^* *2. Since B is an interval, the gap J∖ (J_σ^* * 1∪ J_σ^* * 2) between J_σ^* * 1 and J_σ^* * 2 is contained in B, which yields that
	
    diam(B)   ≥    diam  (J∖ (J_σ^* * 1∪ J_σ^* * 2) )
       =    diam(J)-diam (J_σ^* * 1)-diam (J_σ^* * 2)
       ≥     diam(J_σ^*)  (1-k_ℓ(σ^*)^(1)-k_ℓ(σ^*)^(2) )≥γ· diam(J_σ^*).

	As a result, in both cases, the uniform covering condition (<ref>) holds.



Motivated by Proposition <ref>, we now consider a generalization of the above result.


Let n ≥ 1 and ℋ be a collection of subsets of a metric space (X,d).  Define 
	
    ρ_n(ℋ)=inf{r: There exists a ball B in X of radius r that intersects 
    at least  n+1  elements in ℋ}.

 Here ρ_n(ℋ) is a quantity describing the “gap" between n+1 elements of ℋ.


 Let 𝒥={J_σ : σ∈ D} be a collection of compact subsets of a metric space (X,d), and n ≥ 1.  Define 
    γ_n(𝒥):=inf{ρ_n({J_σ*i: σ∈ R_k, i=1,2,⋯, m})/∑_σ∈ R_kdiam(J_σ):  for some k.
    .  and  R_k⊆ D_k  with  1 ≤ |R_k|≤ n },
  where |R_k| denotes the cardinality of the set R_k. Here γ_n(𝒥) is a quantity describing the relative size of the “gap" between n+1 children of a generation and the size of the parent sets.


Now we give some examples of calculations of these two quantities.

 Let 𝒥 be the collection of closed intervals used in the construction of a Cantor-like set given in (<ref>).  Then 
    γ_1(𝒥)    =   inf{ρ_1({J_σ*i: σ∈ R_k, i=1,2})/diam(J_σ):  for some k and  R_k⊆ D_k  with |R_k|= 1}
       =   inf{ρ_1({J_σ*1, J_σ*2}) /diam(J_σ) :  for σ∈ D }
       =   inf{diam(J_σ) - diam(J_σ*1) - diam(J_σ*2)/diam(J_σ): σ∈ D }
       =   inf{1-diam(J_σ * 1)/diam(J_σ) - diam(J_σ * 2)/diam(J_σ): σ∈ D },
 which agrees with the γ in (<ref>), see Figure <ref>.

	




	Let 𝒥 be the collection of triangles used in (<ref>).  In the following figures, we plot the smallest ball that intersects a certain number of children.  The children that have non-empty intersection with the ball are colored red, while those that have empty intersection are light blue.  
	
	First note that for any σ∈ D, ρ_1({J_σ*1, J_σ*2, J_σ*3})=0 since any pair of children share a vertex.  At the intersection of the two children of J_σ one can construct a ball of arbitrarily small radius.  See Figure <ref>.
	
	

Moreover, ρ_2({J_σ*1, J_σ*2, J_σ*3})>0 because the radius of any ball that intersects all three children of J_σ is bounded below by the radius of the inscribed circle of the removed center triangle.  In other words, ρ_2({J_σ*1, J_σ*2, J_σ*3}) is equal to the radius of the inscribed circle.  See Figure <ref> for illustration.
	
	

Now we may compute γ_n(𝒥) as follows.
	
	Note that for n=1,
    γ_1(𝒥)    =   inf{ρ_1({J_σ*i: σ∈ R_k, i=1,2,3})/diam(J_σ):  for some k and  R_k⊆ D_k  with |R_k|= 1}
       =   inf{ρ_1({J_σ*1, J_σ*2, J_σ*3}) /diam(J_σ) :  for σ∈ D }=0.


On the other hand, when n=2, we have

    γ_2(𝒥)    =   inf{ρ_2({J_σ*i: σ∈ R_k, i=1,2,3})/diam(J_σ):  for some k and  R_k⊆ D_k  with |R_k|≤ 2}.

When |R_k|=1, this is reduced to the same case as Figure <ref>.

When |R_k|=2, we use two parent triangles, and must find the ball with smallest radius that intersects three or more children. See Figure <ref> for a few candidates for the ball with smallest radius.  



For each R_k ⊆ D_k with |R_k| ≤ 2,  ρ_2({J_σ*i: σ∈ R_k, i=1,2,3}) >0.  For some nice 𝒥, one may expect γ_2(𝒥) to also be positive.



	
	Let 𝒥 := {J_σ : σ∈ D} be a collection of compact subsets of (X, d) satisfying MSC(3) and  
    lim_k→∞max{diam(J_σ): σ∈ D_k }=0,
 and let F be the limit set of 𝒥 as given in (<ref>).  If there exists an N such that γ_N(𝒥)>0, then F satisfies the uniform covering condition (<ref>).
	
