S_{\sigma}([0,1]^{d})$ and there are precisely $(2^{d})^{3k-\ell}$ words
$\sigma$, so

$N_{r}(I\cap M)\leqslant N_{r}(I\cap
M_{m})\leqslant(R/r)^{\alpha+\frac{2d}{m-\ell}}.$

We can therefore choose $a$ small enough so that in either case $N_{r}(I\cap
M)\leqslant(R/r)^{\alpha+2\varepsilon}$, as required. ∎

Using this construction, along with the preceding construction for the upper
intermediate dimensions, we can now simultaneously prescribe the upper and
lower intermediate dimensions.

######  Corollary 3.3.12.

Let $0\leqslant\lambda\leqslant\alpha\leqslant d$ and let
$\underline{h},\overline{h}\in\mathcal{H}(\lambda,\alpha)$ satisfy
$\underline{h}(0)=\overline{h}(0)$ and $\underline{h}\leqslant\overline{h}$.
Then there exists a compact perfect set $M$ such that
$\operatorname{\operatorname{dim}_{\mathrm{L}}}M=\lambda$,
$\operatorname{\operatorname{dim}_{\mathrm{A}}}M=\alpha$ and

$\displaystyle\underline{\dim}_{\theta}C$
$\displaystyle=\underline{h}(\theta)$ $\displaystyle\overline{\dim}_{\theta}C$
$\displaystyle=\overline{h}(\theta)$

for all $\theta\in[0,1]$.

###### Proof.

Let $E,F$ be disjoint compact perfect sets such that
$\operatorname{\operatorname{dim}_{\mathrm{L}}}E=\operatorname{\operatorname{dim}_{\mathrm{L}}}F=\lambda$,
$\operatorname{\operatorname{dim}_{\mathrm{A}}}E=\operatorname{\operatorname{dim}_{\mathrm{A}}}F=\alpha$,
$\operatorname{\operatorname{dim}_{\mathrm{H}}}E=\operatorname{\operatorname{dim}_{\mathrm{H}}}F=\underline{h}(0)=\overline{h}(0)$,
and for $\theta\in(0,1]$

$\underline{\dim}_{\theta}F\leqslant\dim_{\theta}E=\underline{h}(\theta)\leqslant\overline{h}(\theta)=\overline{\dim}_{\theta}F.$

For example, such a set $E$ is provided by Theorem 3.3.11 and such a set $F$
is provided by Theorem 3.3.9. Let $M=E\cup F$. Then
$\operatorname{\operatorname{dim}_{\mathrm{L}}}M=\min\\{\operatorname{\operatorname{dim}_{\mathrm{L}}}E,\operatorname{\operatorname{dim}_{\mathrm{L}}}F\\}=\lambda$,
$\operatorname{\operatorname{dim}_{\mathrm{A}}}M=\max\\{\operatorname{\operatorname{dim}_{\mathrm{A}}}E,\operatorname{\operatorname{dim}_{\mathrm{A}}}F\\}=\alpha$,

$\underline{h}(\theta)=\underline{\dim}_{\theta}E\leqslant\underline{\dim}_{\theta}M\leqslant\max\\{\overline{\dim}_{\theta}E,\underline{\dim}_{\theta}F\\}=\underline{h}(\theta),$

and

$\overline{\dim}_{\theta}M=\max\\{\overline{\dim}_{\theta}E,\overline{\dim}_{\theta}F\\}=\overline{h}(\theta)$

for $\theta\in(0,1]$. Thus $M$ satisfies the requirements. ∎

##  Chapter 4 Infinitely generated attractors

###  4.1 Introduction

####  4.1.1 Background

This chapter focuses on infinite IFSs and is based on our joint paper [BF]
with J. M. Fraser. The dimension theory of limit sets of finite iterated
function systems (IFSs) has been studied extensively since Hutchinson’s paper
[Hut]. In a seminal 1996 paper [MU] Mauldin and Urbański extended the theory
to infinite iterated function systems (IIFSs) consisting of countably many
contractions, with the contraction ratios uniformly bounded above by some
$\rho<1$. The dimension theory of IIFSs has been studied further in [MUa, NT,
CN, BFa, KR, Mau] and many other works. Mauldin and Urbański paid particular
attention to (infinite) conformal iterated function systems (CIFSs, defined in
Definition 4.2.2), where the contractions are conformal and are sufficiently
separated. Approximations to one such limit set are shown in Figure 4.1.

Figure 4.1:  First and second level cylinders for an infinitely generated
self-similar set.

There are many similarities, but also many differences, between finite and
infinite iterated function systems. One notable difference is that Hausdorff
and box dimension coincide for the limit set of every finite CIFS, but can
differ for infinite CIFSs, as the presence of infinitely many maps can cause
the limit set to have greater inhomogeneity in space. In particular, Mauldin
and Urbański showed that for a CIFS the Hausdorff dimension can be determined
from a certain topological pressure function defined in (4.2.4) below (see
[MU, Theorem 3.15]). The same authors proved that the upper box and packing
dimensions are given by the maximum of the Hausdorff dimension of the limit
set and the upper box dimension of images of any given point under the maps in
the CIFS (noting that the box dimension of a countable set, unlike the
Hausdorff dimension, can be strictly positive), see [MUa, Theorem 2.11]. They
applied their results to sets of irrational numbers whose continued fraction
expansions have restricted entries, as these are limit sets of an appropriate
CIFS (see Section 4.4).

The following result describes the Assouad type dimensions of the limit set of
a CIFS. The Assouad spectra can display interesting behaviour, such as having
two phase transitions.

######  Theorem 4.1.1 (Banaji–Fraser [BFa]).

Let $F$ be the limit set of an infinite CIFS and let $P$ be the set of fixed
points of the contractions. Then for all $\theta\in(0,1)$,

$\displaystyle\max\\{\dim_{\mathrm{H}}F,\overline{\dim}_{\mathrm{A}}^{\theta}P\\}\leqslant\dim_{\mathrm{A}}^{\theta}F$
$\displaystyle=\overline{\dim}_{\mathrm{A}}^{\theta}F$
$\displaystyle\leqslant\max_{\phi\in[\theta,1]}\frac{(\phi^{-1}-1)\overline{\dim}_{\mathrm{A}}^{\phi}P+(\theta^{-1}-\phi^{-1})\overline{\dim}_{\mathrm{B}}F}{\theta^{-1}-1},$

and these bounds are sharp in general. If we assume the additional separation
condition that $\overline{S_{i}}(V)\cap\overline{S_{j}}(V)=\varnothing$ for
all distinct $i,j\in I$ (using notation from Definition 4.2.2 below), then

$\dim_{\mathrm{A}}F=\max\\{\dim_{\mathrm{H}}F,\dim_{\mathrm{A}}P\\}.$

To prevent this thesis from becoming unreasonably long, we omit the proof of
Theorem 4.1.1 and instead refer the reader to [BFa]. In this chapter we study
the intermediate dimensions of limit sets of infinite iterated function
systems.

####  4.1.2 Structure of chapter and discussion of results

In Section 4.2 we introduce notation, define limit sets, and define the
notions of dimension we will be working with. We introduce and prove basic
properties about the topological pressure function that we will use to obtain
bounds for dimensions of the limit sets. We also define conformal iterated
function systems (CIFSs) and prove geometric consequences of the definition of
a CIFS that we will use in the proof of the main result of this chapter,
Theorem 4.3.4.

In Section 4.3 we prove Theorem 4.3.1, which gives upper bounds for the
Hausdorff, box, intermediate and $\Phi$-intermediate dimensions in terms of
the topological pressure function that hold in the very general setting of
arbitrary IIFSs (without any conformality assumptions or separation
conditions). The proof is an induction argument, using efficient covers at
larger scales to construct efficient covers at smaller scales. In the
conformal setting, Mauldin and Urbański [MU, MUa] proved results for the
Hausdorff and upper box dimensions. We use our upper bound to prove the main
result of this chapter, a simplified version of which we now state.

######  Theorem (See Theorem 4.3.4 for a stronger statement).

If $F$ is the limit set of an (infinite) CIFS (defined in Definition 4.2.2)
and $P$ is the set of fixed points of the contractions then for all
$\theta\in[0,1]$,

$\overline{\dim}_{\theta}F=\max\\{\dim_{\mathrm{H}}F,\overline{\dim}_{\theta}P\\}.$

Our methods also apply to infinite parabolic IFSs. In Example 4.3.7 we
consider an example with intermediate dimensions continuous at $\theta=0$ and
apply a result of Burrell, Falconer and Fraser [BFF] to give an upper bound
for the upper box dimension of orthogonal projections.

In Section 4.4 we apply our results to give a formula in Theorem 4.4.3 for the
upper intermediate dimensions of sets of irrational numbers whose continued
fraction expansions have restricted entries. Recalling the discussion in
Section 1.4, we show that the intermediate dimensions of continued fraction
sets are continuous at $\theta=0$ apply our results and those in [Bura] to
give information related to Hölder distortion and fractional Brownian images
of continued fraction sets. We also obtain similar results in Section 4.4.2
for sets of complex numbers which have complex continued fraction expansions
with restricted entries.

In Section 4.5 we consider the limit sets of ‘generic’ IIFSs, in the same vein
as the seminal paper [Fala] where Falconer considered the generic dimension of
a (finitely-generated) self-affine set by fixing a set of matrices and
randomising the translates in a suitable way. We show that under certain
conditions, the limit set of an IIFS with ‘generic’ translates is somewhere
dense, and so in particular the box and intermediate dimensions equal the
ambient spatial dimension, where ‘generic’ can mean either almost surely with
respect to a natural measure, or comeagre with respect to a natural topology.
This is in stark contrast to the Hausdorff dimension, which Käenmäki and Reeve
[KR] showed satisfies an analogue of Falconer’s affinity dimension formula for
a generic IIFS of affine contractions.

In this chapter, as in [MU, MUa], the separation condition we assume in
Definition 4.2.2 for a CIFS is the open set condition (OSC). Ngai and Tong
[NT] and Chu and Ngai [CN] study the Hausdorff, box and packing dimensions of
the limit sets of IIFSs with overlaps that do not satisfy the OSC but do
satisfy suitable extensions of the weak separation condition. It is therefore
natural to ask (though we will not pursue this) what can be said about the
intermediate or $\Phi$-intermediate dimensions of the limit sets of infinite
iterated function systems with overlaps that do not satisfy the OSC but
perhaps satisfy weaker separation conditions such as the extensions of the
weak separation condition considered in [NT].

###  4.2 Infinite IFSs and pressure functions

We will work with infinite iterated function systems, defined as in [MU] as
follows.

######  Definition 4.2.1.

Let $d\in\mathbb{N}$ and let $X$ be a compact, connected subset of
$\mathbb{R}^{d}$ with more than one point, equipped with the metric induced by
the Euclidean norm $||\cdot||$. We say that an _infinite iterated function
system (IIFS)_ on $X$ is a collection of maps $S_{i}\colon X\to X$, $i\in I$,
where $I$ is a countable index set, such that there exists $\rho\in[0,1)$ such
that

$||S_{i}(x)-S_{i}(y)||\leqslant\rho||x-y||\quad\mbox{for all }x,y\in X\mbox{
and }i\in I.$

The assumption that the maps are uniformly contracting will be important when
defining the limit set. We now introduce some notation. Define
$I_{0}\coloneqq\\{\varnothing\\}$ and
$I^{*}\coloneqq\bigcup_{i=1}^{\infty}I^{n}$. We call elements of $I^{*}$
_finite words_ and elements of $I^{\mathbb{N}}$ _infinite words_. We usually
denote words by the letter $w$, and we write $w=i_{1}\dotsb i_{n}$ and
$w=i_{1}i_{2}\dotsb$ instead of $w=(i_{1},\dotsc,i_{n})$ and
$w=(i_{1},i_{2},\dotsc)$ respectively. We say that a word in $I^{n}$ has
_length_ $n$, and an infinite word has _length_ $\infty$. If $w\in I^{*}\cup
I^{\mathbb{N}}$ and $n\in\mathbb{N}$ does not exceed the length of $w$ then we
write $w|_{n}\coloneqq w_{1}\dotsb w_{n}\in I^{n}$, and
$w|_{0}\coloneqq\varnothing$. If $w\in I_{0}\cup I^{*}\cup I^{\mathbb{N}}$ and
$v\in I_{0}\cup I^{*}$ then we say that $v$ is a _prefix_ of $w$ if there
exists $n\in\\{0,1,2,\dotsc\\}$ such that $v=w|_{n}$. For $w\in I^{n}$ we
define

$S_{w}\coloneqq S_{w_{1}}\circ\dotsb\circ S_{w_{n}},$

and we define $S_{\varnothing}$ to be the identity function on $X$.

Mauldin and Urbański [MU, MUa] study the IIFSs in Definition 4.2.2, where the
contractions are assumed to extend to _conformal_ maps (see (iii) below for a
formal definition), which means that locally they preserve angles. This
assumption is crucial for Mauldin and Urbański’s formulae for the Hausdorff
and upper box dimensions of the limit set (though there has been some
subsequent work on IIFSs consisting of affine contractions [KR, KM]). The
conformality assumption will also be crucial when we prove a formula for the
intermediate dimensions in Section 4.3.2. In one dimension, conformal maps are
simply functions with non-vanishing Hölder continuous derivative. In two
dimensions, they are holomorphic functions with non-vanishing derivative on
their domain. In dimension three and higher, by a theorem of Liouville (1850)
they have a very restricted form: they are Möbius transformations, so can be
composed from homotheties, isometries, reflections in hyperplanes, and
inversions in spheres. Recall that $\mathcal{L}_{d}$ denotes $d$-dimensional
Lebesgue measure.

######  Definition 4.2.2.

A _conformal iterated function system (CIFS)_ is an IIFS (as in Definition
4.2.1) which satisfies the following additional properties:

  1. (i)

(Open set condition (OSC)) The set $X$ has non-empty interior
$U\coloneqq\mathrm{Int}_{\mathbb{R}^{d}}X$, and $S_{i}(U)\subset U$ for all
$i\in I$ and $S_{i}(U)\cap S_{j}(U)=\varnothing$ for all $i,j\in I$ with
$i\neq j$.

  2. (ii)

(Cone condition) $\inf_{x\in
X}\inf_{r\in(0,1)}\mathcal{L}_{d}(B(x,r)\cap\mathrm{Int}_{\mathbb{R}^{d}}X)/r^{d}>0$.

  3. (iii)

(Conformality) There exists an open, bounded, connected subset
$V\subset\mathbb{R}^{d}$ such that $X\subset V$ and such that for each $i\in
I$, $S_{i}$ extends to a $C^{1+\varepsilon}$ diffeomorphism from $V$ to an
open subset of $V$ which is _conformal_ , so for all $x\in V$ the differential
$S_{i}^{\prime}|_{x}$ exists, is non-zero, is a similarity map (so
$||S_{i}^{\prime}|_{x}(y)||=||S_{i}^{\prime}|_{x}||\cdot||y||$ for all
$y\in\mathbb{R}^{d}$), and is $\varepsilon$-Hölder continuous in $x$.
Moreover, there exists $\rho\in(0,1)$ such that $||S_{i}^{\prime}||<\rho$ for
all $i\in I$, where $||\cdot||$ is the supremum norm over $V$.

  4. (iv)

(Bounded distortion property (BDP)) There exists $K>0$ such that
$||S_{w}^{\prime}|_{y}||\leqslant K||S_{w}^{\prime}|_{x}||$ for all $x,y\in V$
and $w\in I^{*}$.

Mauldin and Urbański [MU, (2.8)] use a stronger form of the cone condition,
but note on page 110 that (ii) is sufficiently strong for their aims. Both
forms of this technical condition will be satisfied if $X$ is ‘reasonable,’
for example if $X$ is convex or has a smooth enough boundary. In the published
version of [BF], there is a typo in the BDP (it should read ‘for all $x,y\in
V$’ rather than ‘for all $x,y\in X$’).

For every IIFS, since $|S_{w|_{n}}(X)|\leqslant\rho^{n}|X|$ by the uniform
contractivity, the map

$\pi\colon I^{\mathbb{N}}\to
X,\qquad\pi(w)\coloneqq\bigcap_{n=1}^{\infty}S_{w|_{n}}(X)$

is well-defined and continuous. We are interested in the following set, which
will often be fractal in nature.

######  Definition 4.2.3.

The _limit set_ or _attractor_ of an IIFS is defined by

$F\coloneqq\pi(I^{\mathbb{N}})=\bigcup_{w\in
I^{\mathbb{N}}}\bigcap_{n=1}^{\infty}S_{w|_{n}}(X).$

For $w\in I^{n}$ define $F_{w}=F_{S_{w}}\coloneqq S_{w}(F)$ and
$X_{w}=X_{S_{w}}\coloneqq S_{w}(X)$. Now, $F$ is clearly non-empty and
satisfies the relation

$F=\bigcup_{i\in I}F_{i}.$ (4.2.1)

It is the largest (by inclusion) of many sets which satisfy (4.2.1). If $I$ is
finite then $F$ is compact (and is indeed the only non-empty compact set which
satisfies (4.2.1) by Hutchinson’s application of the Banach contraction
mapping theorem [Hut]), but if $I$ is infinite then $F$ will not generally be
closed. When $I$ is finite, the limit set $F$ equals the closure of the set of
fixed points of all finite compositions of maps in the IFS, and also satisfies
$F=\cap_{n=1}^{\infty}S^{n}(X)$ (where $S(E)\coloneqq\cup_{i\in I}S_{i}(E)$
for $E\subseteq X$), but when $I$ is infinite these sets may strictly contain
$F$. Some cylinder sets in the construction of an infinitely generated self-
similar set are shown in Figure 4.1 on page 4.1.

We define some more quantities that will enable us to define a topological
pressure function for the system. For every IIFS, for $w\in I_{n}$ define

$\displaystyle r_{w}=r_{S_{w}}$ $\displaystyle\coloneqq\inf_{x,y\in X,x\neq
y}\frac{||S_{w}(x)-S_{w}(y)||}{||x-y||};$ $\displaystyle R_{w}=R_{S_{w}}$
$\displaystyle\coloneqq\sup_{x,y\in X,x\neq
y}\frac{||S_{w}(x)-S_{w}(y)||}{||x-y||},$

noting that $0\leqslant r_{w}\leqslant R_{w}\leqslant\rho$. The value $R_{w}$
is the smallest possible Lipschitz constant for $S_{w}$, and these constants
are clearly submultiplicative; $R_{vw}\leqslant R_{v}R_{w}$ for all $v,w\in
I^{*}$. For $n\in\mathbb{N}$ define $M_{n}\coloneqq\\{\,S_{w}:w\in
I^{n}\,\\}$, and for $t\in[0,\infty)$ define

$\phi_{n}(t)\coloneqq\sum_{\sigma\in M_{n}}R_{\sigma}^{t}\in[0,\infty].$
(4.2.2)

Note that as in [NT], we sum over $\sigma\in M_{n}$ instead of $w\in I^{n}$ so
that distinct words $w$ that give rise to the same $S_{w}$ contribute only one
term in the sum (exact overlaps are removed).

######  Lemma 4.2.4.

For every IIFS, for all $t\in(0,\infty)$, using the convention
$\log\infty=\infty$ and $\log 0=-\infty$,

$\frac{1}{n}\log\phi_{n}(t)\xrightarrow[n\to\infty]{}\inf_{n\in\mathbb{N}}\frac{1}{n}\log\phi_{n}(t)\in[-\infty,\infty].$

###### Proof.

By the submultiplicativity of the Lipschitz constants, if $n,m\in\mathbb{N}$
then

$\log\phi_{n+m}\leqslant\log\left(\sum_{\sigma\in M_{n}}\sum_{\tau\in
M_{m}}(R_{\sigma}R_{\tau})^{t}\right)=\log\left(\sum_{\sigma\in
M_{n}}R_{\sigma}^{t}\sum_{\tau\in
M_{m}}R_{\tau}^{t}\right)=\log\phi_{n}+\log\phi_{m}.$

Therefore the sequence $(\log\phi_{n})_{n=1}^{\infty}$ is subadditive, so the
claim follows from Fekete’s lemma. ∎

In light of Lemma 4.2.4 we can make the following definition, which will later
be used when giving bounds and formulae for the different notions of dimension
for the limit sets.

######  Definition 4.2.5.

For an IIFS, define the _(topological) pressure function_
$\overline{P}\colon(0,\infty)\to[-\infty,\infty]$ by

$\overline{P}(t)\coloneqq\lim_{n\to\infty}\frac{1}{n}\log\phi_{n}(t)=\inf_{n\in\mathbb{N}}\frac{1}{n}\log\phi_{n}(t).$

######  Lemma 4.2.6.

For every IIFS, $\overline{P}$ is a decreasing function, and if $0<t<s<\infty$
and $\overline{P}(t)\in\mathbb{R}$ then $\overline{P}(t)>\overline{P}(s)$.

