11institutetext: Louisa F. Barnsley and Michael F. Barnsley 22institutetext:
Australian National University, Canberra, ACT, Australia 22email:
<EMAIL_ADDRESS>
# Blowups and Tops of Overlapping Iterated Function Systems

Louisa F. Barnsley and Michael F. Barnsley

###### Abstract

We review aspects of an important paper by Robert Strichartz concerning
reverse iterated function systems (i.f.s.) and fractal blowups. We compare the
invariant sets of reverse i.f.s. with those of more standard i.f.s. and with
those of inverse i.f.s. We describe Strichartz’ fractal blowups and explain
how they may be used to construct tilings of $\mathbb{R}^{n}$ even in the case
where the i.f.s. is overlapping. We introduce and establish the notion of
“tops” of blowups. Our motives are not pure: we seek to show that a simple
i.f.s. and an idea of Strichartz, can be used to create complicated tilings
that may model natural structures.

Dedicated to Robert Strichartz

Figure 1: Small ferns growing wildly: can fractal geometry model such images?

##  1 Introduction

In “Fractals in the large” strichartz  Robert Strichartz observes that fractal
structure is characterized by repetition of detail at all small scales. He
asks “Why not large scales as well?” He proposes two ways to study large
scaling structures using developments of iterated function systems. Here we
review geometrical aspects of his paper and make a contribution in the area of
tiling theory.

In his first approach, Strichartz defines a reverse iterated function system
(r.i.f.s.) to be a set of $m>1$ expansive maps

$T=\\{t_{i}:M\rightarrow M|i=1,2,\dots,m\\}$

acting on a locally compact discrete metric space $M$, where every point of
$M$ is isolated. Here the large scaling structures are the invariant sets of
$T$, sets $S\subset M$ which obey

$S=\bigcup\limits_{i=1}^{m}t_{i}\left(S\right).$

Why does does Strichartz restrict his definition to functions acting on
discrete metric spaces? (i) He establishes that there are interesting
nontrivial examples. (ii) He shows that such objects (act as a kind of
skeleton to) play a role in his second kind of large scale fractal structure
that he calls a fractal blowup. Probably he had other reasons related to
situations where his approach to analysis on fractals could be explored.

In Section 2 we present notation for iterated function systems (i.f.s.) acting
on $\mathbb{R}^{n}$. We are particularly concerned with notation for chains of
compositions of functions and properties of addresses of points on fractals.
In Section 3 we review Strichartz’ definition and basic theorem concerning
invariant sets of reverse iterated function systems, and we compare them to
the corresponding situation for contractive i.f.s. We describe some kinds of
invariant sets of contractive i.f.s. and consider how they compare to
Strichartz’ large scaling structures. It is a notable feature of Strichartz’
definition that he restricts attention to functions acting on compact discrete
metric spaces. We mention that, if this restriction is lifted, sometimes very
interesting structures, characterized by repetition of structure at large
scales, may be obtained. See for example Figure 2.

In Section 4 we define fractal blowups, Strichartz’ second kind of large scale
fractal structure, and present his characterization of them, when the open set
condition (OSC) is obeyed, as unions of scaled copies of an i.f.s. attractor,
with the scaling restricted to a finite range. We outline the proof of his
characterization theorem using different notation, anticipating fractal tops.
We recall Strichartz’ final theorem on the topic, where he restricts attention
to blowups of an i.f.s. all of the same scaling factor. Here he combines his
two ideas: he reveals that the fractal blowup is in fact a copy of the
original fractal translated by all the points on an invariant set of a
r.i.f.s.

In Section 5 we discuss how tilings of blowups can be extended to overlapping
i.f.s. In tilings  it was shown how, in the overlapping (OSC not obeyed) case,
tilings of blowups can be defined using an artificial recursive system of
“masks”. Here the approach is more natural, but we pay a price–sequences of
tilings are not necessarily nested. Here tilings are defined by using “fractal
tops”, namely attractors with their points labelled by lexicographically
highest addresses. The needed theory of fractal tops is developed in
Subsection 5.1. Then in Subsection 5.2 we use these top addresses to define
and establish the existence of tilings of some blowups for overlapping i.f.s.
The main theorem concerns the relationship between the successive tilings that
may be used to define a tiling of a blowup. In Subsection 5.3 we present an
example involving a tile that resembles a leaf.

Strichartz’ paper has overlap with bandt2 , published about the same time by
Christoph Bandt. Both papers consider the relationship between i.f.s. theory
and self-similar tiling theory. Current work in tiling theory does not
typically use the mapping point of view, but both Bandt and Stricharz do.
Bandt is particularly focused on the open set condition and the algebraic
structure of tilings, but also has a clear understanding of tilings of blowups
when the OSC is obeyed.

Strichartz’ paper also contains measure theory aspects that we do not discuss.
But from the little we have focused on here, much has been learned concerning
the subtlety, the depth, and the elegant simplicity of the mathematical
thinking of Robert Strichartz.

##  2 Preliminaries

Let $\mathbb{N=\\{}1,2,\dots\\}.$ An iterated function system (i.f.s.) is a
set of functions

$F=\\{f_{i}:\mathbb{X\rightarrow X}|i=1,2,\dots,m\\}$

mapping a space $\mathbb{X}$ into itself, with $m\in\mathbb{N}$. An invariant
set of $F$ is $S\subset\mathbb{X}$ such that

$S=F(S):=\bigcup\limits_{i=1}^{m}f_{i}(S)\text{ where
}f_{i}(S)=\\{f_{i}(s)|s\in S\\}.$

We use the same symbol $F$ for the i.f.s. and for its action on $S$, as
defined here.

The i.f.s. $F$ is said to be contractive when $\mathbb{X}$ is equipped with a
metric $d$ such that $d(f_{i}x,f_{i}y)\leq\lambda d(x,y)$ for some
$0<\lambda<1$ and all $x,y\in\mathbb{X}$. If $\mathbb{X=R}^{n},\ $we take $d$
to be the Euclidean metric. A contractive i.f.s. on $\mathbb{R}^{n}$ is
associated with its attractor $A$, the unique non-empty closed and bounded
invariant set of $F$ hutchinson . But Strichartz is also interested in the
case where the underlying space is discrete and the maps are expansive.