 Let γ=γ_N(𝒥)>0.
	For any closed ball B in X with B∩ F≠∅,
	let g(k) be the number of elements σ in D_k  such that B∩ F∩ J_σ≠∅. Then g:ℕ∪{0}→ℕ is monotone increasing with g(0)=1. 
	
	Case 1: If g(k)≤ N for all k=0,1,2,⋯, that is,  for each k, there exists an index set I_k with |I_k|≤ N  such that B∩ F⊆⋃_i∈ I_kJ_σ_i^(k) for some σ_i^(k)∈ D_k. Thus when k is large enough, 
	
    diam(B)>γ·∑_i∈ I_kdiam(J_σ_i^(k))

	due to the fact that 
	
    0≤lim_k→∞∑_i∈ I_kdiam(J_σ_i^(k)) ≤ N·lim_k→∞max{diam(J_σ): σ∈ D_k}=0.
  
	
	Hence, equation  (<ref>) holds for B.
	
	Case 2: There exists k^* ≥ 0 such that g(k^*)≤ N but g(k^*+1)>N.
	
	Since g(k^*) ≤ N, there are g(k^*) many elements σ∈ D_k^* such that B ∩ F ∩ J_σ≠∅.  That is, there exists R_k^*⊆ D_k^* with |R_k^*| ≤ N such that B ∩ F ⊆⋃_σ∈ R_k^* J_σ.  On the other hand, since g(k^*+1)>N, B ∩ F intersects at least N+1 elements of D_k^*+1.  Since B∩ F ⊆⋃_σ∈ R_k^* J_σ, all of these N+1 elements must be children of {J_σ : σ∈ R_k^*}.  Then, by the definition of ρ_N in (<ref>),  
    diam(B) ≥ρ_N( {J_σ*i: σ∈ R_k^*, i=1,2,… m}) ≥γ·∑_σ∈ R_k^* diam(J_σ).

	 As a result, F satisfies the uniform covering condition (<ref>). 

  To show an application of Theorem <ref>, we now consider some examples provided in section <ref>.  Let {_̨ℓ}_ℓ=0^∞ be a sequence in ℳ as defined in (<ref>) and F be the ℱ-limit set generated by (ℱ, {_̨ℓ}_ℓ=0^∞, E_0 ) associated with 𝒥(k⃗)={J_σ: σ∈ D} as defined in Definition <ref>. Let ℋ⊆𝒥_k := {J_σ: σ∈ D_k} for some k ≥0, and consider ρ_8(ℋ). 
	
	We now make the following observation: Suppose there exists a ball B that intersects at least 9 elements of ℋ.  Then diam(B) is greater than or equal to the smallest edge length of the elements in ℋ.  Indeed, by considering the projections to the three coordinate axes, one can see that at least one coordinate contains three non-identical projected images of these 9 elements.  As a result the ball B intersected with these 9 elements will have a diameter at least the length of the smallest side of the three projected images.  This proves our observation. 
	
	
	Let 
    m_ℓ = min{ k_ℓ^(1), k_ℓ^(2)-k_ℓ^(1),1-k_ℓ^(2), k_ℓ^(3), k_ℓ^(4)-k_ℓ^(3),1-k_ℓ^(4), k_ℓ^(5), k_ℓ^(6)-k_ℓ^(5),1-k_ℓ^(6)}
 and 
	
    M_ℓ = max{ k_ℓ^(1), k_ℓ^(2)-k_ℓ^(1),1-k_ℓ^(2), k_ℓ^(3), k_ℓ^(4)-k_ℓ^(3),1-k_ℓ^(4), k_ℓ^(5), k_ℓ^(6)-k_ℓ^(5),1-k_ℓ^(6)}.

For any σ∈ D, direct calculation shows that 
    m_ℓ(σ)≤diam(J_σ*i)diam(J_σ)≤ M_ℓ(σ)

where ℓ(σ) is given in (<ref>). Thus, for any σ = (i_1, i_2, …, i_k) ∈ D_k, we have 
    m_ℓ((i_1)) m_ℓ((i_1,i_2))⋯ m_ℓ((i_1,…,i_k))≤diam(J_σ)diam(J_∅)≤ 	M_ℓ((i_1)) M_ℓ((i_1,i_2))⋯ M_ℓ((i_1,…,i_k)).