###### Proof.

For all $n\in\mathbb{N}$, for all $w\in I^{n}$, $R_{w}\leqslant\rho^{n}<1$, so
$\phi_{n}$ is a decreasing function. Therefore $\overline{P}$ is a decreasing
function. If $0<t<s<\infty$ and $\overline{P}(t)\in\mathbb{R}$ then for all
$n\in\mathbb{N}$, $\phi_{n}(s)\leqslant\rho^{(s-t)n}\phi_{n}(t)$ so
$\frac{1}{n}\log\phi_{n}(s)\leqslant(s-t)\log\rho+\frac{1}{n}\log\phi_{n}(t)$,
hence $\overline{P}(s)\leqslant(s-t)\log\rho+\overline{P}(t)<\overline{P}(t)$,
as required. ∎

######  Definition 4.2.7.

Throughout this chapter, using the convention that $\inf\varnothing=\infty$,
we define the _finiteness parameter_ of the system

$\theta_{S}\coloneqq\inf\\{\,t>0:\overline{P}(t)<\infty\,\\}\in[0,\infty],$

and the quantity

$h\coloneqq\inf\\{\,t>0:\overline{P}(t)<0\,\\}\in[0,\infty].$

We use the letter $h$ because we will see that it is related to the Hausdorff
dimension of the limit set. For all $n\in\mathbb{N}$ and $t\in[0,\infty)$,

$\phi_{n}(t)\leqslant\sum_{w\in I^{n}}R_{w}^{t}\leqslant\sum_{i_{1}\dotsb
i_{n}\in I^{n}}\prod_{k=1}^{n}R_{i_{k}}^{t}.$

Therefore if $\phi_{n}(t)$ is replaced by either $\sum_{w\in I^{n}}R_{w}^{t}$
or $\sum_{i_{1}\dotsb i_{n}\in I^{n}}\prod_{k=1}^{n}R_{i_{k}}^{t}$ in the
definition of the pressure function then the resulting functions would
overestimate $\overline{P}(t)$. Thus the infimal values of $t>0$ for which
these new functions are negative provide upper bounds for $h$. These may be
easier to compute than $h$ itself. For all $n\in\mathbb{N}$, the function
$\frac{1}{n}\log\phi_{n}(t)$ can also be used to give an upper bound, which
will be very good when $n$ is large, by Lemma 4.2.4.

We now establish several geometric facts that hold for all CIFSs and will be
important when proving a formula for the intermediate dimensions of the limit
set in Section 4.3.2. The following geometric property was proved in [MU, page
111], recalling that $K$ is the constant from the bounded distortion
principle.

$\qquad S_{w}(B(x,r))\supseteq
B(S_{w}(x),K^{-1}||S_{w}^{\prime}||r)\quad\mbox{for all }x\in
X,r\in(0,\mathrm{dist}(X,\partial V)],w\in I^{*}.$ (4.2.3)

The following lemma says that the Lipschitz constants are comparable to the
norm of the derivatives of the corresponding map.

######  Lemma 4.2.8.

For every CIFS there exists $D\geqslant 1$ such that for all $w\in I^{*}$,

$D^{-1}||S_{w}^{\prime}||\leqslant r_{w}\leqslant R_{w}\leqslant
D||S_{w}^{\prime}||.$

###### Proof.

The proof is similar to the proofs of some of the consequences of the bounded
distortion principle in [MU, page 110–111]. For the upper bound, note that
$\mathrm{dist}(X,{\mathbb{R}^{d}\setminus V})>0$ since $X$ is compact and
disjoint from the closed set ${\mathbb{R}^{d}\setminus V}$. Let $w\in I^{*}$.
If $B$ is a ball of radius at most $\mathrm{dist}(X,{\mathbb{R}^{d}\setminus
V})$ centred at a point in $X$ and $x,y\in B$ then by the mean value
inequality $||S_{w}(x)-S_{w}(y)||\leqslant||S_{w}^{\prime}||\cdot||x-y||$.
Since $X$ is compact and connected, it can be covered by a finite chain of
open balls $B_{1},\dotsc,B_{q}$ centred at points in $X$ and with radii at
most $\mathrm{dist}(X,\mathbb{R}^{d}\setminus V)/(2K)$ (chain in the sense
that $B_{i}\cap B_{i+1}\neq\varnothing$ for $i=1,2,\dotsc,q-1$). Suppose

$D\geqslant\max\left\\{q,K,\frac{K|X|}{\mathrm{dist}(X,\mathbb{R}^{d}\setminus
V)}\right\\}.$

Then since $D\geqslant q$, the upper bound $R_{w}\leqslant
D||S_{w}^{\prime}||$ holds.

Trivially, $r_{w}\leqslant R_{w}$. For the lower bound, if $x,y\in X$ and
$||x-y||\leqslant\mathrm{dist}(X,\mathbb{R}^{d}\setminus V)$ then by (4.2.3)
and the bijectivity of $S_{w}$,

$||S_{w}(x)-S_{w}(y)||\geqslant K^{-1}||S_{w}^{\prime}||\cdot||x-y||\geqslant
D^{-1}||S_{w}^{\prime}||\cdot||x-y||.$

If $x,y\in X$ and $|X|\geqslant||x-y||>\mathrm{dist}(X,\mathbb{R}^{d}\setminus
V)$ then since

$S_{w}(B(x,\mathrm{dist}(X,\mathbb{R}^{d}\setminus V)))\supseteq
B(S_{w}(x),K^{-1}||S_{w}^{\prime}||\mathrm{dist}(X,\mathbb{R}^{d}\setminus
V)),$

it holds that

$\displaystyle||S_{w}(x)-S_{w}(y)||\geqslant
K^{-1}||S_{w}^{\prime}||\mathrm{dist}(X,\mathbb{R}^{d}\setminus V)$
$\displaystyle\geqslant
K^{-1}||S_{w}^{\prime}||\mathrm{dist}(X,\mathbb{R}^{d}\setminus
V)||x-y||\cdot|X|^{-1}$ $\displaystyle\geqslant
D^{-1}||S_{w}^{\prime}||\cdot||x-y||.$

Therefore the lower bound $r_{w}\geqslant D^{-1}||S_{w}^{\prime}||$ holds, as
required. ∎

Lemma 4.2.9 essentially says that the cone condition holds not just for the
set $X$ itself but also for its images under the conformal map corresponding
to any given finite word. The purpose of Lemma 4.2.9 is to prove Lemma 4.2.10.

######  Lemma 4.2.9.

For all CIFSs, with $D$ as in Lemma 4.2.8,

$\inf_{w\in I^{*}}\inf_{x\in
S_{w}(X)}\inf_{r\in(0,D||S_{w}^{\prime}||)}r^{-d}\cdot\mathcal{L}_{d}(B(x,r)\cap
S_{w}(\mathrm{Int}_{\mathbb{R}^{d}}X))>0.$

###### Proof.

Write $U=\mathrm{Int}_{\mathbb{R}^{d}}X$. Let $n\in\mathbb{N}$ and let $w\in
I^{n}$. The idea is that given any ball centred on $S_{w}(X)$ whose diameter
is not too large we can find a large enough ball centred on $X$ that is mapped
into it under $S_{w}$, a uniform proportion of which intersects $U$ by the
cone condition. The measure of the image of this part under $S_{w}$ is large
enough by conformality and the BDP.

Consider an arbitrary point in $S_{w}(X)$, which we can write as $S_{w}(x)$
for some $x\in X$, and let $r\in(0,D||S_{w}^{\prime}||)$. By the cone
condition there exists $c>0$ such that $\mathcal{L}_{d}(B(x,r)\cap U)/r^{d}>c$
for all $x\in X$ and $r\in(0,1)$. By the upper bound of Lemma 4.2.8,
$S_{w}(B(x,r/(D||S_{w}^{\prime}||)))\subseteq B(S_{w}(x),r)$. Now,
$\mathcal{L}_{d}(B(x,r/(D||S_{w}^{\prime}||))\cap
U)r^{-d}D^{d}||S_{w}^{\prime}||^{d}>c$, so by the inner regularity of the
Lebesgue measure, there exists a compact $C\subset
B(x,r/(D||S_{w}^{\prime}||))\cap U$ such that
$\mathcal{L}_{d}(C)>cr^{d}D^{-d}||S_{w}^{\prime}||^{-d}$. Since $C$ is compact
and disjoint from the closed set
$\mathbb{R}^{d}\setminus(B(x,r/(D||S_{w}^{\prime}||))\cap U)$, it follows that
$\mathrm{dist}(C,\mathbb{R}^{d}\setminus(B(x,r/(D||S_{w}^{\prime}||))\cap
U))>0$. Let $n\in\mathbb{N}$ be large enough so that

$2^{-n}<\min\\{\mathrm{dist}(C,\mathbb{R}^{d}\setminus(B(x,r/(D||S_{w}^{\prime}||))\cap
U))/\sqrt{d},\mathrm{dist}(X,\mathbb{R}^{d}\setminus V)\\}.$

Define $c_{d}\in(0,1)$ by

$c_{d}\coloneqq\frac{\mathcal{L}_{d}(B(0,1))}{\mathcal{L}_{d}([-1,1]^{d})}.$

Then the balls of diameter $2^{-n}$ inside each of the dyadic cubes of
sidelength $2^{-n}$ which intersect $C$ form a disjoint collection of balls
inside $B(x,r/(D||S_{w}^{\prime}||))\cap U$ whose total Lebesgue measure is
greater than $cc_{d}r^{d}D^{-d}||S_{w}^{\prime}||^{-d}$. By (4.2.3), the image
of each of these balls under $S_{w}$ contains a ball of radius
$K^{-1}||S_{w}^{\prime}||2^{-(n+1)}$. These balls are disjoint subsets of
$B(S_{w}(x),r)\cap S_{w}(U)$ whose total Lebesgue measure is greater than
$cc_{d}r^{d}D^{-d}||S_{w}^{\prime}||^{-d}||S_{w}^{\prime}||^{d}K^{-d}=cc_{d}r^{d}D^{-d}K^{-d}$.
Therefore

$\inf_{w\in I^{*}}\inf_{x\in
S_{w}(X)}\inf_{r\in(0,D||S_{w}^{\prime}||)}\mathcal{L}_{d}(B(x,r)\cap
S_{w}(\mathrm{Int}_{\mathbb{R}^{d}}X))/r^{d}\geqslant cc_{d}D^{-d}K^{-d}>0,$

as required. ∎

Lemma 4.2.10 says that it is impossible for too many cylinder sets that are
larger than a given size to cluster together and intersect a region that is
smaller than that size. This will be useful when proving facts about
dimensions, as it shows that a set of a given size cannot cover another set
which intersects too many cylinders that are larger than that given size.
Lemma 4.2.10 and its proof are an application of Lemma 4.2.9 and the OSC, and
are similar to (2.2) in [MUa, Proposition 2.9]. The use of the cone condition
to obtain Lemma 4.2.10 is similar to [GMW, Theorem 4.9].

######  Lemma 4.2.10.

For any CIFS there exists $M\in\mathbb{N}$ such that for all
$z\in\mathbb{R}^{d}$ and $r>0$, if $w_{1},\dotsc,w_{l}$ are distinct words in
$I^{*}$ such that for all $i,j\in\\{1,\dotsc,l\\}$, $w_{i}$ is not a prefix of
$w_{j}$, and for all $i\in\\{1,\dotsc,l\\}$ it holds that $B(z,r)\cap
S_{w_{i}}(X)\neq\varnothing$ and $|S_{w_{i}}(X)|\geqslant r/2$, then
$l\leqslant M$.

###### Proof.

Write $U=\mathrm{Int}_{\mathbb{R}^{d}}X$, and let $k_{d}$ be the
$d$-dimensional Lebesgue measure of a ball in $\mathbb{R}^{d}$ of unit radius.
By Lemma 4.2.9 there exists $c>0$ such that

$\inf_{w\in I^{*}}\inf_{x\in
S_{w}(X)}\inf_{r\in(0,D||S_{w}^{\prime}||\cdot|X|)}\mathcal{L}_{d}(B(x,r)\cap
S_{w}(U))/r^{d}>c.$

For each $i=1,\dotsc,l$ there exists $x_{i}\in X$ such that
$S_{w_{i}}(x_{i})\in B(z,r)$. By the upper bound from Lemma 4.2.8,
$r/2\leqslant|S_{w_{i}}(X)|\leqslant D||S_{w}^{\prime}||\cdot|X|$, so
$\mathcal{L}_{d}(B(S_{w_{i}}(x_{i}),r/2)\cap S_{w}(U))2^{d}r^{-d}>c$. Since no
$w_{i}$ is a prefix of any $w_{j}$, by the OSC the $l$ sets
$B(S_{w_{i}}(x_{i}),r/2)\cap S_{w}(U)$ are disjoint subsets of $B(z,2r)$, each
having $d$-dimensional Lebesgue measure at least $cr^{d}2^{-d}$. Therefore
$lcr^{d}2^{-d}\leqslant\mathcal{L}_{d}(B(z,2r))=k_{d}2^{d}r^{d}$, so if we let
$M\coloneqq k_{d}2^{2d}c^{-1}$ then $l\leqslant M$, as required. ∎

Lemma 4.2.10 shows in particular that for all CIFSs, for all $n\in\mathbb{N}$,

$\\#\\{\,w\in I^{n}:B(z,r)\cap S_{w}(X)\neq\varnothing\textup{ and
}|S_{w}(X)|\geqslant r/2\,\\}\leqslant M,$

so the family $\\{\,S_{w}(X):w\in I^{n}\,\\}$ is _pointwise finite_ in the
sense that each element of $X$ belongs to at most finitely many elements of
this family. Therefore the limit set satisfies

$F=\bigcap_{n=1}^{\infty}\bigcup_{w\in I^{n}}S_{w}(X),$

and so is a Borel subset of $X$ in the class $F_{\sigma\delta}$. Mauldin and
Urbański noted this in [MU, (2.5)] and also showed that the limit set need not
be in the class $G_{\delta}$ (i.e. it need not be a countable intersection of
open sets). Lemma 4.2.8 says that the Lipschitz constants are comparable to
the norm of the derivative of the corresponding map; Lemma 4.2.11 uses this to
show that the sizes of the corresponding cylinder sets are also comparable.

######  Lemma 4.2.11.

For every CIFS there exists $D\geqslant 1$ such that for all $w\in I^{*}$,

$D^{-1}||S_{w}^{\prime}||\leqslant|F_{w}|\leqslant|S_{w}(X)|\leqslant
D||S_{w}^{\prime}||.$

###### Proof.

Lemma 4.2.10 shows in particular that the family $\\{\,S_{i}:i\in I\,\\}$ is
pointwise finite, so since $I$ is infinite, $F$ has positive diameter.
Therefore the result follows from Lemma 4.2.8 if we increase $D$ as required.
∎

For a CIFS, for each $n\in\mathbb{N}$ define
$\psi_{n}\colon[0,\infty)\to\mathbb{R}$ by

$\psi_{n}(t)=\sum_{w\in I^{n}}||S_{w}^{\prime}||^{t}.$

Mauldin and Urbański [MU, Section 3] define the pressure function by

$\lim_{n\to\infty}\frac{1}{n}\log\psi_{n}(t),$ (4.2.4)

showing that this limit always exists in $[-\infty,\infty]$ and proving many
properties about this function. Lemma 4.2.12 shows that this coincides with
our definition for the pressure function $\overline{P}(t)$, and is in
particular independent of the open set $V$.

######  Lemma 4.2.12.

For all CIFSs, for all $t\in(0,\infty)$,

$\frac{1}{n}\log\psi_{n}(t)\to\overline{P}(t)\textup{ as }n\to\infty.$

###### Proof.

The OSC means that there are no exact overlaps, so $\phi_{n}(t)=\sum_{w\in
I^{n}}R_{w}^{t}$ for all $n\in\mathbb{N}$ and $t\in(0,\infty)$. Therefore by
Lemma 4.2.8,

$D^{-t}\phi_{n}(t)\leqslant\psi_{n}(t)\leqslant D^{t}\phi_{n}(t).$

Taking logarithms and dividing through by $n$ gives

$-\frac{1}{n}t\log
D+\frac{1}{n}\log\phi_{n}(t)\leqslant\frac{1}{n}\log\psi_{n}(t)\leqslant\frac{1}{n}t\log
D+\log\phi_{n}(t),$

and the result follows upon taking the limit $n\to\infty$. ∎

One of the properties proved in [MU, Section 3] for a CIFS is that the
finiteness parameter for the pressure function is the same as the value at
which each of the finite-level approximations to the pressure function becomes
finite: $\theta_{S}=\inf\\{\,t>0:\psi_{n}(t)<\infty\,\\}$ for all
$n\in\mathbb{N}$. Note that for a CIFS where each of the maps $S_{i}$ are
similarities, which means that there exists $c_{i}\in(0,1)$ such that
$||S_{i}(x)-S_{i}(y)||=c_{i}||x-y||$ for all $x,y\in X$, the quantity $h$ from
Definition 4.2.7 satisfies the simple form

$h=\inf\left\\{\,t>0:\sum_{i\in I}c_{i}^{t}<1\,\right\\}.$ (4.2.5)

###  4.3 Dimension results

####  4.3.1 General upper bounds

In Theorem 4.3.1 we provide general upper bounds for the Hausdorff, box,
intermediate and $\Phi$-intermediate dimensions of the limit set of an
arbitrary IIFS.

######  Theorem 4.3.1.

For a IIFS with limit set $F$ and notation as above,

  1. (i)

$\dim_{\mathrm{H}}F\leqslant h$

  2. (ii)

$\overline{\dim}_{\mathrm{B}}F\leqslant\max\\{h,\lim_{n\to\infty}\inf\\{\,\overline{\dim}_{\mathrm{B}}P:P\subseteq
X\textup{ and }\forall w\in I^{n},\,P\cap S_{w}(X)\neq\varnothing\,\\}\\}$

  3. (iii)

For all $\theta\in[0,1]$,

$\overline{\dim}_{\theta}F\leqslant\max\\{h,\lim_{n\to\infty}\inf\\{\,\overline{\dim}_{\theta}P:P\subseteq
X\textup{ and }\forall w\in I^{n},\,P\cap S_{w}(X)\neq\varnothing\,\\}\\}$

  4. (iv)

If $\Phi$ is monotonically admissible (recall Definition 2.1.1 from page
2.1.1) then

$\overline{\dim}^{\Phi}F\leqslant\max\\{h,\lim_{n\to\infty}\inf\\{\,\overline{\dim}^{\Phi}P:P\subseteq
X\textup{ and }\forall w\in I^{n},\,P\cap S_{w}(X)\neq\varnothing\,\\}\\}$

Since $\dim_{\mathrm{P}}F\leqslant\overline{\dim}_{\mathrm{B}}F$ always holds,
(ii) also gives an upper bound for the packing dimension. Note that the above
upper bounds hold even when there are overlaps of cylinders, and for
contractions which are not differentiable and do not satisfy any bi-Lipschitz
or bounded distortion condition. However, in some such cases $h$ can
overestimate $\dim_{\mathrm{H}}F$ significantly, and may even be infinite.

###### Proof.

All the bounds are trivial if $h=\infty$, so assume $h<\infty$.

(i). The proof is similar to the proof of the first part of [MU, Theorem
3.15]. Let $s>h$. By Lemma 4.2.6 and the definition of $h$,
$\overline{P}(s)<0$. Therefore there exists $N\in\mathbb{N}$ such that
$\frac{1}{n}\log\phi_{n}(s)<\overline{P}(s)/2$, so
$\phi_{n}(s)<e^{n\overline{P}(s)/2}$, for all $n\geqslant N$. Therefore

$\sum_{\sigma\in
M_{n}}|\sigma(X)|^{s}\leqslant|X|^{s}\phi_{n}(s)<|X|^{s}e^{n\overline{P}(s)/2}\xrightarrow[n\to\infty]{}0.$

But $\\{\,\sigma(X):\sigma\in M_{n}\,\\}$ forms a $|X|\rho^{n}$-cover of $F$,
and $|X|\rho^{n}\to 0$ as $n\to\infty$, so this means that the $s$-dimensional
Hausdorff measure of $F$ is 0. Thus $\dim_{\mathrm{H}}F\leqslant s$. Letting
$s\to h^{+}$ gives $\dim_{\mathrm{H}}F\leqslant h$, as required.

(ii). follows from the case $\theta=1$ of (iii).