We use addresses to describe compositions of maps. Addresses are defined in
terms of the indices of the maps of $F$. Let
$\Sigma=\left\\{1,2,\dots,m\right\\}^{\mathbb{N}}$, the set of strings of the
form $\mathbf{j}=j_{1}j_{2}\dots$ where each $j_{i}$ belongs to
$\left\\{1,2,\dots,m\right\\}$. We write
$\Sigma_{n}=\left\\{1,2,\dots,m\right\\}^{n}$ and let
$\Sigma_{\mathbb{N}}=\cup_{n=1}^{\infty}\Sigma_{n}.$ The address
$\mathbf{j}\in\Sigma$ truncated to length $n$ is denoted by
$\mathbf{j}|n=j_{1}j_{2}\dots j_{n}\in\Sigma_{\mathbb{N}},$ and we define

$\displaystyle f_{\mathbf{j}|n}$ $\displaystyle=f_{j_{1}}f_{j_{2}}\dots
f_{j_{n}}=f_{j_{1}}\circ f_{j_{2}}\circ\dots\circ f_{j_{n}},$ $\displaystyle
f_{-\mathbf{j}|n}$ $\displaystyle=f_{j_{1}}^{-1}f_{j_{2}}^{-1}\dots
f_{j_{n}}^{-1}=f_{j_{1}}^{-1}\circ f_{j_{2}}^{-1}\circ\dots\circ
f_{j_{n}}^{-1}.$

Define a metric $d$ on $\Sigma$ by
$d(\mathbf{j},\mathbf{k})=2^{-\max\\{n|j_{m}=k_{m},m=1,2,...,n\\}}$ for
$\mathbf{j}\neq\mathbf{k}$, so that $\left(\Sigma,d\right)$ is a compact
metric space.

The forward orbit of a point $x$ under (the semigroup generated by) $F$ is

$\\{f_{\mathbf{j}|n}(x)|\mathbf{j}\in\Sigma,n\in\mathbb{N\\}}\text{.}$

Here we do not allow $\mathbf{j}|n$ to be the empty set, so $x$ is not
necessarily an element of its forward orbit under the i.f.s. Indeed, $x$ is a
member of its forward orbit if and only if $x$ is a fixed point of one of the
composite maps $f_{\mathbf{j}|n}$.

Now let $F$ be a contractive IFS of invertible maps on $\mathbb{R}^{n}$. Then
a continuous surjection $\pi:\Sigma\rightarrow A$ is defined by

$\pi(\mathbf{j})=\lim_{N\rightarrow\infty}f_{\mathbf{j}|N}(x)=\lim_{N\rightarrow\infty}f_{j_{1}}f_{j_{2}}\dots
f_{j_{N}}(x).$

The limit is independent of $x$. The convergence is uniform in $\mathbf{j}$
over $\Sigma,$ and uniform in $x$ over any compact subset of $\mathbb{R}^{n}$.
We say $\mathbf{j}\in\Sigma$ is an address of the point $\pi(\mathbf{j})\in
A$.

We define $i:\Sigma\rightarrow\Sigma$ by $i(\mathbf{j)=}ij_{1}j_{2}\dots$ But
we may also write $k_{1}k_{2}...k_{l}\mathbf{j}$ to mean the address
$k_{1}k_{2}\dots k_{l}j_{1}j_{2}\dots\in\Sigma.$ Let
$\sigma:\Sigma\rightarrow\Sigma$ be the shift operator defined by
$\sigma(\mathbf{j)=}j_{2}j_{3}\dots$ . It is well-known that

$f_{i}\circ\pi=\pi\circ i\mathbf{\ }\text{and
}\pi\circ\sigma\left(\mathbf{j}\right)=f_{j_{1}}^{-1}\circ\pi\left(\mathbf{j}\right)$

for all $i\in\\{1,2,...m\\},\mathbf{j\in}\Sigma$.

A notable shift invariant subset of $\Sigma$ is the set of disjunctive
addresses $\Sigma_{dis}$. An address $\mathbf{j}\in\Sigma$ is disjunctive
when, for each finite address $i_{1}i_{2}i_{3}\dots
i_{k}\in\left\\{1,2,\dots,m\right\\}^{k}$, there is $l\in\mathbb{N}$ so that
$j_{l+1}...j_{l+k}=i_{1}i_{2}i_{3}\dots i_{k}$. The set of disjunctive
addresses $\Sigma_{dis}\subset\Sigma$ is totally invariant under the shift,
according to $\sigma\left(\Sigma_{dis}\right)=\Sigma_{dis}.$ A point $a\in A$
is disjunctive if there is a disjunctive address $\mathbf{j}\in\Sigma$ such
that $\pi(\mathbf{j})=a$. Disjunctive points play a role in the structure of
attractors. For example, if the i.f.s. obeys the open set condition (OSC) and
its attractor has non-empty interior, then all the disjunctive points belong
to the interior of the attractor bandt . Recall that $F$ obeys the OSC when
there exists a nonempty open set $O$ such that $\cup f_{i}(O)\subset O$ and
$f_{i}(O)\cap f_{j}(O)=\emptyset$ whenever $i\neq j.$

##  3 Reverse iterated function systems

In his first approach to large scaling structures, Strichartz defines a
reverse iterated function system (r.i.f.s.) to be a set of $m>1$ expansive
maps

$T:=\\{t_{i}:M\rightarrow M|i=1,2,\dots,m\\}$

acting on a locally compact discrete (i.e. every point is isolated) metric
space $M$. We write $T$ and $t_{i}$ in place of $F$ and $f_{i}$ to distinguish
this special kind of i.f.s. A mapping $t_{i}:M\rightarrow M$ is said to be
expansive if there is a constant $r>1$ such that $d(t_{i}x,t_{i}y)\geq
rd(x,y)$ for all $x\neq$ $y$ in $M.$ An expansive mapping is necessarily one-
to-one and has at most one fixed point.

In this case Strichartz’s large scaling structures are the invariant sets of
r.i.f.s.; that is, sets $S\subset M$ which obey

$S=T(S)=\bigcup\limits_{i=1}^{m}t_{i}\left(S\right).$

By requiring that $M$ is discrete, Strichartz restricts the possible invariant
sets to be discrete.

Let $P$ be the fixed points of $\\{t_{\mathbf{i}|k}:M\rightarrow
M|k\in\mathbb{N},\mathbf{i}\in\Sigma\\}$. Contrast Theorem 3.1 with Theorem
3.2.

######  Theorem 3.1 (Strichartz)

A set is invariant for a r.i.f.s. if and only if it is a finite union of
forward orbits of points in $P$. In particular, invariant sets exist if and
only if $P$ is nonempty, and there are at most a finite number of invariant
sets.

