# Local Dimensions of Self-similar Measures Satisfying the Finite Neighbour
Condition

Kathryn E. Hare and Alex Rutar  Dept. of Pure Mathematics, University of
Waterloo, Waterloo, Canada N2L 3G1<EMAIL_ADDRESS>Mathematical
Institute, North Haugh, St Andrews, Fife KY16 9SS, Scotland  ar339@st-
andrews.ac.uk

###### Abstract.

We study sets of local dimensions for self-similar measures in $\mathbb{R}$
satisfying the finite neighbour condition, which is formally stronger than the
weak separation condition but satisfied in all known examples. Under a mild
technical assumption, we establish that the set of attainable local dimensions
is a finite union of (possibly singleton) compact intervals. The number of
intervals is bounded above by the number of non-trivial maximal strongly
connected components of a finite directed graph construction depending only on
the governing iterated function system. We also explain how our results allow
computations of the sets of local dimensions in many explicit cases. This
contextualizes and generalizes a vast amount of prior work on sets of local
dimensions for self-similar measures satisfying the weak separation condition.

###### Key words and phrases:

iterated function system, self-similar, local dimension, multifractal
analysis, weak separation condition

###### 2020 Mathematics Subject Classification:

28A80

KEH was supported by NSERC Grant 2016-03719. AR was supported by this grant as
well as EPSRC Grant EP/V520123/1

This paper is in final form and no version of it will be submitted for
publication elsewhere.

###### Contents

  1. 1 Introduction
    1. 1.1 Organization of the paper
    2. 1.2 Some questions
    3. 1.3 Notation
    4. 1.4 Acknowledgements
  2. 2 Graph-directed matrix product systems
    1. 2.1 Basic definitions
    2. 2.2 Irreducible matrix product systems
    3. 2.3 Attainable Lyapunov exponents
    4. 2.4 Density of periodic paths
  3. 3 Iterated function systems and their matrix product systems
    1. 3.1 The transition graph and the finite neighbour condition
    2. 3.2 Transition matrices
    3. 3.3 Maximal loop classes and irreducibility
  4. 4 Sets of local dimensions of self-similar measures
    1. 4.1 Basic results about local dimensions and periodic points
    2. 4.2 Sets of local dimensions for simple and irreducible loop classes
  5. 5 Examples of IFS satisfying the finite neighbour condition
    1. 5.1 Bernoulli convolutions
    2. 5.2 Testud measures
    3. 5.3 Other examples
      1. 5.3.1 Cantor-like measures
      2. 5.3.2 An example of Lau and Wang
      3. 5.3.3 A non-equicontractive finite type example
      4. 5.3.4 An example with the set of lower local dimensions not equal to the set of upper local dimensions
      5. 5.3.5 A Pisot reciprocal Bernoulli convolution with a non-simple non-essential loop class

##  1\. Introduction

A natural question when studying Borel probability measures on the real line,
in particular those which are not absolutely continuous with respect to
Lebesgue measure, is to quantify the singularity of the measure. The Hausdorff
dimension of the measure provides one coarse measurement. A more fine-grained
approach is through the local dimensions of the measure $\mu$ at points $x$ in
its support, namely, the quantities

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

In this paper, we are interested in determining properties of the set of
attainable local dimensions for a given measure.

Our focus is on the invariant measures associated with an iterated function
system (IFS) of similarities on $\mathbb{R}$, also known as self-similar
measures. These measures are simple to describe (see Eq. 3.1 for the
definition), yet exhibit rich and complex behaviour. Historically, such
measures have been of great interest.

Investigation of the sets of local dimensions of self-similar measures is
related to multifractal analysis, in which one studies dimensional properties
of the level sets of the local dimension function. A heuristic relationship,
known as the multifractal formalism [13], implies (when it is satisfied) that
the set of local dimensions is a closed interval. The multifractal formalism
holds if the IFS satisfies the classical open set condition (OSC) and there
are simple formulas for the endpoints of the interval of attainable local
dimensions [1, 20]. But when the OSC fails to hold, the situation is much more
complicated and less is known.

In [15], Hu and Lau discovered that when $\mu$ is the 3-fold convolution of
the classical middle-third Cantor measure, the set of local dimensions of
$\mu$ consists of a closed interval along with an isolated point.
Generalizations of this example were studied in [11, 22], for example, while
Testud [23] gave an example of a Cantor-like measure, but with some of the
similarities in the IFS having negative contraction factors, whose set of
local dimensions was the union of two disjoint (non-trivial) intervals.
Another much studied family of self-similar measures which fail the OSC are
the Bernoulli convolutions. These are the measures associated with the IFS
$\\{\rho x,\rho x+1-\rho\\}$ where $1/2<\rho<1$. (See [24] for more background
on Bernoulli convolutions.) It was shown by Feng [4] that when $\rho$ is the
reciprocal of a simple Pisot number, such as the Golden mean, the set of local
dimensions of the corresponding uniform Bernoulli convolution is, again, a
closed interval. However, all biased Bernoulli convolutions (regardless of the
choice of $\rho$) and unbiased Bernoulli convolutions with contraction ratio
greater than the reciprocal of the Golden mean have an isolated point in their
set of local dimensions [8]. We refer the reader to Section 5 for more
discussion on these important examples.

Convolutions of the middle-third Cantor measure and the Bernoulli convolutions
with contraction factor the reciprocal of a Pisot number are all examples of
self-similar measures associated with IFSs that satisfy the weak separation
condition (WSC) [16]. This separation condition is similar to the open set
condition but allows exact overlaps [25]. For such measures, the second author
recently established the existence of a directed transition graph that encodes
the local behaviour of the measure, and related the multifractal analysis of
the measure with connectivity properties of the graph [21]. One corollary of
this earlier work is that when the transition graph is strongly connected, the
set of attainable local dimensions of the measure is a closed interval.

In this paper, we significantly extend this local dimension result beyond the
strongly connected case to obtain a more thorough understanding of sets of
attainable local dimensions. We specialize slightly to the case where the
transition graph is finite, which we call the finite neighbour condition. This
separation condition is closely related to the generalized finite type
condition defined by Lau and Ngai [17]. The finite neighbour condition is
equivalent to the weak separation condition when the support of the measure is
an interval [12]. Our main contribution is to establish under the finite
neighbour condition, and a weak technical assumption, that the set of local
dimensions is a finite union of (possibly singleton) intervals. Moreover, the
number of intervals is bounded above by the number of non-trivial maximal
strongly connected components of the transition graph.

Our research generalizes and contextualizes the prior analysis of sets of
local dimensions for overlapping iterated function systems satisfying the weak
separation condition. We should emphasize that, in contrast with much of the
earlier work on this problem, we do not require the IFS to have similarities
with commensurable contraction factors. Moreover, we are not aware of any
examples of self-similar measures in $\mathbb{R}$, satisfying the weak
separation condition, to which our results do not apply.

###  1.1. Organization of the paper

The main content of the paper is separated into two conceptual components:
analysis of a graph-theoretic symbolic case, and specialization to self-
similar measures.

First, in Section 2, we introduce a general weighted matrix product system.
This symbolic formalism can be thought of as a weighted generalization of the
matrix-valued functions on shift space studied by past authors [2, 5, 6].
Under an irreducibility hypothesis similar to [5], and using modified versions
of the techniques contained therein, we establish in Theorem 2.10 that the
corresponding sets of Lyapunov exponents form a closed interval. We also
establish in Proposition 2.13 the density of Lyapunov exponents at special
types of paths for which local dimension computations are particularly
straightforward; this is useful in the computation of sets of local dimensions
for specific examples.

In Section 3, we review the details of the transition graph construction from
[21] with a particular focus on self-similar measures $\mu$ on $\mathbb{R}$
that satisfy the finite neighbour condition (see Definition 3.6). This
construction establishes the existence of a finite directed graph such that
infinite paths in the graph correspond (almost) injectively to points in the
support of $\mu$. In fact, this directed graph construction is our motivation
for studying the general matrix product systems. The $\mu$-measure of a rich
set of intervals (generating the topology on the support of $\mu$) is
determined by products of non-negative matrices. The weights in the matrix
product system allow us to handle non-equicontractive IFS.

Then, in Section 4, we apply the results from the symbolic case to the study
of the sets of local dimensions for these measures. The relevant transition
graph can be decomposed into finitely many non-trivial strongly connected
components, which we refer to as maximal loop classes. Any infinite path in
the graph is eventually in exactly one maximal loop class, so maximal loop
classes correspond to particular subsets of the support of $\mu$. Under a
technical assumption - that each maximal loop class satisfies either a
simplicity or irreducibility hypothesis (see Definition 3.10) - we relate the
local dimensions at points corresponding to a maximal loop class to the
Lyapunov exponents of the associated matrix product system. This allows us to
establish in Corollary 4.8 that the set of local dimensions at points
corresponding to such a maximal loop class forms a closed interval.
Consequently, in Corollary 4.11 we deduce that the set of attainable local
dimensions of the measure is a finite union of intervals, some of which could
be degenerate, with the number of intervals bounded above by the number of
maximal loop classes. The same results hold for upper local dimensions as
well.

Lastly, in Section 5, we illustrate these ideas with examples, including those
mentioned above.

###  1.2. Some questions

  1. (1)

We do not know if every self-similar measure in $\mathbb{R}$ that satisfies
the weak separation condition also satisfies our formally stronger finite
neighbour condition, or if every measure satisfying the finite neighbour
condition satisfies the required technical assumption. If not, it would be of
interest to extend the analysis.

  2. (2)

Our results establish that the sets of local dimensions and sets of upper
local dimensions coincide. However, the set of lower local dimensions can be
different, as seen in Remark 4.9. In that example, the set of lower local
dimensions is still, however, a finite union of intervals corresponding to
maximal loop classes. It is of interest to determine if similar results hold
for sets of lower local dimensions.

###  1.3. Notation

The reals $\operatorname{{\mathbb{R}}}$ are a metric space with the usual
Euclidean metric, and $\operatorname{{\mathbb{N}}}$ is the set of natural
numbers beginning at 1. The set $B(x,r)$ is a closed ball centred at $x$ with
ratius $r$. Given a set $E\subseteq\operatorname{{\mathbb{R}}}$, we write
$\operatorname{diam}(E)=\sup\\{|x-y|:x,y\in E\\}$.

Given a set $X$, we write $\\#X$ to denote the cardinality of $X$. Given two
real-valued functions $f(z),g(z)$ defined on some index set $Z$, we write
$f\succcurlyeq g$ (resp. $f\preccurlyeq g$) if there exists some $c>0$ such
that $f(z)\geq cg(z)$ (resp. $f(z)\leq cg(z)$) for each $z\in Z$. We say
$f\asymp g$ if $f\succcurlyeq g$ and $f\preccurlyeq g$.

If $M$ is a square matrix, we denote by $\operatorname{sp}M$ the spectral
radius of $M$. All matrices in this document are non-negative.

###  1.4. Acknowledgements

The authors would like to thank K. G. Hare for many helpful conversations.

##  2\. Graph-directed matrix product systems

###  2.1. Basic definitions

Let $G$ be a finite directed graph with vertex set $V(G)$ and edge set $E(G)$.
We will assume that $G$ is _strongly connected_ , which means that there is a
directed path connecting any two vertices. Each vertex $v$ has a _dimension_
$d(v)\in\operatorname{{\mathbb{N}}}$, and to each edge $e=(v_{1},v_{2})$ we
associate a non-negative $d(v_{1})\times d(v_{2})$ _transition matrix_ $T(e)$
and a _weight_ $W(e)\in(0,1)$. We let

$d_{\max}=\max_{v\in G}d(v).$

Let $\Sigma$ denote the set of all infinite paths $(e_{i})_{i=1}^{\infty}$ in
$G$ and let $\Sigma^{*}$ denote the set of all finite paths in $G$. A path is
a _cycle_ if it begins and ends at the same vertex. The _length_ of a finite
path is the number of edges it contains. We say a path $\eta\in\Sigma^{*}$ is
a _prefix_ of a (finite or infinite) path $\gamma$ if
$\gamma=\eta\gamma^{\prime}$ for some path $\gamma^{\prime}$. Given
$\gamma=(e_{n})_{n=1}^{\infty}\in\Sigma$, we write
$\gamma|n=(e_{1},\ldots,e_{n})\in\Sigma^{*}$ to denote the unique prefix of
length $n$.

Given $\eta=(e_{1},\ldots,e_{n})\in\Sigma^{*}$, we write

$W(\eta)=W(e_{1})\cdots W(e_{n})$

and if $\eta\in\Sigma^{*}$ has length at least 1,
$\eta^{-}=(e_{1},\ldots,e_{n-1})$. For convenience, let

$\displaystyle W_{\min}$ $\displaystyle=\min\\{W(e):e\in E(G)\\}>0,$
$\displaystyle W_{\max}$ $\displaystyle=\max\\{W(e):e\in E(G)\\}<1.$

If $\eta$ is the empty path, we say $W(\eta)=1$. Similarly, we write

$T(\eta)=T(e_{1})\cdots T(e_{n})\text{ for
}\eta=(e_{1},\ldots,e_{n})\in\Sigma^{*}.$

We equip $\Sigma$ with the topology induced by the metric

$d(\gamma,\xi)=\inf\\{W(\eta):\eta\text{ a prefix of }\gamma\text{ and
}\xi\\}.$

With this topology, $\Sigma$ is a compact totally disconnected metric space.

We refer to this data as a _graph-directed matrix product system_ or, in
short, a _matrix product system_. Typically, we will denote this by
$\mathcal{G}$.

######  Definition 2.1.

Given an infinite path $\gamma=(e_{j})_{j=1}^{\infty}\in\Sigma$, we define the
_lower Lyapunov exponent_ by

$\underline{\lambda}(\mathcal{G},\gamma)=\liminf_{n\to\infty}\frac{\log\left\lVert
T(e_{1})\cdots T(e_{n})\right\rVert}{\log W(e_{1})\cdots W(e_{n})}.$

The _upper Lyapunov exponent_ is defined similarly; when the values coincide,
we call this value the _Lyapunov exponent_ of the path $\gamma$, and denote it
by $\lambda(\mathcal{G},\gamma)$. Typically, we omit writing $\mathcal{G}$
when it is clear from the context.

For any $t>0$, denote

$\Sigma_{t}=\\{\eta\in\Sigma^{*}:W(\eta)<t\leq W(\eta^{-}),\left\lVert
T(\eta)\right\rVert>0\\},$

which is the set of paths with non-zero transition matrix and weight
approximately $t$.

###  2.2. Irreducible matrix product systems

It is clear that the geometric properties of the metric space $\Sigma$ are
determined completely from the edge weights. However, in order to say
meaningful things about products of matrices and Lyapunov exponents, we
require a stronger form of irreducibility than the graph $G$ being strongly
connected.

######  Definition 2.2.

We say that the matrix product system is _irreducible_ if there exists a
finite family of paths $\mathcal{H}\subset\Sigma^{*}$ such that for any
vertices $v_{1},v_{2}$, $1\leq i\leq d(v_{1})$, and $1\leq j\leq d(v_{2})$,
there exists a path $\gamma\in\mathcal{H}$ from vertex $v_{1}$ to $v_{2}$ such
that $T(\gamma)_{i,j}>0$.