(iii). The proof is motivated by the proof of [MUa, Lemma 2.8], which gives a
result for the box dimension in the less general setting of a CIFS. We will
consider $\delta\in\left(\frac{1}{n+1},\frac{1}{n}\right]$ and induct on $n$.
The idea is that if we fix a large enough $q\in\mathbb{N}$, the level-$q$
cylinders with size $\lesssim\delta$ can be covered efficiently using a cover
of a set $P$ corresponding to level $q$, and the cylinders with size
$\gtrsim\delta$ can be covered efficiently using images of efficient covers of
$F$ with larger diameters that are assumed to exist by the inductive
hypothesis, and the fact that $\overline{P}(s)<0$ if $s>h$.

By definition $\overline{\dim}_{0}=\dim_{\mathrm{H}}$, so since we can take
$P$ to be a countable set (with Hausdorff dimension 0) for all
$n\in\mathbb{N}$, the case $\theta=0$ follows from (i). Henceforth suppose
$\theta\in(0,1]$. Let

$s>\max\\{h,\lim_{n\to\infty}\inf\\{\,\overline{\dim}_{\theta}P:P\subseteq
X\textup{ and }\forall w\in I^{n},\,P\cap S_{w}(X)\neq\varnothing\,\\}\\}.$

Since $s>h$, it holds that $\overline{P}(s)<0$. Therefore there exists
$Q\in\mathbb{N}$ such that $\frac{1}{q}\log\phi_{q}(s)<\overline{P}(s)/2$ for
all $q\geqslant Q$. Fix $q\geqslant Q$ large enough such that
$\phi_{q}(s)\leqslant 1/2$ and $\rho^{q}<1/4$, so $R_{w}<1/4$ for all words
$w$ of length at least $q$. By the definition of $s$, increasing $q$ further
if necessary, we may assume there exists a subset $P_{q}\subseteq X$ such that
$P_{q}\cap S_{w}(X)\neq\varnothing$ for all $w\in I^{q}$, and
$\overline{\dim}_{\theta}P_{q}<s$. Therefore there exists $A>0$ such that for
all $\delta\in(0,1]$ there exists a cover $\\{V_{j}\\}$ of $P_{q}$ such that
$\delta\leqslant|V_{j}|\leqslant\delta^{\theta}$ for all $j$, and
$\sum_{j}|V_{j}|^{s}\leqslant A$. Let $A_{d,1+2|X|}$ be as in (1.3.3). Fix any
$B>\frac{A_{d,1+2|X|}A}{1-\phi_{q}(s)}$ large enough so that for all
$\delta\in(1/2,1]$ there exists a cover $\\{U_{i}^{\delta}\\}_{i}$ of $F$ such
that $\delta\leqslant|U_{i}|\leqslant\delta^{\theta}$ for all $i$, and
$\sum_{i}|U_{i}|^{s}\leqslant B$. It suffices to show that
$\overline{\dim}_{\theta}F\leqslant s$, which follows from the following
claim.

Claim: For all $n\in\mathbb{N}$, for all
$\delta\in\left(\frac{1}{n+1},\frac{1}{n}\right]$ there exists a cover
$\\{U_{i}^{\delta}\\}_{i}$ of $F$ such that
$\delta\leqslant|U_{i}|\leqslant\delta^{\theta}$ for all $i$, and
$\sum_{i}|U_{i}|^{s}\leqslant B$.

Proof of claim: We prove the claim by induction on $n$. The claim holds for
$n=1$ by the definition of $B$. Let $n\in\mathbb{N}$, $n>1$, and assume the
claim holds for $1,2,\dotsc,n-1$. Let
$\delta\in\left(\frac{1}{n+1},\frac{1}{n}\right]$. By the definition of $A$
there exists a cover $\\{V_{j}\\}$ of $P_{q}$ such that
$\delta\leqslant|V_{j}|\leqslant\delta^{\theta}$ for all $j$, and
$\sum_{j}|V_{j}|^{t}\leqslant A$. By the definition of $A_{d,1+2|X|}$, for all
$j$ there exist $V_{j,1},\dotsc,V_{j,A_{d,1+2|X|}}\subseteq\mathbb{R}^{d}$,
each of diameter

$\max\\{\delta,|\mathcal{S}_{|X|\delta}(V_{j})|/(1+2|X|)\\},$

such that

$\mathcal{S}_{|X|\delta}(V_{j})\subseteq\bigcup_{k=1}^{A_{d,1+2|X|}}V_{j,k}.$

By the triangle inequality,

$|\mathcal{S}_{\delta|X|}(V_{j})|\leqslant|V_{j}|+2|X|\delta\leqslant(1+2|X|)|V_{j}|\leqslant(1+2|X|)\delta^{\theta},$

so $\delta\leqslant|V_{j,k}|\leqslant\delta^{\theta}$ and
$|V_{j,k}|\leqslant|V_{j}|$ for all $j,k$. Recalling that $M_{q}$ is the set
of maps corresponding to words of length $q$, let
$C_{\delta}\coloneqq\\{\,\tau\in M_{q}:|F_{\sigma}|\leqslant|X|\delta\,\\}$.
Since $\\{V_{j}\\}$ covers $P_{q}$, $\\{\mathcal{S}_{|X|\delta}(V_{j})\\}$
covers $\cup_{\tau\in C_{\delta}}F_{\tau}$, so
$\cup_{k=1}^{A_{d,1+2|X|}}V_{j,k}$ covers $\cup_{\tau\in C_{\delta}}F_{\tau}$.

Now suppose $\sigma\in M_{q}\setminus C_{\delta}$, so
$|X|\delta<|F_{\sigma}|\leqslant|X|R_{\sigma}$, so $\delta/R_{\sigma}<1$, and
since $R_{\sigma}<1/4$,

$\frac{\delta}{R_{\sigma}}\geqslant\frac{1}{(n+1)R_{\sigma}}>\frac{4}{n+1}>\frac{1}{n}.$

Therefore by the inductive assumption there exists a cover
$\\{U_{i}^{\delta/R_{\sigma}}\\}$ of $F$ such that
$\delta/R_{\sigma}\leqslant|U_{i}^{\delta/R_{\sigma}}|\leqslant(\delta/R_{\sigma})^{\theta}$
for all $i$ and $\sum_{i}|U_{i}^{\delta/R_{\sigma}}|^{s}\leqslant B$. For each
$i$, let $W_{\sigma,i}$ be a set with diameter

$|W_{\sigma,i}|=\max\\{|S_{\sigma}(U_{i}^{\delta/R_{\sigma}})|,\delta\\}$
(4.3.1)

such that $S_{\sigma}(U_{i}^{\delta/R_{\sigma}})\subseteq W_{\sigma,i}$. Since
$\\{S_{\sigma}(U_{i}^{\delta/R_{\sigma}})\\}_{i}$ covers $F_{\sigma}$, also
$\\{W_{\sigma,i}\\}_{i}$ covers $F_{\sigma}$. By the definition of
$R_{\sigma}$, $|S_{\sigma}(U_{i}^{\delta/R_{\sigma}})|\leqslant
R_{\sigma}|U_{i}^{\delta/R_{\sigma}}|$ for all $j$, and also
$\delta=R_{\sigma}\delta/R_{\sigma}\leqslant
R_{\sigma}|U_{i}^{\delta/R_{\sigma}}|$, so by (4.3.1),

$\delta\leqslant|W_{\sigma,i}|\leqslant
R_{\sigma}|U_{i}^{\delta/R_{\sigma}}|\leqslant
R_{\sigma}(\delta/R_{\sigma})^{\theta}\leqslant\delta^{\theta}.$ (4.3.2)

The last inequality (which is crucial to the argument) holds since
$R_{\sigma}<1$.

Now, $\\{V_{j,k}\\}\cup\\{W_{\sigma,i}\\}$ is a cover of $F$ and the diameter
of each of these sets lies in the interval $[\delta,\delta^{\theta}]$.
Moreover, since $|V_{j,k}|\leqslant|V_{j}|$ and by (4.3.2),

$\displaystyle\sum_{j}\sum_{k=1}^{A_{d,1+2|X|}}|V_{j,k}|^{s}+\sum_{\sigma\in
M_{q}\setminus C_{\delta}}\sum_{i}|W_{\sigma,i}|^{s}$ $\displaystyle\leqslant
A_{d,1+2|X|}\sum_{j}|V_{j}|^{s}+\sum_{\sigma\in M_{q}\setminus
C_{\delta}}R_{\sigma}^{s}\sum_{i}|U_{i}^{\delta/R_{\sigma}}|^{s}$
$\displaystyle\leqslant A_{d,1+2|X|}A+B\phi_{q}(s)$ $\displaystyle\leqslant
B,$

so the claim holds by induction.

(iv). By Lemma 2.2.10 from page 2.2.10 we may assume without loss of
generality that $\Phi$ is invertible. Then (iv) holds by almost exactly the
same proof as (iii), with $\overline{\dim}_{\theta}$, $\delta^{\theta}$ and
$(\delta/R_{w})^{\theta}$ replaced by $\overline{\dim}^{\Phi}$,
$\Phi^{-1}(\delta)$ and $\Phi^{-1}(\delta/R_{w})$ respectively throughout. In
place of (4.3.2), the key inequality
$R_{w}\Phi^{-1}(\delta/R_{w})\leqslant\Phi^{-1}(\delta)$ holds since
$\Phi(\delta)/\delta\searrow 0$ monotonically as $\delta\to 0^{+}$ by
assumption. ∎

####  4.3.2 Precise formulae for conformal iterated function systems

In order to use the upper bounds in Theorem 4.3.1 to prove a simpler formula
for intermediate dimensions of the limit set of a CIFS in Theorem 4.3.4 as the
maximum of the Hausdorff dimension and the intermediate dimensions of the
fixed points, we need further lemmas. The following lemma and proof are
similar to [MUa, Proposition 2.9] for the box dimension.

######  Lemma 4.3.2.

Fix any CIFS, let $n\in\mathbb{N}$, and assume that $P$ and $Q$ are both
subsets of $\cup_{w\in I^{n}}S_{w}(X)$ satisfying

$\displaystyle 0<\inf_{w\in I^{n}}\\#(P\cap S_{w}(X))\leqslant\sup_{w\in
I^{n}}\\#(P\cap S_{w}(X))<\infty,$ $\displaystyle 0<\inf_{w\in I^{n}}\\#(Q\cap
S_{w}(X))\leqslant\sup_{w\in I^{n}}\\#(Q\cap S_{w}(X))<\infty.$

Then $\overline{\dim}_{\theta}P=\overline{\dim}_{\theta}Q$ for all
$\theta\in[0,1]$, and $\overline{\dim}^{\Phi}P=\overline{\dim}^{\Phi}Q$ for
all monotonically admissible functions $\Phi$. The same holds with
$\overline{\dim}$ replaced by $\underline{\dim}$ throughout.

###### Proof.

The idea is to use an efficient cover of $Q$ at scale $\delta$ to construct an
efficient cover of $P$ at scale $\delta$. Elements of $P$ in cylinders of size
$\lesssim\delta$ can be covered using the cover of the element of $Q$ in the
same cylinder, and the conditions of a CIFS (via Lemma 4.2.10) mean that each
element of the cover can intersect only a bounded number of cylinders that are
larger than the covering set in question.

Since for each $n\in\mathbb{N}$, $\\{\,S_{w}:w\in I^{n}\,\\}$ forms a CIFS
with the same limit set, we may henceforth assume without loss of generality
that $n=1$. Let $A_{d,3}$ be as in (1.3.3) on page 1.3.3 and let $M$ be as in
Lemma 4.2.10. Let

$C\coloneqq\max\Big{\\{}\sup_{w\in I^{n}}\\#(P\cap S_{w}(X)),\sup_{w\in
I^{n}}\\#(Q\cap S_{w}(X))\Big{\\}}.$

If $\theta=0$ then $\overline{\dim}_{\theta}P=\overline{\dim}_{\theta}Q=0$
because $P$ and $Q$ are countable, so henceforth assume that $\theta\in(0,1]$.

Claim: Given any $\delta>0$, if $\\{U_{j}\\}$ is a cover of $Q$ such that
$\delta\leqslant|U_{j}|\leqslant\delta^{\theta}$ for all $j$ then there exists
a cover $\\{V_{m}\\}$ of $P$ such that
$\delta\leqslant|V_{m}|\leqslant\delta^{\theta}$ for all $m$, and

$\sum_{m}|V_{m}|^{s}\leqslant(A_{d,3}+MC)\sum_{j}|U_{j}|^{s}$

for all $s\geqslant 0$.

Proof of claim: For each $j$, if $i\in I$ is such that
$|S_{i}(X)|\leqslant|U_{j}|$ and $S_{i}(X)\cap U_{j}\neq\varnothing$, then

$S_{i}(X)\subseteq\mathcal{S}_{|U_{j}|}(U_{j})\subseteq\bigcup_{l=1}^{A_{d,3}}\mathcal{S}_{|U_{j}|}(U_{j})_{l},$

where $\mathcal{S}_{|U_{j}|}(U_{j})$ is the neighbourhood set, which has
diameter $3|U_{j}|$. By Lemma 4.2.10 there exist $i_{1},\dotsc,i_{M}\in I$,
not necessarily distinct, such that $S_{i_{k}}(X)\cap U_{j}\neq\varnothing$
for $k=1,\dotsc,M$, and such that if $i\in I\setminus\\{i_{1},\dotsc,i_{M}\\}$
and $|S_{i}(X)|>|U_{j}|$ then $S_{i}(X)\cap U_{j}=\varnothing$. If
$k=1,\dotsc,M$ then we can cover $P\cap S_{i_{k}}(X)$ by $C$ balls
$\\{B_{j,k,p}\\}_{p=1}^{C}$, each of diameter $|U_{j}|$. Since $\\{U_{j}\\}$
covers $Q$,

$P\subseteq\bigcup_{j}\left(\bigcup_{l=1}^{A_{d,3}}\mathcal{S}_{|U_{j}|}(U_{j})_{l}\cup\bigcup_{k=1}^{M}\bigcup_{p=1}^{C}B_{j,k,p}\right).$

Each element of this cover of $P$ has diameter in the interval
$[\delta,\delta^{\theta}]$ by construction. Moreover,

$\sum_{j}\left(\sum_{l=1}^{A_{d,3}}|\mathcal{S}_{|U_{j}|}(U_{j})_{l}|^{s}+\sum_{k=1}^{M}\sum_{p=1}^{C}|B_{j,k,p}|^{s}\right)=(A_{d,3}+MC)\sum_{j}|U_{j}|^{s},$

proving the claim.

The claim shows that
$\overline{\dim}_{\theta}P\leqslant\overline{\dim}_{\theta}Q$ and
$\underline{\dim}_{\theta}P\leqslant\underline{\dim}_{\theta}Q$. The reverse
inequalities hold by symmetry, so
$\overline{\dim}_{\theta}P=\overline{\dim}_{\theta}Q$ and
$\underline{\dim}_{\theta}P=\underline{\dim}_{\theta}Q$, as required. If
$\Phi$ is a monotonically admissible function then by Lemma 2.2.10 on page
2.2.10 we may assume without loss of generality that $\Phi$ is invertible.
Then the same proof works with $\delta^{\theta}$, $\overline{\dim}_{\theta}$
and $\underline{\dim}_{\theta}$ replaced by $\Phi^{-1}(\delta)$,
$\overline{\dim}^{\Phi}$ and $\underline{\dim}^{\Phi}$ respectively
throughout. ∎

Lemma 4.3.3 shows that the upper intermediate dimensions of a set of points
corresponding to the $n$-th level cylinders are either all bounded above by
the finiteness parameter, and hence the Hausdorff dimension of the limit set,
or they all equal the upper intermediate dimensions of the level-1 fixed
points. We will combine this lemma with the upper bounds in Theorem 4.3.1
(which considers arbitrarily deep levels) to prove that the dimensions in fact
depend only on the level-1 fixed points (and the Hausdorff dimension). Mauldin
and Urbański prove in [MUa, Lemma 2.10] that the upper box dimension of the
level-1 iterates of a given point is greater than or equal to the finiteness
parameter $\theta_{S}$, and deduce that it equals the box dimension of the
$n$-th level iterates for all $n\in\mathbb{N}$. The intermediate dimensions,
on the other hand, will _not_ always exceed the finiteness parameter, so we
cannot make the same conclusion for the intermediate dimensions in Lemma
4.3.3.

######  Lemma 4.3.3.

Consider a CIFS, and suppose that for each $n\in\mathbb{N}$,
$P_{n}\subseteq\cup_{w\in I^{n}}S_{w}(X)$ is any set satisfying $0<\inf_{w\in
I^{n}}\\#(P_{n}\cap S_{w}(X))\leqslant\sup_{w\in I^{n}}\\#(P_{n}\cap
S_{w}(X))<\infty$. Then

  1. (i)

for all $\theta\in[0,1]$, either
$\overline{\dim}_{\theta}P_{n}\leqslant\theta_{S}\leqslant h$ for all
$n\in\mathbb{N}$ or
$\overline{\dim}_{\theta}P_{n}=\overline{\dim}_{\theta}P_{1}$ for all
$n\in\mathbb{N}$.

  2. (ii)

If $\Phi$ is monotonically admissible then either
$\overline{\dim}^{\Phi}P_{n}\leqslant\theta_{S}\leqslant h$ for all
$n\in\mathbb{N}$ or $\overline{\dim}^{\Phi}P_{n}=\overline{\dim}^{\Phi}P_{1}$
for all $n\in\mathbb{N}$.

###### Proof.

(i). This is true for the Hausdorff dimension because each $P_{n}$ is
countable, so henceforth suppose $\theta\in(0,1]$. By Lemma 4.3.2,
$\overline{\dim}_{\theta}P_{n}$ does not depend on the particular set $P_{n}$,
so
$\overline{\dim}_{\theta}P_{1}\leqslant\overline{\dim}_{\theta}P_{2}\leqslant\dotsb$.
By Lemmas 4.3.2 and 4.2.10, we can henceforth fix $x\in X$ and assume without
loss of generality that $P_{n}\coloneqq\\{\,S_{w}(x):w\in I^{n}\,\\}$ for all
$n\in\mathbb{N}$. It suffices to prove that
$\overline{\dim}_{\theta}P_{n}\leqslant\max\\{\theta_{S},\overline{\dim}_{\theta}P_{1}\\}$
for all $n\in\mathbb{N}$, which follows from the following claim.

Claim: For all $s>\max\\{\theta_{S},\overline{\dim}_{\theta}P_{1}\\}$, for all
$n\in\mathbb{N}$ there exists $B_{n}\in(0,\infty)$ such that for all
$\delta\in(0,1]$ there exists a cover $\\{U_{j}^{\delta,n}\\}_{j}$ of $P_{n}$
such that $\delta\leqslant|U_{j}^{\delta,n}|\leqslant\delta^{\theta}$ for all
$i$ and $\sum_{j}|U_{j}^{\delta,n}|^{s}\leqslant B_{n}$.

Proof of claim: Fix $s>\max\\{\theta_{S},\overline{\dim}_{\theta}P_{1}\\}$. We
prove the claim by induction on $n$. Suppose $n>1$ and assume the claim holds
for $1,2,\dotsc,n-1$. Let $\delta\in(0,1]$. By the definition of
$A_{d,1+2|X|}$ in (1.3.3), for all $j$ there exist
$U_{j,1}^{\delta,n-1},\dotsc,U_{j,A_{d,1+2|X|}}^{\delta,n-1}\subseteq\mathbb{R}^{d}$,
each of diameter

$\max\left\\{\delta,\frac{|\mathcal{S}_{|X|\delta}(U_{j}^{\delta,n-1})|}{1+2|X|}\right\\},$

such that

$\mathcal{S}_{|X|\delta}(U_{j}^{\delta,n-1})\subseteq\bigcup_{k=1}^{A_{d,1+2|X|}}U_{j,k}^{\delta,n-1}.$

By the triangle inequality,

$|\mathcal{S}_{\delta|X|}(U_{j}^{\delta,n-1})|\leqslant|U_{j}^{\delta,n-1}|+2|X|\delta\leqslant(1+2|X|)|U_{j}^{\delta,n-1}|\leqslant(1+2|X|)\delta^{\theta}.$

Therefore $\delta\leqslant|U_{j,k}^{\delta,n-1}|\leqslant\delta^{\theta}$ and
$|U_{j,k}^{\delta,n-1}|\leqslant|U_{j}^{\delta,n-1}|$ for all $j,k$.