EXAMPLE 1 Let $M=\mathbb{Z}$, $T=\\{t_{i}:M\rightarrow
M;t_{1}(x)=2x,t_{2}(x)=2x-1\\}.$ It is readily verified that $M$ is invariant
for this r.i.f.s, $T.$ It consists of the forward orbits of the fixed points
of $t_{1}$ and $t_{2}.$

EXAMPLE 2 Strichartz presents the following example of a r.i.f.s. Let $M$ be
the set of integer lattice points $\mathbb{Z}^{2}$ in the plane, lying between
or on the lines $y=\rho x$ and $y=\rho x+1$ where $\rho+\rho^{2}=1,$
$\rho=\left(\sqrt{5}-1\right)/2.$ The r.i.f.s. comprises the two maps

$t_{1}(x,y)=(-x-y,-x),t_{2}(x,y)=(1-x-y,1-x).$

These maps are expansive on $M$, even though when viewed as transformations
acting on $\mathbb{R}^{2},$ they contract pairs of points that lie on any
straight line with slope $-1/\rho$. The fixed point of $t_{1}$ is $(0,0)$ and
of $t_{2}$ is $(0,1),$ both of which lie in $M.$ The union of the forward
orbits of these two points is $M$. So this unlikely looking set of discrete
points is invariant under the r.i.f.s.

This example yields, by projection onto the line $y=\rho x,$ an example of a
quasi-periodic linear tiling using tiles of lengths $\rho$ and $1+\rho.$
Strichartz also points out that by projection onto the perpendicular line
$y=-x/\rho$ of a natural measure on $M$ one obtains, after renormalizing, the
unique self-similar measure on $[0,1]$ associated with the overlapping i.f.s.
$f_{1}(x)=\rho x$, $f_{2}(x)=\rho x+1$ with equal probabilities.

Theorem 3.1 leads one to wonder: What are the invariant sets of an i.f.s.?
Usually the focus is on compact invariant sets, namely attractors. The
following Theorem is simply a list of some of the invariant sets of a
contractive i.f.s. The wealth of such invariants here stands in sharp contrast
to Theorem 3.1.

######  Theorem 3.2 (Some Invariant Sets of an i.f.s.)

Let $F$ be a contractive i.f.s. of invertible maps on $\mathbb{R}^{n}.$ If
$S\subset\mathbb{R}^{n}$ is invariant and bounded, then either
$\overline{S}=\emptyset,$ or $\overline{S}=A.$ The followings sets are
invariant.

  1. 1.

The attractor $A,$ and the whole space $\mathbb{R}^{n}.$

  2. 2.

The forward orbit under $F$ of any periodic point $p\in P.$

  3. 3.

The set of disjunctive points of $A$.

  4. 4.

The orbit of any $x\in\mathbb{R}^{n}$ under the free group generated by the
maps of $F$ and their inverses.

  5. 5.

The union of any collection of invariant sets.

There are other invariant sets. For example, let $A$ be a Sierpinski triangle,
the attractor of an i.f.s. $F_{sierp}$ in the usual way. Let $B$ be the union
of the sides of all triangles in $A.$ Then $B$ is an invariant set for
$F_{sierp}$. It is not covered by Theorem 3.2.

We note that the invariant set in (4) is also invariant under the inverse
i.f.s.

$F^{-1}:=\left\\{f_{i}^{-1}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}|i=1,\dots,m\right\\}.$

The orbit under $F^{-1}$ of the attractor $A$ is invariant under $F^{-1}.$
This set may be referred to as the fast basin of $A$ with respect to $F$, see
fastbasin . It is an example of a set which is “invariant in the large”,
admitted when Strichartz’ constraint, that the underlying space is discrete
and locally compact, is lifted.

Figure 2 illustrates the fast basin associated with (left) a Sierpinski
triangle i.f.s. and (right) a different i.f.s. of three similitudes of scaling
factor $1/2.$ Fast basins are interesting from a computational point of view,
because they comprise the points $x$ in $\mathbb{R}^{n}$ for which there is an
address $\mathbf{j\in}\Sigma_{\mathbb{N}}$ such that $f_{\mathbf{j}}(x)\in A$.

Figure 2: Two examples of invariant sets of inverse iterated function systems.
The left image illustrates part of the fast basin of a Sierpinski triangle
i.f.s. The right image illustrates the fast basin of an i.f.s. whose attractor
is illustrated in red. These unbounded sets are ”invariant in the large” but
are not discrete.

##  4 Strichartz’ fractal blowups

Strichartz uses r.i.f.s. to analyze the structure of what he christened
“fractal blowups”. These structures have been used to develop differential
operators on unbounded fractals, see for example strichartz2 ; teplyaev .

Let $F$ be an i.f.s. of similitudes. The maps take the form

$f_{j}\left(x\right)=r_{j}U_{j}x+b_{j}$

where $0<r_{j}<1,$ $b_{j}\in\mathbb{R}^{n}$ and $U_{j}$ is an orthogonal
transformation. It is convenient to write $r_{j}=r^{a_{j}}$ where
$r=\max\left\\{r_{j}\right\\}$, so that $1\leq a_{j}<a_{\max}$ . A blowup
$\mathcal{A}$ of $A$ is the union of an increasing sequence of sets

$A=A_{0}\subset A_{1}\subset A_{2}\subset\dots$ (1)

where $A_{j}=f_{-\mathbf{k}|j}\left(A\right)$ for some fixed
$\mathbf{k}\in\Sigma$ and all $j\in\mathbb{N}$. We have

$\mathcal{A=A}\left(\mathbf{k}\right)=\bigcup\limits_{j=1}^{\infty}f_{-\mathbf{k}|j}\left(A\right).$
(2)

Strichartz starts with a more general definition of a blowup, but restricts
consideration to the one given here.

######  Theorem 4.1 (Strichartz)

Let $\mathcal{A}\left(\mathbf{k}\right)$ be a blowup of $A$ of the form in
Equation (2) and assume $F$ satisfies the OSC. Then
$\mathcal{A}\left(\mathbf{k}\right)$ is the union of sets $\mathcal{G}_{n}$
which are similar to $A$ with the contraction ratios bounded from above and
below, and the number of sets $\mathcal{G}_{n}$ that intersect any ball of
radius $R$ is at most a multiple of $R^{n}.$ In particular the union
$\mathcal{A}\left(\mathbf{k}\right)=\cup_{n=1}^{\infty}\mathcal{G}_{n}$ is
locally finite, and the intersection of $\mathcal{A}\left(\mathbf{k}\right)$
with any compact set is equal to the intersection of
$\cup_{n=1}^{N}\mathcal{G}_{n}$ with that compact set for large $N$.