######  Remark 2.3.

Equivalently, for each $1\leq i,j\leq m=\\#V(G)$, define $M_{i,j}=T(e)$ if
there is an edge $e$ from vertex $v_{i}$ to $v_{j}$, and let $M_{i,j}=0$
otherwise. The matrix product system is irreducible if and only if the block
matrix

$M=\begin{pmatrix}M_{1,1}&\cdots&M_{1,m}\\\ \vdots&\ddots&\vdots\\\
M_{m,1}&\cdots&M_{m,m}\end{pmatrix}$

is irreducible, i.e. there exists some $r>0$ such that $\sum_{k=1}^{r}M^{k}$
is a strictly positive matrix.

Of course, irreducible systems are necessarily strongly connected.

######  Remark 2.4.

Our irreducibility criterion is very similar to the one assumed by Feng [5].
However, since our weights depend on the edge rather than the source vertex,
we find it more natural to speak of infinite paths in a graph rather than
words in a sequence space. One may equivalently think of the graph as a
subshift of finite type determined by a weighted adjacency matric.

For the remainder of this section, unless otherwise stated, our matrix product
system is irreducible.

Irreducibility is essential for obtaining the following estimates, which we
will use frequently.

######  Lemma 2.5.

There are constants $A,B>0$ such that for any paths
$\eta_{1},\eta_{2}\in\Sigma^{*}$, there exists some $\gamma\in\mathcal{H}$
such that $\eta_{1}\gamma\eta_{2}$ is a path and

$A\left\lVert T(\eta_{1})\right\rVert\left\lVert
T(\eta_{2})\right\rVert\leq\left\lVert
T(\eta_{1}\gamma\eta_{2})\right\rVert\leq B\left\lVert
T(\eta_{1})\right\rVert\left\lVert T(\eta_{2})\right\rVert$

###### Proof.

By the irreducibility assumption, for any $v_{1},v_{2}\in V(G)$, $1\leq i\leq
d(v_{1})$, and $1\leq j\leq d(v_{2})$, there exists a path
$\gamma=\gamma(v_{1},v_{2},i,j)\in\mathcal{H}$ from $v_{1}$ to $v_{2}$ such
that $T(\gamma)_{i,j}>0$. Let

$\displaystyle C$
$\displaystyle=\min\\{T(\gamma(v_{1},v_{2},i,j))_{i,j}:v_{1},v_{2}\in
V(G),1\leq i\leq d(v_{1}),1\leq j\leq d(v_{2})\\}.$

If $\eta_{1},\eta_{2}$ are arbitrary paths, by the pidgeonhole principle,
there exists some $k,i,j,\ell$ such that
$d_{\max}^{2}T(\eta_{1})_{k,i}\geq\left\lVert T(\eta_{1})\right\rVert$ and
$d_{\max}^{2}T(\eta_{2})_{j,\ell}\geq\left\lVert T(\eta_{2})\right\rVert$.
Assume $\eta_{1}$ ends at vertex $v_{1}$, $\eta_{2}$ begins at vertex $v_{2}$,
and take $\gamma=\gamma(v_{1},v_{2},i,j)\in\mathcal{H}$. Then
$\eta_{1}\gamma\eta_{2}$ is a path and

$\left\lVert T(\eta_{1}\gamma\eta_{2})\right\rVert\geq
T(\eta_{1})_{k,i}T(\gamma)_{i,j}T(\eta_{2})_{j,\ell}\geq\frac{C}{d_{\max}^{4}}\left\lVert
T(\eta_{1})\right\rVert\left\lVert T(\eta_{2})\right\rVert.$

The lower bound follows by taking $A=C/d_{\max}^{4}$.

To obtain the upper bound, we simply note that

$\left\lVert T(\eta_{1}\gamma\eta_{2})\right\rVert\leq\left\lVert
T(\eta_{1})\right\rVert\left\lVert T(\gamma)\right\rVert\left\lVert
T(\eta_{2})\right\rVert$

and it suffices to take $B=\max\\{\left\lVert
T(\gamma)\right\rVert:\gamma\in\mathcal{H}\\}$. ∎

In the following lemma, we do not formally need the irreducibility hypothesis:
it suffices to know that if $\eta$ is any path in $\mathcal{G}$, then
$T(\eta)$ is not the zero matrix.

######  Lemma 2.6.

There are constants $A,r>0$ such that for any $t_{1},t_{2}\in(0,1)$,
$\eta_{1}\in\Sigma_{t_{1}}$, and $\eta_{2}\in\Sigma_{t_{2}}$, there are paths
$\phi$ and $\psi$ such that $\eta_{1}\phi\eta_{2}\psi\in\Sigma_{t_{1}t_{2}r}$
and

$\left\lVert T(\eta_{1}\phi\eta_{2}\psi)\right\rVert\leq A\left\lVert
T(\eta_{1})\right\rVert\left\lVert T(\eta_{2})\right\rVert.$

###### Proof.

Take $r=\min\\{W(\eta):\eta\in\mathcal{H}\\}$. By the irreducibility
hypothesis, there exists some $\phi\in\mathcal{H}$ such that $\left\lVert
T(\eta_{1}\phi\eta_{2})\right\rVert>0$.

Moreover, for any path $\eta$ with $\left\lVert T(\eta)\right\rVert>0$, there
exists an edge $e$ such that $\eta e$ is a path and $\left\lVert T(\eta
e)\right\rVert>0$. Since $t_{1}t_{2}W_{\min}^{-2}\geq
W(\eta_{1}\phi\eta_{2})\geq t_{1}t_{2}r$, repeatedly applying this
observation, there exists $\psi$ such that
$\eta_{1}\phi\eta_{2}\psi\in\Sigma_{t_{1}t_{2}r}$. Note that $W(\psi)\geq
rW_{\min}^{2}$. Thus $\left\lVert T(\eta_{1}\phi\eta_{2}\psi)\right\rVert\leq
A\left\lVert T(\eta_{1})\right\rVert\left\lVert T(\eta_{2})\right\rVert$ where

$A=\max\\{\left\lVert T(\eta)\right\rVert:\eta\in E(\mathcal{G}),W(\eta)\geq
rW_{\min}^{2}\\}\cdot\max\\{\left\lVert
T(\eta):\eta\in\mathcal{H}\right\rVert\\}$

as required. ∎

######  Lemma 2.7.

There are constants $A,B>0$ such that for any path $\eta\in\Sigma^{*}$, there
exists some $\phi\in\mathcal{H}$ such that $\eta\phi$ is a cycle and

$A\left\lVert T(\eta)\right\rVert\leq\operatorname{sp}T(\eta\phi)\leq
B\left\lVert T(\eta)\right\rVert.$

###### Proof.

Let $C$ be the minimal strictly positive coefficient of any $T(\phi)$ for
$\phi\in\mathcal{H}$. Suppose $T(\eta)_{i,\ell}$ is the maximal coordinate of
$T(\eta)$. Get $\phi\in\mathcal{H}$ such that $\eta\phi$ is a cycle and
$T(\phi)_{\ell,i}\geq C>0$. Then

$\displaystyle\operatorname{Tr}T(\eta\phi)$ $\displaystyle\geq
T(\eta\phi)_{i,i}\geq T(\eta)_{i,\ell}T(\phi)_{\ell,i}\geq\left\lVert
T(\eta)\right\rVert\frac{C}{d_{\max}^{2}}.$

Since the trace of a matrix is the sum of its eigenvalues,

$\operatorname{Tr}(T(\eta\phi))\leq d_{\max}\operatorname{sp}T(\eta\phi).$

Thus with $A=C/d_{\max}^{3}$, we have $\operatorname{sp}(T(\eta\phi))\geq
A\left\lVert T(\eta)\right\rVert$.

Conversely, we have

$\operatorname{sp}(T(\eta\phi))\leq\left\lVert
T(\eta\phi)\right\rVert\leq\left\lVert T(\eta)\right\rVert\left\lVert
T(\phi)\right\rVert\leq B\left\lVert T(\eta)\right\rVert$

where $B=\max\\{\left\lVert T(\gamma)\right\rVert:\gamma\in\mathcal{H}\\}$. ∎

###  2.3. Attainable Lyapunov exponents

The main goal of this subsection is to determine the possible values of
Lyapunov exponents of paths in the matrix product system.

We begin with notation. Put

(2.1) $\displaystyle\alpha_{\min}(\mathcal{G})=\alpha_{\min}$
$\displaystyle:=\lim_{t\to 0}\min_{\eta\in\Sigma_{t}}\frac{\log\left\lVert
T(\eta)\right\rVert}{\log t},$
$\displaystyle\alpha_{\max}(\mathcal{G})=\alpha_{\max}$
$\displaystyle:=\lim_{t\to 0}\max_{\eta\in\Sigma_{t}}\frac{\log\left\lVert
T(\eta)\right\rVert}{\log t}.$

We will first show that $\alpha_{\min}$ and $\alpha_{\max}$ are well defined
and take real values. This will use the following standard submultiplicativity
result, which is a slightly modified version of, for example, [14, Thm.
7.6.1].

######  Lemma 2.8.

Let $f:(0,1)\to\operatorname{{\mathbb{R}}}$ be measurable and suppose there
exists $c>0$ and $r>0$ such that $f(t_{1}t_{2}r)\leq c\cdot f(t_{1})f(t_{2})$
for all $t_{1},t_{2}\in(0,1)$. Then

$\lim_{t\to 0}\frac{\log f(t)}{\log t}=\inf_{t>0}\frac{\log f(t)}{\log t}.$

Note the similarity of the following lemma with [5, Lem. 2.3].

######  Lemma 2.9.

The limits defining $\alpha_{\min}$ and $\alpha_{\max}$ exist and take real
values.

###### Proof.

We will first prove that the limit

$\lim_{t\to 0}\min_{\eta\in\Sigma_{t}}\frac{\log\left\lVert
T(\eta)\right\rVert}{\log t}=\lim_{t\to
0}\frac{\log\max_{\eta\in\Sigma_{t}}\left\lVert T(\eta)\right\rVert}{\log t}$

exists. Set $f(t)=\max_{\eta\in\Sigma_{t}}\left\lVert T(\eta)\right\rVert$.
Let $t_{1},t_{2}>0$ be arbitrary. If $\eta\in\Sigma_{t_{1}t_{2}W_{\min}}$, we
may write $\eta=\eta_{1}\eta_{2}\gamma$ where $\eta_{1}\in\Sigma_{t_{1}}$ and
$\eta_{2}\in\Sigma_{t_{2}}$ and $W(\gamma)\geq W_{\min}^{2}$. In particular,
with $c=\max\\{\left\lVert T(\psi)\right\rVert:W(\psi)\geq W_{\min}^{2}\\}$,
we have $\left\lVert T(\eta)\right\rVert\leq c\left\lVert
T(\eta_{1})\right\rVert\left\lVert T(\eta_{2})\right\rVert$ and therefore
$f(t_{1}t_{2}W_{\min})\leq c\cdot f(t_{1})f(t_{2})$. Applying Lemma 2.8 with
$c$ as above and $r=W_{\min}$, we have our desired result.

We now show that $\lim_{t\to 0}\max_{\eta\in\Sigma_{t}}\frac{\log
T(\eta)}{\log t}$ exists. Set $g(t)=\min_{\eta\in\Sigma_{t}}\left\lVert
T(\eta)\right\rVert$. Let $t_{1},t_{2}>0$ and let $\eta_{1}\in\Sigma_{t_{1}}$
and $\eta_{2}\in\Sigma_{t_{2}}$ be arbitrary. Note that $\eta_{1}\eta_{2}$
need not be a path, and even if it were, it need not hold that
$\eta_{1}\eta_{2}\in\Sigma_{t_{1}t_{2}}$. By Lemma 2.6, there exists some
$c,r>0$ (not depending on $\eta_{1}$ and $\eta_{2}$) and paths $\phi$ and
$\psi$ such that $\eta_{1}\phi\eta_{2}\psi$ is an admissible path in
$\Sigma_{rt_{1}t_{2}}$ and

(2.2) $g(rt_{1}t_{2})\leq c\left\lVert T(\eta_{1})\right\rVert\left\lVert
T(\eta_{2})\right\rVert.$

Now taking the minimum over all $\eta_{1}\in\Sigma_{t_{1}}$ and
$\eta_{2}\in\Sigma_{t_{2}}$ yields $g(rt_{1}t_{2})\leq cg(t_{1})g(t_{2})$.
Thus $g$ satisfies Lemma 2.8.

To see that $\alpha_{\min},\alpha_{\max}\in\operatorname{{\mathbb{R}}}$, let
$a$ be the smallest strictly positive entry in any $T(e)$ for $e\in E(G)$. Let
$b=\min\\{\left\lVert T(e)\right\rVert:e\in E(G)\\}$. Then if
$\eta\in\Sigma_{t}$ is any path of length $n$, we have that

$\frac{\log b}{\log W_{\min}}\leq\frac{\log\left\lVert
T(\eta)\right\rVert}{\log t}\leq\frac{n\log a}{(n-1)\log W_{\max}}$

so that $\alpha_{\min},\alpha_{\max}$ are real-valued. ∎

Of course, if $\eta\in\Sigma_{t}$, then $W(\eta)\asymp t$. Consequently,

$\alpha_{\min}=\lim_{t\to 0}\min_{\eta\in\Sigma_{t}}\frac{\log\left\lVert
T(\eta)\right\rVert}{\log W(\eta)}\text{ and }\alpha_{\max}=\lim_{t\to
0}\max_{\eta\in\Sigma_{t}}\frac{\log\left\lVert T(\eta)\right\rVert}{\log
W(\eta)}.$

We are now ready to prove the following result about the set of attainable
Lyapunov exponents. We remind the reader that the Lyapunov exponent of the
path $\gamma$, $\lambda(\gamma)$, was defined in Definition 2.1.

Our proof follows [5, Lem. 2.3 and Prop. 3.2].

######  Theorem 2.10.

Let $\mathcal{G}$ be a matrix product system satisfying the irreducibility
hypothesis.

  1. (1)

If $\gamma\in\Sigma$ is any path, then
$\underline{\lambda}(\gamma),\overline{\lambda}(\gamma)\in[\alpha_{\min},\alpha_{\max}]$.

  2. (2)

For any $\alpha\in[\alpha_{\min},\alpha_{\max}]$ and $\xi\in\Sigma^{*}$ with
$T(\xi)$ non-zero, there exists some $\gamma=(e_{m})_{m=1}^{\infty}\in\Sigma$
and a sequence $(m_{j})_{j=1}^{\infty}$ such that $\lambda(\gamma)=\alpha$,
$\lim_{j\to\infty}m_{j+1}/m_{j}=1$, and for each
$j\in\operatorname{{\mathbb{N}}}$, $\xi$ is a prefix of
$(e_{m_{j}},e_{m_{j}+1},\ldots)$.

###### Proof.

For (i), if $\gamma\in\Sigma$ is arbitrary, then

$\underline{\lambda}(\gamma)=\lim_{k\to\infty}\frac{\log\left\lVert
T(\gamma|{n_{k}})\right\rVert}{W(\gamma|{n_{k}})}$

for some subsequence $n_{k}$. But if $\gamma|n_{k}\in\Sigma_{t_{n_{k}}}$, then
$W(\gamma|{n_{k}})\asymp t_{n_{k}}$ and

$\alpha_{\min}\leq\lim_{k\to\infty}\frac{\log\left\lVert
T(\gamma|{n_{k}})\right\rVert}{\log t_{n_{k}}}\leq\alpha_{\max}$

from the existence of the limits defining $\alpha_{\min}$ and $\alpha_{\max}$.
The upper Lyapunov exponent result is identical, giving (i).