Let $C_{\delta}\coloneqq\\{\,w\in I^{n-1}:|X_{w}|\leqslant|X|\delta\,\\}$. If
$w\in C_{\delta}$ then there exists $p_{w}\in S_{w}(X)\cap P_{n-1}$, and there
exists $j$ such that $p_{w}\in U_{j}^{\delta,n-1}$, so the neighbourhood set
$\mathcal{S}_{|X|\delta}(U_{j}^{\delta,n-1})$ covers $S_{w}(X)$. Thus

$P_{n}\cap S_{w}(X)\subseteq
S_{w}(X)\subseteq\mathcal{S}_{|X|\delta}(U_{j}^{\delta,n-1})\subseteq\bigcup_{k=1}^{A_{d,1+2|X|}}U_{j,k}^{\delta,n-1}.$
(4.3.3)

If, on the other hand, $w\in I^{n-1}\setminus C_{\delta}$, then
$|X|\delta<|X_{w}|\leqslant|X|R_{w}$ and $\delta/R_{w}<1$. Consider the cover
$\\{U_{l}^{\delta/R_{w},1}\\}_{l}$ of $P_{1}$ whose existence is guaranteed by
the base case $n=1$. For each $l$, let $W_{w,l}$ be a set with diameter
$|W_{w,l}|=\max\\{|S_{w}(U_{l}^{\delta/R_{w},1}\cap X)|,\delta\\}$ such that
$S_{w}(U_{l}^{\delta/R_{w},1}\cap X)\subseteq W_{w,l}$. Since
$P_{n}\coloneqq\\{\,S_{w}(x):w\in I^{n}\,\\}$, the sets
$\\{S_{w}(U_{l}^{\delta/R_{w},1}\cap X)\\}_{l}$ cover $P_{n}\cap S_{w}(X)$, so
$\\{W_{w,l}\\}_{l}$ covers $P_{n}\cap S_{w}(X)$. By the definition of $R_{w}$,
for all $l$ we have $|S_{w}(U_{l}^{\delta/R_{w},1}\cap X)|\leqslant
R_{w}|U_{l}^{\delta/R_{w},1}\cap X|\leqslant R_{w}|U_{l}^{\delta/R_{w},1}|$,
and also $\delta=R_{w}\delta/R_{w}\leqslant R_{w}|U_{l}^{\delta/R_{w},1}|$, so

$\delta\leqslant|W_{w,l}|\leqslant R_{w}|U_{l}^{\delta/R_{w},1}|\leqslant
R_{w}(\delta/R_{w})^{\theta}\leqslant\delta^{\theta}.$ (4.3.4)

Now, $\\{U_{j,k}^{\delta,n-1}\\}\cup\\{W_{w,l}\\}$ covers $P_{n}$ and the
diameter of each element of this cover lies in the interval
$[\delta,\delta^{\theta}]$. Moreover, since
$|U_{j,k}^{\delta,n-1}|\leqslant|U_{j}^{\delta,n-1}|$ for all $j,k$, and by
(4.3.4),

$\displaystyle\sum_{j}\sum_{k=1}^{A_{d,1+2|X|}}|U_{j,k}^{\delta,n-1}|^{s}$
$\displaystyle+\sum_{w\in I^{n}\setminus C_{\delta}}\sum_{l}|W_{w,l}|^{s}$
$\displaystyle\leqslant
A_{d,1+2|X|}\sum_{j}|U_{j}^{\delta,n-1}|^{s}+\sum_{w\in I^{n}\setminus
C_{\delta}}R_{w}\sum_{l}|U_{l}^{\delta/R_{w},1}|^{s}$ $\displaystyle\leqslant
A_{d,1+2|X|}B_{n-1}+B_{1}\phi_{n}(s),$

recalling the definition of $\phi_{n}$ from (4.2.2). Therefore letting
$B_{n}\coloneqq A_{d,1+2|X|}B_{n-1}+B_{1}\phi_{n}(s)$, since $s>\theta_{S}$,
$\phi_{n}(s)<\infty$, so $B_{n}<\infty$, and the claim holds by induction.

(ii). By Lemma 2.2.10 we may assume that $\Phi$ is invertible. Then (ii) holds
by the same proof as (i), with $\overline{\dim}_{\theta}$, $\delta^{\theta}$
and $(\delta/R_{w})^{\theta}$ replaced by $\overline{\dim}^{\Phi}$,
$\Phi^{-1}(\delta)$ and $\Phi^{-1}(\delta/R_{w})$ respectively. In place of
(4.3.4), $R_{w}\Phi^{-1}(\delta/R_{w})\leqslant\Phi^{-1}(\delta)$ holds since
$\Phi(\delta)/\delta\searrow 0$ monotonically as $\delta\to 0^{+}$ by
assumption. ∎

Mauldin and Urbański [MU, Theorem 3.15] show that the Hausdorff dimension of
the limit set $F$ of a CIFS is $h$. In fact, this is true even if the cone
condition (ii) is not assumed (see [URM, Theorem 19.6.4]), but in this chapter
we do use the cone condition in the proof of Lemma 4.2.9 (and hence Lemmas
4.2.10 and 4.3.2). We now use the fact that $h=\dim_{\mathrm{H}}F$, together
with the upper bounds in Theorem 4.3.1 and Lemmas 4.3.3 and 4.3.2, to prove
the main result of this chapter, Theorem 4.3.4. This gives the following
simple formulae for other dimensions of the limit set as the maximum of the
Hausdorff dimension of the limit set and the corresponding dimension of any
set $P$ satisfying certain conditions. From Lemma 4.2.10 we see that examples
for the set $P$ include $\\{\,S_{i}(x):i\in I\,\\}$ for any given $x\in X$, as
in [MUa], or the set of fixed points in $X$ of the contractions $S_{i}$. There
is a typo in the definition of $P$ in the published version of [BF].

######  Theorem 4.3.4.

For all CIFSs with limit set $F$ and notation as above, for all subsets
$P\subseteq\cup_{i\in I}S_{i}(X)$ which satisfy $0<\inf_{i\in I}\\#(P\cap
S_{i}(X))\leqslant\sup_{i\in I}\\#(P\cap S_{i}(X))<\infty$,

  1. (i)

$\overline{\dim}_{\mathrm{B}}F=\dim_{\mathrm{P}}F=\max\\{h,\overline{\dim}_{\mathrm{B}}P\\}$
(very similar to Mauldin and Urbański [MUa, Theorem 2.11] but with a more
general condition on $P$)

  2. (ii)

$\overline{\dim}_{\theta}F=\max\\{h,\overline{\dim}_{\theta}P\\}$ for all
$\theta\in[0,1]$

  3. (iii)

$\overline{\dim}^{\Phi}F=\max\\{h,\overline{\dim}^{\Phi}P\\}$ if $\Phi$ is
monotonically admissible

###### Proof.

(i). follows from the case $\theta=1$ of (ii) and the fact that
$\overline{\dim}_{\mathrm{B}}F=\dim_{\mathrm{P}}F$ by [MU, Theorem 3.1].

(ii). For each $n\in\mathbb{N}$ let $P_{n}\coloneqq\\{\,x\in
X:x=S_{w}(x)\mbox{ for some }w\in I^{n}\,\\}$, so $P_{n}\subseteq
F\subseteq\cup_{w\in I^{n}}S_{w}(X)$. Then by Lemmas 4.3.2 and 4.3.3 (i),
$\overline{\dim}_{\theta}P_{n}\leqslant\max\\{h,\overline{\dim}_{\theta}P\\}$
for all $n\in\mathbb{N}$. Therefore by Theorem 4.3.1 (iii),

$\overline{\dim}_{\theta}F\leqslant\max\\{h,\lim_{n\to\infty}\overline{\dim}_{\theta}P_{n}\\}\leqslant\max\\{h,\max\\{h,\overline{\dim}_{\theta}P\\}\\}=\max\\{h,\overline{\dim}_{\theta}P\\}.$

But $P_{n}\subseteq F$ so since $\overline{\dim}_{\theta}$ is monotonic for
subsets, by Lemma 4.3.2,
$\overline{\dim}_{\theta}P=\overline{\dim}_{\theta}P_{1}\leqslant\overline{\dim}_{\theta}F$,
and by [MU, Theorem 3.15],
$h=\dim_{\mathrm{H}}F\leqslant\overline{\dim}_{\theta}F$, so
$\max\\{h,\overline{\dim}_{\theta}P\\}\leqslant\overline{\dim}_{\theta}F$.
Therefore $\overline{\dim}_{\theta}F=\max\\{h,\overline{\dim}_{\theta}P\\}$,
as required.

(iii) is similar to (ii). ∎

A consequence is the following bounds for the lower versions of the dimensions
in Theorem 4.3.4.

######  Corollary 4.3.5.

For all CIFSs with limit set $F$ and notation as above, for all subsets
$P\subseteq\cup_{i\in I}S_{i}(X)$ which satisfy $0<\inf_{i\in I}\\#(P\cap
S_{i}(X))\leqslant\sup_{i\in I}\\#(P\cap S_{i}(X))<\infty$,

  1. (i)

$\max\\{h,\underline{\dim}_{\mathrm{B}}P\\}\leqslant\underline{\dim}_{\mathrm{B}}F\leqslant\max\\{h,\overline{\dim}_{\mathrm{B}}P\\}$

  2. (ii)

$\max\\{h,\underline{\dim}_{\theta}P\\}\leqslant\underline{\dim}_{\theta}F\leqslant\max\\{h,\overline{\dim}_{\theta}P\\}$
for all $\theta\in[0,1]$

  3. (iii)

$\max\\{h,\underline{\dim}^{\Phi}P\\}\leqslant\underline{\dim}^{\Phi}F\leqslant\max\\{h,\overline{\dim}^{\Phi}P\\}$
if $\Phi$ is monotonically admissible.

###### Proof.

We prove (ii); (i) and (iii) are similar. By Theorem 4.3.4 (ii)
$\underline{\dim}_{\theta}F\leqslant\overline{\dim}_{\theta}F=\max\\{h,\overline{\dim}_{\theta}P\\}$.
If $P_{1}$ is the set of fixed points in $X$ of the maps $\\{S_{i}\\}_{i\in
I}$ then by Lemma 4.3.2,
$\underline{\dim}_{\theta}P=\underline{\dim}_{\theta}P_{1}\leqslant\underline{\dim}_{\theta}F$,
and by [MU, Theorem 3.15],
$h=\dim_{\mathrm{H}}F\leqslant\underline{\dim}_{\theta}F$, so
$\max\\{h,\underline{\dim}_{\theta}P\\}\leqslant\underline{\dim}_{\theta}F$,
as required. ∎

######  Question 4.3.6.

Are the bounds in Corollary 4.3.5 sharp or can they be improved in general?

Very recently, the author and Rutar [BRa] have calculated a formula for
$\underline{\dim}_{\mathrm{B}}F$, which interestingly depends on more refined
properties of the covering function $r\mapsto N_{r}(P)$ than merely
$\underline{\dim}_{\mathrm{B}}P$ and $\overline{\dim}_{\mathrm{B}}P$. The box
dimension of $F$ exists if and only if the bounds in part (i) of Question
4.3.6 coincide, and if this is not the case then the sharp upper bound for
$\underline{\dim}_{\mathrm{B}}F$ in terms of $h$,
$\underline{\dim}_{\mathrm{B}}P$ and $\overline{\dim}_{\mathrm{B}}P$ is
strictly smaller than $\max\\{h,\overline{\dim}_{\mathrm{B}}P\\}$.

If the fixed points are arranged to be at $\frac{1}{\log n}$ then by Falconer,
Fraser and Kempton [FFKa, Example 1], the intermediate dimensions of the set
of fixed points will be $1$ for all $\theta\in(0,1]$. However, the contraction
ratios can be made small enough (and tending to $0$ rapidly enough) so that
$h<1$. In this case, the limit set and its closure will have the same
Hausdorff dimension as they differ by a countable set (namely the images of
the point 0 under maps corresponding to finite words) by Mauldin and Urbański
[MU, Lemma 2.1]. Therefore by Theorem 2.5.1 from page 2.5.1, the
$\Phi$-intermediate dimensions can be used to ‘recover the interpolation’
between the Hausdorff and box dimensions of the limit set $F$ in the sense
that for all $s\in[h,1]$ there exists an admissible function $\Phi_{s}$ with
$\dim^{\Phi_{s}}F=s$. Theorem 4.3.4 (iii) can help to find these functions. It
is possible for the intermediate dimensions to be discontinuous at $\theta=0$
even when the box dimension is less than 1. Indeed, consider a countable
compact subset $P\subset\mathbb{R}$ with box and Assouad dimension equal and
strictly between 0 and 1, so by [FFKa, Proposition 2.4],
$\dim_{\theta}P=\dim_{\mathrm{B}}P$ for all $\theta\in(0,1)$. We may then
choose a set of similarity maps whose fixed points form the set $P$ and whose
contraction ratios are small enough that the system forms a CIFS with the
Hausdorff dimension of the limit set being smaller than $\dim_{\mathrm{B}}P$.
Then the intermediate dimensions of the limit set will be discontinuous at
$\theta=0$ by Theorem 4.3.4.

Our results are also relevant to the dimension theory of infinite parabolic
iterated function systems. In such a system, each map $S\colon X\to X$ still
satisfies $||S(x)-S(y)||<||x-y||$ for all $x,y\in X$, but finitely many of the
maps may contain a _parabolic fixed point_ $p\in X$, meaning that $S(p)=p$ but
the derivative of $S$ (or an extension of $S$) at $p$ has norm 1. The other
countably many maps must be uniformly contracting, and are called
_hyperbolic_. Manneville–Pomeau maps of the form ${x\mapsto x-x^{q}}$ for
fixed $q>1$ are examples of functions with a parabolic fixed point at $0$. The
theory of parabolic IFSs has been developed by Mauldin and Urbański in [MUb],
and they have also been studied in [MUc, BFa], [Fra, Section 9.2], and many
other works. Given an infinite parabolic IFS as defined in [MUb, Section 2],
one can associate an ‘induced’ uniformly contracting infinite CIFS (see [MUb,
Theorem 5.2]). It is clear that if $F$ is the limit set of the parabolic IFS
and $F^{*}$ is the limit set of the induced CIFS then $F^{*}\subseteq F$ with
$F\setminus F^{*}$ countable, and $F$ and $F^{*}$ have the same closure.
Therefore if $\dim$ is Hausdorff/box/intermediate/Assouad dimension, then
$\dim F=\dim F^{*}$. In particular, Theorem 4.3.4 can be applied directly to
the induced system to give information about the corresponding dimension of
$F$. This is only relevant for systems consisting of infinitely many
hyperbolic maps (and finitely many parabolic maps), because the limit sets of
finite parabolic IFSs have equal Hausdorff and upper box dimensions (see [Urb,
Remark 6.6] and [MUc]). In [BFa, Section 6], however, we use inducing to
calculate the Assouad spectrum of a class of ‘parabolic Cantor sets’ (see
[Urb]), which are generated by finite parabolic IFSs.

####  4.3.3 An example and a first application

We use Proposition 3.2.10 from page 3.2.10 and a result of Burrell, Falconer
and Fraser to give an application of Theorem 4.3.4 to orthogonal projections.
Dimension theory of orthogonal projections has a long history in fractal
geometry, see [FFJ, Shm]. There has been particular interest in orthogonal
projections of dynamically defined sets, where one can often obtain more
precise information than is provided by the general projection theorems, see
[HS, Shm]. The following example falls into this category. Recall the
definition

$G_{p,d}\coloneqq\\{\,x/||x||^{2}:x\in\\{1^{p},2^{p},3^{p},\dotsc\\}^{d}\,\\}.$

######  Example 4.3.7.

Let $p>0$ and consider a set of contracting similarity maps on
$\mathbb{R}^{2}$ with fixed points lying in the set $G_{p,2}$ from Proposition
3.2.10, with no two maps having the same fixed point. Assume the contraction
ratios are small enough that the system forms a CIFS, with limit set $F$, say,
and small enough that $\dim_{\mathrm{H}}F<1$. Then by Theorem 4.3.4 and
Corollary 4.3.5,

$\dim_{\theta}F=\max\left\\{\dim_{\mathrm{H}}F,\frac{2\theta}{p+\theta}\right\\},$

which is continuous at $\theta=0$. Therefore by Burrell, Falconer and Fraser’s
Theorem 1.4.3 from page 1.4.3 there exists $c<1$ such that
$\overline{\dim}_{\mathrm{B}}\pi(F)\leqslant c$ for every orthogonal
projection $\pi\colon\mathbb{R}^{2}\to\mathbb{R}$, and
$\overline{\dim}_{\mathrm{B}}\pi(F)=c$ for almost every orthogonal projection
$\pi$ (with respect to the natural measure on projective space). This
conclusion is perhaps most interesting when $p$ is very close to 0 (and so
$\dim_{\mathrm{B}}F$ is very close to 2) and $\dim_{\mathrm{H}}F$ is very
close to 1.

More generally, if $1\leqslant k<d$ are integers and the contraction ratios
lie on $G_{p,d}$ and $\dim_{\mathrm{H}}F<k$, then
$\overline{\dim}_{\mathrm{B}}\pi(F)<k$ for every orthogonal projection
$\pi\colon\mathbb{R}^{d}\to\mathbb{R}^{k}$.

###  4.4 Continued fraction sets

####  4.4.1 Real continued fractions

In this section we apply Theorem 4.3.4 to give information about sets of
irrational numbers whose continued fractions have restricted entries, as in
the following definition.

######  Definition 4.4.1.

For a non-empty, proper subset $I\subset\mathbb{N}$, define

$F_{I}\coloneqq\left\\{\,z\in(0,1)\setminus\mathbb{Q}:z=\frac{1}{b_{1}+\frac{1}{b_{2}+\frac{1}{\ddots}}},b_{n}\in
I\mbox{ for all }n\in\mathbb{N}\,\right\\}.$

The following lemma shows why our general results can be applied in this
setting.

######  Lemma 4.4.2.

Working in $\mathbb{R}$, letting $X\coloneqq[0,1]$ and $V\coloneqq(-1/8,9/8)$,

  1. (i)

If $1\notin I$ then $\\{\,S_{b}(x)\coloneqq 1/(b+x):b\in I\,\\}$ is a CIFS
with limit set $F_{I}$.

  2. (ii)

If $1\in I$ then $\\{\,S_{b}(x)\coloneqq 1/(b+x):b\in I,b\neq
1\,\\}\cup\left\\{\,S_{1b}(x)\coloneqq\frac{1}{b+\frac{1}{1+x}}:b\in
I\,\right\\}$ is a CIFS with limit set $F_{I}$.

###### Proof.

(i) is verified in [MUa, page 4997], and (ii) can be verified similarly,
noting that when $1\in I$ the different CIFS is needed to ensure that the
system is uniformly contractive, because the derivative of $x\to 1/(1+x)$ at
$x=0$ is $-1$. ∎

It follows from [MU, Theorem 3.15] that $\dim_{\mathrm{H}}F_{I}=h$; the
Hausdorff dimension of such limit sets has been studied in [MUa, KZ, HUa, Ing,
CLU] and other works. It follows from [MUa, Theorem 2.11] that
$\overline{\dim}_{\mathrm{B}}F_{I}=\max\\{h,\overline{\dim}_{\mathrm{B}}\\{\,1/b:b\in
I\,\\}\\}$. In Theorem 4.4.3 we apply Theorem 4.3.4 to give information about
the intermediate dimensions of $F_{I}$.

######  Theorem 4.4.3.

Using the notation in Definition 4.4.1, for all non-empty proper
$I\subset\mathbb{N}$,

  1. (i)

For all $\theta\in[0,1]$,

$\displaystyle\overline{\dim}_{\theta}F_{I}=\max\\{h,\overline{\dim}_{\theta}\\{\,1/b:b\in
I\,\\}\\};$ $\displaystyle\max\\{h,\underline{\dim}_{\theta}\\{\,1/b:b\in
I\,\\}\\}\leqslant\underline{\dim}_{\theta}F_{I}\leqslant\max\\{h,\overline{\dim}_{\theta}\\{\,1/b:b\in
I\,\\}\\}.$

  2. (ii)

The maps $\theta\mapsto\overline{\dim}_{\theta}F_{I}$ and
$\theta\mapsto\underline{\dim}_{\theta}F_{I}$ are continuous at $\theta=0$.

###### Proof.

(i) Case 1: Assume $1\notin A$. Then the result follows from Theorem 4.3.4
(ii) and Corollary 4.3.5 (ii) if we take $P=\\{\,S_{b}(0):b\in
I\,\\}=\\{\,1/b:b\in I\,\\}$.