	
	Let R_k⊆ D_k for some k.  Suppose |R_k|≤8.  Then for any σ∈ R_k, by the observation 
    ρ_8({J_σ*i: σ∈ R_k , i=1,2, … 20})∑_σ∈ R_k diam(J_σ)
       ≥    smallest diameter of J_σ*i8 ·max{ diam(J_σ): σ∈ R_k}
       ≥   min_σ=(i_1, i_2, … , i_k) ∈ R_k, i_k+1=1, … , 20{m_ℓ((i_1)) m_ℓ((i_1,i_2))⋯ m_ℓ((i_1,…,i_k+1))  diam(J_∅)M_ℓ((i_1)) M_ℓ((i_1,i_2))⋯ M_ℓ((i_1,…,i_k)) diam(J_∅)}
       ≥   18 (∏_i=1^∞ m_iM_i) lim inf_i →∞ m_i,
  
where the last inequality follows from 0 ≤ m_i ≤ M_i for each i.	
	
	
	
	
		
		
 Using this observation, we show that the -limit set in Example <ref> satisfies the uniform covering condition.  In this example, 
	 
    m_ℓ = 
    	 a_ℓ   ℓ even
    
    	 b_ℓ   ℓ odd and 
    	 M_ℓ = 
    	 b_ℓ   ℓ even
    
    	 a_ℓ   ℓ odd
 where
	
 
    a_ℓ = k_ℓ^(3)= 1/3 - (-1)^ℓ/6(ℓ +1)^2 and  b_ℓ= 1-2 k_ℓ^(3)= 1/3 + (-1)^ℓ/3(ℓ+1)^2.
  
	
	One may show that the product ∏_i=1^∞ m_iM_i is convergent, whose numerical value is 0.369761… and lim inf_i →∞ m_i =1/3.  Thus γ_8(𝒥) >0.  Therefore, by Theorem <ref>, F satisfies the uniform covering condition.
		
		
			7
		
		barnsley  Barnsley, M.F., Fractals Everywhere, Academic Press Professional, Inc. San Diego, CA, USA (1988)
		
		vvar1 Barnsley, M.F., Hutchinson, J.E., Stenflo, O.,  V-variable fractals: fractals with partial similarity. Adv. Math. 218, 2051–2088 (2008)
		
		vvar2 Barnsley, M.F., Hutchinson, J.E., Stenflo, O.,  V-variable fractals: dimension results. Forum Math. 24, 445–470 (2012)
		
		BV Barnsley, M.F., Vince, A., Developments in fractal geometry. Bull. Math. Sci. 3, 299-348 (2013)
		
		Beardon Beardon, A. On the Hausdorff dimension of general Cantor sets. Mathematical Proceedings of the Cambridge Philosophical Society, 61(3), 679-694 (1965)
		
		Bedford Bedford, T. Applications of dynamical systems to fractal-A study of cookie-cutter Cantor sets. Fractal Geometry and Analysis, Canada: Kluwer, 1-44 (1991)
			
		hutch Hutchinson, J., Fractals and Self-Similarity Indiana University Math. J. 30 (5), 713-747 (1981)
		
		inhom Holland, M. and Zhang, Y., Dimension results for inhomogeneous Moran set constructions. Dyn. Syst., 28(2):222–250, (2013)
		
		Falconer1Falconer, K. Random fractals. Math. Proc. Cambridge Philos. Soc. 100 (3), 559-582, (1986)
		
		Falconer Falconer, K. Fractal Geometry: Mathematical Foundations and Applications, (2014)
		
		Li Li, W., Xiao, D., A note on generalized Moran set. Acta Math. Sci. (Suppl.) 18 88-93 (1998)
		
		Mandelbrot Mandelbrot, B. B. The fractal geometry of nature. San Francisco, California: W. H. Freeman and Co. (1982)
		
		mauldin Mauldin, R. D., Urbanski, M., Dimensions and Measures in Infinite Iterated Function Systems, Proc. London Math. Soc., 73(3):105 (1996)
		
		mcmullen McMullen, Curt. The Hausdorff dimension of general Sierpiński carpets. Nagoya Math. J. 96, 1–9. (1984)
		 
		Moran Moran P.A.P. Additive functions of intervals and Hausdorff measure. Proc. Camb. Philos. Soc, 42, 15-23 (1946)
		
		peres  Peres, Y., Solomyak B. Problems on self-similar sets and self-affine sets: An update, Fractal Geometry and Stochastics II, Progress in Probability, 46, 95-106, (2000)
		
		PW  Pesin, Y., Weiss, H., On the Dimension of Deterministic and Random Cantor-like Sets, Symbolic Dynamics, and the Eckmann-Ruelle Conjecture. Comm. Math. Phys., 182: 105. (1996)
		
		Su Hua, S., Rao, H., Wen, Z. et al. On the structures and dimensions of Moran Sets, Sci. China Ser. A-Math. 43: 836 (2000)
		Wen Wen, Z. Moran Sets and Moran Classes, Chinese Science bulletin, 46 (22), 1849-1856 (2001)
		
	
  
  