Now for (ii), given $\alpha\in[\alpha_{\min},\alpha_{\max}]$, let $s\in[0,1]$
be such that $\alpha=s\alpha_{\min}+(1-s)\alpha_{\max}$. For each
$n\in\operatorname{{\mathbb{N}}}$, choose
$\phi_{n},\psi_{n}\in\Sigma_{2^{-n}}$ with the property that

$\frac{\log\left\lVert T(\phi_{n})\right\rVert}{\log
W(\phi_{n})}=u_{n}\rightarrow\alpha_{\min}\text{ and }\frac{\log\left\lVert
T(\psi_{n})\right\rVert}{\log W(\psi_{n})}=v_{n}\rightarrow\alpha_{\max}.$

Let $\\{A_{n}\\}_{n=1}^{\infty}$, $\\{B_{n}\\}_{n=1}^{\infty}$ be sequences of
natural numbers given by

$\displaystyle A_{n}$ $\displaystyle=[sn]$ $\displaystyle B_{n}$
$\displaystyle=[(1-s)n],$

where $[x]$ denotes the integer part of $x$. Then define a sequence by

$\underbrace{\phi_{1},\ldots,\phi_{1}}_{A_{1}},\underbrace{\psi_{1},\ldots,\psi_{1}}_{B_{1}},\ldots,\underbrace{\phi_{n},\ldots,\phi_{n}}_{A_{n}},\underbrace{\psi_{n},\ldots,\psi_{n}}_{B_{n}},\ldots$

and relabel it $\\{\eta_{n}\\}_{n=1}^{\infty}$, i.e. $\eta_{1}=\phi_{1}$,
$\eta_{A_{1}}=\phi_{1}$, $\eta_{A_{1}+1}=\psi_{1}$, etc.

Now since $\xi\in\Sigma^{*}$ has $T(\xi)$ non-zero, by repeatedly applying
Lemma 2.5, there are constants $C_{1},C_{2}>0$ such that for each
$n\in\operatorname{{\mathbb{N}}}$ there are paths
$\nu_{n}^{(1)},\nu_{n}^{(2)}$ in $\mathcal{H}$ such that with
$\nu_{n}:=\nu_{n}^{(1)}\xi\nu_{n}^{(2)}$,

$\gamma:=(\eta_{1},\nu_{1},\eta_{2},\nu_{2},\ldots)$

is an infinite path and

(2.3) $C_{1}^{m}\prod_{i=1}^{m}\left\lVert
T(\eta_{i})\right\rVert\leq\left\lVert
T(\eta_{1}\nu_{1}\ldots\eta_{m}\nu_{m})\right\rVert\leq
C_{2}^{m}\prod_{i=1}^{m}\left\lVert T(\eta_{i})\right\rVert.$

Since $\mathcal{H}$ is a finite set, there also exists $D_{1},D_{2}>0$ such
that

(2.4) $D_{1}^{m}\prod_{i=1}^{m}W(\eta_{i})\leq
W(\eta_{1}\nu_{1}\ldots\eta_{m}\nu_{m})\leq
D_{2}^{m}\prod_{i=1}^{m}W(\eta_{i}).$

For notation, let $\\{n_{k}\\}_{k=1}^{\infty}$ be the indices such that
$n_{k}$ is the index of the edge preceding the first edge of $\phi_{k+1}$ in
repetition $A_{k+1}$.

Let $(m_{j})_{j=1}^{\infty}$ be the sequence of indices such that $\xi$ is a
prefix and fix some $j\in\operatorname{{\mathbb{N}}}$. For any
$\ell\in\operatorname{{\mathbb{N}}}$, since $\phi_{\ell}$ and $\psi_{\ell}$
are in $\Sigma_{2^{-\ell}}$, there exists some $a,b>0$ such that
$a\ell\leq|\phi_{\ell}|\leq b\ell$ where $|\phi_{\ell}|$ is the number of
edges in $\phi_{\ell}$. Moreover, $(\gamma_{m_{j}},\gamma_{m_{j}+1},\ldots)$
has prefix $\xi\zeta_{1}\phi_{\ell}\zeta_{2}\xi$ or
$\xi\zeta_{1}\psi_{\ell}\zeta_{2}\xi$ where $\ell$ is chosen suitably and
$\zeta_{1},\zeta_{2}\in\mathcal{H}$ have bounded length. Thus there exists
some $M>0$ such that $m_{j+1}-m_{j}\leq M+b\ell$. On the other hand, it always
holds that $m_{j}\geq\sum_{i=1}^{\ell-1}ai$. It follows that
$\lim_{j\to\infty}m_{j+1}/m_{j}=1$, as claimed

We now prove that $\overline{\lambda}(\gamma)\leq\alpha$; the lower bound
$\underline{\lambda}(\gamma)\geq\alpha$ will follow by a similar argument. To
this end, let $n$ be a large number of edges and let $k$ be maximal such that
$n\geq n_{k}$ (that is, $k$ is the maximal number of completed blocks
$A_{i},B_{i}$ which occur before edge $n$). There exist constants
$C_{3},C_{4}>0$ such that

$C_{3}^{n_{k+1}-n_{k}}\left\lVert
T(\gamma_{1}\ldots\gamma_{n_{k+1}})\right\rVert\leq\left\lVert
T(\gamma|n)\right\rVert\leq C_{4}^{n_{k+1}-n_{k}}\left\lVert
T(\gamma_{1}\ldots\gamma_{n_{k}})\right\rVert$

and

$W(\gamma_{1}\ldots\gamma_{n_{k+1}})\leq W(\gamma|n)\leq
W(\gamma_{1}\ldots\gamma_{n_{k}}).$

Since the number of edges contained in $\gamma|n$ is at least
$\sum_{i=1}^{k}(A_{i}+B_{i})$ and at most $\sum_{i=1}^{k+1}(A_{i}+B_{i})$, we
deduce from Eq. 2.3 and Eq. 2.4 that

$\displaystyle\frac{\log\left\lVert T(\gamma|n)\right\rVert}{\log
W(\gamma|n)}$ $\displaystyle\qquad\leq\frac{(n_{k+1}-n_{k})\log
C_{3}+\sum_{i=1}^{k+1}\bigl{(}(A_{i}+B_{i})\log C_{1}+A_{i}\log\left\lVert
T(\phi_{i})\right\rVert+B_{i}\log\left\lVert
T(\psi_{i})\right\rVert\bigr{)}}{\sum_{i=1}^{k}\bigl{(}(A_{i}+B_{i})\log
D_{2}+A_{i}\log W(\phi_{i})+B_{i}\log W(\psi_{i})\bigr{)}}.$

Since each $\phi_{i},\psi_{i}\in\Sigma_{t_{2^{-i}}}$, we have
$W(\phi_{i})\asymp W(\psi_{i})\asymp 2^{-i}$. Recall, also, that
$A_{i},B_{i}\asymp i$. Therefore

$\Bigl{\lvert}\frac{\sum_{i=1}^{k+1}(A_{i}+B_{i})\log
C_{1}}{\sum_{i=1}^{k}(A_{i}\log W(\phi_{i})+B_{i}\log
W(\psi_{i}))}\Bigr{\rvert}\preccurlyeq\frac{\sum_{i=1}^{k+1}i}{\sum_{i=1}^{k}i^{2}}\preccurlyeq\frac{1}{k}\rightarrow
0$

and a similar statement holds with the numerator replaced by
$\sum_{i=1}^{k}(A_{i}+B_{i})\log D_{2}$. Moreover, since $(n_{k+1}-n_{k})\log
C_{3}\asymp(k+1)(A_{k+1}+B_{k+1})$, we also have

$\Bigl{\lvert}\frac{(n_{k+1}-n_{k})\log C_{3}}{\sum_{i=1}^{k}(A_{i}\log
W(\phi_{i})+B_{i}\log W(\psi_{i}))}\Bigr{\rvert}\rightarrow 0$

We thus have that

$\displaystyle\limsup_{n}\frac{\log\left\lVert T(\gamma|n)\right\rVert}{\log
W(\gamma|n)}$
$\displaystyle\leq\limsup_{k}\frac{\sum_{i=1}^{k+1}(A_{i}\log\left\lVert
T(\phi_{i})\right\rVert+B_{i}\log\left\lVert
T(\psi_{i})\right\rVert)}{\sum_{i=1}^{k}(A_{i}\log W(\phi_{i})+B_{i}\log
W(\psi_{i}))}$ $\displaystyle=\limsup_{k}\frac{\sum_{i=1}^{k+1}(A_{i}u_{i}\log
W(\phi_{i})+B_{i}v_{i}\log W(\psi_{i}))}{\sum_{i=1}^{k}(A_{i}\log
W(\phi_{i})+B_{i}\log W(\psi_{i}))}$
$\displaystyle=\limsup_{k}\frac{\sum_{i=1}^{k+1}(iA_{i}u_{i}+iB_{i}v_{i})}{\sum_{i=1}^{k}(iA_{i}+iB_{i})}$
$\displaystyle=\limsup_{k}\frac{\sum_{i=1}^{k+1}i^{2}(su_{i}+(1-s)v_{i})}{\sum_{i=1}^{k}i^{2}}.$

Fix $\epsilon>0$. Since $\lim_{i\to\infty}(su_{i}+(1-s)v_{i})=\alpha$, for
large enough $N$, $su_{i}+(1-s)v_{i}\leq\alpha+\epsilon$ for all $i\geq N$.
Thus

$\limsup_{n}\frac{\log\left\lVert T(\gamma|n)\right\rVert}{\log
W(\gamma|n)}\leq\limsup_{k}\frac{\sum_{i=N}^{k+1}i^{2}(\alpha+\epsilon)}{\sum_{i=1}^{k}i^{2}}\leq\alpha+\epsilon.$

Similar reasoning shows that

$\liminf_{n}\frac{\log\left\lVert T(\gamma|n)\right\rVert}{\log
W(\gamma|n)}\geq\alpha-\epsilon.$

As $\epsilon>0$ was arbitrary, it follows that $\lambda(\gamma)=\alpha$, as
claimed. ∎

The following result now follows directly from Theorem 2.10.

######  Corollary 2.11.

Let $\mathcal{G}$ be an irreducible matrix product system. Then the set of
attainable Lyapunov exponents is the compact interval
$[\alpha_{\min},\alpha_{\max}]$.

###  2.4. Density of periodic paths

An interesting class of paths are the so-called _periodic paths_ , which are
the paths in $\Sigma$ of the form

$\gamma=(\theta,\theta,\ldots)$

where $\theta$ is a cycle. We denote them by $\mathcal{P}$. We refer to
$\theta$ as a _period_ of the path.

The Lyapunov exponent of a periodic path always exists and has a simple
formula.

######  Proposition 2.12.

Let $\gamma$ be a periodic path with period $\theta$. Then the Lyapunov
exponent of $\gamma$ exists and is given by

$\lambda(\gamma)=\frac{\log\operatorname{sp}T(\theta)}{\log W(\theta)}.$

###### Proof.

Assume that $\theta=(\theta_{1},...,\theta_{k})$. For any positive integer $k$
and $j=1,\ldots,k$,

$\left\lVert
T(\theta^{n}\theta_{1}\cdots\theta_{j})\right\rVert\leq\left\lVert
T(\theta^{n})\right\rVert\left\lVert
T(\theta_{1}\cdots\theta_{j})\right\rVert$

and

$\left\lVert T(\theta^{n+1})\right\rVert\leq\left\lVert
T(\theta^{n}\theta_{1}\cdots\theta_{j})\right\rVert\left\lVert
T(\theta_{j+1}\cdots\theta_{k})\right\rVert.$

Consequently, there is some $A,B>0$, depending only on $\theta$, such that

$A\left\lVert T(\theta^{n+1})\right\rVert\leq\left\lVert
T(\theta^{n}\theta_{1}\cdots\theta_{j})\right\rVert\leq B\left\lVert
T(\theta^{n})\right\rVert.$

The result follows directly from the fact that

$\lim_{n\to\infty}\frac{\log\left\lVert
T(\theta^{n})\right\rVert}{n}=\log\operatorname{sp}(T(\theta)).$

∎

######  Proposition 2.13.

The set $\\{\lambda(\gamma):\gamma\in\mathcal{P}\\}$ is dense in
$[\alpha_{\min},\alpha_{\max}]$.

###### Proof.

It suffices to show that if $\gamma\in\Sigma$ is an arbitrary path such that
$\lambda(\gamma)$ exists, there exists a sequence of periodic paths
$\\{y_{n}\\}_{n=1}^{\infty}$ such that
$\lim_{n\to\infty}\lambda(y_{n})=\lambda(\gamma)$.

By Lemma 2.7, there are constants $A,B>0$ such that for any
$k\in\operatorname{{\mathbb{N}}}$, there is a path $\eta_{k}\in\mathcal{H}$
such that $(\gamma|k)\eta_{k}=:\theta_{k}$ is a cycle and

$A\left\lVert T(\gamma|k)\right\rVert\leq\operatorname{sp}T(\theta_{k})\leq
B\left\lVert T(\gamma|k)\right\rVert.$

Let $\overline{\theta}_{k}=(\theta_{k},\theta_{k},\ldots)$. This is a periodic
path with period $\theta_{k}$, so that

$\lambda(\overline{\theta_{k}})=\frac{\log\operatorname{sp}(T(\theta_{k}))}{\log
W(\theta_{k})}$

by Proposition 2.12. Also, $W(\gamma|k)\asymp W(\theta_{k})$. Hence

$\displaystyle\limsup_{k\to\infty}\lambda(\overline{\theta_{k}})$
$\displaystyle\leq\limsup_{k\to\infty}\frac{\log A\left\lVert
T(\gamma|k)\right\rVert}{\log W(\gamma|k)}=\lambda(\gamma)$

and the lower bound follows identically. Thus
$\lim_{k\to\infty}\lambda(\overline{\theta_{k}})=\lambda(\gamma)$ and we have
density, as claimed. ∎

##  3\. Iterated function systems and their matrix product systems

We now turn to studying iterated function systems of similarities. In this
section, we will describe how the dynamics of associated self-similar sets and
measures can be encoded with a matrix product system.

###  3.1. The transition graph and the finite neighbour condition

We begin with notation and terminology. By an iterated function system (IFS),
$\\{S_{i}\\}_{i=1}^{m},$ we mean a finite set of similarities

(3.1) $S_{i}(x)=r_{i}x+d_{i}:\mathbb{R}\rightarrow\mathbb{R}\text{ for each
}i=1,...,m$

with $0<\left|r_{i}\right|<1$ and $m\geq 1$. We say that the IFS is
_equicontractive_ if $r_{1}=\cdots=r_{m}>0$.

Each IFS generates a unique non-empty, compact set $K$ satisfying

$K=\bigcup_{i=1}^{m}S_{i}(K),$

known as the associated _self-similar set_. We will assume $K$ is not a
singleton. By translating the $d_{i}$ as necessary, without loss of generality
we may assume that the convex hull of $K$ is $[0,1]$.