Case 2: Assume $1\in A$. Since the map $x\mapsto 1/(1+x)$ is bi-Lipschitz on
$[0,1]$ and $\overline{\dim}_{\theta}$ is stable under bi-Lipschitz maps,

$\overline{\dim}_{\theta}\left\\{\,\frac{1}{1+\frac{1}{b}}\,\right\\}=\overline{\dim}_{\theta}\\{\,1/b:b\in
I\,\\}.$

Since the removal of finitely many points from a set does not change its
dimension,

$\overline{\dim}_{\theta}\\{\,1/b:b\in I,b\neq
1\,\\}=\overline{\dim}_{\theta}\\{\,1/b:b\in I\,\\}.$

It is clear from the definition that $\overline{\dim}_{\theta}$ is finitely
stable, so the equality for $\overline{\dim}_{\theta}F_{I}$ follows from
Theorem 4.3.4 (ii) if we take

$P\coloneqq\left\\{\,\frac{1}{1+\frac{1}{b}}\,\right\\}\cup\\{\,1/b:b\in
I,b\neq 1\,\\}.$

The lower bound holds since

$\max\\{h,\underline{\dim}_{\theta}\\{\,1/b:b\in
I\,\\}\\}=\max\\{\dim_{\mathrm{H}}F_{I},\underline{\dim}_{\theta}\\{\,1/b:b\in
I,b\neq 1\,\\}\\}\leqslant\underline{\dim}_{\theta}F_{I}$

by [MU, Theorem 3.15].

(ii) For all $\theta\in(0,1]$,

$\displaystyle\overline{\dim}_{\theta}F_{I}$
$\displaystyle=\max\\{h,\overline{\dim}_{\theta}\\{\,1/b:b\in I\,\\}\\}$ by
(i)
$\displaystyle\leqslant\max\\{h,\overline{\dim}_{\theta}\\{\,1/b:b\in\mathbb{N}\,\\}\\}$
$\displaystyle=\max\\{h,\theta/(1+\theta)\\}$ by [FFKa, Proposition 3.1]
$\displaystyle\xrightarrow[\theta\to
0^{+}]{}\max\\{h,0\\}=h=\dim_{\mathrm{H}}F_{I}=\overline{\dim}_{0}F_{I}$
$\displaystyle\text{by \cite[cite]{[\@@bibref{}{Mauldin1996iifs}{}{},
Theorem\leavevmode\nobreak\ 3.15]}},$

so $\theta\mapsto\overline{\dim}_{\theta}F_{I}$ is continuous at $\theta=0$.
Since
$\dim_{\mathrm{H}}F_{I}\leqslant\underline{\dim}_{\theta}F_{I}\leqslant\overline{\dim}_{\theta}F_{I}$
for all $\theta\in[0,1]$ it follows that
$\theta\mapsto\underline{\dim}_{\theta}F_{I}$ is also continuous at
$\theta=0$. ∎

The following example is similar to [MUa, Theorem 6.2]; we consider a nice
family of subsets $I$ which result in the upper and lower intermediate
dimensions coinciding.

######  Corollary 4.4.4.

Fix $p>1$ and for $l\in\mathbb{N}$, $l\geqslant 2$ define

$I_{p,l}\coloneqq\\{\,\lfloor n^{p}\rfloor:n\geqslant l\\}.$

Then the intermediate dimensions of the continued fraction set exist and are
given by

$\dim_{\theta}F_{I_{p,l}}=\max\left\\{\dim_{\mathrm{H}}F_{I_{p,l}},\frac{\theta}{p+\theta}\right\\}.$
(4.4.1)

Moreover, there exists $q\in\mathbb{N}$ such that for all $l\geqslant q$ we
have

$\dim_{\mathrm{H}}F_{I_{p,l}}<\dim_{\mathrm{B}}F_{I_{p,l}}=\frac{1}{p+1}.$

###### Proof.

Since $I_{p,l}$ is bi-Lipschitz equivalent to a cofinite subset of
$\\{\,i^{-p}:i\in\mathbb{N}\,\\}$, [FFKa, Proposition 3.1] gives

$\dim_{\theta}\\{\,1/b:b\in
I_{p,l}\,\\}=\dim_{\theta}\\{\,i^{-p}:i\in\mathbb{N}\,\\}=\frac{\theta}{p+\theta}$

for $\theta\in[0,1]$. Therefore the bounds in Theorem 4.4.3 (i) coincide and
(4.4.1) holds. It was shown in [MU, Section 3] that
$\theta_{S}=\inf\\{\,t>0:\psi_{1}(t)<\infty\,\\}$. But there exists
$C\geqslant 1$ such that $1/(Cb^{2})\leqslant||S_{b}^{\prime}||\leqslant
C/b^{2}$ for all $b\in\mathbb{N}$. Therefore, since

$\sum_{n=1}^{\infty}((n^{p})^{-2})^{1/(2p)}=\sum_{n=1}^{\infty}n^{-1}=\infty;$
$\sum_{n=1}^{\infty}((n^{p})^{-2})^{s}=\sum_{n=1}^{\infty}n^{-2p/s}<\infty\qquad\mbox{for
all }s>1/(2p),$

it follows that the finiteness parameter $\theta_{I_{p,l}}=1/(2p)$ and so
$\dim_{\mathrm{H}}F_{I_{p,l}}>1/(2p)$. But in [MUa, Theorem 1.5] Mauldin and
Urbański showed that $\theta_{I_{p,l}}$ is the infimum of the Hausdorff
dimension of cofinite subsystems, so $\dim_{\mathrm{H}}F_{I_{p,l}}\to 1/(2p)$
as $l\to\infty$. Since $1/(2p)<1/(p+1)=\dim_{\mathrm{B}}\\{\,1/b:b\in
I_{p,l}\,\\}$, for all sufficiently large $l$ we have
$\dim_{\mathrm{H}}F_{I_{p,l}}<\dim_{\mathrm{B}}F_{I_{p,l}}=1/(p+1)$. ∎

The graph of the intermediate dimensions of the continued fraction set from
Corollary 4.4.4 in the case $p=2$ and $l$ large enough that
$1/4<\dim_{\mathrm{H}}F_{I_{p,l}}<\dim_{\mathrm{B}}F_{I_{p,l}}=1/3$ is the
black curve in Figure 4.2. Note that the intermediate dimensions of the graph
of the popcorn function in Figure 3.3 on page 3.3 has a similar form.

Recall the discussion in Section 1.4.2. Example 4.4.5 shows that the
intermediate dimensions can give better information about Hölder exponents
than either the Hausdorff or box dimensions.

######  Example 4.4.5.

Let $p,q$ be such that $1<p<q<2p-1<\infty$. As in Corollary 4.4.4 there exists
$l\in\mathbb{N}$ large enough so that if $I_{p,l}\coloneqq\\{\,\lfloor
n^{p}\rfloor:n\geqslant l\\}$ then

$1/(2p)<h_{p}<1/(q+1)<1/(p+1)$

where $h_{p}$ is the Hausdorff dimension of the continued fraction set
$\dim_{\mathrm{H}}F_{I_{p,l}}$, and then
$\dim_{\theta}F_{I_{p,l}}=\max\left\\{h_{p},\frac{\theta}{p+\theta}\right\\}$.
Similarly, if $I_{q}$ is a subset of $\mathbb{N}$ whose symmetric difference
with $\\{\,\lfloor n^{q}\rfloor:n\in\mathbb{N}\\}$ is finite then
$\dim_{\theta}F_{I_{q}}=\max\left\\{h_{q},\frac{\theta}{q+\theta}\right\\}$,
where $h_{q}\coloneqq\dim_{\mathrm{H}}F_{I_{q}}$. If $I_{q}$ is also such that
$h_{q}\in\left(\frac{ph_{p}}{q-qh_{p}+ph_{p}},\frac{1}{q+1}\right)$ and
$f\colon F_{I_{q}}\to\mathbb{R}$ is an $\alpha$-Hölder map such that
$f(F_{I_{q}})\supseteq F_{I_{p,l}}$ then (1.4.7) gives the best upper bound
for $\alpha$ when $\theta=\frac{qh_{q}}{1-h_{q}}$, when

$\alpha^{-1}h_{q}=\alpha^{-1}\dim_{\theta}F_{I_{q}}\geqslant\overline{\dim}_{\theta}f(F_{I_{q}})\geqslant\dim_{\theta}F_{I_{p,l}}=\frac{\theta}{p+\theta}=\frac{qh_{q}}{p-ph_{q}+qh_{q}},$

and so

$\alpha\leqslant\frac{p-ph_{q}+qh_{q}}{q}.$

Using the Hausdorff dimension merely gives that $\alpha\leqslant h_{q}/h_{p}$,
and the box dimension merely gives $\alpha\leqslant\frac{p+1}{q+1}$. The
intermediate dimensions of the two sets and the upper bound with a certain
choice of parameters are plotted in Figure 4.2.

Figure 4.2:  Graph of the intermediate dimensions of the real continued
fraction sets in Example 4.4.5 and the upper bound for $\alpha$ against
$\theta$ in the case $p=2$, $q=2.9$, $h_{p}\approx 0.26$, $h_{q}\approx 0.22$.

In the following corollary of Theorem 4.4.3 we apply results of Burrell [Bura]
to give some consequences of the continuity of the intermediate dimensions of
continued fraction sets for dimensions of images of $F_{I}$ under
index-$\alpha$ fractional Brownian motion. Perhaps the most interesting part
of Corollary 4.4.6 is the sufficiency of the condition $\alpha>h$ for the
upper box dimension of the image to be strictly less than 1; this is an
example of how the intermediate dimensions can be used to obtain information
about the box dimension of sets.

######  Corollary 4.4.6.

Let $\alpha\in(0,1)$ and let $B_{\alpha}\colon\mathbb{R}\to\mathbb{R}$ denote
index-$\alpha$ fractional Brownian motion. Then for all non-empty, proper
subsets $I\subset\mathbb{N}$, recalling that $h=\dim_{\mathrm{H}}F_{I}$,

  1. (i)

The maps $\theta\mapsto\overline{\dim}_{\theta}B_{\alpha}(F_{I})$ and
$\theta\mapsto\underline{\dim}_{\theta}B_{\alpha}(F_{I})$ are almost surely
continuous at $\theta=0$.

  2. (ii)

If $\alpha>h$ then almost surely

$h/\alpha=\dim_{\mathrm{H}}B_{\alpha}(F_{I})\leqslant\overline{\dim}_{\mathrm{B}}B_{\alpha}(F_{I})<1.$

  3. (iii)

If $\alpha\leqslant h$ then almost surely

$\dim_{\mathrm{H}}B_{\alpha}(F_{I})=\dim_{\mathrm{B}}B_{\alpha}(F_{I})=1.$

###### Proof.

(i). follows immediately from Theorem 4.4.3 (ii) and Burrell [Bura, Corollary
3.5].

(ii). The set $F_{I}$ is the limit set of a CIFS so it is Borel (see the
discussion after Lemma 4.2.10), so Kahane’s general results [Kah, Chapter 18]
give $h/\alpha=\dim_{\mathrm{H}}B_{\alpha}(F_{I})$ almost surely. The middle
inequality is a general property of the dimensions, and
$\overline{\dim}_{\mathrm{B}}B_{\alpha}(F_{I})<1$ almost surely by Theorem
4.4.3 (ii) and Burrell [Bura, Corollary 3.7].

(iii) follows from Kahane [Kah, Chapter 18] since $F_{I}$ is Borel. ∎

Note that since
$\dim_{\mathrm{P}}B_{\alpha}(F_{I}),\underline{\dim}_{\mathrm{B}}B_{\alpha}(F_{I})\in(\dim_{\mathrm{H}}B_{\alpha}(F_{I}),\overline{\dim}_{\mathrm{B}}B_{\alpha}(F_{I}))$,
if $\alpha>h$ then almost surely $\dim_{\mathrm{P}}B_{\alpha}(F_{I})<1$ and
$\underline{\dim}_{\mathrm{B}}B_{\alpha}(F_{I})<1$. On the other hand, if
$\alpha\leqslant h$ then almost surely
$\dim_{\mathrm{P}}B_{\alpha}(F_{I})=\underline{\dim}_{\mathrm{B}}B_{\alpha}(F_{I})=1$.

####  4.4.2 Complex continued fractions

In this section we study sets of complex numbers which have a complex
continued fraction expansion with restricted entries. For a non-empty
$I\subseteq\\{\,m+ni:m\in\mathbb{N},n\in\mathbb{Z}\,\\}$, define

$F_{I}\coloneqq\left\\{\,z\in\mathbb{C}:z=\frac{1}{b_{1}+\frac{1}{b_{2}+\frac{1}{\ddots}}},b_{n}\in
I\mbox{ for all }n\in\mathbb{N}\,\right\\}.$

If $1\notin I$ then it can be verified, as in [MU, Section 6], that if
$1\notin I$ then $\\{\,S_{b}(z)\coloneqq 1/(b+z):b\in I\,\\}$ is a CIFS with
limit set $F_{I}$, with $X\subset\mathbb{C}$ being the closed disc centred at
1/2 with radius 1/2, and $V=B(1/2,3/4)$. If $1\in I$ then $S_{1}$ is not
uniformly contracting but it is straightforward to verify that

$\\{\,S_{b}(z)\coloneqq{1/(b+z)}:{b\in I,}{b\neq
1}\,\\}\cup\left\\{\,S_{1b}(z)\coloneqq{\frac{1}{b+\frac{1}{1+z}}}:{b\in
I}\,\right\\}$

is a CIFS with the same limit set. By [MU, Theorem 3.15], the Hausdorff
dimension can be determined by the topological pressure function, and has been
studied in [HU] with estimates given in [MU, Section 6] and [GM, Pri, FN,
Ing].

######  Theorem 4.4.7.

Using the notation above, for all $\theta\in[0,1]$,

  1. (i)

For all $\theta\in[0,1]$,

$\displaystyle\overline{\dim}_{\theta}F_{I}=\max\\{h,\overline{\dim}_{\theta}\\{\,1/b:b\in
I\,\\}\\};$ $\displaystyle\max\\{h,\underline{\dim}_{\theta}\\{\,1/b:b\in
I\,\\}\\}\leqslant\underline{\dim}_{\theta}F_{I}\leqslant\max\\{h,\overline{\dim}_{\theta}\\{\,1/b:b\in
I\,\\}\\}.$

  2. (ii)

The maps $\theta\mapsto\overline{\dim}_{\theta}F_{I}$ and
$\theta\mapsto\underline{\dim}_{\theta}F_{I}$ are continuous at $\theta=0$.

###### Proof.

We sketch the proof as it is similar to the proof of Theorem 4.4.3.

(i). follows from Theorem 4.3.4.

(ii). follows from (i) since $\\{\,1/b:b\in I\,\\}$ is the disjoint union of
bi-Lipschitz copies of two subsets of $G_{1,2}$, whose intermediate dimensions
are continuous by Proposition 3.2.10. ∎

######  Corollary 4.4.8.

For $p\in(1,\infty)$ and $R\in[0,\infty)$ let

$I_{p,R}\coloneqq\\{\,\lfloor m^{p}\rfloor+\lfloor n^{p}\rfloor
i:{n,m\in\mathbb{N}}\,\\}\setminus{B(0,R)}.$

Then

$\dim_{\theta}F_{I_{p,R}}=\max\left\\{\dim_{\mathrm{H}}F_{I_{p,R}},\frac{2\theta}{p+\theta}\right\\},$
(4.4.2)

and for all $R$ sufficiently large,
$\dim_{\mathrm{H}}F_{I_{p,R}}<\dim_{\mathrm{B}}F_{I_{p,R}}=2/(p+1)$.

###### Proof.

The set $\\{\,1/b:{b\in I}\,\\}$ is bi-Lipschitz equivalent to a cofinite
subset of the set $G_{p,2}$ from Proposition 3.2.10, so

$\dim_{\theta}\\{\,1/b:b\in
I\,\\}=\dim_{\theta}G_{p,2}=\frac{2\theta}{p+\theta}.$

Therefore the bounds in Theorem 4.4.7 (i) coincide and (4.4.2) holds. For all
$t\geqslant 0$, writing $\simeq$ to mean up to multiplication by a positive,
finite function of $t$, $p$ and $R$ and/or addition by a real-valued function
of $t$, $p$ and $R$, and using the convention that $a\infty=\infty+c=\infty$
for $a\in(0,\infty)$ and $c\in\mathbb{R}$, we have

$\displaystyle\psi_{1}(t)$ $\displaystyle=\sum_{b\in I_{p,R}}||S_{b}||^{t}$
$\displaystyle\simeq\sum_{b\in I_{p,R}}|b|^{-2t}$ (Koebe distortion theorem)
$\displaystyle=\sum_{n=0}^{\infty}\sum_{\begin{subarray}{c}b\in I_{p,R}\\\
2^{n}\leqslant|b|<2^{n+1}\end{subarray}}|b|^{-2t}$
$\displaystyle\simeq\sum_{n=0}^{\infty}\\#\\{\,b\in
I_{p,R}:2^{n}\leqslant|b|<2^{n+1}\,\\}(2^{n})^{-2t}$
$\displaystyle\simeq\sum_{n=0}^{\infty}(2^{n/p})^{2}(2^{n})^{-2t}$
$\displaystyle=\sum_{n=0}^{\infty}4^{n(p^{-1}-t)}.$

Therefore the finiteness parameter $\theta_{I_{p,R}}=1/p$. By [MUa, Theorem
1.5],

$\dim_{\mathrm{H}}F_{I_{p,R}}\xrightarrow[R\to\infty]{}\frac{1}{p}<\frac{2}{p+1},$

as required. ∎

Again there are consequences for fractional Brownian images.

######  Corollary 4.4.9.

Let $\alpha\in(0,1)$ and let $B_{\alpha}\colon\mathbb{C}\to\mathbb{C}$ denote
index-$\alpha$ fractional Brownian motion (identifying $\mathbb{C}$ with
$\mathbb{R}^{2}$). Then for all non-empty
$I\subseteq\\{\,m+ni:m\in\mathbb{N},n\in\mathbb{Z}\,\\}$,

  1. (i)

The maps $\theta\mapsto\overline{\dim}_{\theta}B_{\alpha}(F_{I})$ and
$\theta\mapsto\underline{\dim}_{\theta}B_{\alpha}(F_{I})$ are almost surely
continuous at $\theta=0$.

  2. (ii)

If $\alpha>(\dim_{\mathrm{H}}F_{I})/2$ then almost surely

$(\dim_{\mathrm{H}}F_{I})/\alpha=\dim_{\mathrm{H}}B_{\alpha}(F_{I})\leqslant\overline{\dim}_{\mathrm{B}}B_{\alpha}(F_{I})<2.$

  3. (iii)

If $\alpha\leqslant(\dim_{\mathrm{H}}F_{I})/2$ then almost surely
$\dim_{\mathrm{H}}B_{\alpha}(F_{I})=\dim_{\mathrm{B}}B_{\alpha}(F_{I})=2$.

###### Proof.

This follows from Theorem 4.4.7 in a similar way to how Corollary 4.4.6
follows from Theorem 4.4.3. ∎

Since
$\dim_{\mathrm{P}}B_{\alpha}(F_{I}),\underline{\dim}_{\mathrm{B}}B_{\alpha}(F_{I})\in(\dim_{\mathrm{H}}B_{\alpha}(F_{I}),\overline{\dim}_{\mathrm{B}}B_{\alpha}(F_{I}))$,
if $\alpha>(\dim_{\mathrm{H}}F_{I})/2$ then almost surely
$\dim_{\mathrm{P}}B_{\alpha}(F_{I})<2$ and
$\underline{\dim}_{\mathrm{B}}B_{\alpha}(F_{I})<2$, whereas if
$\alpha\leqslant(\dim_{\mathrm{H}}F_{I})/2$ then almost surely
$\dim_{\mathrm{P}}B_{\alpha}(F_{I})=\underline{\dim}_{\mathrm{B}}B_{\alpha}(F_{I})=2$.

###  4.5 Generic attractors

####  4.5.1 Background and motivation

Often, and especially in non-conformal settings and in the presence of
overlaps, it is difficult to compute the dimension of a particular IFS
attractor. In a seminal paper from 1988 [Fala] Falconer introduced the idea of
studying the generic dimension of a (finitely generated) self-affine set by
fixing a set of matrices and then randomising the translations in a suitable
way. It turns out that, for Lebesgue almost every choice of translations, the
Hausdorff and box dimension of the associated self-affine set are given by the
affinity dimension: a dimension formula depending only on the matrices.