###### Proof

We outline a proof for the case of a single scaling factor $0<r<1\ $with
$f_{i}(x)=rU_{i}x+b_{i}$. At heart, our proof is the same as Strichartz, but
we introduce notation that helps with our generalization to overlapping i.f.s.
in Section 5.

Since $F$ satisfies the OSC, there is a bounded open set $\mathcal{O}$ such
that $A\subset$ $\overline{\mathcal{O}},$ $f_{i}(\mathcal{O)\subset O}$ for
all $i,$ $f_{i}(\mathcal{O)\cap}$ $f_{j}(\mathcal{O)=\emptyset}$ for all
$i\neq j.$ Note that the latter condition implies that the sets in
$\left\\{f_{j_{1}j_{2}...j_{l}}(\mathcal{O)}|j_{1}j_{2}...j_{l}\in\Sigma_{l}\right\\}$
are disjoint.

Define a collection of sets

$\Pi_{S}(\mathbf{k|}n):=\left\\{f_{-\mathbf{k}|n}f_{\mathbf{m|}n}(S)|\mathbf{m}\in\Sigma\right\\}$

where $S$ may be $\mathcal{O}$, $\overline{\mathcal{O}},$ or $A.$ Observe that

$\Pi_{A}(\mathbf{k|}1)\subset\Pi_{A}(\mathbf{k|}2)\subset...$

and

$f_{-\mathbf{k}|l}\left(A\right)=\bigcup\limits_{n=1}^{\l}\Pi_{A}(\mathbf{k|}n).$

Also

$\Pi_{\mathcal{O}}(\mathbf{k|}\left(n+1\right))\backslash\Pi_{\mathcal{O}}(\mathbf{k|}n)=\left\\{f_{-\mathbf{k}|n+1}f_{\mathbf{m|}\left(n+1\right)}(\mathcal{O})|\mathbf{m}\in\Sigma_{n+1},k_{n+1}\neq
m_{1}\right\\}$

consists of $m^{n}(m-1)$ disjoint open sets. It follows that
$\left\\{\Pi_{\mathcal{O}}(\mathbf{k|}n)|n=1,2,\dots\right\\}$ is a nested
increasing sequence of disjoint open sets, whose closed union contains
$\mathcal{A}\left(\mathbf{k}\right)$. The closure of each open set contains a
copy of $A.$ Since each open set has volume bounded below by a positive
constant, local finiteness is assured.

A general case of a Strichartz style blowup is captured by defining

$\displaystyle\Pi_{S}\left(\mathbf{k|}j\right)$
$\displaystyle=f_{-\mathbf{k}|l}(\\{f_{\mathbf{m}}(S)|\eta^{-}(\mathbf{m})<\eta(\mathbf{k}|l)\leq\eta(\mathbf{m}),\mathbf{m}\in\Sigma_{\mathbb{N}}\\})$
$\displaystyle\Pi_{S}\left(\mathbf{k}\right)$
$\displaystyle=\bigcup\limits_{j\in\mathbb{N}}\Pi_{S}\left(\mathbf{k|}j\right)$

where

$\displaystyle\eta^{-}(m_{1}m_{2}\dots m_{n})$
$\displaystyle=a_{m_{1}}+a_{m_{2}}+\dots+a_{n-1}$
$\displaystyle\eta(m_{1}m_{2}\dots m_{n})$
$\displaystyle=a_{m_{1}}+a_{m_{2}}+\dots+a_{n}$

These formulas provide a specific form to Strichartz’ stopping time argument.
Using these more general expressions one obtains, for fixed $\mathbf{k}$, an
increasing union of copies of $A$ scaled by factors that lie between
$r^{a_{\max}}$ and $r.$ See for example tilings ; polygon . The argument
concerning local finiteness is essentially the same as above.

Strichartz unites his two ideas, reverse i.f.s. and blowups, by considering
the case where $f_{j}\left(x\right)=rx+b_{j}$ for all $j,$ and studying the
blowup $\mathcal{A}\left(\overline{\mathbf{1}}\right)$ where
$\overline{\mathbf{1}}\mathbf{=}$ $111\dots,$ that is

$\mathcal{A}\left(\overline{\mathbf{1}}\right)=\cup_{n=1}^{\infty}(f_{1}^{-1})^{n}A$

######  Theorem 4.2 (Strichartz combines r.i.f.s. and blowups)

Let $f_{j}x=rx+b_{j}.$ Then
$\mathcal{A}\left(\overline{\mathbf{1}}\right)=A+D$ where $D$ is an invariant
set for the r.i.f.s.

$t_{j}\left(x\right)=r^{-1}(x+b_{j}-b_{1}),j=1,2,...,m$

Specifically, $D$ is the forward orbit of $0$, the fixed point of $t_{1}.$

That is, $\mathcal{A}\left(\overline{\mathbf{1}}\right)$ is the Minkowski sum
of the attractor of the i.f.s. and an invariant set of a r.i.f.s.

##  5 Tops tilings

In this Section we study tilings of fractal blowups in the case of overlapping
i.f.s. attractors. First, in Subsection 5.1 we give relevant theory of fractal
tops. In Subsection 5.2 we show how fractal tops may be used to generate
tilings of fractal blowups for overlapping i.f.s. The approach here is
distinct from the one in tilings . In Subsection 5.3 we illustrate fractal
tops for an i.f.s. of two maps, with overlapping attractor that looks like a
leaf, suggesting applications to modelling of complicated real-world images.

###  5.1 Fractal tops

Let $F$ be a strictly contractive i.f.s. acting on a complete metric space
$\mathbb{X}$, with maps $f_{i}$ and attractor $A.$ We assume that there are
two or more maps, at least two of which have different fixed points. Also all
of the maps are invertible.

###### Lemma 1

Let $C$ be a closed subset of $\Sigma.$ Let $\mathbf{j=}$
$\max\\{\mathbf{k}\in C\\}.$ Then
$\mathbf{j}=j_{1}\max\\{\mathbf{m}\in\Sigma|(j_{1}\mathbf{m)\in}C\\}.$

###### Proof

$C$ is the union of the three closed sets $\\{\mathbf{k}\in
C|k_{1}>j_{1}\\}$,$\\{\mathbf{k}\in C|k_{1}=j_{1}\\},$ and $\\{\mathbf{k}\in
C|k_{1}<j_{1}\\}.$ The maximum over $C$ is the maximum of the maxima over
these three sets. But the set $\\{\mathbf{k}\in C|k_{1}>j_{1}\\}$ is empty,
because if not then $\max\\{\mathbf{k}\in C\\}\geq\max\\{\mathbf{k}\in
C|k_{1}>j_{1}\\}>\mathbf{j}$ which is a contradiction. If
$\max\\{\mathbf{k}\in C\\}=\max\\{\mathbf{k}\in C|k_{1}<j_{1}\\}$ then
$\mathbf{j>j}$, again a contradiction.