Given probabilities $\mathbf{p}=(p_{i})_{i=1}^{m}$ where $p_{i}>0$ and
$\sum_{i=1}^{m}p_{i}=1$, there exists a unique Borel probability measure
$\mu_{\bm{p}}$ satisfying

(3.2) $\mu_{\bm{p}}(E)=\sum_{i=1}^{m}p_{i}\mu_{\bm{p}}(S_{i}^{-1}(E))$

for any Borel set $E\subseteq K$. This non-atomic measure $\mu_{\bm{p}}$ is
known as an associated self-similar measure and has as its support the self-
similar set $K$.

Given $\sigma=(\sigma_{1},\ldots,\sigma_{j})\in\\{1,...,m\\}^{j}$, we denote

$\sigma^{-}=(\sigma_{1},\ldots,\sigma_{j-1})\text{,
}S_{\sigma}=S_{\sigma_{1}}\circ\cdots\circ S_{\sigma_{j}}\text{ and
}r_{\sigma}=r_{\sigma_{1}}\cdots r_{\sigma_{j}}.$

For $t>0,$ put

$\Lambda_{t}=\\{\sigma:|r_{\sigma}|<t\leq|r_{\sigma^{-}}|\\}.$

The elements of $\Lambda_{t}$ are called the _words of generation $t$_. We
remark that in the literature it is more common to see this defined by the
rule $|r_{\sigma}|\leq t<|r_{\sigma^{-}}|$, but this essentially equivalent
choice is more convenient for our purposes.

The notions of net intervals and neighbour sets were first introduced in [3]
and extended in [12, 21]. We summarize the key ideas here.

Let $h_{1},\ldots,h_{s(t)}$ be the collection of distinct elements of the set
$\\{S_{\sigma}(0),S_{\sigma}(1):\sigma\in\Lambda_{t}\\}$ listed in strictly
ascending order and set

$\mathcal{F}_{t}=\\{[h_{j},h_{j+1}]:1\leq j<s(t)\text{ and
}(h_{j},h_{j+1})\cap K\neq\emptyset\\}.$

The elements of $\mathcal{F}_{t}$ are called the _net intervals of generation_
$t$. Note that $[0,1]$ is the (unique) net interval of any generation $t>1$
and denote by

$\mathcal{F}=\bigcup_{t>0}\mathcal{F}_{t}$

the set of all net intervals.

Given a net interval $\Delta$, we denote by $T_{\Delta}$ the unique similarity
$T_{\Delta}(x)=rx+a$ with $r>0$ such that

$T_{\Delta}([0,1])=\Delta.$

Of course, here $r=\operatorname{diam}(\Delta)$ and $a$ is the left endpoint
of $\Delta$.

######  Definition 3.1.

We will say that a similarity $f(x)=Rx+a$ is a _neighbour_ of
$\Delta\in\mathcal{F}_{t}$ if there exists some $\sigma\in\Lambda_{t}$ such
that $S_{\sigma}(K)\cap\Delta^{\circ}\neq\emptyset$ and
$f=T_{\Delta}^{-1}\circ S_{\sigma}$. In this case, we also say that
$S_{\sigma}$ _generates_ the neighbour $f$.

The _neighbour set_ of $\Delta$ is the maximal set

$\mathcal{V}_{t}(\Delta)=\\{f_{1},\ldots,f_{k}\\}$

where each $f_{i}=T_{\Delta}^{-1}\circ S_{\sigma_{i}}$ is a distinct neighbour
of $\Delta$. We denote by

$R_{\max}(\Delta):=\max\\{|R|:x\mapsto Rx+a\in\mathcal{V}_{t}(\Delta)\\}$

the maximum contraction factor of any neighbour of $\Delta$.

When the generation is implicit, we will often write $\mathcal{V}(\Delta)$.
Since $K=\bigcup_{\sigma\in\Lambda_{t}}S_{\sigma}(K)$, every net interval
$\Delta$ has a non-empty neighbour set.

######  Remark 3.2.

As explained in [21, Rem. 2.2], for an equicontractive IFS $\\{\lambda
x+d_{i}\\}_{i\in\mathcal{I}}$ with $0<\lambda<1$, our notion of neighbour set
is closely related to Feng’s neighbour and characteristic vector construction
[3]. Instead of normalizing by some global factor of the form $\lambda^{n}$,
we normalize locally with respect to $\operatorname{diam}(\Delta)$. In this
case, the words of generation $\lambda^{n-1}$ are the words of length $n$ and
the net intervals of generation $n$ (in Feng’s notation) have diameter
comparable to $\lambda^{n}$.

This is important since, outside the equicontractive case, there is no uniform
notion of an integer-valued generation.

We now discuss some illustrative examples of this construction.

Assume $\Delta\in\mathcal{F}_{t}$ has neighbour set $\\{f_{1},\ldots,f_{k}\\}$
and for each $i$, let $S_{\sigma_{i}}$ generate the neighbour $f_{i}$. The
_transition generation_ of $\Delta$, denoted $\operatorname{tg}(\Delta)$, is
given by

$\operatorname{tg}(\Delta)=\max\\{|r_{\sigma_{i}}|:1\leq i\leq k\\}.$

It is straightforward to verify that
$\operatorname{tg}(\Delta)=R_{\max}(\Delta)\operatorname{diam}(\Delta)$. The
_children_ of (_parent_) $\Delta$ are the net intervals of generation
$\operatorname{tg}(\Delta)$ contained in $\Delta$. We remark that if there is
only one child, $\Delta_{1}$, then
$\mathcal{V}(\Delta)\neq\mathcal{V}(\Delta_{1})$. Given $\Delta=[a,b],$ with
child $\Delta_{1}=[a_{1},b_{1}]$, we define the position index
$q(\Delta,\Delta_{1})=(a_{i}-a)/\operatorname{diam}\Delta$. The position index
will enable us to distinguish children with the same neighbour set.

The children of a net interval are locally determined by the neighbour set of
the net interval in the following sense.

######  Theorem 3.3 ([21], Thm. 2.8).

Let $\\{S_{i}\\}_{i=1}^{m}$ be an arbitrary IFS. Then for any
$\Delta\in\mathcal{F}_{t}$ with children $(\Delta_{1},\ldots,\Delta_{n})$ in
$\mathcal{F}_{\operatorname{tg}(\Delta)}$, the index $n$, neighbour sets
$\mathcal{V}(\Delta_{i})$, position indices $q(\Delta,\Delta_{i})$, and ratios
$\operatorname{tg}(\Delta_{i})/\operatorname{tg}(\Delta)$ depend only on
$\mathcal{V}(\Delta)$.

Thus much of the important information about the IFS is captured in the
behaviour of the neighbour sets. This motivates the construction of the
directed transition graph, $\mathcal{G}(\\{S_{i}\\}_{i=1}^{m})$, defined as
follows. The vertex set of $\mathcal{G}$, denoted $V(\mathcal{G})$, is the set
of distinct neighbour sets,
$V(\mathcal{G})=\\{\mathcal{V}(\Delta):\Delta\in\mathcal{F}\\}$. For each
parent/child pair of net intervals, $\Delta\in\mathcal{F}_{t}$ and
$\Delta_{i}\in\mathcal{F}_{\operatorname{tg}(\Delta)}$, we introduce an edge
$e=(\mathcal{V}_{t}(\Delta),\mathcal{V}_{\operatorname{tg}(\Delta)}(\Delta_{i}),q(\Delta,\Delta_{i}))$.
Here $\mathcal{V}_{t}(\Delta)$ is the source vertex and
$\mathcal{V}_{\operatorname{tg}(\Delta)}(\Delta_{i})$ is the target vertex. We
write $E(\mathcal{G})$ for the set of all edges. By Theorem 3.3, this
construction is well-defined since it depends only on the neighbour set of
$\Delta$.

An _(admissible) path_ $\eta$ in $\mathcal{G}$ is a sequence of edges
$\eta=(e_{1},\ldots,e_{n})$ in $\mathcal{G}$ where the target of $e_{i}$ is
the source of $e_{i+1}$. A path in $\mathcal{G}$ is a _cycle_ if it begins and
ends at the same vertex. We denote by $\Sigma_{0}$ the set of infinite paths
beginning at the root vertex $\mathcal{V}([0,1])$, and $\Sigma_{0}^{*}$ the
set of finite paths beginning at $\mathcal{V}([0,1])$.

Nested sequences of net intervals are in correspondence with finite paths in
$\Sigma_{0}^{*}$. Given $\Delta\in\mathcal{F}_{t}$, consider the sequence
$(\Delta_{0},\ldots,\Delta_{n})$ where $\Delta_{0}=[0,1]$,
$\Delta_{n}=\Delta$, and each $\Delta_{i}$ is a child of $\Delta_{i-1}$. By
the _symbolic representation_ of $\Delta$, we mean the finite path
$\eta=(e_{1},\ldots,e_{n})$ in $\mathcal{G}$ where

$e_{i}=\bigl{(}\mathcal{V}(\Delta_{i-1}),\mathcal{V}(\Delta_{i}),q(\Delta_{i-1},\Delta_{i})\bigr{)}\text{
for each }i=1,...,n.$

Conversely, if $\eta=(e_{1},\ldots,e_{n})$ is any finite path, we say that
$\eta$ is _realized_ by $(\Delta_{i})_{i=0}^{n}$ if each $\Delta_{i}$ is a
child of $\Delta_{i-1}$ and each
$e_{i}=(\mathcal{V}(\Delta_{i-1}),\mathcal{V}(\Delta_{i}),q(\Delta_{i-1},\Delta_{i}))$.
We denote the symbolic representation of $\Delta$ by $[\Delta]$.

######  Definition 3.4.

Given some $x\in K$, we say that an infinite path $\gamma\in\Sigma_{0}$ is a
_symbolic representation_ of $x$ if

$\\{x\\}=\bigcap_{i=1}^{\infty}\Delta_{i}$

where for each $n$, $[\Delta_{n}]$ is the symbolic representation of the
length $n$ prefix of $\gamma$, denoted by $\gamma|n$. We say that $x$ is an
_interior point_ of $K$ if $x$ has a unique symbolic representation.

If $x$ is not an interior point, then $x$ must be an endpoint of two distinct
net intervals at any sufficiently small scale.

######  Definition 3.5.

Let $\mathcal{G}$ be the transition graph of an IFS. We define the _edge
weight_ , $W:E(\mathcal{G})\to(0,1)$ by the rule that if edge $e$ has source
$\mathcal{V}(\Delta_{1})$ and target $\mathcal{V}(\Delta_{2})$, then
$W(e)=\operatorname{tg}(\Delta_{2})/\operatorname{tg}(\Delta_{1})$.

This function is well-defined by Theorem 3.3. We extend $W$ to finite paths by
putting $W(\eta)=W(e_{1})\cdots W(e_{n})$ when $\eta=(e_{1},\ldots,e_{n})$ .

An important observation is that if $\Delta\in\mathcal{F}_{t}$ is any net
interval with symbolic representation $\eta$, then $W(\eta)\asymp t$, with
constants of comparability not depending on $\Delta$. While the above choice
of the weight for an edge is not unique with this property, a straightforward
argument shows that any such function must agree with $W$ on any cycle.

######  Definition 3.6.

We say that the IFS $\\{S_{i}\\}_{i=1}^{m}$ satisfies the _finite neighbour
condition_ if its transition graph is a finite graph.

Equivalently, there are only finitely many neighbours. We also say that the
associated self-similar measure satisfies the finite neighbour condition, even
though this condition does not depend on the choice of probabilities.

The finite neighbour condition was introduced in [12] and explored in more
detail in [21, Sec. 5]. In [12] it was shown that the finite neighbour
condition is equivalent to the generalized finite type condition holding with
respect to the invariant open set $(0,1)$ (see [17] for the original
definition of GFT) and hence satisfies the weak separation condition [17]. In
particular, all IFS that satisfy the open set condition or the finite type
condition with respect to $(0,1)$ (see [3] for the definition of finite type)
satisfy the finite neighbour condition. For simplicity, throughout the
remainder of this document, whenever we say that an IFS satisfies the finite
type condition, we always mean with respect to $(0,1)$.

This includes examples such as the iterated function systems

$\\{\rho x,\rho x+(1-\rho)\\}$

where $\rho$ is the reciprocal of a Pisot number. Here, the associated self-
similar measures are the much studied Bernoulli convolutions (c.f., [4], [24]
and the many references cited therein), or the overlapping Cantor-like IFS

$\\{x/d+i(d-1)/md:i=0,1,...,m\\}$

where $d\geq 3$ is a natural number (see [11, 22]). For example, in the case
of the Bernoulli convolution with $\rho$ the reciprocal of the Golden mean,
there are six neighbour sets. These are listed in Section 5.1 and the
transition graph is given in Fig. 1.

A non-equicontractive example is given by the IFS $\\{\rho
x,rx+\rho(1-r),rx+1-r\\}$ where $\rho,r>0$ satisfy $\rho+2r-\rho r\leq 1$.
This was introduced in [18] where it was shown to satisfy the WSC. In fact,
this IFS satisfies the finite neighbour condition (see [17] or [21, Sec.
5.3]). Note that it does not satisfy the open set condition (due to the
existence of exact overlaps) and does not necessarily have commensurable
contraction factors, so it cannot be of finite type. See Fig. 3 for its
transition graph and Section 5.3.2 for more details about its structure. Other
examples of IFS satisfying the finite neighbour condition can also be found in
Section 5.

In [12, Thm. 4.4] it was proven, under the assumption that the self-similar
set is an interval, that the finite neighbour condition is equivalent to the
weak separation condition. It is unknown if the two properties coincide for
IFS in $\mathbb{R}$. Further details on these various separation conditions
for IFS can be found in [12].

###  3.2. Transition matrices

We now show how one can encode the measure of net intervals through the so-
called transition matrices.

Fix a total order on the set of all neighbours
$\\{f:f\in\mathcal{V}(\Delta),\Delta\in\mathcal{F}\\}$. Let $e\in
E(\mathcal{G})$ be an edge, say
$e=(\mathcal{V}(\Delta_{1}),\mathcal{V}(\Delta_{2}),q(\Delta_{1},\Delta_{2}))$.
Assume the neighbour sets are given by
$\mathcal{V}(\Delta_{1})=\\{f_{1},\ldots,f_{k}\\}$ and
$\mathcal{V}(\Delta_{2})=\\{g_{1},\ldots,g_{n}\\}$ where $f_{1}<\cdots<f_{k}$
and $g_{1}<\cdots<g_{n}$. We define the _transition matrix_ $T(e)$ as the non-
negative $k\times n$ matrix given by

(3.3) $T(e)_{i,j}=p_{\ell}$

if there exists an index $\ell\in\\{1,...,m\\}$ such that $f_{i}$ is generated
by $\sigma$ and $g_{j}$ is generated by $\sigma\ell$; otherwise, set
$T(e)_{i,j}=0$. Note that this definition is slightly different than the
original definition; see [21, Sec. 5.2] for more detail concerning this.

It is clear from Theorem 3.3 that this definition depends only on the edge
$e$. If $\eta=(e_{1},\ldots,e_{n})$ is a path, we define

$T(\eta)=T(e_{1})\cdots T(e_{n}).$

We refer to these matrices as transition matrices, as well.