Käenmäki and Reeve [KR] extended Falconer’s approach to the theory of
infinitely generated self-affine sets. Here one needs to randomise infinitely
many translations, and do so in a manner which outputs a bounded set. As such,
the natural space to draw from is $V^{\mathbb{N}}$ for some bounded set
$V\subseteq\mathbb{R}^{d}$ with positive $d$-dimensional Lebesgue measure
$\mathcal{L}_{d}(V)>0$. For convenience from now on we assume $V=[0,1)^{d}$.
The space $V^{\mathbb{N}}$ carries a natural infinite product probability
measure

$\mu=\prod_{i\in\mathbb{N}}\mathcal{L}_{d}|_{V}.$

Käenmäki and Reeve proved that if one fixes an infinite collection of strictly
contracting matrices on $\mathbb{R}^{d}$ and randomises the translations
according to $\mu$, then the Hausdorff dimension of the associated attractor
is almost surely given by the natural extension of the affinity dimension to
the infinite case. In comparison with Falconer’s result, this notably omits
the box dimension. There is good reason for this since the box dimension of an
infinitely generated attractor depends much more sensitively on the
translations themselves, as we have seen above.

We show here that almost surely the box dimension is $d$, that is, the ambient
spatial dimension. In fact we show more. We show that for an arbitrary IIFS
(not necessarily consisting of affine maps), the associated attractor is
generically somewhere dense, and therefore the box and intermediate dimensions
are all generically equal to $d$. Moreover, _generically_ can refer to either
$\mu$-almost surely, or for a comeagre set of translates (topologically
generic). If one equips $V^{\mathbb{N}}$ with a topological group structure as
the infinite product of the group $V$ under addition mod 1 with the product
topology, then being _prevalent_ in the sense of [EN, OY, Chr, HSY] is the
same as holding $\mu$-almost surely.

####  4.5.2 Results

Fix an IIFS $\\{S_{i}\\}_{i\in\mathbb{N}}$ defined on $[0,2]^{d}$ with the
property that $S_{i}([0,2]^{d})\subseteq[0,1]^{d}$ for all $i\in\mathbb{N}$.
For $t=(t_{1},t_{2},\dots)\in V^{\mathbb{N}}$ let $F_{t}$ denote the attractor
of the IIFS $\\{S_{i}+t_{i}\\}_{i\in\mathbb{N}}$ with $S_{i}+t_{i}$ defined on
$[0,2]^{d}$ by $(S_{i}+t_{i})(x)=S_{i}(x)+t_{i}$. We assume throughout that
the contraction ratios of the maps $S_{i}$ only accumulate at zero. The
assumption that $[0,2]^{d}$ maps into $[0,1]^{d}$ is to ensure the maps can be
composed with translations in a well-defined way.

Write $\mathrm{fix}(g)$ to denote the unique fixed point of a contraction $g$
on $[0,2]^{d}$. We use the following simple lemma to relate random
translations to random fixed points. Fixed points are useful because they
necessarily belong to the attractor.

######  Lemma 4.5.1.

Let $u\in V$, $g$ be a contraction on $[0,2]^{d}$ with
$g([0,2]^{d})\subseteq[0,1]^{d}$, $q\in[0,2]^{d}$, and $\delta>0$. Then

$\mathrm{fix}(g+u)\in B(q,\delta)\Leftrightarrow u\in
B(q-g(\mathrm{fix}(g+u)),\delta).$

###### Proof.

By the definition of a fixed point,

$\displaystyle\mathrm{fix}(g+u)\in B(q,\delta)$
$\displaystyle\Leftrightarrow||\mathrm{fix}(g+u)-q||<\delta$
$\displaystyle\Leftrightarrow||g(\mathrm{fix}(g+u))+u-q||<\delta$
$\displaystyle\Leftrightarrow u\in B(q-g(\mathrm{fix}(g+u)),\delta),$

as required. ∎

######  Theorem 4.5.2.

For $\mu$-almost all $t\in V^{\mathbb{N}}$, the attractor $F_{t}$ is somewhere
dense in $[0,2]^{d}$, so for all $\theta\in(0,1]$,

$\dim_{\theta}F_{t}=\dim_{\mathrm{B}}F_{t}=d.$

###### Proof.

Let $z\in[0,1]^{d}$ be an accumulation point of the set
$\\{\mathrm{fix}(S_{i})\\}_{i}$. We prove that $F_{t}$ is almost surely dense
in the explicit (unit) square $V+z$. The idea is that for infinitely many
$i\in\mathbb{N}$ the fixed point of $S_{i}$ will be close to $z$ and the
contraction ratio of $S_{i}$ will be small, so if the fixed point of
$S_{i}+t_{i}$ is far away from a given point $q\in V+z$ then $t_{i}$ must be
far away from $q-z$, which we use to bound the measure. We have

$\displaystyle\left\\{\,t\in V^{\mathbb{N}}\ :\ F_{t}\text{ is nowhere
dense}\,\right\\}\subseteq\left\\{\,t\in V^{\mathbb{N}}\ :\ F_{t}\text{ is not
dense in $V+z$}\,\right\\}$
$\displaystyle\subseteq\bigcup_{q\in(V+z)\cap\mathbb{Q}^{d}}\bigcup_{\delta\in\mathbb{Q}^{+}}\left\\{\,t\in
V^{\mathbb{N}}\ :\ \forall i\in\mathbb{N},\,\mathrm{fix}(S_{i}+t_{i})\notin
B(q,\delta)\,\right\\}$
$\displaystyle=\bigcup_{q\in(V+z)\cap\mathbb{Q}^{d}}\bigcup_{\delta\in\mathbb{Q}^{+}}\left\\{\,t\in
V^{\mathbb{N}}\ :\ \forall i\in\mathbb{N},\,t_{i}\notin
B(q-S_{i}(\mathrm{fix}(S_{i}+t_{i})),\delta)\,\right\\}\quad\text{(Lemma\leavevmode\nobreak\
\ref{genericlma})}$
$\displaystyle=\bigcup_{q\in(V+z)\cap\mathbb{Q}^{d}}\bigcup_{\delta\in\mathbb{Q}^{+}}\bigcap_{N=1}^{\infty}T_{q,\delta,N}$

where for $N\in\mathbb{N}$,

$T_{q,\delta,N}\coloneqq\left(\prod_{i=1}^{N}\\{\,t\in
V:||q-S_{i}(\mathrm{fix}(S_{i}+t))-t||\geqslant\delta\,\\}\right)\times\left(\prod_{j=N+1}^{\infty}V\right).$

For each $i\in\mathbb{N}$, $\\{\,t\in
V:||q-S_{i}(\mathrm{fix}(S_{i}+t))-t||\geqslant\delta\,\\}$ is a Borel set as
it is the preimage of $[\delta,\infty)$ by a continuous function, so each
$T_{q,\delta,N}$ is $\mu$-measurable. By definition of $z$ we can find
infinitely many $i\in\mathbb{N}$ such that the maximum of
$||\mathrm{fix}(S_{i})-z||$ and the contraction ratio of $S_{i}$ is less than
$\delta/10$. For all such $i\in\mathbb{N}$, if $t\in B(q-z,\delta/2)\cap V$
then

$\displaystyle||q-S_{i}(\mathrm{fix}(S_{i}+t))-t||$
$\displaystyle\leqslant||q-z-t||+||z-\mathrm{fix}(S_{i})||+||\mathrm{fix}(S_{i})-S_{i}(\mathrm{fix}(S_{i}+t))||$
$\displaystyle<\frac{\delta}{2}+\frac{\delta}{10}+2\frac{\delta}{10}$
$\displaystyle<\delta.$

Therefore for infinitely many $i\in\mathbb{N}$,

$\mathcal{L}_{d}\Big{(}\\{t\in
V:||q-S_{i}(\mathrm{fix}(S_{i}+t))-t||\geqslant\delta\\}\Big{)}\leqslant
1-2^{-d}\left(\frac{\min\\{\delta,1/2\\}}{2}\right)^{d}k_{d}<1,$

where $k_{d}$ is the $d$-dimensional Lebesgue measure of a ball in
$\mathbb{R}^{d}$ of unit radius. This uniform bound away from 1 is independent
of $i$, so for all $q$ and $\delta$, $\mu(T_{q,\delta,N})\to 0$ as
$N\to\infty$, so $\bigcap_{i=1}^{\infty}T_{q,\delta,N}$ is $\mu$-measurable
(as the countable intersection of $\mu$-measurable sets) with
$\mu\left(\bigcap_{i=1}^{\infty}T_{q,\delta,N}\right)=0$. Therefore

$\bigcup_{q\in(V+z)\cap\mathbb{Q}^{d}}\bigcup_{\delta\in\mathbb{Q}^{+}}\bigcap_{i=1}^{\infty}T_{q,\delta,N}$

is a countable union of $\mu$-measurable sets with $\mu$-measure 0, so it is
itself $\mu$-measurable with $\mu$-measure 0, which proves the result. ∎

Next, we establish a topological result. We endow $V^{\mathbb{N}}$ with the
Hilbert cube metric

$d(t,s)=\left(\sum_{i=1}^{\infty}\frac{||t_{i}-s_{i}||^{2}}{i^{2}}\right)^{1/2},$

noting that this generates the product topology on $V^{\mathbb{N}}$. Recall
that a subset of $V^{\mathbb{N}}$ is called _residual_ or _comeagre_
(topologically generic) if it contains a countable intersection of open dense
sets. Note that $V$ is homeomorphic to the separable complete metric space
$[0,\infty)$, so $V$ is a Polish space, hence the countable product
$V^{\mathbb{N}}$ is also a Polish space. The Baire Category Theorem therefore
implies that $V^{\mathbb{N}}$ is a Baire space, meaning that residual subsets
are dense.

######  Theorem 4.5.3.

For a residual set of $t\in V^{\mathbb{N}}$, the attractor $F_{t}$ is
somewhere dense in $[0,2]^{d}$, so for all $\theta\in(0,1]$,

$\dim_{\theta}F_{t}=\dim_{\mathrm{B}}F_{t}=d.$

###### Proof.

Let $z\in[0,1]^{d}$ be as in the proof of Theorem 4.5.2. Then

$\displaystyle\left\\{\,t\in V^{\mathbb{N}}\ :\ F_{t}\text{ is somewhere
dense}\,\right\\}$ $\displaystyle\supseteq\left\\{\,t\in V^{\mathbb{N}}\ :\
F_{t}\text{ is dense in $V+z$}\,\right\\}$
$\displaystyle=\bigcap_{q\in(V+z)\cap\mathbb{Q}^{d}}\bigcap_{\delta\in\mathbb{Q}^{+}}\left\\{\,t\in
V^{\mathbb{N}}\ :\ \exists i\in\mathbb{N},\,\mathrm{fix}(S_{i}+t_{i})\in
B(q,\delta)\,\right\\}$
$\displaystyle=\bigcap_{q\in(V+z)\cap\mathbb{Q}^{d}}\bigcap_{\delta\in\mathbb{Q}^{+}}\left\\{\,t\in
V^{\mathbb{N}}\ :\ \exists i\in\mathbb{N},\,t_{i}\in
B(q-S_{i}(\mathrm{fix}(S_{i}+t_{i})),\delta)\,\right\\}$ (Lemma 4.5.1)
$\displaystyle\eqqcolon\bigcap_{q\in(V+z)\cap\mathbb{Q}^{d}}\bigcap_{\delta\in\mathbb{Q}^{+}}T_{q,\delta}.$

Fix $q\in(V+z)\cap\mathbb{Q}^{d}$ and $\delta\in\mathbb{Q}^{+}$. The set
$T_{q,\delta}$ is immediately seen to be open since
$\mathrm{fix}(S_{i}+t_{i})$ is continuous in $t_{i}$ and we use open balls.
Moreover, it is also dense since an element $t\in V^{\mathbb{N}}$ may be
approximated arbitrarily well in the metric $d$ within the set $T_{q,\delta}$
by replacing $t_{i}$ with $q-z$ for sufficiently large $i$. ∎

######  Remark 4.5.4.

In the above setting, all the contraction ratios were bounded above by 1/2,
but we can easily avoid this. Indeed, fix any $c>0$ and fix any IIFS
$\\{S_{i}\\}_{i\in\mathbb{N}}$ of contractions defined on $[0,1+c]^{d}$
satisfying $S_{i}([0,1+c]^{d})\subseteq[0,1]^{d}$ for all $i\in\mathbb{N}$.
Then the contraction ratios are bounded above by $\frac{1}{1-c}$ and we assume
they accumulate only at 0. We can let $V\coloneqq[0,c)^{d}$, and again
$V^{\mathbb{N}}$ can be equipped with a natural probability measure by taking
the infinite product of the Lebesgue measure on $V$ and then normalising.
Moreover, $V^{\mathbb{N}}$ can be equipped with a topological group structure
by taking the infinite product (with the product topology) of the group $V$
under addition _mod $c$_ on each of the $d$ coordinates. If
$t=(t_{1},t_{2},\dotsc)\in V^{\mathbb{N}}$ then
$\\{S_{i}+t_{i}\\}_{i\in\mathbb{N}}$ is an IIFS of contractions on
$[0,1+c]^{d}$, with limit set $F_{t}$, say. Similar proofs show that under the
assumptions of either Theorem 4.5.2 or Theorem 4.5.3, $F_{t}$ is somewhere
dense in $[0,1+c]^{d}$, so the same conclusions about dimensions hold.

Under the assumptions of either Theorem 4.5.2 or Theorem 4.5.3, either in the
setting of those theorems or in the more general setting of Remark 4.5.4, the
attractor $F_{t}$ is somewhere dense. This means that if $\dim$ is any notion
of dimension which is stable under closure, for example any of the
$\Phi$-intermediate or Assouad type dimensions, then $\dim F_{t}=d$. If all of
the $S_{i}$ are bi-Lipschitz, then by a result of Mauldin and Urbański [MU,
Theorem 3.1] the upper box and packing dimensions of the attractor coincide,
so under the assumptions of Theorem 4.5.2 or 4.5.3, it holds that
$\dim_{\mathrm{P}}F_{t}=d$.

##  Chapter 5 Bedford–McMullen carpets

###  5.1 Introduction

####  5.1.1 Self-affine carpets

In this chapter, which is based on [BK] (joint with I. Kolossváry), we
consider _self-affine sets_ , which are the attractors of IFSs consisting of
affine contractions. The dimension theory of self-affine sets (surveyed in
[Fald]) is broadly divided into two strands of research. The former is
concerned with verifying that in many ‘generic’ situations the Hausdorff and
box dimensions coincide with Falconer’s affinity dimension [BHR, Fala]. This
chapter, however, is concerned with the exceptional theory, where the
dimensions can take different values, and the intermediate dimensions are
therefore relevant. In particular, self-affine sets in the plane for which the
matrices of the defining contractions are diagonal are often called ‘carpets,’
and various models with different levels of generality have been studied [Bed,
McM, LG, Bar, FW, KP]. The three-dimensional versions are often called
sponges, and surprisingly a class of such sets appear in the paper [DS] as the
first examples of expanding repellers whose Hausdorff dimension is not
attained as the Hausdorff dimension of any ergodic invariant measure; it is
not known whether a repeller with this property exists in the plane. There are
many interesting open problems about self-affine sets, such as whether the box
dimension of every self-affine set exists.

In this chapter, we work with the simplest model of self-affine carpets, which
were originally introduced independently by Bedford [Bed] and McMullen [McM].
A Bedford–McMullen carpet $\Lambda$ is a subset of the plane, but can also be
viewed as an invariant subset of the 2-torus $[0,1)^{2}$ under the toral
endomorphism $(x,y)\mapsto(mx\mod 1,ny\mod 1)$. Figure 5.1 shows a simple
example of a Bedford–McMullen carpet with distinct Hausdorff and box
dimension. The three shaded rectangles show the image of $[0,1]^{2}$ under the
three maps in the IFS, and the attractor is also shown. For this carpet,
$\dim_{\mathrm{H}}\Lambda\approx
1.34968<1.36907\approx\dim_{\mathrm{B}}\Lambda$. These carpets can be realised
as cross-sections of invariant sets of discrete dynamical systems given by
iterating three-dimensional horseshoe maps, showing that dynamically invariant
sets can have distinct Hausdorff and box dimensions. We refer the interested
reader to the survey [Frac] for background on the dimension theory of these
carpets. Our main objective is to determine a formula for the intermediate
dimensions of all Bedford–McMullen carpets. In the process, we uncover new
interesting features about the form of the intermediate dimensions and make an
unexpected connection to multifractal analysis and bi-Lipschitz equivalence of
these carpets.

Figure 5.1: A Bedford–McMullen carpet with non-uniform fibres. The images of
$[0,1]^{2}$ under the iterated function system generating the carpet are
shaded. Figure 5.2: The graph of the intermediate dimensions of the
Bedford–McMullen carpet from Figure 5.1 is plotted in blue, using the formula
obtained in this chapter. Note that there are countably many phase transitions
(one at each negative integer power of $\gamma$, the first three of which are
labelled). Certain bounds which were obtained previously are plotted in
orange.

These carpets are constructed by splitting $[0,1]^{2}$ into $m$ columns of
equal width and $n$ rows of equal height for some integers $n>m\geqslant 2$
and considering maps of the form

$f_{(i,j)}(\underline{x})\coloneqq\begin{pmatrix}1/m&0\\\
0&1/n\end{pmatrix}\begin{pmatrix}x\\\ y\end{pmatrix}+\begin{pmatrix}(i-1)/m\\\
(j-1)/n\end{pmatrix}$

for the index set
$(i,j)\in\mathcal{A}\subseteq\\{1,\dotsc,m\\}\times\\{1,\dotsc,n\\}$. The
attractor

$\Lambda=\bigcup_{(i,j)\in\mathcal{A}}f_{(i,j)}(\Lambda)$

of the IFS $\mathcal{F}=\\{f_{(i,j)}\\}_{(i,j)\in\mathcal{A}}$ is called a
_Bedford–McMullen carpet_. Fix notation

$\gamma\coloneqq\frac{\log n}{\log m}>1.$

For the remainder of the chapter, we index the maps of $\mathcal{F}$ by
$i\in\\{1,\dotsc,N\\}$. Let $1<M\leqslant m$ denote the number of non-empty
columns and $\mathbf{N}\coloneqq(N_{1},\dotsc,N_{M})$, where
$N_{\hat{\jmath}}$ is the number of maps $f_{i}$ that map $[0,1]^{2}$ to the
$\hat{\jmath}$-th non-empty column. Observe that $N=N_{1}+\dotsb+N_{M}$.

Let $\mathcal{P}_{M}$ denote the set of probability vectors on
$\\{1,\dotsc,M\\}$. The _entropy_ of $\mathbf{q}\in\mathcal{P}_{M}$ is

$H(\mathbf{q})=-\sum_{\hat{\jmath}=1}^{M}q_{\hat{\jmath}}\log
q_{\hat{\jmath}},$

where we use the convention that $0\log 0=0$. We introduce

$\mathbf{P}=(P_{1},\dotsc,P_{M})\coloneqq\left(\frac{N_{1}}{N},\dotsc,\frac{N_{M}}{N}\right)\quad\mbox{
and }\quad\mathbf{Q}\coloneqq\left(\frac{1}{M},\dotsc,\frac{1}{M}\right).$

For $\mathbf{q}\in\mathcal{P}_{M}$, it holds that $H(\mathbf{q})\leqslant\log
M$ with equality if and only if $\mathbf{q}=\mathbf{Q}$. For the entire
chapter, we also introduce

$\underline{t}\coloneqq\frac{1}{M}\sum_{\hat{\jmath}=1}^{M}\log
N_{\hat{\jmath}}\quad\mbox{ and }\quad\overline{t}\coloneqq\log
N-H(\mathbf{P}).$ (5.1.1)

We say that $\Lambda$ has _uniform (vertical) fibres_ if and only if
$\mathbf{P}=\mathbf{Q}$, in other words each non-empty column has the same
number of maps.

Bedford and McMullen showed that the Hausdorff and box dimensions of $\Lambda$
are equal to

$\displaystyle\dim_{\mathrm{H}}\Lambda$ $\displaystyle=\frac{1}{\log
m}\log\Bigg{(}\sum_{\hat{\jmath}=1}^{M}N_{\hat{\jmath}}^{\log m/\log
n}\Bigg{)},$ (5.1.2) $\displaystyle\dim_{\mathrm{B}}\Lambda$
$\displaystyle=\frac{\log N}{\log n}+\left(1-\frac{\log m}{\log
n}\right)\frac{\log M}{\log m}.$ (5.1.3)

In particular, $\dim_{\mathrm{H}}\Lambda=\dim_{\mathrm{B}}\Lambda$ if and only
if $\Lambda$ has uniform fibres. In this case,
$\theta\mapsto\dim_{\theta}\Lambda$ is just a constant function. Therefore, we
assume throughout that the carpet has non-uniform fibres, in which case the
appropriate dimensional Hausdorff measure of $\Lambda$ is infinite [Per].
Using the concavity of the logarithm function, an immediate consequence of
non-uniform fibres is that $\underline{t}<\log(N/M)$. Carpets with uniform and
non-uniform fibres are shown in [Frac, Figure 15.1].