Since $\pi:\Sigma\rightarrow A$ is continuous and onto, it follows that
$\pi^{-1}(x)$ is closed for all $x\in A.$ Lemma 1 tells us that a map
$\tau:A\rightarrow\Sigma$ and subset $\Sigma_{top}\subset\Sigma$ are well-
defined by

$\tau(x):=\max\\{\mathbf{k\in}\Sigma|\pi(\mathbf{k)=}x\\},\Sigma_{top}:=\tau(A).$

Conventionally the maximum here is with respect to lexicographical ordering.
We refer loosely to these objects and the ideas around them as relating to the
top of $A.$ Formally, the top of $A$ is the graph of $\tau,$ namely
$\left\\{(x,\tau(x))|x\in A\right\\}$.

Top addresses of points in $A,$ namely points in $\Sigma_{top},$ can be
calculated by following the orbits of the shift map
$\sigma:\Sigma_{top}\rightarrow\Sigma_{top}.$ Simply partition $A$ into
$A_{1}=f_{1}(A),$ $A_{2}=f_{2}(A)\backslash A_{1},$
$A_{3}=f_{3}(A)\backslash(A_{1}$ $\cup
A_{2}),\dots,A_{m}=f_{m}(A)\backslash\cup_{n\neq m}A_{n}$. Define the orbit
$\left\\{x_{n}\right\\}_{n=1}^{\infty}$ of $x=x_{1}\in A,$ under the tops
dynamical system, by $x_{n+1}=f_{i_{n}}^{-1}(x_{n})$ where $i_{n}$ is the
unique index such that $x_{n}\in A_{i_{n}}$.

A version of the following observation can be found in tops . See also bandt .

###### Theorem 5.1

The set of top addresses is shift invariant, according to
$\Sigma_{top}=\sigma\left(\Sigma_{top}\right)$ where $\sigma$ is the left
shift.

###### Proof

First we show that $\sigma\left(\tau(A)\right)\subset\tau(A).$ If
$\mathbf{j}\in\tau(A),$ then

$\displaystyle\mathbf{j}$
$\displaystyle=\max\\{\mathbf{k}\in\Sigma|\pi(\mathbf{k})=\pi(\mathbf{j})\\}\text{(by
definition)}$
$\displaystyle=\max\\{j_{1}\mathbf{l}\in\Sigma|\pi(j_{1}\mathbf{l})=\pi(\mathbf{j})\\}\text{
(by Lemma \ref{lemma1})}$
$\displaystyle=j_{1}\max\\{\mathbf{l}\in\Sigma|f_{j_{1}}(\pi(\mathbf{l}))=f_{j_{1}}(\pi(\sigma\mathbf{j}))\\}$
$\displaystyle=j_{1}\max\\{\mathbf{l}\in\Sigma|\pi(\mathbf{l})=\pi(\sigma\left(\mathbf{j}\right))\\}\text{
(since }f_{j_{1}}\text{ is invertible)}$
$\displaystyle=j_{1}\tau(\pi(\sigma\left(\mathbf{j}\right))).$

Hence $\sigma(\mathbf{j})=\tau(\pi(\sigma\left(\mathbf{j}\right))).$ Hence
$\left\\{\sigma\left(\mathbf{j}\right)\mathbf{|j}\in\tau(A)\right\\}=\left\\{\tau(\pi(\sigma\left(\mathbf{j}\right)))\mathbf{|j}\in\tau(A)\right\\}$
which implies $\sigma\left(\tau(A)\right)\subset\tau(A).$

We also have $1\left(\Sigma\right)\subset\Sigma$ so
$\tau(\pi(1\left(\Sigma\right)))\subset\tau(\pi(\Sigma))=\tau(A).$ But
$\tau(\pi(1\left(\Sigma\right))))=1\left(\tau(\pi(\Sigma))\right)$ by a
similar argument to the proof of Lemma 1, so $1\tau(A)\subset\tau(A).$
Applying $\sigma$ to both sides, we obtain
$\tau(A)\subset\sigma\left(\tau(A)\right).$

It appears that the shift space $\Sigma_{top}$ is not of finite type in
general, and graph directed constructions cannot be used in general. This is a
topic of ongoing research.

Define $\Sigma_{top,n}$ to be the elements of $\Sigma_{top}$ truncated to the
first $n$ elements. That is,

$\Sigma_{top,n}=\\{\left(\mathbf{k}|n\right)|\mathbf{k\in}\Sigma_{top}\\}.$

Define $\pi_{top}:$ $\Sigma_{top}\rightarrow A$ to be the restriction of
$\pi:\Sigma\rightarrow A$ to $\Sigma_{top}.$ Extend the definition of
$\pi_{top}$ so that it acts on truncated top addresses according to:

$\pi_{top}\left(\mathbf{k|}n\right)=\\{x\in f_{\mathbf{k}|n}(A)|x\notin
f_{\mathbf{c}|n}(A)\text{ for all }\mathbf{c}|n>\mathbf{k}|n\\}$

for all $\mathbf{k\in}\Sigma_{top}$ and all $n\in\mathbb{N}$. We will make use
of the following observation.

###### Lemma 2

If $\mathbf{k}\in\Sigma_{top},$ then
$f_{k_{1}}\left(\pi_{top,n-1}(\sigma\left(\mathbf{k|}n\right))\right)\supset$
$\pi_{top,n}(\mathbf{k|}n)$ for all $n\in\mathbb{N}$.

###### Proof

We need to compare the sets

$\\{f_{k_{1}...k_{n}}(x)|f_{k_{1}...k_{n}}(x)\notin
f_{l_{1}l_{2}...l_{n}}(A)\text{ for all }l_{1}...l_{n}>k_{1}...k_{n}\\}$

and

$\\{f_{k_{1}...k_{n}}(x)|f_{k_{1}}f_{k_{2}...k_{n}}(x)\notin
f_{k_{1}}f_{l_{2}...l_{n}}(A)\text{ for all }l_{2}...l_{n}>k_{2}...k_{n}\\}.$

The condition in the latter expression is less restrictive.

The sets of truncated top addresses $\Sigma_{top,n}$ have an interesting
structure. Any addresses in $\Sigma_{top,n}$ can be truncated on the left or
on the right to obtain an address in $\Sigma_{top,n-1}.$ The following Lemma
is readily verified.