Recall that if $\sigma^{\prime}$ generates any neighbour of $\Delta_{2}$, then
necessarily $\sigma^{\prime}=\sigma\ell$ for some $\sigma$ which generates a
neighbour of $\Delta_{1}$; thus, every column of $T(e)$ has a positive entry.
More generally, if $\eta$ is a path, then $T(\eta)$ has a positive entry in
every column. However, it may not hold that each row of $T(\eta)$ has a
positive entry.

We continue to use the notation $\left\lVert T\right\rVert=\sum_{i,j}T_{ij}$
for the matrix $1$-norm of a non-negative matrix $T$.

The following relationship between the measure of net intervals and transition
matrices is known.

######  Proposition 3.7 ([21], Cor. 5.5).

Suppose $\Delta$ is a net interval with symbolic representation $\eta$. Then

$\mu_{\bm{p}}(\Delta)\asymp\left\lVert T(\eta)\right\rVert,$

with constants of comparability not depending on the choice of $\Delta$.

Thus the transition matrices encode the distribution of $\mu_{\bm{p}}$ on net
intervals.

We conclude this subsection by mentioning the following straightforward
property of transition matrices.

######  Lemma 3.8.

Let $\gamma$ be a finite path. Fix $n\in\operatorname{{\mathbb{N}}}$ and let
$\gamma^{\prime}=(\gamma_{n+j})_{j=0}^{\infty}$. We have $\left\lVert
T(\gamma^{\prime})\right\rVert\asymp\left\lVert T(\gamma)\right\rVert$ with
constant of comparability depending only on $n$.

###### Proof.

Write $\gamma=\eta\gamma^{\prime}$ where $\eta$ is a path of length $n$. Since
every transition matrix has a non-zero entry in each column, a straightforward
calculation shows that there exists some constant $a=a(\eta)$ such that
$\left\lVert T(\eta\gamma^{\prime})\right\rVert\geq a(\eta)\left\lVert
T(\gamma^{\prime})\right\rVert$. On the other hand, $\left\lVert
T(\eta\gamma^{\prime})\right\rVert\leq\left\lVert
T(\eta)\right\rVert\left\lVert T(\gamma^{\prime})\right\rVert$. But there are
only finitely many paths $\eta$ of length $n$, giving the result. ∎

###  3.3. Maximal loop classes and irreducibility

From this point on we will assume that $\mathcal{G}$ is the matrix product
system corresponding to an IFS that satisfies the finite neighbour condition.

Let $\mathcal{L}$ be an induced subgraph of $\mathcal{G}$ (i.e. $\mathcal{L}$
is the graph consisting of the vertices $V(\mathcal{L})$ and any edge $e\in
E(\mathcal{G})$ such that $e$ connects two vertices in $\mathcal{L}$). Of
course, the induced subgraph naturally inherits a matrix product system from
the full graph.

######  Definition 3.9.

We say that the subgraph $\mathcal{L}$ is a _loop class_ if for any vertices
$v_{1},v_{2}\in V(\mathcal{L})$, there is a non-empty directed path connecting
$v_{1}$ and $v_{2}$, and we call $\mathcal{L}$ _maximal_ if it is maximal with
this property.

Two maximal loop classes neccessarily have disjoint vertex sets, but not all
vertices need to belong to a maximal loop class. However, given any symbolic
representation $\gamma=(\gamma_{i})_{i=0}^{\infty}$, there is a unique maximal
loop class $\mathcal{L}$ in which $\gamma$ is _eventually_ , meaning there
exists some $N$ such that $\gamma^{\prime}:=(\gamma_{i})_{i=N}^{\infty}$ is an
element of $\Sigma(\mathcal{L})$.

We will let

$K_{\mathcal{L}}=\\{x\in K:x\text{ has a symbolic representation that is
eventually in }\mathcal{L}\\}.$

Every element of $K$ belongs to at least one set $K_{\mathcal{L}}$ for a
maximal loop class $\mathcal{L}$, and at most two such sets.

Abusing notation slightly, given $\gamma$ which is eventually in
$\mathcal{L}$, we write

$\lambda(\mathcal{L},\gamma)=\lim_{n\to\infty}\frac{\log\left\lVert
T(\gamma|n)\right\rVert}{\log W(\gamma|n)}.$

By Lemma 3.8, for $k\geq N$ we have $\left\lVert
T(\gamma|k)\right\rVert\asymp\left\lVert T(\gamma^{\prime}|k-N)\right\rVert$
where $\gamma^{\prime}$ is as above. Since, also, $W(\gamma|k)\asymp
W(\gamma^{\prime}|k-N)$, we have
$\lambda(\mathcal{L},\gamma)=\lambda(\mathcal{L},\gamma^{\prime})$ (and
similarly for upper and lower Lyapunov exponents), where $\gamma^{\prime}$ is
a path in $\mathcal{L}$, justifying our notation.

We are primarily interested in three types of maximal loop classes.

######  Definition 3.10.

  1. (1)

We say that a maximal loop class is _irreducible_ if the corresponding matrix
product system is irreducible.

  2. (2)

We say that a maximal loop class is _simple_ if all cycles share the same edge
set.

  3. (3)

We say that a maximal loop class is an _essential class_ if any vertex
reachable from the maximal loop class by a directed path is also in the
maximal loop class.

For example, the IFS $\\{\rho x,\rho x+1-\rho\\}$ where $\rho$ is the
reciprocal of the Golden ratio has an essential class with three elements, and
two other singleton maximal loop classes. These loop classes are all
irreducible; for more details, see Section 5.1. Other examples are also given
in Section 5.

Note that irreducibility is a statement about the IFS and does not depend on
the choice of (non-zero) probabilities.

Any IFS satisfying the weak separation condition (such as those satisfying the
finite neighbour condition) has a unique essential class by [21, Prop. 3.3].
In fact, the finite neighbour condition can be characterized by the property
that the associated transition graph has a finite essential class [21, Thm.
5.3].

Moreover, the essential class is always irreducible; this is essentially shown
in [21, Lem. 3.9] (or [5, Lem. 6.4] in the equicontractive case), but we
include a self-contained proof here as an illustrative example:

######  Proposition 3.11.

Let $\mathcal{G}$ be the transition graph of an IFS satisfying the finite
neighbour condition with essential class $\mathcal{G}_{\operatorname{ess}}$.
Then $\mathcal{G}_{\operatorname{ess}}$ is irreducible.

###### Proof.

It suffices to show that for any
$v_{1},v_{2}\in\mathcal{G}_{\operatorname{ess}}$, $1\leq i\leq d(v_{1})$,
$1\leq j\leq d(v_{2})$, there exists a path $\eta$ from $v_{1}$ to $v_{2}$
such that $T(\eta)_{i,j}>0$. Let $\Delta\in\mathcal{F}$ be some net interval
with $\mathcal{V}(\Delta)=v_{1}$, let $\Delta_{0}\in\mathcal{F}_{t_{0}}$ be a
net interval with neighbour set in the essential class such that
$R_{\max}(\Delta_{0})$ is maximal and $\\#\mathcal{V}(\Delta_{0})$ maximal
among such net intervals. Let $\phi$ be a path from $\mathcal{V}(\Delta_{0})$
to $v_{2}$. Let $\xi$ generate neighbour $i$ of $\Delta$ and let
$\sigma\in\mathcal{I}^{*}$ have prefix $\xi$, $r_{\sigma}>0$ and
$S_{\sigma}([0,1])\subseteq\Delta$; such a $\sigma$ must necessarily exist
since $\Delta^{\circ}\cap S_{\xi}(K)\neq\emptyset$. Write
$\mathcal{V}(\Delta_{0})=\\{f_{1},\ldots,f_{k}\\}$ with $f_{1}<\cdots<f_{k}$
where each neighbour $f_{j}$ is generated by some word
$\omega_{j}\in\Lambda_{t_{0}}$. We have
$\Delta_{0}=[S_{\tau_{1}}(z_{1}),S_{\tau_{2}}(z_{2})]$ where
$\tau_{1},\tau_{2}\in\Lambda_{t_{0}}$ and $z_{1},z_{2}\in\\{0,1\\}$.

We now show that $S_{\sigma}(\Delta_{0})$ is indeed a net interval. Note that
the words $\sigma\tau_{j}$ and $\sigma\omega_{j}$ are in
$\Lambda_{r_{\sigma}t_{0}}$ by direct computation. Suppose for contradiction
$S_{\sigma}(\Delta_{0})$ is not a net interval. Without loss of generality,
let $\omega_{1}$ have
$|r_{\omega_{1}}|=R_{\max}(\Delta_{0})\operatorname{diam}(\Delta_{0})$. Since
$S_{\omega_{1}}(K)\cap\Delta_{0}^{\circ}\neq\emptyset$, we have
$S_{\sigma\omega_{1}}(K)\cap S_{\sigma}(\Delta_{0})^{\circ}\neq\emptyset$ so
there exists some net interval $\Delta^{\prime}\subseteq
S_{\sigma}(\Delta_{0})$ where the inclusion is proper and
$S_{\sigma\omega_{1}}(K)\cap(\Delta^{\prime})^{\circ}\neq\emptyset$. But then
$R_{\max}(\Delta^{\prime})>R_{\max}(\Delta_{0})$, contradicting the choice of
$\Delta_{0}$. Thus $\Delta_{1}:=S_{\sigma}(\Delta_{0})$ is indeed a net
interval.

Moreover, $\Delta_{1}$ has neighbours generated by the words $S_{\sigma}\circ
S_{\omega_{i}}$, and since $r_{\sigma}>0$,
$T_{\Delta_{1}}^{-1}=T_{\Delta_{0}}^{-1}\circ S_{\sigma}^{-1}$ and thus
$\mathcal{V}(S_{\sigma}(\Delta_{0}))\supseteq\mathcal{V}(\Delta_{0})$.
Equality then follows by the maximality of $\\#\mathcal{V}(\Delta_{0})$.

As $\xi$ is a prefix of $\sigma$, write $\sigma=\xi\tau$ for some
$\tau\in\mathcal{I}^{*}$. Let $\Delta$ have symbolic representation
$\phi_{0}$. Since $\Delta_{1}\subseteq\Delta$, there exists $\eta_{1}$ such
that $\Delta_{1}$ has symbolic representation $\phi_{0}\eta_{1}$. Since each
neighbour of $\Delta_{1}$ is generated by a word
$\sigma\omega_{j}=\xi\tau\omega_{j}$, by definition of the transition matrix
and choice of $\xi$, row $i$ of the matrix $T(\eta_{1})$ is strictly positive.
But then $\eta:=\eta_{1}\phi$ is an admissible path from $v_{1}$ to $v_{2}$,
and since every column of a transition matrix has a positive entry, row $i$ of
the matrix $T(\eta)$ is strictly positive as well. ∎

######  Remark 3.12.

In fact, as we argued in the above proof, the essential class satisfies a
somewhat stronger form of irreducibility: for any
$v_{1},v_{2}\in\mathcal{G}_{\operatorname{ess}}$ and $1\leq j\leq d(v_{2})$,
there exists a path $\eta$ from $v_{1}$ to $v_{2}$ such that row $j$ of
$T(\eta)$ is strictly positive. This property is closely related to the key
feature of the quasi-product structure under the weak separation condition
demonstrated by Feng and Lau [7]. Moreover, under somewhat stronger hypotheses
(satisfied, for example, when the attractor $K$ is an interval), the path
$\eta$ can be chosen such that $T(\eta)$ is a positive matrix (see [11] for a
proof in the equicontractive case, but the general case follows similarly).

##  4\. Sets of local dimensions of self-similar measures

We continue to use the notation of the previous section. In particular, we
assume that $\mathcal{G}$ is the matrix product system associated with an IFS
that satisfies the finite neighbour condition.

###  4.1. Basic results about local dimensions and periodic points

The following notion is a well-studied way of quantifying the singularity of
the measure $\mu$ with respect to Lebesgue measure at a point
$x\in\operatorname{supp}\mu=K$.

######  Definition 4.1.

Let $x\in K$ be arbitrary. Then the _lower local dimension of $\mu$ at $x$_ is
given by

$\underline{\dim}_{\operatorname{loc}}\mu(x)=\liminf_{t\to
0}\frac{\log\mu(B(x,t))}{\log t}$

and the _upper local dimension_ is given similarly with the limit infimum
replaced by the limit supremum. When the values of the upper and lower local
dimension agree, we call the shared value the _local dimension of $\mu$ at
$x$_.

Intuitively, the multifractal analysis of self-similar sets satisfying the
finite neighbour condition is related to the multifractal analysis of the
corresponding matrix product system. However, the exact relationship is
somewhat more complicated to establish: while the Lyapunov exponent of a path
$\gamma$ depends only on the single sequence of edges determining $\gamma$,
the local dimension of $\mu_{\bm{p}}$ at a point $x\in K$ can also depend on
net intervals which are adjacent to net intervals containing $x$. This happens
when $x$ is the shared boundary point of two distinct net intervals, but it
can also happen when $x$ is approximated very well by boundary points (so that
balls $B(x,r)$ overlap significantly with neighbouring net intervals, for many
values of $r$).

A point $x\in K$ is said to be _periodic_ if it has a symbolic representation
that is eventually a periodic path. For such points, this issue with overlaps
is easy to resolve. A boundary point of a net interval is a periodic point and
all elements of a simple loop class are periodic points. Indeed, for each
simple loop class there is a cycle $\theta$ such that all elements in the loop
class have a symbolic representation of the form $\gamma_{0}\overline{\theta}$
where $\overline{\theta}$ is the infinite periodic path with cycle $\theta$.
If $x$ has two distinct symbolic representations, then $x$ is necessarily the
endpoint of a net interval so the finite neighbour condition ensures that both
symbolic representations are periodic points. Note that a periodic point can
be an interior point (in the sense of Definition 3.4), but every non-periodic
point is interior.

By [21, Prop. 3.15] (see also [11, Prop. 2.7]), we have the following simple
formula for the local dimension of a periodic point:

######  Proposition 4.2 ([21], Prop. 3.15).

Suppose $x$ is an interior, periodic point with unique symbolic representation
$\gamma$ which is eventually in the loop class $\mathcal{L}$. Let $\theta$ be
any period of $\gamma$ and let $\overline{\theta}\in\Sigma(\mathcal{L})$
denote the path formed by repeating $\theta$ infinitely. Then the local
dimension exists at $x$ and is given by

$\dim_{\operatorname{loc}}\mu_{\bm{p}}(x)=\frac{\log\operatorname{sp}(T(\theta))}{\log
W(\theta)}=\lambda(\mathcal{L},\overline{\theta}).$

Otherwise, $x$ has two distinct symbolic representations with periods
$\theta_{1}$ and $\theta_{2}$ and

$\dim_{\operatorname{loc}}\mu_{\bm{p}}(x)=\min\left\\{\frac{\log\operatorname{sp}(T(\theta_{1}))}{\log
W(\theta_{1})},\frac{\log\operatorname{sp}(T(\theta_{2}))}{\log
W(\theta_{2})}\right\\}.$

######  Corollary 4.3.

If $x\in K_{\mathcal{L}}$ is a periodic point, then
$\dim_{\operatorname{loc}}\mu_{\bm{p}}(x)$ belongs to

$\bigcup_{i=1}^{m}\\{\lambda(\mathcal{L}_{i},\gamma):\gamma\in\Sigma(\mathcal{L}_{i})\\}$

where $\mathcal{L}_{1},\ldots,\mathcal{L}_{m}$ is a complete list of the
maximal loop classes in $\mathcal{G}$.