####  5.1.2 Previous results

Previous papers on the topic [FFKa, Frac, Kol] have established crude bounds
for the intermediate dimensions and speculated about the possible form. The
question of determining the intermediate dimensions of all Bedford–McMullen
carpets was explicitly asked in [Falg, FFKa, Frac, Kol]. Loosely speaking, the
results of Falconer, Fraser and Kempton [FFKa] concentrate on the behaviour of
$\dim_{\theta}\Lambda$ for $\theta$ close to 0, while the results of
Kolossváry [Kol] concentrate on the behaviour for
$\theta\geqslant\gamma^{-1}$.

A linear lower bound for the intermediate dimensions of Bedford–McMullen
carpets with non-uniform fibres was obtained in [FFKa] which shows that
$\underline{\dim}_{\theta}\Lambda>\dim_{\mathrm{H}}\Lambda$ for every
$\theta\in(0,1]$. For many carpets, but not all, a lower bound in [Kol]
performs better than the linear bound and general bounds such as Corollary
3.2.9 from page 3.2.9 for large values of $\theta$. The lower bound depicted
in Figure 5.2 is the best combination of these results. Falconer, Fraser and
Kempton [FFKa, Proposition 4.1] show an upper bound of the form
$\overline{\dim}_{\theta}\Lambda\leqslant\dim_{\mathrm{H}}\Lambda+c/(-\log\theta)$
for an explicit $c>0$ and $\theta$ sufficiently small. In particular, this
implies that $\theta\mapsto\underline{\dim}_{\theta}\Lambda$ and
$\theta\mapsto\overline{\dim}_{\theta}\Lambda$ are continuous also at
$\theta=0$. Hence, the results of Burrell, Falconer and Fraser [BFF, Section
6] and Burrell [Bura, Section 3] can be applied. For example, if
$\dim_{\mathrm{H}}\Lambda<1$ then $\overline{\dim}_{\mathrm{B}}\pi(\Lambda)<1$
for every orthogonal projection $\pi$ from $\mathbb{R}^{2}$ onto a
1-dimensional subspace, regardless of the value of $\dim_{\mathrm{B}}\Lambda$.
For almost every projection, $\underline{\dim}_{\theta}\pi(\Lambda)$ and
$\overline{\dim}_{\theta}\pi(\Lambda)$ are continuous at $\theta=0$, and if
$\gamma\notin\mathbb{Q}$ then this holds for _every_ orthogonal projection.
Furthermore, if $B_{h}\colon\mathbb{R}^{2}\to\mathbb{R}^{2}$ is index-$h$
fractional Brownian motion, then
$\theta\mapsto\underline{\dim}_{\theta}B_{h}(\Lambda)$ and
$\theta\mapsto\overline{\dim}_{\theta}B_{h}(\Lambda)$ are almost surely
continuous, and if $h>(\dim_{\mathrm{H}}\Lambda)/2$ then almost surely
$\overline{\dim}_{\mathrm{B}}B_{h}(\Lambda)<2$.

A cover of $\Lambda$ is constructed in [Kol] using just the two extreme scales
to obtain an explicit upper bound of the form
$\overline{\dim}_{\theta}\Lambda\leqslant\dim_{\mathrm{B}}\Lambda-\Delta(\theta)$
for $\theta\geqslant\gamma^{-1}$, where $\Delta(\theta)\searrow 0$ as
$\theta\to 1$ and has a strictly positive derivative at $\theta=1$. This bound
was used to show that $\overline{\dim}_{\theta}\Lambda$ is not concave for the
whole range of $\theta$ in general, already hinting at richer behaviour than
previously witnessed in other examples. Figure 5.2 shows this upper bound.

####  5.1.3 Summary of results

The formal statements are presented in Section 5.2. In Theorem 5.2.1 we state
an explicit formula for
$\underline{\dim}_{\theta}\Lambda=\overline{\dim}_{\theta}\Lambda$ for all
$\theta$, thus fully resolving the problem of calculating the intermediate
dimensions of all Bedford–McMullen carpets $\Lambda$. For illustration see
Figure 5.2, where $\theta\mapsto\dim_{\theta}\Lambda$ is plotted for the
carpet from Figure 5.1. Central to the formula is a large deviations rate
function for which we give three additional equivalent characterisations in
Proposition 5.2.11: one in terms of a pressure-like function, another as a
certain probability vector with an entropy maximising property, and finally a
relationship to the multifractal spectrum of the uniform self-affine measure
on $\Lambda$. In the proof of Theorem 5.2.1, we construct a cover of $\Lambda$
that uses an increasing number of different scales in the permissible range as
$\theta\to 0$. We also show in Corollary 5.2.2 that using more than two scales
is necessary.

In Corollary 5.2.3, we prove all the features suggested by the plot in Figure
5.2 about the form of the intermediate dimensions for all carpets. Namely,
$\theta\mapsto\dim_{\theta}\Lambda$ is strictly increasing and has phase
transitions at all negative integer powers of $\gamma$. Between consecutive
phase transitions the intermediate dimensions are analytic and strictly
concave. Moreover, for $\theta$ small enough $\dim_{\theta}\Lambda$ behaves
like $\dim_{\mathrm{H}}\Lambda+c(\log\theta)^{-2}$. In particular, the
derivative tends to $+\infty$ as $\theta\to 0$. No previous family of sets has
shown such rich and complex behaviour. Some illustrative examples are
presented in Section 5.2.5.

We show in Theorem 5.2.5 that two different carpets with non-uniform fibres
have equal intermediate dimensions for every $\theta\in[0,1]$ if and only if
the multifractal spectra of the uniform Bernoulli measure on the two carpets
are equal. If, in addition, it is assumed that the two carpets are defined on
the same grid, then Theorem 5.2.5 provides further equivalent conditions for
their intermediate dimensions to be the same: a certain condition on the rate
functions appearing in the formula or certain relationships between the
parameters of the carpets, or the equality of the intermediate dimensions on
any one open interval of $[\gamma^{-1},1]$.

Our main application relates to bi-Lipschitz equivalence. It is known [RYZ,
Corollary 1.1] that the equality of these multifractal spectra is necessary
for two carpets to be bi-Lipschitz equivalent if it is assumed that the two
carpets are defined on the same grid and are totally disconnected. Since bi-
Lipschitz maps preserve intermediate dimensions, Theorem 5.2.5 implies that
both of these assumptions can be dropped, see Corollary 5.2.10. In Proposition
5.2.14 we construct two carpets which are not bi-Lipschitz equivalent by
Corollary 5.2.10, but where this does not follow from [RYZ]. This is the first
instance where intermediate dimensions are used to show that two sets are not
bi-Lipschitz equivalent, but where this fact does not follow from any other
notion of dimension or existing result. For this example, we also use the
intermediate dimensions to give estimates on the Hölder distortion of the two
carpets.

For comparison, we mention that the calculation of the Assouad spectrum of
Bedford–McMullen carpets [FY] is not as involved as the proof of Theorem 5.2.1
for the intermediate dimensions. Indeed, the intermediate dimensions are more
subtle in that they depend on all the $N_{i}$ individually, as does the
Hausdorff dimension. This is in contrast to the Assouad spectrum (and indeed
the lower spectrum and the box, packing, Assouad, quasi-Assouad, lower and
quasi-lower dimensions) which depend only on
$m,n,N,M,\max_{1\leqslant\hat{\imath}\leqslant M}N_{\hat{\imath}}$ and
$\min_{1\leqslant\hat{\imath}\leqslant M}N_{\hat{\imath}}$, see [Frac]. The
Assouad spectrum also has just one phase transition at $\theta=\gamma^{-1}$,
which occurs when the spectrum reaches the Assouad dimension and thus is
constant for $\theta\in(\gamma^{-1},1)$.

###  5.2 Results and examples

####  5.2.1 Main result: formula for intermediate dimensions

Recalling (5.1.1), let $t\in(\underline{t},\overline{t})$. This is a non-empty
interval because the fact that there are non-uniform fibres implies that
$\underline{t}<\log(N/M)<\overline{t}$. Let $X_{1},X_{2},\dotsc,X_{J},\dotsc$
be a sequence of independent and identically distributed random variables
taking values in the set $\\{\log N_{1},\dotsc,\log N_{M}\\}$, with

$\mathds{P}(X_{1}=\log
N_{\hat{\imath}})=\frac{1}{M}\cdot\\#\\{\,\hat{\jmath}\in\\{1,\dotsc,M\\}:N_{\hat{\jmath}}=N_{\hat{\imath}}\,\\}.$
(5.2.1)

Then $\underline{t}$ is the expectation of $X_{1}$. The large deviations rate
function of the average $\frac{1}{J}\sum_{i=1}^{J}X_{i}$ is

$I(t)=\sup_{\lambda\in\mathbb{R}}\bigg{\\{}\lambda
t-\log\bigg{(}\frac{1}{M}\sum_{\hat{\jmath}=1}^{M}N_{\hat{\jmath}}^{\lambda}\bigg{)}\bigg{\\}},$
(5.2.2)

noting that $\frac{1}{M}\sum_{\hat{\jmath}=1}^{M}N_{\hat{\jmath}}^{\lambda}$
is the expectation of $e^{\lambda X_{1}}$. For
$t\in[\underline{t},\max_{1\leqslant\hat{\imath}\leqslant M}\log
N_{\hat{\imath}})$, differentiating shows that the supremum in the definition
of $I(t)$ is attained at the unique $\lambda\geqslant 0$ satisfying
$t=\sum_{\hat{\imath}=1}^{M}\frac{N_{\hat{\imath}}^{\lambda}}{\sum_{\hat{\jmath}=1}^{M}N_{\hat{\jmath}}^{\lambda}}\log
N_{\hat{\imath}}$. This allows $I(t)$ to be calculated numerically. On this
interval, $I(t)$ is real analytic (as the Legendre transform of an analytic
function). The derivative $I^{\prime}(t)>0$ is the value of $\lambda$ at which
the supremum is attained and $I(t)$ is strictly increasing for
$t\in[\underline{t},\max\log N_{\hat{\imath}}]$. Moreover,
$I^{\prime\prime}(t)>0$, so the function is strictly convex on this interval.
Some particular values of interest are

$\displaystyle I(\underline{t})$ $\displaystyle=0,\qquad\qquad$ $\displaystyle
I^{\prime}(\underline{t})=0,$ $\displaystyle I(\overline{t})$
$\displaystyle=\log M-H(\mathbf{P}),\qquad\qquad$ $\displaystyle
I^{\prime}(\overline{t})=1,$

see [DZ, Lemma 2.2.5]. Moreover,

$I(\max_{1\leqslant\hat{\imath}\leqslant M}\log N_{\hat{\imath}})=\log
M-\log\\#\\{\,\hat{\jmath}\in\\{1,\dotsc,M\\}:N_{\hat{\jmath}}=\max_{1\leqslant\hat{k}\leqslant
M}N_{\hat{k}}\,\\},$

and

$I^{\prime}(t)\to\infty\quad\mbox{ as
}t\to(\max_{1\leqslant\hat{\imath}\leqslant M}\log N_{\hat{\imath}})^{-}.$

For $s\in\mathbb{R}$, we define the function
$T_{s}\colon\mathbb{R}\to\mathbb{R}$ by

$T_{s}(t)\coloneqq\left(s-\frac{\log M}{\log m}\right)\log n+\gamma I(t).$
(5.2.3)

For $\ell\in\mathbb{N}$ we denote the composition by
$T_{s}^{\ell}\coloneqq\underbrace{T_{s}\circ\dotsb\circ T_{s}}_{\ell\mbox{
times}}$, and $T_{s}^{0}$ denotes the identity function. We use the sequences
$(t_{\ell})_{\ell=1}^{\infty}=(t_{\ell}(s))_{\ell=1}^{\infty}$ defined by

$t_{\ell}(s)\coloneqq T_{s}^{\ell-1}\left(\left(s-\frac{\log M}{\log
m}\right)\log n\right).$ (5.2.4)

Note that these depend only on $s$ and the carpet, but not on $\theta$.
Observe that $T_{s}(\underline{t})=t_{1}(s)$ for all $s\in\mathbb{R}$. We are
now ready to state the main result of this chapter (see Section 5.4 for the
rather involved proof).

######  Theorem 5.2.1.

Let $\Lambda$ be a Bedford–McMullen carpet with non-uniform fibres. For all
$\theta\in(0,1)$, $\dim_{\theta}\Lambda$ exists and is given in the following
way. For fixed $\theta\in(0,1)$ let $L=L(\theta)\coloneqq
1+\lfloor\frac{-\log\theta}{\log\gamma}\rfloor$, so
$\gamma^{-L}<\theta\leqslant\gamma^{-(L-1)}$. Then there exists a unique
solution $s=s(\theta)\in(\dim_{\mathrm{H}}\Lambda,\dim_{\mathrm{B}}\Lambda)$
to the equation

$\gamma^{L}\theta\log
N-(\gamma^{L}\theta-1)t_{L}(s)+\gamma(1-\gamma^{L-1}\theta)(\log
M-I(t_{L}(s)))-s\log n=0,$ (5.2.5)

and $s(\theta)=\dim_{\theta}\Lambda$.

In the case $L=1$ the formula (5.2.5) simplifies to

$\dim_{\mathrm{B}}\Lambda-\frac{1}{\log
n}\left(\frac{1}{\theta}-1\right)I(t_{1}(s))-s=0.$

If $\theta=\gamma^{-(L-1)}$ for some $L\in\mathbb{N}$ with $L\geqslant 2$,
then it becomes

$\dim_{\mathrm{B}}\Lambda-\frac{1}{\log
m}\left(1-\frac{1}{\gamma}\right)I(t_{L-1}(s))-s=0.$

Theorem 5.2.1 and Corollary 5.2.3 below fully resolve [Frac, Problem 15.8.1]
and the questions about Bedford–McMullen carpets in Falconer’s survey paper
[Falg, Section 14.8], and indeed provide more information. In particular, this
is the first time it has been shown that the intermediate dimensions of
Bedford–McMullen carpets exist for $\theta\in(0,1)$. Tools used in the proof
include the method of types (see [Kola]) and a variant of a mass distribution
principle for the intermediate dimensions, see Proposition 1.4.2 from page
1.4.2. The proof of the upper bound involves the construction of an explicit
cover using scales
$\delta,\delta^{\gamma},\delta^{\gamma^{2}},\dotsc,\delta^{\gamma^{L-1}}$ and
$\delta^{1/\theta},\delta^{1/(\gamma\theta)},\dotsc,\delta^{1/(\gamma^{L-1}\theta)}$.
This cover consists of _approximate squares_ , which we define in Section
5.4.1. We decide which parts of each approximate square to cover at which
scale depending on how the different parts of the symbolic representation of
the approximate square relate to each other. The proof simplifies when
$\theta\geqslant 1/\gamma$ (where we just use the smallest and largest
permissible scales), and when $\theta=\gamma^{-k}$ for $k\in\mathbb{N}$ (where
we use scales $\delta,\delta^{\gamma},\dotsc,\delta^{\gamma^{k}}$ due to
scales ‘lining up’). Note that the cover jumps from using $2k$ scales when
$\theta\in(\gamma^{-k},\gamma^{-(k-1)})$ to using $2k+2$ scales when
$\theta\in(\gamma^{-(k+1)},\gamma^{-k})$ (and uses only $k$ scales when
$\theta=\delta^{-k}$), which gives an indication of why one might expect a
phase transition at $\theta=\gamma^{-k}$.

For sets whose intermediate dimensions have previously been calculated such as
spirals [BFFa], sequences [FFKa, Section 3.1], and concentric spheres and
topologist’s sine curves [Tan], only the two extreme scales were used in the
cover. Several of the results in this thesis, however, use many different
scales to obtain the upper bound. In particular, we see from (4.3.2) on page
4.3.2 that many different scales in the interval $[\delta,\delta^{\theta}]$
are generally used in the construction of the cover for infinitely generated
self-similar sets. Moreover, it is clear from the construction in Theorem
3.3.11 on page 3.3.11 that there are inhomogeneous Moran sets for which every
cover approximating the intermediate dimensions arbitrarily closely would
require an unbounded number of scales as $\delta$ tends to zero. This answers
a question of Falconer [Falg, Section 14.8]. Corollary 5.2.2, which we prove
in Section 5.5, shows that for Bedford–McMullen carpets with non-uniform
fibres, more than two scales are needed when $\theta$ is small.

######  Corollary 5.2.2.

Let $\Lambda$ be a Bedford–McMullen carpet with non-uniform fibres. There
exist $\theta_{0},\varepsilon,\delta_{0}>0$ such that for all
$\theta\in(0,\theta_{0})$ and all covers $\\{U_{i}\\}$ of $\Lambda$ that uses
at most two scales, both of which are less than $\delta_{0}$, we have
$\sum_{i}|U_{i}|^{\dim_{\theta}\Lambda+\varepsilon}\geqslant 1$.

We now continue with corollaries of Theorem 5.2.1 about the form of the graph
of the function $\theta\mapsto\dim_{\theta}\Lambda$ that do not follow from
the general theory, give a rather unexpected connection to multifractal
analysis and bi-Lipschitz equivalence of two carpets, and provide equivalent
formulations of the rate function $I(t)$.

####  5.2.2 Form of the intermediate dimensions

We assume that the carpet has non-uniform fibres, otherwise,
$\theta\mapsto\dim_{\theta}\Lambda$ is just a constant function. We denote the
left and right derivatives at $\theta$ by

$\partial_{-}\dim_{\theta}\Lambda\coloneqq\lim_{h\to
0^{+}}\frac{\dim_{\theta}\Lambda-\dim_{\theta-h}\Lambda}{h}\;\text{ and
}\;\partial_{+}\dim_{\theta}\Lambda\coloneqq\lim_{h\to
0^{+}}\frac{\dim_{\theta+h}\Lambda-\dim_{\theta}\Lambda}{h}.$

######  Corollary 5.2.3.

Let $\Lambda$ be a Bedford–McMullen carpet with non-uniform fibres. Then the
function $\theta\mapsto\dim_{\theta}\Lambda$ has the following properties:

  1. (i)

it is real analytic on the interval $(\gamma^{-L},\gamma^{-(L-1)})$ for all
$L\in\mathbb{N}$;

  2. (ii)

$\partial_{-}\dim_{\theta}\Lambda$ exists at every $\theta\in(0,1]$ and
$\partial_{+}\dim_{\theta}\Lambda$ exists at every $\theta\in(0,1)$;

  3. (iii)

it is strictly increasing and has phase transitions at every negative integer
power of $\gamma$. More precisely, there exists $C_{0}>0$ depending only on
$\Lambda$ such that for all $\theta\in(0,1)$,

$C_{0}<\partial_{-}\dim_{\theta}\Lambda\leqslant\partial_{+}\dim_{\theta}\Lambda$

with equality if and only if for all $L\in\mathbb{N}$ we have
$\theta\neq\gamma^{-L}$. Moreover,
$\frac{\partial_{+}\dim_{\gamma^{-L}}\Lambda}{\partial_{-}\dim_{\gamma^{-L}}\Lambda}$
converges to a constant in $(1,\infty)$ as $L\to\infty$;

  4. (iv)

there exist $C\in[1,\infty)$ and $\theta_{0}>0$ depending only on $\Lambda$
such that for all $\theta\in(0,\theta_{0}]$,

$\dim_{\mathrm{H}}\Lambda+\frac{C^{-1}}{(\log\theta)^{2}}\leqslant\dim_{\theta}\Lambda\leqslant\dim_{\mathrm{H}}\Lambda+\frac{C}{(\log\theta)^{2}};$

  5. (v)

it is strictly concave on the interval $[\gamma^{-L},\gamma^{-(L-1)}]$ for all
$L\in\mathbb{N}$.

In [FFKa, Proposition 4.1] for small enough $\theta$ the upper bound
$\overline{\dim}_{\theta}\Lambda\leqslant\dim_{\mathrm{H}}\Lambda+c(-\log\theta)^{-1}$
was proved for a constant $c$ depending only on $\Lambda$. Corollary 5.2.3
(iv) shows that although this bound is not sharp, we do indeed have that
$\frac{\dim_{\theta}\Lambda-\dim_{\mathrm{H}}\Lambda}{\theta}\to\infty$ as
$\theta\to 0^{+}$.