###### Lemma 3

Let $n>1.$ If $i_{1}i_{2}\dots i_{n-1}i_{n}\in\Sigma_{top,n}$, then both
$i_{2}\dots i_{n-1}i_{n}\ $and $i_{1}i_{2}\dots i_{n-1}\ $belong to
$\Sigma_{top,n-1}.$

###  5.2 Top blowups and tilings

Here we are particularly interested in the overlapping case, where the OSC
does not hold. We show that natural partitions of fractal blowups, that we
call tilings, may still be obtained.

Throughout this subsection, $F$ is an i.f.s. with

$f_{j}\left(x\right)=rU_{j}x+b_{j}$ (3)

where $b_{j}\in\mathbb{R}^{n}$ and $U_{j}$ is an orthogonal transformation. We
assume that there are two or more maps, at least two of which have distinct
fixed points. We have in mind the situation where $A$ is homeomorphic to a
ball, although this is not required by Theorems 5.2 and 5.3.

As in Section 4, but restricted to $\mathbf{i}\in\Sigma_{top},$ fractal
blowups are well defined by

$\mathcal{A}_{n}=\mathcal{A}\left(\mathbf{i|}n\right)=\bigcup\limits_{l=1}^{n}f_{-\mathbf{i}|l}\left(A\right)\text{
and
}\mathcal{A=A}\left(\mathbf{i}\right)=\bigcup\limits_{l=1}^{\infty}f_{-\mathbf{i}|l}\left(A\right).$

The unions are of increasing nested sequences of sets so
$\mathcal{A}_{n}=f_{\mathbf{i}|n}^{-1}\left(A\right)$ and
$\mathcal{A}=\cup\mathcal{A}_{n}.$ Note that
$\mathcal{A}\left(\mathbf{i|}n\right)$ is related to
$\mathcal{A}\left(\mathbf{j|}n\right)$ by the isometry
$\left(f_{-\mathbf{j}|n}\right)\left(f_{-\mathbf{i}|n}\right)^{-1}$. But
possible relationships between $\mathcal{A}\left(\mathbf{i}\right)$ and
$\mathcal{A}\left(\mathbf{j}\right)$ are quite subtle because inverse limits
are involved.

Under conditions on $F$ and $\mathbf{i}$, stated in Theorems 5.2 and 5.3, we
can define a generalized tilings of $\mathcal{A}\left(\mathbf{i}\right)$ with
the aid of the following two definitions:

$\displaystyle\Pi_{top}(\mathbf{i}|k)$
$\displaystyle:=\left\\{f_{-\mathbf{i}|k}\left(\left\\{x\in\pi_{top}\left(\mathbf{t}|\left(k+1\right)\right)\right\\}\right)|\mathbf{t\in}\Sigma_{top}\right\\},$
$\displaystyle\Pi_{top}(\mathbf{i})$
$\displaystyle:=\lim_{k\rightarrow\infty}\Pi_{top}(\mathbf{i}|k),\text{ when
this limit is well defined.}$

For example, the limit is well defined when
$\Pi_{top}(\mathbf{i}|k)\subset\Pi_{top}(\mathbf{i}|k+1)$ for all $k,$ as
occurs when the OSC holds. As we will show, it is also well defined in some
more complicated situations.

We call each set
$f_{-\mathbf{i}|k}\left(\left\\{x\in\pi_{top}\left(\mathbf{t}|k\right)\right\\}\right)$
a tile, and we call the collection of disjoint sets
$\left\\{f_{-\mathbf{i}|k}\left(\left\\{x\in\pi_{top}\left(\mathbf{t}|k\right)\right\\}\right)|\mathbf{t\in}\Sigma_{top}\right\\}$
a partial tiling. The partial tilings
$\left\\{f_{-\mathbf{i}|k}\left(\left\\{x\in\pi_{top}\left(\mathbf{t}|k\right)\right\\}\right)|\mathbf{t\in}\Sigma_{top}\right\\}$
are well defined. However, $\Pi_{top}(\mathbf{i})$ may not be well defined,
because there may not be any simple relationship between successive partial
tilings. But when it is well defined, we call it a tiling.

The tiles in the partial tiling
$\left\\{f_{-\mathbf{i}|k}\left(\left\\{x\in\pi_{top}\left(\mathbf{t}|k\right)\right\\}\right)|\mathbf{t\in}\Sigma_{top}\right\\}$
may be referred to by their addresses. It is convenient to define

$tile(i_{1}i_{2}...i_{k}.t_{1}t_{2}...t_{k})=f_{-\mathbf{i}|k}\left(\left\\{x\in\pi_{top}\left(\mathbf{t}|k\right)\right\\}\right)$

for all $\mathbf{i}|k$ and all $\mathbf{t}|k\in\Sigma_{top}.$ We also define
$tile(\varnothing)=A,$ corresponding to $k=0.$

###### Lemma 4

This concerns the sequence of tilings $\Pi_{top}(\mathbf{i}|n).$ If
$i_{n}p_{1}p_{2}...p_{n-1}$ $\in\Sigma_{top,n},$ then
$tile(i_{1}i_{2}...i_{n-1}.p_{1}p_{2}...p_{n-1})\subset
tile(i_{1}i_{2}...i_{n}.j_{1}j_{2}...j_{n})$ implies
$i_{n}p_{1}p_{2}...p_{n-1}=j_{1}j_{2}j_{3}..j_{n}$.

###### Proof

Suppose $tile(i_{1}i_{2}...i_{n-1}.p_{1}p_{2}...p_{n-1})\subset
tile(i_{1}i_{2}...i_{n}.j_{1}j_{2}...j_{n}).$ Then applying
$\left(f_{-\mathbf{i}|\left(n-1\right)}\right)^{-1}$ to both sides we obtain
$\pi_{top,n-1}(p_{1}p_{2}...p_{n-1})\subset
f_{i_{n}}^{-1}(\pi_{top,n}(j_{1}j_{2}...j_{n}))$ which is equivalent to

$f_{i_{n}}\pi_{top,n-1}(p_{1}p_{2}...p_{n-1})\subset\pi_{top,n}(j_{1}j_{2}...j_{n}).$

But $\pi_{top,n}(i_{n}p_{1}p_{2}...p_{n-1})\subset
f_{i_{n}}\pi_{top,n-1}(p_{1}p_{2}...p_{n-1})$ by Lemma 2, so

$\pi_{top,n}(i_{n}p_{1}p_{2}...p_{n-1})\subset\pi_{top,n}(j_{1}j_{2}...j_{n}).$

This implies $i_{n}p_{1}p_{2}...p_{n-1}=j_{1}j_{2}j_{3}..j_{n}$ because
otherwise $\pi_{top,n}(i_{n}p_{1}p_{2}...p_{n-1})$ and
$\pi_{top,n}(j_{1}j_{2}...j_{n})$ are disjoint subsets of $A$.