More generally, when the Lyapunov exponent exists or the local dimension
exists, we can relate the two notions.

######  Proposition 4.4.

Suppose $x\in K$ is an interior point with exactly one symbolic representation
$\gamma$ that is eventually in the maximal loop class $\mathcal{L}$.

  1. (1)

Then

$\alpha_{\min}(\mathcal{L})\leq\underline{\lambda}(\mathcal{L},\gamma)\leq\overline{\dim}_{\operatorname{loc}}\mu_{\bm{p}}(x)\leq\overline{\lambda}(\mathcal{L},\gamma)\leq\alpha_{\max}(\mathcal{L}).$

  2. (2)

If $\lambda(\mathcal{L},\gamma)$ exists, then
$\overline{\dim}_{\operatorname{loc}}\mu_{\bm{p}}(x)=\lambda(\gamma)$.

  3. (3)

If $\dim_{\operatorname{loc}}\mu_{\bm{p}}(x)$ exists, then
$\underline{\lambda}(\gamma)=\dim_{\operatorname{loc}}\mu_{\bm{p}}(x)$.

###### Proof.

By assumption, $x$ belongs to $K_{\mathcal{L}}$ and $\mathcal{L}$ is unique
with this property.

  1. (1)

We have already seen that
$\overline{\lambda}(\mathcal{L},\gamma)\leq\alpha_{\max}(\mathcal{L})$ and
$\underline{\lambda}(\mathcal{L},\gamma)\geq\alpha_{\min}(\mathcal{L})$ in
Theorem 2.10.

For any $t>0$, we have $B(x,t)\supseteq\Delta(\gamma|t)$ and therefore

$\displaystyle\overline{\dim}_{\operatorname{loc}}\mu_{\bm{p}}(x)=\limsup_{t\to
0}\frac{\log\mu_{\bm{p}}(B(x,t))}{\log t}$ $\displaystyle\leq\limsup_{t\to
0}\frac{\log\left\lVert T(\gamma|t)\right\rVert}{\log
t}=\overline{\lambda}(\mathcal{L},\gamma).$

If $x$ is not a boundary point of some net interval, since there are only
finitely many neighbour sets, there exists some $\rho>0$ and a monotonically
increasing sequence $(n_{k})_{k=1}^{\infty}$ with $n_{1}>N$ such that for each
$k\in\operatorname{{\mathbb{N}}}$ we have $B(x,\rho
W(\gamma|n_{k}))\subseteq\Delta(\gamma|n_{k})$. Since $\rho
W(\gamma|n_{k})\asymp W(\gamma|n_{k})$ we have

$\displaystyle\overline{\dim}_{\operatorname{loc}}\mu_{\bm{p}}(x)=\limsup_{t\to
0}\frac{\log\mu_{\bm{p}}(B(x,t))}{\log t}$
$\displaystyle\geq\limsup_{k\to\infty}\frac{\log\mu_{\bm{p}}(B(x,\rho
W(\gamma|n_{k})))}{\log\rho W(\gamma|n_{k})}$
$\displaystyle\geq\limsup_{k\to\infty}\frac{\log\left\lVert
T(\gamma|n_{k})\right\rVert}{\log W(\gamma|n_{k})}$
$\displaystyle\geq\underline{\lambda}(\mathcal{L},\gamma).$

as required.

Otherwise, $x$ is a boundary point with a unique symbolic representation. In
this case, for suitable $\rho>0$ and large $n$, $B(x,\rho W(\gamma|n))\cap
K_{\mathcal{L}}\subseteq\Delta(\gamma|n)$, so we can argue similarly.

  2. (2)

This is immediate from (i) as
$\underline{\lambda}(\mathcal{L},\gamma)=\overline{\lambda}(\mathcal{L},\gamma)$.

  3. (3)

The same argument as (i) shows that
$\underline{\dim}_{\operatorname{loc}}\mu_{\bm{p}}(x)\leq\underline{\lambda}(\mathcal{L},\alpha)$,
from which the result follows.

∎

###  4.2. Sets of local dimensions for simple and irreducible loop classes

We begin by noting that periodic points are dense in the set of local
dimensions.

######  Proposition 4.5.

Let $\mathcal{L}$ be an irreducible maximal loop class. Then the set of local
dimensions at interior periodic points is dense in
$[\alpha_{\min}(\mathcal{L}),\alpha_{\max}(\mathcal{L})]$.

###### Proof.

This follows by slightly modifying the proof of Proposition 2.13 by choosing
the paths $\eta_{k}$ such that they are also interior paths. Thus the
corresponding point in $K_{\mathcal{L}}$ is interior periodic and has local
dimension equal to the symbolic local dimension by Proposition 4.4. Then the
result follows from Corollary 2.11, which states that
$\\{\underline{\lambda}(\mathcal{L},\gamma):\gamma\text{ eventually in
}\mathcal{L}\\}=[\alpha_{\min}(\mathcal{L}),\alpha_{\max}(\mathcal{L})]$. ∎

Our next result establishes a converse to Proposition 4.4.

######  Theorem 4.6.

Let $\mathcal{L}$ be an irreducible, maximal loop class that is not simple.
Then

$[\alpha_{\min}(\mathcal{L}),\alpha_{\max}(\mathcal{L})]\subseteq\\{\dim_{\operatorname{loc}}\mu_{\bm{p}}(x):x\in
K_{\mathcal{L}},x\text{ interior}\\}.$

###### Proof.

Since $\mathcal{L}$ is not simple, there exists a path
$\xi\in\Sigma^{*}(\mathcal{L})$ such that if $\xi$ is realized by
$(\Delta_{i})_{i=0}^{m}$, then $\Delta_{m}\subseteq\Delta_{0}^{\circ}$. Let
$\alpha\in[\alpha_{\min}(\mathcal{L}),\alpha_{\max}(\mathcal{L})]$ be
arbitrary and by Theorem 2.10 get some
$\gamma=(e_{n})_{n=1}^{\infty}\in\Sigma(\mathcal{L})$ and a sequence
$(n_{j})_{j=1}^{\infty}$ with $\lim_{j}n_{j+1}/n_{j}=1$ such that
$\lambda(\gamma)=\alpha$ and for each $j$, $\xi$ is a prefix of
$(e_{n_{j}},e_{n_{j}+1},\ldots)$. Let $\zeta_{0}$ be such that
$\zeta_{0}\gamma$ is a path in $\mathcal{G}$ beginning at the root vertex
$\mathcal{V}([0,1])$. By the choice of $\xi$, there exists a unique interior
point $x$ with symbolic representation $\zeta_{0}\gamma$. We will show that
$\dim_{\operatorname{loc}}\mu_{\bm{p}}(x)=\alpha$.

We first note that $B(x,t)\supseteq\Delta_{t}(x)$ where $\Delta_{t}(x)$ is the
unique net interval in generation $t$ containing $x$. Thus if
$\zeta_{0}e_{1}\ldots e_{n}$ is the symbolic representation of
$\Delta_{t}(x)$, then

$\mu_{\bm{p}}(B(x,t)))\geq\mu_{\bm{p}}(\Delta_{t}(x))\asymp\left\lVert
T(\zeta_{0}e_{1}\ldots e_{n})\right\rVert\asymp\left\lVert T(e_{1}\ldots
e_{n})\right\rVert$

by Lemma 3.8. Hence for some constant $C>0$ we have

$\frac{\log\mu_{\bm{p}}(B(x,t))}{\log t}\leq\frac{\log C\left\lVert
T(e_{1}\ldots e_{n})\right\rVert}{\log t}.$

Since $\log t\asymp W(e_{1}\ldots e_{n})$, it follows that

$\overline{\dim}_{\operatorname{loc}}\mu_{\bm{p}}(x)\leq\alpha.$

To obtain the other inequality, we use the special properties of the path
$\gamma$. For each $k\in\operatorname{{\mathbb{N}}}$, let $\Delta^{(k)}$ be
the net interval with symbolic representation $\zeta_{0}e_{1}\ldots
e_{n_{k}-1}$. Let $\rho>0$ be such that for each
$k\in\operatorname{{\mathbb{N}}}$,

$B(x,\rho\operatorname{diam}(\Delta^{(k)}))\subseteq\Delta^{(k)}.$

Given $t>0$ sufficiently small, let $k$ be such that
$\rho\operatorname{diam}(\Delta^{(k+1)})\leq
t<\rho\operatorname{diam}(\Delta^{(k)})$, so that

$B(x,t)\subseteq
B(x,\rho\operatorname{diam}(\Delta^{(k)}))\subseteq\Delta^{(k)}.$

This ensures that $\mu_{\bm{p}}(B(x,t))\leq\mu_{\bm{p}}(\Delta^{(k)})$. We
also have

$\displaystyle t\geq\rho\operatorname{diam}(\Delta^{(k+1)})$
$\displaystyle\asymp W(\zeta_{0}e_{1}\ldots
e_{n_{k+1}-1})=W(\zeta_{0}e_{1}\ldots e_{n_{k}-1})W(e_{n_{k}}\ldots
e_{n_{k+1}-1})$ $\displaystyle\geq
W_{\min}^{n_{k+1}-n_{k}}W(\zeta_{0}e_{1}\ldots e_{n_{k}-1})\asymp
W_{\min}^{n_{k+1}-n_{k}}\operatorname{diam}(\Delta^{(k)}).$

Combining these observations, we see that there exist positive constants
$C_{1},C_{2}$ such that

$\displaystyle\frac{\log\mu_{\bm{p}}(B(x,t))}{\log t}$
$\displaystyle\geq\frac{\log\mu_{\bm{p}}(\Delta^{(k)})}{\log t}\geq\frac{\log
C_{1}+\log\left\lVert T(e_{1}\ldots
e_{n_{k}-1})\right\rVert}{\log\rho+\log\operatorname{diam}(\Delta^{(k+1)})}$
$\displaystyle\geq\frac{\log C_{1}+\log\left\lVert T(e_{1}\ldots
e_{n_{k}-1})\right\rVert}{(n_{k+1}-n_{k})\log
C_{2}+\log\operatorname{diam}(\Delta^{(k)})}$

But $\log\operatorname{diam}(\Delta^{(k)})\asymp-n_{k}$ and
$\lim_{k}(n_{k+1}-n_{k})/n_{k}=0$ by choice of $\gamma$, so that

$\displaystyle\underline{\dim}_{loc}\mu_{\bm{p}}(x)$
$\displaystyle\geq\liminf_{k\to\infty}\frac{\log\left\lVert T(e_{1}\ldots
e_{n_{k}-1})\right\rVert}{\log\operatorname{diam}(\Delta^{(k)})}$
$\displaystyle\geq\liminf_{n\to\infty}\frac{\log\left\lVert T(e_{1}\ldots
e_{n})\right\rVert}{\log W(e_{1}\ldots e_{n})}=\alpha.$

We have thus shown that $\dim_{\operatorname{loc}}\mu_{\bm{p}}(x)=\alpha$, as
required. ∎

######  Remark 4.7.

If $\mathcal{L}$ is a simple loop class with interior points, it is clear that
the conclusions of the theorem also hold and $\alpha_{\min}=\alpha_{\max}$.

The preceding theorem gives us strong information about the set of attainable
local dimensions:

######  Corollary 4.8.

Let $\mathcal{L}$ be an irreducible, maximal loop class that is not simple.
Then

$\displaystyle[\alpha_{\min}(\mathcal{L}),\alpha_{\max}(\mathcal{L})]$
$\displaystyle=\\{\dim_{\operatorname{loc}}\mu_{\bm{p}}(x):x\in
K_{\mathcal{L}},x\text{ interior}\\}$
$\displaystyle=\\{\overline{\dim}_{\operatorname{loc}}\mu_{\bm{p}}(x):x\in
K_{\mathcal{L}},x\text{ interior}\\}.$

###### Proof.

This follows by combining Theorem 4.6 and Proposition 4.4, noting that the set
of local dimensions is contained in the set of upper local dimensions. ∎

######  Remark 4.9.

When $\mathcal{L}$ is the essential class, this result was shown in [21].
Moreover, in that case, $[\alpha_{\min},\alpha_{\max}]$ is also the set of
lower local dimensions.

However, outside the essential class, the same statement need not hold for the
lower local dimension in place of the upper local dimension; the set of lower
local dimensions can be strictly larger. Consider the example from Section
5.3.4; this example and the result here is treated in [11, Sec. 6]. In that
example, with our notation, if $\mathcal{L}$ is the irreducible maximal loop
class not equal to the essential class, then

$[\alpha_{\min}(\mathcal{L}),\alpha_{\max}(\mathcal{L})]=\left[\frac{\log
7}{\log 4},\frac{\log 14}{\log 4}\right],$

while

$\\{\underline{\dim}_{loc}\mu_{\bm{p}}(x):x\in K_{\mathcal{L}}\text{
interior}\\}=\left[\frac{1}{2},\frac{\log 14}{\log 4}\right].$

Here $1/2$ is also the local dimension of a boundary point (that is not
interior).

It would be interesting to know if the set of lower local dimensions at
interior points is always an interval.

If $\mathcal{L}$ is a loop class that contains interior points, then the set
of local dimensions at these interior points is given by the interval
$[\alpha_{\min}(\mathcal{L}),\alpha_{\max}(\mathcal{L})]$. If $\mathcal{L}$
does not contain any interior points, then $\mathcal{L}$ is a simple loop
class. In this situation, it may hold that every $x\in K_{\mathcal{L}}$ has
two symbolic representations, and the local dimension is always given by the
symbolic representation of the adjacent path which is not eventually in
$\mathcal{L}$. This motivates the following definition:

######  Definition 4.10.

We say that a loop class $\mathcal{L}$ is _non-degenerate_ if $\mathcal{L}$ is
not simple, or if $\mathcal{L}$ is simple with period $\theta$ and there
exists some $x\in K_{\mathcal{L}}$ such that

$\dim_{\operatorname{loc}}\mu_{\bm{p}}(x)=\lambda(\overline{\theta}).$

We say that $\mathcal{L}$ is _degenerate_ otherwise.

We emphasize that, unlike simplicity or irreducibility, degeneracy depends on
the choice of probabilities. For an example of this phenomenon, see Section
5.3.3.

The point is that if $\mathcal{L}$ is a degenerate loop class, then the local
dimension at any point $x\in K_{\mathcal{L}}$ is given by the Lyapunov
exponent of a path not in $\mathcal{L}$. We now have the following corollary,
which holds under the assumptions that all maximal loop classes are either
irreducible or simple.