####  5.2.3 Multifractal analysis and bi-Lipschitz equivalence

In this section, and in Section 5.5.3 where we prove the results in this
section, it is convenient to change notation. Here, the parameters
$(M_{0},N_{1},\dotsc,N_{M_{0}},R_{1},\dotsc,R_{M_{0}})$ will define a
Bedford–McMullen carpet. Now $M_{0}$ denotes the number of different values
that the number of maps in a non-empty column can take. Note that
$M_{0}\geqslant 2$, since $M_{0}=1$ corresponds to the uniform fibre case. We
write $N_{1},\dotsc,N_{M_{0}}$ for the actual values that the number of maps
in a non-empty column can take, and we order them as
$N_{1}>N_{2}>\dotsb>N_{M_{0}}$. For each
$\hat{\imath}\in\\{1,\dotsc,M_{0}\\}$, we write $R_{\hat{\imath}}$ for the
number of columns containing exactly $N_{\hat{\imath}}$ maps. As with the
previous notation, we write $M=\sum_{\hat{\imath}=1}^{M_{0}}R_{\hat{\imath}}$
for the number of non-empty columns, and
$N=\sum_{\hat{\imath}=1}^{M_{0}}R_{\hat{\imath}}N_{\hat{\imath}}$ for the
total number of maps. For example, for the carpet in Figure 5.1, the number of
maps in a non-empty column is either $1$ or $2$, so $N_{1}=2$ and $N_{2}=1$;
each corresponds to just one column, so $R_{1}=R_{2}=1$, and
$M_{0}=\\#\\{1,2\\}=2$.

A central problem in multifractal analysis is to examine the way a Borel
measure $\mu$ is spread over its support $\mathrm{supp}\,\mu$. For a survey of
this topic, we refer the reader to [Fale, Chapter 17]. The _local dimension of
$\mu$ at $x$_ is

$\dim_{\mathrm{loc}}(\mu,x)=\lim_{r\to 0}\frac{\log\mu(B(x,r))}{\log r}$

if the limit exists, which approximately measures the rate of decay of
$\mu(B(x,r))$ as a power law $r^{\alpha}$. The measure $\mu$ is _exact
dimensional_ if $\dim_{\mathrm{loc}}(\mu,x)$ is equal to a specific $\alpha$
for $\mu$-almost all $x$. However, $\dim_{\mathrm{loc}}(\mu,x)$ can still
potentially take up a whole spectrum of different $\alpha$. This motivates the
definition of the _fine_ or _Hausdorff multifractal spectra_

$f_{\mu}(\alpha)\coloneqq\dim_{\mathrm{H}}\\{\,x\in\mathrm{supp}\,\mu:\dim_{\mathrm{loc}}(\mu,x)=\alpha\,\\}.$

Concentrating on the self-affine setting, given a self-affine iterated
function system $\mathcal{S}=\\{S_{1},\dotsc,S_{N}\\}$, meaning that all
$S_{i}\colon\mathbb{R}^{d}\to\mathbb{R}^{d}$ are contracting affine maps, and
a probability vector $\mathbf{p}$ with strictly positive entries, the _self-
affine measure_ $\mu_{\mathbf{p}}$ is the unique probability measure supported
on the attractor of $\mathcal{S}$ satisfying

$\mu_{\mathbf{p}}(A)=\sum_{i=1}^{N}p_{i}\mu_{\mathbf{p}}(S_{i}^{-1}A)\;\text{
for all Borel sets }A\subset\mathbb{R}^{d}.$

It is known that all self-affine measures are exact dimensional [BKa, Fen];
this was resolved earlier in [KP] for those supported on Bedford–McMullen
carpets. Moreover, the dimension satisfies a Ledrappier–Young type formula, as
per the strand of research initiated in [LY, LYa]. The fine multifractal
spectrum of self-affine measures on Bedford–McMullen carpets is also known
[BM, JR, Kin] and (under the separation condition assumed in [Kin]) has been
generalised to higher dimensions in [Olsa]. When $\mathbf{p}=(1/N,\dotsc,1/N)$
is the uniform vector, we simply write $\nu=\mu_{\mathbf{p}}$ and call it the
_uniform self-affine measure_. In this case, define the function
$\beta_{\nu}(\xi)$ for $\xi\geqslant 0$ by

$m^{-\beta_{\nu}(\xi)}N^{-\xi}\sum_{\hat{\imath}=1}^{M_{0}}R_{\hat{\imath}}N_{\hat{\imath}}^{\gamma^{-1}+(1-\gamma^{-1})\xi}=1.$
(5.2.6)

Note that because of the minus sign before $\beta_{\nu}(\xi)$ (which in some
papers is erroneously omitted), $\beta_{\nu}(\xi)$ is a convex function.
Define

$\alpha_{\text{min}}\coloneqq\frac{\log N}{\log m}-\frac{1-\gamma^{-1}}{\log
m}\log N_{1};\qquad\alpha_{\text{max}}\coloneqq\frac{\log N}{\log
m}-\frac{1-\gamma^{-1}}{\log m}\log N_{M_{0}}.$

Then by [JR, Theorem 1], the multifractal spectrum is

$f_{\nu}(\alpha)=\inf_{\xi}(\alpha\xi+\beta_{\nu}(\xi))=-\beta_{\nu}^{*}(-\alpha)\qquad\mbox{for
all}\qquad\alpha\in(\alpha_{\text{min}},\alpha_{\text{max}}),$ (5.2.7)

where
$\beta_{\nu}^{*}(\alpha^{\prime})\coloneqq\sup_{\xi^{\prime}}(\alpha^{\prime}\xi^{\prime}-\beta_{\nu}(\xi^{\prime}))$
is the Legendre transform of $\beta_{\nu}$ (defined in the same way as for the
rate function in (5.2.2)).

Another quantity that is closely related to the multifractal spectrum is the
$L^{q}$ spectrum of a measure $\mu$ (see [Falb, Chapter 11]). It is a function
$T_{\mu}\colon\mathbb{R}\to\mathbb{R}$ which quantifies the global fluctuation
of $\mu$, and its value at $q=0$ describes the box dimension of the support of
the measure. The $L^{q}$ spectrum can be defined by

$T_{\mu}(q)\coloneqq\lim_{\delta\to 0^{+}}\frac{\log
T_{\delta}(\mu,q)}{-\log\delta}$ (5.2.8)

if the limit exists, where

$T_{\delta}(\mu,q)\coloneqq\sup\left\\{\,\sum_{i}(\mu(B_{i}))^{q}:B_{i}\mbox{
disjoint balls of radius }\delta\mbox{ centred in
}\mathrm{supp}(\mu)\,\right\\}.$

The $L^{q}$ spectrum of self-similar measures has been studied in [CM], [Falb,
Chapter 11], and more recently (under the exponential separation condition) in
Shmerkin’s groundbreaking paper [Shma]. For self-similar measures satisfying
the open set condition, the limit (5.2.8) exists and the multifractal
formalism is satisfied, meaning that the Legendre transform of the $L^{q}$
spectrum equals the multifractal spectrum [Ols]. Now let $\nu$ be the uniform
self-affine measure on a Bedford–McMullen carpet with non-uniform fibres, and
let $\nu_{x}$ be the measure obtained by projecting $\nu$ orthogonally onto
the $x$-axis. Note that $\nu_{x}$ is a homogeneous self-similar measure with
all contraction ratios equal to $1/m$, satisfying the open set condition. For
$1\leqslant\hat{\imath}\leqslant M_{0}$, the weight $N_{\hat{\imath}}/N$
occurs with multiplicity $R_{\hat{\imath}}$. Therefore for $q$ in an open
neighbourhood of $1$, $T_{\nu_{x}}(q)$ satisfies

$\sum_{\hat{\imath}=1}^{M_{0}}R_{\hat{\imath}}\left(\frac{N_{\hat{\imath}}}{N}\right)^{q}\left(\frac{1}{m}\right)^{T_{\nu_{x}}(q)}=1.$

A direct calculation shows that

$T_{\nu_{x}}(q)=-\frac{\log N}{\log
m}q+\frac{\log\sum_{\hat{\imath}=0}^{M_{0}}R_{\hat{\imath}}(N_{\hat{\imath}})^{q}}{\log
m}.$

The $L^{q}$ spectra of self-affine measures have been studied in [Kola, FW,
Falc]. For self-affine measures supported on Bedford–McMullen carpets, the
limit in (5.2.8) still exists but the multifractal formalism does not
generally hold. Applying a result of Feng and Wang [FW, Theorem 2] shows that
$T_{\nu}(q)$ satisfies

$N\cdot N^{-q}m^{-T_{\nu_{x}}(q)}n^{-(T_{\nu}(q)-T_{\nu_{x}}(q))}=1.$

A direct manipulation shows that for $q$ in an open neighbourhood of $1$,

$\displaystyle\begin{split}T_{\nu}(q)&=\frac{\log N}{\log n}-q\frac{\log
N}{\log m}+\left(1-\frac{\log m}{\log
n}\right)\frac{\log\sum_{\hat{\imath}=1}^{M_{0}}R_{\hat{\imath}}(N_{\hat{\imath}})^{q}}{\log
m}\\\ &=\left(1-\frac{\log m}{\log
n}\right)\beta_{\nu}\left(\frac{\log(n^{q}/m)}{\log(n/m)}\right).\end{split}$
(5.2.9)

Before describing the connection between the multifractal spectra and the
intermediate dimensions, we observe that the grid on which a carpet can be
defined is not unique, and in fact by iterating the IFS one can see that every
carpet can be defined on infinitely many grids. For example, by iterating the
IFS, the carpet from Figure 5.1 can be defined on a $2\times 3$ grid or on a
$4\times 9$ grid (though of course many carpets on a $4\times 9$ grid cannot
be realised on a $2\times 3$ grid). Theorem 5.2.4 gives information about the
grids on which a carpet can be defined, and is proved in Section 5.5.3. It
demonstrates for example that since $2$ and $3$ are multiplicatively
independent, a carpet with non-uniform fibres defined on a $2\times 4$ grid
cannot be defined on a $3\times 9$ grid (even though $\log 4/\log 2=\log
9/\log 3$).

######  Theorem 5.2.4.

  1. (i)

If a Bedford–McMullen carpet $\Lambda$ with non-uniform fibres can be defined
on both a $m\times n$ grid and on a $m^{\prime}\times n^{\prime}$ grid, then
$\log n/\log m=\log n^{\prime}/\log m^{\prime}$ and $\log n/\log
n^{\prime}\in\operatorname{{\mathbb{Q}}}$.

  2. (ii)

Consider two carpets $\Lambda_{1}$ and $\Lambda_{2}$ with non-uniform fibres
which are defined by IFSs $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ on grids of
size $m_{1}\times n_{1}$ and $m_{2}\times n_{2}$ respectively. Then they can
be realised on the same grid if and only if

$\frac{\log n_{1}}{\log n_{2}}=\frac{\log m_{1}}{\log
m_{2}}\in\operatorname{{\mathbb{Q}}}.$

In fact, we will see below that if two carpets with non-uniform fibres are bi-
Lipschitz equivalent then they can be defined on the same grid. The same is
true even if we merely assume the carpets have the same intermediate
dimensions, or support uniform Bernoulli measures with equal multifractal
spectra.

We make some remarks about part (ii). The reverse implication is immediate.
Indeed, if $\log n_{1}/\log n_{2}=\log m_{1}/\log
m_{2}=a/b\in\operatorname{{\mathbb{Q}}}$, then the $b$-th iterate of
$\mathcal{S}_{1}$ and the $a$-th iterate of $\mathcal{S}_{2}$ are both defined
on the same grid of size $m_{1}^{b}\times n_{1}^{b}$. It is straightforward to
see that the rate function $I(t)$ of $\mathcal{S}_{1}$ and the rate function
$I^{(b)}(t)$ of the $b$-th iterate of $\mathcal{S}_{1}$ are related by
$I^{(b)}(bt)=bI(t)$. For the forward implication, if both carpets can be
realised on the same grid, then the fact that $\log n_{1}/\log m_{1}=\log
n_{2}/\log m_{2}$ was noted by Fraser and Yu using the Assouad spectrum in
[FY, Theorem 3.3], and also follows from the intermediate dimension formula,
noting that geometric quantities such as dimensions of course do not change by
taking an iterate of the system. The fact that $n$ and $n^{\prime}$ must be
multiplicatively dependent (as must $m$ and $m^{\prime}$) follows from Lemma
5.5.3 on page 5.5.3, and is related to work of Meiri and Peres [MP, Theorem
1.2]. This is in turn related to Furstenberg’s $\times 2,\times 3$ principle
(which suggests that expansions in multiplicatively independent bases should
have no common structure). Other work along these lines includes [Fur, Wu,
Shma], but there are many challenging open problems, such as whether there
exists a non-atomic measure on the torus that is $\times 2$\- and $\times
3$-invariant but not Lebesgue.

Our next result, which we prove in Section 5.5.3 using Theorem 5.2.1, gives a
direct connection between the intermediate dimensions and the multifractal and
$L^{q}$ spectra of the uniform self-affine measure.

######  Theorem 5.2.5.

Let $\Lambda$ and $\Lambda^{\prime}$ be two Bedford–McMullen carpets with non-
uniform fibres, and denote the corresponding uniform self-affine measures by
$\nu$ and $\nu^{\prime}$. Then the following are equivalent:

  1. (i)

$\dim_{\theta}\Lambda=\dim_{\theta}\Lambda^{\prime}$ for every
$\theta\in[0,1]$;

  2. (ii)

$f_{\nu}(\alpha)=f_{\nu^{\prime}}(\alpha)$ for all
$\alpha\in(\alpha_{\text{min}},\alpha_{\text{max}})$.

Moreover, if (i), (ii) hold, then both carpets can be defined on the same
grid.

Now assume that $\Lambda$ and $\Lambda^{\prime}$ are defined on the same
$m\times n$ grid to begin with, with parameters
$\\{M_{0},N_{1},\dotsc,N_{M_{0}},R_{1},\dotsc,R_{M_{0}}\\}$ and
$\\{M^{\prime}_{0},N^{\prime}_{1},\dotsc,N^{\prime}_{M^{\prime}_{0}},R^{\prime}_{1},\dotsc,R^{\prime}_{M^{\prime}_{0}}\\}$,
respectively. Denote the corresponding rate functions defined in (5.2.2) by
$I(t)$ and $I^{\prime}(t)$. Let $\underline{t}$ and $\overline{t}$ be as
defined previously, for the carpet $\Lambda$. Let
$(a,b)\subset(\gamma^{-1},1)$ be a (non-empty) open interval. Then each of
(i), (ii) is equivalent to each of the following:

  1. (iii)

$\dim_{\theta}\Lambda=\dim_{\theta}\Lambda^{\prime}$ for every
$\theta\in(a,b)$;

  2. (iv)

$T_{\nu}(q)=T_{\nu^{\prime}}(q)$ for all $q\in\mathbb{R}$.

  3. (v)

$I(t)=I^{\prime}(t-\gamma\log(M^{\prime}/M))$ for all
$t\in(\underline{t},\overline{t})$;

  4. (vi)

$M_{0}=M^{\prime}_{0}$, furthermore,
$N_{\hat{\imath}}/N^{\prime}_{\hat{\imath}}=(R^{\prime}_{\hat{\imath}}/R_{\hat{\imath}})^{\gamma}=(M^{\prime}/M)^{\gamma}$
for all $\hat{\imath}=1,\dotsc,M_{0}$.

We make several comments about Theorem 5.2.5.

######  Remark 5.2.6.

  1. 1.

For carpets defined on the same grid, the equivalence of (ii) and the explicit
condition (vi) was proved by Rao, Yang and Zhang in [RYZ, Theorem 1.2], using
[JR].

  2. 2.

In Step 4 of the proof of Proposition 5.2.11 in Section 5.3, we use scaling
properties of Legendre transforms to establish a direct link between the
multifractal spectrum of the uniform Bernoulli measure and the rate function
$I(t)$. Since $I(t)$ appears in the intermediate dimension formula, this
indicates why the link between the intermediate dimensions and multifractal
spectrum in Theorem 5.2.5 is to be expected.

  3. 3.

For carpets defined on the same grid with $M=M^{\prime}$, (vi) is simply
saying that the column sequence of one carpet is a permutation of the column
sequence of the other.

  4. 4.

Equality of the $L^{q}$ dimensions and the coarse multifractal spectra can be
added to the above equivalences in Theorem 5.2.5 if the carpets are defined on
the same grid, since these quantities can be obtained from the $L^{q}$
spectrum by dividing by $1-q$ or taking the Legendre transform respectively.

  5. 5.

If (iv) holds then other notions of dimension of $\nu$ and $\nu^{\prime}$
which can be deduced from their $L^{q}$ spectra must be equal, such as exact
(Hausdorff/packing/entropy) dimension, correlation dimension (Rényi entropy),
Frostman dimension, and box dimension (in the sense of [FFK]).

  6. 6.

The formulae in [Frac] and (vi) can be used to show that equality of
intermediate dimensions implies equality of other notions of dimensions of
sets such as packing, Assouad, quasi-Assouad, lower, quasi-lower or modified
lower dimensions, or the Assouad spectrum or lower spectrum for any fixed
$\theta\in(0,1)$.

  7. 7.

Since the multifractal spectrum is analytic (as the Legendre transform of an
analytic function), if $I\subseteq(\alpha_{\text{min}},\alpha_{\text{max}})$
is an open interval, then (ii) holds for all $\alpha\in I$ if and only if it
holds for all $\alpha\in(\alpha_{\text{min}},\alpha_{\text{max}})$. Similarly,
if $J\subseteq(\underline{t},\overline{t})$ is an open interval then (v) holds
for all $t\in J$ if and only if it holds for all
$t\in(\underline{t},\overline{t})$.

######  Question 5.2.7.

In the statement of Theorem 5.2.5, can $(a,b)$ be taken to be an arbitrary
non-empty open subinterval of $(0,1)$?

Since the proof strategy of Lemma 5.5.3 on page 5.5.3 does not seem to work
under the assumption that the $L^{q}$ spectra are equal, we ask the following
question.

######  Question 5.2.8.

Do there exist two Bedford–McMullen carpets with non-uniform fibres which
cannot be realised on the same grid but whose uniform Bernoulli measures have
the same $L^{q}$ spectra?

Turning now to bi-Lipschitz equivalence, recall that two metric spaces
$(X,d_{X})$ and $(Y,d_{Y})$ are _bi-Lipschitz equivalent_ if there is a bi-
Lipschitz map $f\colon X\to Y$. In our setting $X$ and $Y$ are two
Bedford–McMullen carpets with the Euclidean distance. The following open
problem seems challenging:

######  Question 5.2.9.

Find an explicit necessary and sufficient condition that determines, given two
iterated function systems each generating a Bedford–McMullen carpet, whether
or not the two carpets are bi-Lipschitz equivalent.

Partial progress towards Question 5.2.9 has been made in [LLM, RYZ, YZ], all
of which assume some disconnectivity property. Fraser and Yu [FY] used the
Assouad spectrum to show that $\gamma$ is a bi-Lipschitz invariant within the
class of Bedford–McMullen carpets, a fact which is also evident from observing
the form of the intermediate dimensions. Moreover, the _gap sequence_ of a set
is a topological quantity which has been shown to be bi-Lipschitz invariant
[RRY], and which is known for Bedford–McMullen carpets [MXX, LMR]. Using the
fact that the intermediate dimensions are stable under bi-Lipschitz maps, we
obtain the following necessary condition for bi-Lipschitz equivalence as an
immediate corollary of Theorem 5.2.5.

######  Corollary 5.2.10.

Let $\Lambda$ and $\Lambda^{\prime}$ be two Bedford–McMullen carpets with non-
uniform fibres which are bi-Lipschitz equivalent, and let $\nu$ and
$\nu^{\prime}$ be the corresponding uniform Bernoulli measures. Then
$f_{\nu}(\alpha)=f_{\nu^{\prime}}(\alpha)$ for all
$\alpha\in(\alpha_{\text{min}},\alpha_{\text{max}})$ and
$T_{\nu}(q)=T_{\nu^{\prime}}(q)$ for all $q\in\mathbb{R}$, and both carpets
can be defined on the same $m\times n$ grid, on which condition (vi) above
holds.

This strengthens [RYZ, Corollary 1.1], where it is assumed that $\Lambda$ and
$\Lambda^{\prime}$ are totally disconnected and defined on the same grid. In
Proposition 5.2.14, we construct two carpets which we know are not bi-
Lipschitz equivalent by Corollary 5.2.10, but where [RYZ, Corollary 1.1] does
not apply. Corollary 5.2.10 also shows in particular that if two carpets
defined on the same grid with the same number of non-empty columns are bi-
Lipschitz equivalent then the column sequence of one must be a permutation of