We say that $\mathbf{i\in}\Sigma_{top}$ is reversible when, for each
$n\in\mathbb{N}$ there exists $\mathbf{j=j}_{n}\mathbf{\in}\Sigma_{top}$ such
that $j_{1}=i_{n},j_{2}=i_{n-1},...,j_{n}=i_{1}.$ Note that $\mathbf{j}$
depends on $n.$ The address $\overline{1}=11111\dots$ is reversible and
belongs to $\Sigma_{top}$ in all cases.

EXAMPLE 3 For the i.f.s.
$\left\\{\mathbb{R};f_{1}(x)=2x/3;f_{2}(x)=2x/3+1/3\right\\},$ the strings
$\overline{1}$ and $\overline{2}$ both belong to $\Sigma_{top}$ and are
reversible. Figure 3 and Figure 4 illustrate two ways of looking at the
development of top addresses. Figure 5 (a) illustrates the sets in
$\Sigma_{top,n}$ for $n=0,1,2,3,4,5$. We usually use lexicographic ordering to
define top addresses, but Figure 4 uses standard ordering.

Figure 3: This compares the development of the top addresses for an i.f.s. of
two maps in the cases (a) where each scaling is 1/3 (b) each scaling is 1/2
(c) each scaling is 2/3. Figure 4: One way of illustrating the top of the
attractor of an i.f.s. See Example 3. The ordering here is not
lexicographical, so 2 is greater than 1.

EXAMPLE 4 For the i.f.s.
$\left\\{\mathbb{R};f_{1}(x)=2x/3;f_{2}(x)=1-2x/3\right\\},$ each of the
strings $\overline{1},\overline{2},\overline{12},\overline{21}$ belongs to
$\Sigma_{top}$ and is reversible. Figure 5(b) illustrates the sets in
$\Sigma_{top,n}$ for $n=0,1,2,3,4,5$. Here it appears that all addresses are
reversible.

Figure 5: See Examples 3 and 4.

Let us define a new tile to be a tile at level $n+1$ that is not contained in
any tile at level $n.$ Also, a child or child tile, is a tile at level $n+1$
that is contained in a tile, its parent at level $n.$

###### Theorem 5.2

Let $F$ be an invertible contractive i.f.s. on $\mathbb{R}^{n}$, as defined in
Equation 3. Let $\mathbf{i\in}\Sigma_{top}$ be reversible. Each tile in
$\Pi_{top}(\mathbf{i}|k+1)$ is either (i) a nonempty subset, the child of a
tile in $\Pi_{top}(\mathbf{i}|k),$ of the form
$tile(i_{1}...i_{k+1}.i_{k+1}p_{1}...p_{k}),$ or (ii) a nonempty set of the
form $tile(i_{1}...i_{k+1}.q_{1}q_{2}...q_{k}q_{k+1})$ where $q_{1}\neq
i_{k+1},$ a new tile. Each tile in $\Pi_{top}(\mathbf{i}|k)$ contains exactly
one child in $\Pi_{top}(\mathbf{i}|k+1)$.

###### Proof

We can write

$\displaystyle\Pi_{top}(\mathbf{i}|k+1)$
$\displaystyle=\left\\{tile(i_{1}i_{2}...i_{k+1}.i_{k+1}p_{2}...p_{k+1})|i_{k+1}p_{2}...p_{k+1}\in\Sigma_{top,k+1}\right\\}$
$\displaystyle\cup\left\\{tile(i_{1}i_{2}...i_{k+1}.j_{1}j_{2}...j_{k+1})|j_{1}j_{2}...j_{k+1}\in\Sigma_{top,k+1},j_{1}\neq
i_{k+1}\right\\}$

Each tile in the first set is a subset of a tile in $\Pi_{top}(\mathbf{i}|k),$
and it is non-empty because $\mathbf{i}$ is reversible. (By reversibility, the
set of top addresses
$\\{i_{k+1}p_{1}p_{2}...p_{k}\in\Sigma_{top,k+1}|p_{1}p_{2}...p_{k}\in\Sigma_{top,k}\\}$
is nonempty.)

Consider any tile $tile(i_{1}i_{2}...i_{k+1}.p_{1}p_{2}...p_{k+1})$ in the
second set. By Lemma 4: if
$tile(i_{1}i_{2}...i_{k}.p_{1}p_{2}...p_{k})\subset$
$tile(i_{1}i_{2}...i_{k+1}.i_{k+1}p_{2}...p_{k+1}),$ then
$i_{k+1}p_{1}p_{2}...p_{k}=j_{1}j_{2}j_{3}..j_{k+1}$ which is not possible
because $j_{1}\neq i_{k+1}$. So no tile in the second set is contained in a
tile in the first set. That is to say, the tiles in the second set, which have
non-cancelling addresses, are “new” and do not contain any tile in the first
set.

This says that every tile at level $k$ has a unique child at level $k+1,$
either equal to its parent, or smaller but not empty; also, there are new
tiles at level $k+1$ which do not have predecessors at level $k$, because
$\mathcal{A}_{k+1}\neq\mathcal{A}_{k}.$ Each tile in $\Pi_{top}(\mathbf{i}|k)$
contains a child in $\Pi_{top}(\mathbf{i}|k+1)$. One deduces that
$\mathcal{A}_{k+1}\backslash\cup\left\\{\text{children of tiles at level
}k\right\\}$ is tiled by new tiles.