######  Corollary 4.11.

Let $\\{S_{i}\\}_{i\in\mathcal{I}}$ be an IFS satisfying the finite neighbour
condition with maximal loop classes $\\{\mathcal{L}_{i}\\}_{i=1}^{\ell}$.
Suppose each $\mathcal{L}_{i}$ is either irreducible or simple. Let
$\\{\mathcal{L}_{i}\\}_{i=1}^{\ell^{\prime}}$ denote the non-degenerate
maximal loop classes. Then

$\\{\overline{\dim}_{\operatorname{loc}}\mu_{\bm{p}}(x):x\in
K\\}=\\{\dim_{loc}\mu_{\bm{p}}(x):x\in
K\\}=\bigcup_{i=1}^{\ell^{\prime}}[\alpha_{\min}(\mathcal{L}_{i}),\alpha_{\max}(\mathcal{L}_{i})].$

###### Proof.

If $x$ is a periodic point, then
$\dim_{\operatorname{loc}}\mu_{\bm{p}}(x)=\lambda(\mathcal{L},\gamma)$ for
some path $\gamma$ in some non-degenerate loop class $\mathcal{L}$ according
to Proposition 4.2. Otherwise, $x$ must be an interior point of some
$K_{\mathcal{L}}$ where $\mathcal{L}$ is not simple. By Corollary 4.8,
$\overline{\dim}_{\operatorname{loc}}\mu_{\bm{p}}(x)=\alpha$ for some
$\alpha\in[\alpha_{\min}(\mathcal{L}),\alpha_{\max}(\mathcal{L})]$.

On the other hand, if $\mathcal{L}$ is a non-simple loop class, then Theorem
4.6 shows that each
$\alpha\in[\alpha_{\min}(\mathcal{L}),\alpha_{\max}(\mathcal{L})]$ is attained
as a local dimension. If $\mathcal{L}$ is a simple loop class, then
$\alpha_{\min}(\mathcal{L})=\alpha_{\max}(\mathcal{L})$ and this value is
attained as a local dimension precisely when $\mathcal{L}$ is non-degenerate.
∎

######  Remark 4.12.

The authors do not know if this result continues to hold without the
irreducibility assumption. However, we are not aware of any examples in
$\operatorname{{\mathbb{R}}}$ satisfying the weak separation condition which
do not satisfy the hypotheses for Corollary 4.11.

##  5\. Examples of IFS satisfying the finite neighbour condition

Throughout this section, for $\mathcal{L}$ a maximal loop class and
$\mu_{\bm{p}}$ a self similar measure, we will write

$\displaystyle\mathcal{D}(\mathcal{L})$
$\displaystyle=\\{\lambda(x):x\in\Sigma(\mathcal{L})\\}$
$\displaystyle\mathcal{D}(\mu_{\bm{p}})$
$\displaystyle=\\{\dim_{\operatorname{loc}}\mu_{\bm{p}}(x):x\in\operatorname{supp}\mu_{\bm{p}}\\}.$

###  5.1. Bernoulli convolutions

One much studied example of an equicontractive IFS of finite type is the IFS
with two contractions,

(5.1) $\\{\rho x,\rho x+1-\rho\\},$

with $\rho$ the reciprocal of the Golden mean. Feng [4] (see also [10, 11])
computed the neighbour sets (or characteristic vectors in his terminology)
with respect to the original net interval construction.

In our slightly modified setting, there six neighbour sets. These are:

$\displaystyle v_{1}$ $\displaystyle=\\{x\\}$ $\displaystyle v_{2}$
$\displaystyle=\\{x\cdot(1+\rho)\\}$ $\displaystyle v_{3}$
$\displaystyle=\\{x\cdot(1+\rho)-\rho\\}$ $\displaystyle v_{4}$
$\displaystyle=\\{x\cdot(2+\rho),x(2+\rho)-(1+\rho)\\}$ $\displaystyle v_{5}$
$\displaystyle=\\{x\cdot(3+2\rho)-(1+\rho)\\}$ $\displaystyle v_{6}$
$\displaystyle=\\{x\cdot(1+\rho),x\cdot(1+\rho)-\rho\\}$

The weight function is given by $W(e)=\rho$ for all edges $e$. The essential
class has $V(\mathcal{G}_{\operatorname{ess}})=\\{v_{4},v_{5},v_{6}\\}$ and
there are two other maximal loop classes, $\mathcal{L}_{1}$ and
$\mathcal{L}_{2}$, which are the simple loops with vertex sets
$V(\mathcal{L}_{1})=\\{v_{2}\\}$ and $V(\mathcal{L}_{2})=\\{v_{3}\\}$. Both
simple loop classes are non-degenerate since 0 and 1 are interior points. We
have $K_{\operatorname{ess}}=(0,1)$, $K_{\mathcal{L}_{1}}=\\{0\\}$, and
$K_{\mathcal{L}_{2}}=\\{1\\}$. See Fig. 1 for the transition graph as well as
the associated transition matrices.

$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{6}$$e_{1}$$e_{2}$$e_{3}$$e_{4}$$e_{5}$$e_{6}$$e_{7}$$e_{8}$$e_{9}$$e_{10}$$e_{11}$$e_{12}$

Edge | Weight | Transition Matrix
---|---|---
$e_{1}$ | $\rho$ | $\begin{pmatrix}p\end{pmatrix}$
$e_{2}$ | $\rho$ | $\begin{pmatrix}1-p\end{pmatrix}$
$e_{3}$ | $\rho$ | $\begin{pmatrix}1-p&p\end{pmatrix}$
$e_{4}$ | $\rho$ | $\begin{pmatrix}p\end{pmatrix}$
$e_{5}$ | $\rho$ | $\begin{pmatrix}1-p&p\end{pmatrix}$
$e_{6}$ | $\rho$ | $\begin{pmatrix}1-p\end{pmatrix}$
$e_{7}$ | $\rho$ | $\begin{pmatrix}1-p&p\end{pmatrix}$
$e_{8}$ | $\rho$ | $\begin{pmatrix}1-p&p\end{pmatrix}$
$e_{9}$ | $\rho$ | $\begin{pmatrix}p\\\ 1-p\end{pmatrix}$
$e_{10}$ | $\rho$ | $\begin{pmatrix}p&0\\\ 1-p&p\end{pmatrix}$
$e_{11}$ | $\rho$ | $\begin{pmatrix}1-p&p\\\ 0&1-p\end{pmatrix}$
$e_{12}$ | $\rho$ | $\begin{pmatrix}p&0\\\ 0&1-p\end{pmatrix}$

Figure 1. Transition graph for the Golden mean Bernoulli convolution

Since the essential class is always irreducible and the non-essential maximal
loop classes are simple, the set of local dimensions is a union of a possibly
non-singleton interval along with at most two isolated points. The
corresponding sets of Lyapunov exponents are

$\mathcal{D}(\mathcal{L}_{1}):=\left\\{\frac{\log p}{\log\rho}\right\\}\text{
and
}\mathcal{D}(\mathcal{L}_{2})=\left\\{\frac{\log(1-p)}{\log\rho}\right\\}.$

These are also the local dimensions at $0$ and $1$ respectively since $0$ and
$1$ are interior points, so we have
$\mathcal{D}(\mathcal{L}_{1}),\mathcal{D}(\mathcal{L}_{2})\subseteq\mathcal{D}(\mu_{\bm{p}})$.

Now, for $p\leq 1/2$, note that the transition matrix of the cycle
$(e_{11},e_{12})$ has spectral radius $(1-p)^{2}$ and weight $\rho^{2}$. The
Lyapunov exponent corresponding to this path is $\frac{\log(1-p)}{\log\rho}$
so that
$\mathcal{D}(\mathcal{L}_{2})\subseteq\mathcal{D}(\mathcal{G}_{\operatorname{ess}})$.
Similarly if $p\geq 1/2$, then the cycle $(e_{10},e_{12})$ has corresponding
Lyapunov exponent $\frac{\log p}{\log\rho}$ and
$\mathcal{D}(\mathcal{L}_{1})\subseteq\mathcal{D}(\mathcal{G}_{\operatorname{ess}})$.
In particular, when $p=1/2$, then $\mathcal{D}(\mu_{\bm{p}})$ is a closed
interval, and when $p\neq 1/2$, $\mathcal{D}(\mu_{\bm{p}})$ is a closed
interval along with at most a singleton point.

When $p\neq 1/2$, we know in general, by a short argument in [8], that
$\mathcal{D}(\mu_{\bm{p}})$ must contain an isolated point corresponding to
either $x=0$ or $x=1$, so $\mathcal{D}(\mu_{\bm{p}})$ is precisely a closed
interval along with an isolated point.

###  5.2. Testud measures

Consider the IFS given by the maps

$\displaystyle S_{1}(x)$ $\displaystyle=\frac{x}{4}$ $\displaystyle S_{2}(x)$
$\displaystyle=\frac{x}{4}+\frac{1}{4}$ $\displaystyle S_{3}(x)$
$\displaystyle=\frac{x}{4}+\frac{1}{2}$ $\displaystyle S_{4}(x)$
$\displaystyle=\frac{x}{4}+\frac{3}{4}$ $\displaystyle S_{5}(x)$
$\displaystyle=-\frac{x}{4}+\frac{1}{4}$ $\displaystyle S_{6}(x)$
$\displaystyle=-\frac{x}{4}+\frac{1}{2}$

This example is treated in [23, Sec. 6.2]. For each $i$, we have
$S_{i}([0,1])=[j/4,(j+1)/4]$ for some $j\in\\{0,1,2,3\\}$. There are two
neighbour sets,

$\displaystyle v_{1}$ $\displaystyle=\\{x\\}$ $\displaystyle v_{2}$
$\displaystyle=\\{-x+1,x\\}.$

The transition graph is given in Fig. 2, and there is the essential class
$\mathcal{G}_{\operatorname{ess}}$ with vertex set $\\{v_{2}\\}$ and a non-
simple irreducible maximal loop class $\mathcal{L}$ with vertex set
$\\{v_{1}\\}$.

Every cycle in $\mathcal{L}$ is a concatenation of the edges $e_{1}$ and
$e_{2}$; since the corresponding transition matrices are singletons, we have

$\displaystyle\mathcal{D}(\mathcal{L})$
$\displaystyle=\Bigl{\\{}\frac{\log(p_{3}^{n}p_{4}^{m})}{-\log 4^{n+m}}:n\geq
0,m\geq 0,n+m\geq 1\Bigr{\\}}$
$\displaystyle=\Bigl{[}\frac{\log\max\\{p_{3},p_{4}\\}}{-\log
4},\frac{\log\min\\{p_{3},p_{4}\\}}{-\log 4}\Bigr{]}.$

Similarly, the cycles in $\mathcal{G}_{\operatorname{ess}}$ are arbitrary
concatenations of edges in $\\{e_{5},e_{6},e_{7},e_{8}\\}$. Now, under the
assumption that $p_{1}=p_{4}+p_{5}$ and $p_{2}=p_{3}+p_{6}$, if
$\eta=(e_{i_{1}},e_{i_{2}},\ldots,e_{i_{k}})$ is any path in the essential
class with $n$ edges in $\\{e_{5},e_{8}\\}$ and $m$ edges in
$\\{e_{6},e_{7}\\}$, one may show that
$\operatorname{sp}T(\eta)=p_{1}^{n}p_{2}^{m}$. Thus

$\displaystyle\mathcal{D}(\mathcal{G}_{\operatorname{ess}})$
$\displaystyle=\Bigl{\\{}\frac{\log(p_{1}^{n}p_{2}^{m})}{-\log 4^{n+m}}:n\geq
0,m\geq 0,n+m\geq 1\Bigr{\\}}$
$\displaystyle=\Bigl{[}\frac{\log\max\\{p_{1},p_{2}\\}}{-\log
4},\frac{\log\min\\{p_{1},p_{2}\\}}{-\log 4}\Bigr{]}.$

We therefore have

$\displaystyle\mathcal{D}(\mu_{\bm{p}})=\Bigl{[}\frac{\log\max\\{p_{1},p_{2}\\}}{-\log
4},\frac{\log\min\\{p_{1},p_{2}\\}}{-\log
4}\Bigr{]}\cup\Bigl{[}\frac{\log\max\\{p_{3},p_{4}\\}}{-\log
4},\frac{\log\min\\{p_{3},p_{4}\\}}{-\log 4}\Bigr{]}.$

In particular, if $p_{1}<p_{2}<p_{3}<p_{4}$, then $\mathcal{D}(\mu_{\bm{p}})$
is a disjoint union of two non-trivial closed intervals.

Note that the other examples treated in [23] can be analysed similarly.

$v_{1}$$v_{2}$$e_{1}$$e_{2}$$e_{3}$$e_{4}$$e_{8}$$e_{7}$$e_{6}$$e_{5}$

Edge | Weight | Transition Matrix
---|---|---
$e_{1}$ | $1/4$ | $\begin{pmatrix}p_{3}\end{pmatrix}$
$e_{2}$ | $1/4$ | $\begin{pmatrix}p_{4}\end{pmatrix}$
$e_{3}$ | $1/4$ | $\begin{pmatrix}p_{5}&p_{1}\end{pmatrix}$
$e_{4}$ | $1/4$ | $\begin{pmatrix}p_{6}&p_{2}\end{pmatrix}$
$e_{5}$ | $1/4$ | $\begin{pmatrix}p_{4}&0\\\ p_{5}&p_{1}\end{pmatrix}$
$e_{6}$ | $1/4$ | $\begin{pmatrix}p_{3}&0\\\ p_{6}&p_{2}\end{pmatrix}$
$e_{7}$ | $1/4$ | $\begin{pmatrix}p_{2}&p_{6}\\\ 0&p_{3}\end{pmatrix}$
$e_{8}$ | $1/4$ | $\begin{pmatrix}p_{1}&p_{5}\\\ 0&p_{4}\end{pmatrix}$

Figure 2. Transition graph for the Testud IFS

###  5.3. Other examples

####  5.3.1. Cantor-like measures

Consider the family of IFS given by

$\bigl{\\{}S_{j}(x)=\frac{x}{r}+\frac{j}{mr}(r-1):0\leq j\leq m\bigr{\\}}$

for integers $m,r$ satisfying $2\leq r\leq m$. This family includes a rescaled
version of the three-fold convolution of the middle-third Cantor measure,
which was the earliest example of a self-similar measure known to exhibit
isolated points in the set of local dimensions [15]. The transition graph
consists of an essential class along with two simple maximal loop classes
$\mathcal{L}_{1}$ and $\mathcal{L}_{2}$, where $K_{\mathcal{L}_{1}}=\\{0\\}$,
$K_{\mathcal{L}_{2}}=\\{1\\}$, and
$K_{\mathcal{G}_{\operatorname{ess}}}=(0,1)$. The loops $\mathcal{L}_{1}$ and
$\mathcal{L}_{2}$ consist of single edges with $1\times 1$ transition
matrices, and

$\displaystyle\mathcal{D}(\mathcal{L}_{1})$ $\displaystyle=\frac{\log
p_{0}}{-\log r}$ $\displaystyle\mathcal{D}(\mathcal{L}_{1})$
$\displaystyle=\frac{\log p_{m}}{-\log r}.$

For appropriately chosen probabilities, these singletons contribute the
isolated points in the set of local dimensions for the self-similar measure
$\mu_{\bm{p}}$ and the essential class contributes a closed interval of
dimensions. See [11] for more details.

####  5.3.2. An example of Lau and Wang

By nature of the definition, an IFS of finite type must have logarithmically
commensurable contraction factors. Here is an example of an IFS satisfying the
finite neighbour condition which does not have commensurable contraction
factors.

The IFS $\\{\rho x,rx+\rho(1-r),rx+1-r\\}$ where $\rho+2r-\rho r\leq 1$ was
seen to satisfy the WSC in [18], but it is not of finite type when $\rho$ and
$r$ are non-commensurable. For simplicity, we consider the case $\rho=1/3$ and
$r=1/4$; for a more general treatment, this family was studied in [21, Sec.
5.2].

There are 5 neighbour sets given by

$\displaystyle v_{1}$ $\displaystyle=\\{x\\}$ $\displaystyle v_{2}$
$\displaystyle=\\{4x/3\\}$ $\displaystyle v_{3}$
$\displaystyle=\\{3x/2-1/2\\}$ $\displaystyle v_{4}$
$\displaystyle=\\{3x,4x-3\\}$ $\displaystyle v_{5}$
$\displaystyle=\\{x,3x\\}.$

The transition graph and transition matrices are given in Fig. 3. One can see
that there is only one maximal loop class, which is the essential class; thus,
the set of local dimensions is a closed interval. For more details on the
computations of the set of attainable local dimensions, we refer the reader to
[21].

$v_{1}$$v_{3}$$v_{2}$$v_{5}$$v_{4}$$e_{1}$$e_{9}$$e_{5}$$e_{3}$$e_{8}$$e_{11}$$e_{12}$$e_{2}$$e_{4}$$e_{6}$$e_{7}$$e_{10}$

Edge | Weight | Transition Matrix
---|---|---
$e_{1}$ | $1/4$ | $\begin{pmatrix}p_{3}\end{pmatrix}$
$e_{2}$ | $1/3$ | $\begin{pmatrix}p_{1}\end{pmatrix}$
$e_{3}$ | $1/4$ | $\begin{pmatrix}p_{2}\end{pmatrix}$
$e_{4}$ | $1/3$ | $\begin{pmatrix}p_{2}&p_{1}\end{pmatrix}$
$e_{5}$ | $1/3$ | $\begin{pmatrix}p_{1}\end{pmatrix}$
$e_{6}$ | $1/4$ | $\begin{pmatrix}p_{2}\end{pmatrix}$
$e_{7}$ | $1/3$ | $\begin{pmatrix}p_{2}&p_{1}\end{pmatrix}$
$e_{8}$ | $1/4$ | $\begin{pmatrix}p_{3}\end{pmatrix}$
$e_{9}$ | $1/4$ | $\begin{pmatrix}p_{2}\end{pmatrix}$
$e_{10}$ | $1/3$ | $\begin{pmatrix}1\\\ p_{1}\end{pmatrix}$
$e_{11}$ | $1/3$ | $\begin{pmatrix}0&1\\\ p_{2}&p_{1}\end{pmatrix}$
$e_{12}$ | $3/4$ | $\begin{pmatrix}0&1\\\ p_{3}&0\end{pmatrix}$

Figure 3. Transition graph for the example of Lau and Wang

####  5.3.3. A non-equicontractive finite type example

Here is an example which satisfies the finite type condition without equal
contraction ratios.

Take $\rho=(\sqrt{5}-1)/2$, the reciprocal of the Golden mean. Consider the
IFS given by the maps

$\displaystyle S_{1}(x)$ $\displaystyle=\rho x$ $\displaystyle S_{2}(x)$
$\displaystyle=\rho^{2}x+\rho-\rho^{2}$ $\displaystyle S_{3}(x)$
$\displaystyle=\rho^{2}x+(1-\rho^{2})$

with probabilities $(p_{i})_{i=1}^{3}$. This IFS has 7 neighbour sets given by

$\displaystyle v_{1}$ $\displaystyle=\\{x\\}$ $\displaystyle v_{2}$
$\displaystyle=\\{(2+\rho)x\\}$ $\displaystyle v_{3}$
$\displaystyle=\\{x,(1+\rho)x-(1+\rho)\\}$ $\displaystyle v_{4}$
$\displaystyle=\\{(2+\rho)x,(3+2\rho)x-(1+\rho)\\}$ $\displaystyle v_{5}$
$\displaystyle=\\{(1+\rho)x-\rho,(2+\rho)x,(2+\rho)x-(1+\rho)\\}$
$\displaystyle v_{6}$ $\displaystyle=\\{(1+\rho)x,(2+\rho)x,(2+\rho)x-1\\}$

and

$v_{7}=\\{(2+\rho)x,(2+\rho)x-(1+\rho),(3+2\rho)x-2(1+\rho),(3+2\rho)x-(1+\rho)\\}.$

There are three simple non-essential maximal loop classes, with vertex sets
$V(\mathcal{L}_{1})=\\{v_{1}\\}$, $V(\mathcal{L}_{2})=\\{v_{2}\\}$, and
$V(\mathcal{L}_{3})=\\{v_{3}\\}$. The essential class has vertex set
$V(\mathcal{G}_{\operatorname{ess}})=\\{v_{4},v_{5},v_{6},v_{7}\\}$. The
transition graph and transition matrices are given in Fig. 4.

A direct computation shows that
$\mathcal{D}(\mathcal{L}_{1})=\mathcal{D}(\mathcal{L}_{3})=\frac{\log
p_{3}}{2\log\rho}$ and $\mathcal{D}(\mathcal{L}_{2})=\frac{\log
p_{1}}{\log\rho}$. Thus $\mathcal{D}(\mu_{\bm{p}})$ consists of a possibly
non-singleton interval along with at most two isolated points. Both
$K_{\mathcal{L}_{1}}$ and $K_{\mathcal{L}_{2}}$ contain interior points, so
they are non-degenerate. However, every point in $x\in K_{\mathcal{L}_{3}}$
has two symbolic representations of the form

$(\underbrace{e_{1},\ldots,e_{1}}_{n},e_{3},e_{6},e_{6},\ldots)\text{ and
}(\underbrace{e_{1},\ldots,e_{1}}_{n},e_{1},e_{2},e_{4},e_{4},\ldots)$

for some $n\geq 0$. Thus for any $x\in K_{\mathcal{L}_{3}}$, we have

$\dim_{\operatorname{loc}}\mu_{\bm{p}}(x)=\min\Bigl{\\{}\frac{\log
p_{1}}{\log\rho},\frac{\log p_{3}}{2\log\rho}\Bigr{\\}}$

and when the minimum is not attained at $\frac{\log p_{3}}{2\log\rho}$,
$\mathcal{L}_{3}$ is a degenerate loop class. However, this does not impact
the set of possible local dimensions.

Suppose in particular that the probabilities satisfy $p_{1}^{2}>p_{2}$ and
$p_{3}>p_{2}$. Then the cycle $(e_{10},e_{11})$ in the essential class has
$\operatorname{sp}T(e_{10},e_{11})=p_{1}^{2}$ and $W(e_{10},e_{11})=\rho^{2}$,
so

$\mathcal{D}(\mathcal{L}_{2})\subseteq\mathcal{D}(\mathcal{G}_{\operatorname{ess}}).$

Similarly, the cycle $(e_{12},e_{13})$ in the essential class has
$\operatorname{sp}T(e_{12},e_{13})=p_{3}$ and $W(e_{12},e_{13})=\rho^{2}$ so

$\mathcal{D}(\mathcal{L}_{1})=\mathcal{D}(\mathcal{L}_{3})\subseteq\mathcal{D}(\mathcal{G}_{\operatorname{ess}})$

Therefore
$\mathcal{D}(\mu_{\bm{p}})=\mathcal{D}(\mathcal{G}_{\operatorname{ess}})$ is a
closed interval for such probabilities.

$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{6}$$v_{7}$$e_{1}$$e_{2}$$e_{3}$$e_{4}$$e_{5}$$e_{6}$$e_{7}$$e_{8}$$e_{9}$$e_{10}$$e_{11}$$e_{12}$$e_{13}$

Edge | Weight | Transition Matrix
---|---|---
$e_{1}$ | $\rho^{2}$ | $\begin{pmatrix}p_{3}\end{pmatrix}$
$e_{2}$ | $\rho$ | $\begin{pmatrix}p_{1}\end{pmatrix}$
$e_{3}$ | $\rho$ | $\begin{pmatrix}p_{2}&p_{1}\end{pmatrix}$
$e_{4}$ | $\rho$ | $\begin{pmatrix}p_{1}\end{pmatrix}$
$e_{5}$ | $\rho$ | $\begin{pmatrix}p_{2}&p_{1}\end{pmatrix}$
$e_{6}$ | $\rho$ | $\begin{pmatrix}0&1\\\ p_{3}&0\end{pmatrix}$
$e_{7}$ | $\rho$ | $\begin{pmatrix}0&1&0\\\ p_{2}&0&p_{1}\end{pmatrix}$
$e_{8}$ | $\rho$ | $\begin{pmatrix}0&1&0\\\ p_{2}&0&p_{1}\end{pmatrix}$
$e_{9}$ | $\rho$ | $\begin{pmatrix}0&1\\\ p_{2}&p_{1}\\\ p_{3}&0\end{pmatrix}$
$e_{10}$ | $\rho$ | $\begin{pmatrix}0&1&0\\\ 0&p_{1}&0\\\ p_{2}&0&p_{1}\end{pmatrix}$
$e_{11}$ | $\rho$ | $\begin{pmatrix}0&0&1\\\ 0&p_{1}&0\\\ p_{3}&0&0\end{pmatrix}$
$e_{12}$ | $\rho$ | $\begin{pmatrix}0&0&1&0\\\ p_{2}&0&0&p_{1}\\\ 0&p_{3}&0&0\end{pmatrix}$
$e_{13}$ | $\rho$ | $\begin{pmatrix}0&1&0\\\ 0&0&1\\\ p_{3}&0&0\\\ p_{2}&0&p_{1}\end{pmatrix}$

Figure 4. Transition graph for the non-equicontractive finite type example

####  5.3.4. An example with the set of lower local dimensions not equal to
the set of upper local dimensions

The IFS with $S_{i}(x)=x/4+d_{i}/12$ for $d_{i}=i$ when $i=0,1,...,5$,
$d_{6}=8$, $d_{7}=9$, is known to be of finite type [11, 19] and satisfies the
finite neighbour condition. The essential class is a single vertex with four
outgoing edges, and there are two additional loop classes: a simple loop class
with one vertex, along with a non-simple irreducible loop class with three
vertices.

This example is notable since the set of lower local dimensions in the non-
essential irreducible loop class need not coincide with the set of upper local
dimensions (see Remark 4.9).

####  5.3.5. A Pisot reciprocal Bernoulli convolution with a non-simple non-
essential loop class

Another interesting example is the Bernoulli convolution with parameter
$\rho$, where $\rho$ is the reciprocal of the Pisot root of $x^{3}-x^{2}-1$.
This finite type IFS has five maximal loop classes: the essential class with
46 elements, another irreducible loop class with 23 elements, and three
additional simple loop classes. For more details on this IFS, we refer the
reader to [9].

## References

  * [1] Robert Cawley and R. Daniel Mauldin, _Multifractal decompositions of Moran fractals_ , Adv. Math. 92 (1992), no. 2, 196–236.
  * [2] De-Jun Feng, _Lyapunov exponents for products of matrices and multifractal analysis. Part I: Positive matrices_ , Isr. J. Math. 138 (2003), no. 1, 353–376.
  * [3] by same author, _Smoothness of the $L^{q}$-spectrum of self-similar measures with overlaps_, J. Lond. Math. Soc. 68 (2003), no. 01, 102–118.
  * [4] by same author, _The limited Rademacher functions and Bernoulli convolutions associated with Pisot numbers_ , Adv. Math. 195 (2005), no. 1, 24–101.
  * [5] by same author, _Lyapunov exponents for products of matrices and multifractal analysis. Part II: General matrices_ , Isr. J. Math. 170 (2009), no. 1, 355–394.
  * [6] De-Jun Feng and Ka-Sing Lau, _The Pressure Function for Products of Non-negative Matrices_ , Math. Res. Lett. 9 (2002), no. 3, 363–378.
  * [7] by same author, _Multifractal formalism for self-similar measures with weak separation condition_ , J. Math. Pures Appl. 92 (2009), no. 4, 407–428.
  * [8] Kathryn E. Hare and Kevin G. Hare, _Local Dimensions of Overlapping Self-Similar Measures_ , Real Anal. Exchange 44 (2019), no. 2, 247\.
  * [9] Kathryn E. Hare, Kevin G. Hare, and Kevin R. Matthews, _Local dimensions of measures of finite type: Appendix_ , arXiv:1504.00510 [math] (2015).
  * [10] by same author, _Local dimensions of measures of finite type_ , J. Fractal Geom. 3 (2016), no. 4, 331–376.
  * [11] Kathryn E. Hare, Kevin G. Hare, and Michael K.S. Ng, _Local dimensions of measures of finite type II: Measures without full support and with non-regular probabilities_ , Can. J. Math. 70 (2018), no. 4, 824–867.
  * [12] Kathryn E. Hare, Kevin G. Hare, and Alex Rutar, _When the Weak Separation Condition implies the Generalized Finite Type Condition_ , Proc. Amer. Math. Soc. (to appear).
  * [13] H. George E. Hentschel and Itamar Procaccia, _The infinite number of generalized dimensions of fractals and strange attractors_ , Physica D: Nonlinear Phenomena 8 (1983), no. 3, 435–444.
  * [14] Einar Hille and Ralph S. Phillips, _Functional analysis and semi-groups_ , American Mathematical Society, Providence, R.I., 1957.
  * [15] Tian-You Hu and Ka-Sing Lau, _Multifractal Structure of Convolution of the Cantor Measure_ , Adv. Appl. Math. 27 (2001), no. 1, 1–16.
  * [16] Ka-Sing Lau and Sze-Man Ngai, _Multifractal Measures and a Weak Separation Condition_ , Adv. Math. 141 (1999), no. 1, 45–96.
  * [17] by same author, _A generalized finite type condition for iterated function systems_ , Adv. Math. 208 (2007), no. 2, 647–671.
  * [18] Ka-Sing Lau and Xiang-Yang Wang, _Iterated function systems with a weak separation condition_ , Studia Math. 161 (2004), no. 3, 249–268.
  * [19] Sze-Man Ngai and Yang Wang, _Hausdorff dimension of self-similar sets with overlaps_ , J. Lond. Math. Soc. 63 (2001), no. 3, 655–672.
  * [20] Norbert Patzschke, _Self-Conformal Multifractal Measures_ , Adv. Appl. Math. 19 (1997), no. 4, 486–513.
  * [21] Alex Rutar, _Geometric and Combinatorial Properties of Self-similar Multifractal Measures_ , arXiv:2008.00197 [math] (2020).
  * [22] Pablo Shmerkin, _A Modified Multifractal Formalism for a Class of Self-similar Measures with Overlap_ , Asian J. Math. 9 (2005), no. 3, 323–348.
  * [23] Benoît Testud, _Phase transitions for the multifractal analysis of self-similar measures_ , Nonlinearity 19 (2006), no. 5, 1201–1217.
  * [24] Péter P. Varjú, _Recent progress on Bernoulli convolutions_ , Apollo Cambridge Repository (2018).
  * [25] Martin P.W. Zerner, _Weak Separation Properties for Self-Similar Sets_ , Proc. Amer. Math. Soc. 124 (1996), no. 11, 3529–3539.