In the special case $\mathbf{i=}\overline{\mathbf{1}},$ also considered by
Strichartz in Theorem 4.2, we have:

###### Theorem 5.3

Let $F$ be an invertible contractive i.f.s. on $\mathbb{R}^{n}$, as defined in
Equation 3. Then $\Pi_{top}(\overline{\mathbf{1}})$ is a well defined tiling
of $\mathcal{A}(\overline{\mathbf{1}}):$ specifically
$\Pi_{top}(\overline{\mathbf{1}}|k)\subset\Pi_{top}(\overline{\mathbf{1}}|k+1),$
and

$\Pi_{top}(\overline{\mathbf{1}})=\bigcup\limits_{k=1}^{\infty}\Pi_{top}(\overline{\mathbf{1}}|k).$

Each tile $\Pi_{top}(\overline{\mathbf{1}}|k)$ (for all $k\in\mathbb{N}$) in
$\Pi_{top}(\overline{\mathbf{1}})$ can be written
$tile((\overline{1}|k)|t_{1}t_{2}...t_{k})$ for some
$t_{1}t_{2}...t_{k}\in\Sigma_{top,k}$ for some $k,$ with $t_{1}\neq 1.$ The
tile $A$ corresponds to $k=0.$

###### Proof

The result follows from the observation that in this case all children are
exact copies of their parents. To see this simply note that
$f_{1}^{-1}\pi_{top}(1t_{1}t_{2}\dots t_{k})=\pi_{top}(t_{1}t_{2}\dots t_{k})$
for all $1t_{1}t_{2}\dots t_{k}\in\Sigma_{top,k+1}.$

For future work, one can consider the case where $A$ is homeomorphic to a
ball. By introducing a stronger notion of reversibility (see also tilings ),
that requires the tops dynamical system orbit of a reversible point
$\mathbf{i}\in\Sigma_{top}$ to be contained in a compact set $A^{\prime}$
contained in the interior of $A,$ one can ensure that new tiles are located
further and further away from $A.$ This means that new tiles have only
finitely many successive generations of children (one child at each subsequent
generation) before children are identical to their parents. Hence, given any
ball $B$ of finite radius, the set of tiles in $\Pi_{top}(\mathbf{i}|k)$ that
have nonempty intersection with $B$ remains constant for all large enough $k.$
In such cases one $\Pi_{top}(\mathbf{i}|k)\cap B$ is constant for all $k$
sufficiently large, and so the tiling $\Pi_{top}(\mathbf{i})$ is well defined.
We note that if $\mathbf{i}$ is disjunctive then
$\mathcal{A}(\mathbf{i})=\mathbb{R}^{n},$ see tilings .

We conjecture that if $A$ is homeomorphic to a ball and if
$\mathbf{i\in\Sigma}_{top}$ is both reversible and disjunctive (relative to
the top), then $\Pi_{top}(\mathbf{i})$ is a well defined tiling of
$\mathbb{R}^{n}.$

###  5.3 A leafy example of a two-dimensional top tiling

For a two-dimensional affine transformation
$f:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ we write

$\text{ }f=\begin{bmatrix}a&b&e\\\ c&d&g\end{bmatrix}\text{ for
}f(x,y)=(ax+by+e,cx+dy+g)\text{ }$

where $a,b,c,d,e,g\in\mathbb{R}$. We consider the i.f.s. defined by the two
similitudes

$f_{1}=\begin{bmatrix}0.7526&-.2190&.2474\\\
0.2190&0.7526&-.0726\end{bmatrix},f_{2}=\begin{bmatrix}-0.7526&0.2190&1.0349\\\
0.2190&0.7526&0.0678\end{bmatrix}$ (4)

The attractor, $L$=leaf, illustrated in Figure 6, is made of two overlapping
copies of itself. The copy illustrated in black is associated with $f_{1}.$
The point with top address $\overline{\mathbf{1}}=111\dots$ is represented by
the tip of the stem of the leaf. The stem is actually arranged in an infinite
spiral, not visible in the picture. In all tiling pictures, the colors of the
tiles were obtained by overlaying the tiling on a colorful photograph: the
color of each tile is the color of a point beneath it. In this way, if the
tiles were very small, the tiling would look like a mozaic representation of
the underlying picture.

Figure 6: The overlapping attractor of an i.f.s. of two similitudes, each with
the same scaling factor.

Figure 7 illustrates the top of $L$ at depths $n\in\\{1,2,\dots,6\\}$ labelled
by the addresses in $\Sigma_{n,top}$ .

Figure 7: Successsive fractal tops.

Figure 8 illustrates the successive blowups $\Pi_{top,n}(\overline{1}|n)$ for
$n=1,2,\dots,6$ for the i.f.s. in Equation (5.2). See also Figure 9 where the
successive images are illustrated in their correct relative positions.

Figure 8: This shows the sequence of tops $\Pi(111...|n)$ for $n=0,1,\dots,6$
for the leaf i.f.s. In each case the tip of the stem is at the origin. Figure
9: This illustrates the relationship between the successive partial tilings in
Figure 8.

Figure 10 shows a patch of a leaf tiling, illustrating its complexity. Figure
11 illustrates a patch of a top tiling obtained using an i.f.s. of four maps.

Figure 10: Patch of a leaf tiling. Figure 11: Fractal top for an i.f.s of
three maps looks both random and somewhat natural, but is not the real thing,
compare with Figure 1.

ACKNOWLEDGEMENTS: We thank both Brendan Harding and Giorgio Mantica for
careful reading and corrections.

## References

  * (1) C. Bandt, S.Graf, Self-similar sets VII. A characterization of self-similar fractals with positive Hausdorff measure, Proc. Amer. Math. Soc. 114 (1992), 995-1001.
  * (2) C. Bandt, Self-similar tilings and patterns described by mappings, Mathematics of Aperiodic Order (ed. R. Moody) Proc. NATO Advanced Study Institute C489, Kluwer, (1997) 45-83.
  * (3) C. Bandt, M. F. Barnsley, M. Hegland, A. Vince, Old wine in fractal bottles I: Orthogonal expansions on self-referential spaces via fractal transformations, Chaos, Solitons and Fractals, 91 (2016), 478-489.
  * (4) M. F. Barnsley, A. Vince, Fractal tilings from iterated function systems, Discrete and Computational Geometry, 51 (2014), 729-752.
  * (5) M. F. Barnsley, A. Vince, Self-similar polygonal tilings, Amer. Math. Monthly, 124 (2017), 905-921.
  * (6) M. F. Barnsley, K. Leśniak, M. Rypka, Basic topological structure of fast basins, Fractals 26 (2018), no. 1, 1850011.
  * (7) M. F. Barnsley, M. F., Theory and Applications of Fractal Tops, in: Levy-Vehel, J., Lutton, E., (eds.)Fractals in Engineering: New Trends in Theory and Applications. Springer, London (2005).
  * (8) J. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713-747.
  * (9) R. S. Strichartz, Fractals in the large, Canad. J. Math., 50 (1998), 638-657.
  * (10) R. S. Strichartz, Differential Equations on Fractals, Princeton University Press, Princeton, New Jersey (2006).
  * (11) A. Teplyaev, Spectral analysis on infinite Sierpinski gaskets, J. Funct. Anal. 129 (1998), 357-567.