24.5cm16.2006cm 1.5cm1cm0cm0cm

# Piecewise linear iterated function systems on the line of overlapping
construction

R. Dániel Prokaj  R. Dániel Prokaj, Budapest University of Technology and
Economics, Hungary<EMAIL_ADDRESS>and  Károly Simon  Károly Simon,
Budapest University of Technology and Economics, MTA-BME Stochastics Research
Group, P.O. Box 91, 1521 Budapest, Hungary<EMAIL_ADDRESS>
(Date: 28th August 2024)

###### Abstract.

In this paper we consider Iterated Function Systems (IFS) on the real line
consisting of continuous piecewise linear functions. We assume some bounds on
the contraction ratios of the functions, but we do not assume any separation
condition. Moreover, we do not require that the functions of the IFS are
injective, but we assume that their derivatives are separated from zero. We
prove that if we fix all the slopes but perturb all other parameters, then for
all but a set of less than full packing dimensional exceptional set, the
Hausdorff dimension of the attractor is equal to the exponent which comes from
the most natural system of covers of the attractor.

###### Contents

  1. 1 Introduction
    1. 1.1 Attractors on the line
    2. 1.2 Introducing Continuous Piecewise Linear IFSs
    3. 1.3 Notations
    4. 1.4 The steps of our argument
    5. 1.5 The organization of the paper
  2. 2 Preliminaries and main results
    1. 2.1 Parametrizing CPLIFSs
    2. 2.2 Our main results
  3. 3 Self-similar and self-similar graph-directed IFSs on $\mathbb{R}$
    1. 3.1 Self-similar IFSs on the line
      1. 3.1.1 Some important results about self-similar IFSs
    2. 3.2 Self-similar graph-directed IFSs on the line
  4. 4 The preparation for the proof of the Main Theorem
    1. 4.1 The generated self-similar IFS
    2. 4.2 Associated self-similar and graph-directed self-similar IFS for a regular CPLIFS
    3. 4.3 Constructing a subshift in $\mathscr{A}^{\mathbb{N}}$
    4. 4.4 An ergodic measure on $\Lambda^{\mathscr{S}_{\mathcal{F}}}$ supported by $\Lambda^{\mathcal{F}}$
      1. 4.4.1 Auxiliary measures
      2. 4.4.2 The entropy and Lyapunov exponent of $\mathfrak{m}$
  5. 5 The proof of Theorems 2.2, 2.1 assuming the Main Proposition
  6. 6 The proof of the Main Proposition
  7. 7 Appendix

##  1\. Introduction

###  1.1. Attractors on the line

Iterated Function Systems (IFS) on the line consist of finitely many strictly
contracting self-mappings of $\mathbb{R}$. It was proved by Hutchinson [9]
that for every IFS $\mathcal{F}=\left\\{f_{k}\right\\}_{k=1}^{m}$ there is a
unique non-empty compact set $\Lambda^{\mathcal{F}}$ which is called the
attractor of the IFS $\mathcal{F}$ and defined by

(1.1)
$\Lambda^{\mathcal{F}}=\bigcup\limits_{k=1}^{m}f_{k}(\Lambda^{\mathcal{F}}).$

It is easy to see that for every IFS $\mathcal{F}$ there exists a unique
"smallest" non-empty compact interval $I^{\mathcal{F}}$ which is sent into
itself by all the mappings of $\mathcal{F}$:

(1.2) $I^{\mathcal{F}}:=\bigcap\left\\{J:J\subset\mathbb{R}\mbox{ compact
interval with }f_{k}(J)\subset J\mbox{, for all }k\in[m]\right\\},$

where $[m]:=\left\\{1,\dots,m\right\\}$. It is easy to see that

(1.3)
$\Lambda^{\mathcal{F}}=\bigcap\limits_{n=1}^{\infty}\bigcup\limits_{(i_{1},\dots,i_{n})\in[m]^{n}}I_{i_{1}\dots
i_{n}}^{\mathcal{F}},$

where $I_{i_{1}\dots i_{n}}^{\mathcal{F}}:=f_{i_{1}\dots
i_{n}}(I^{\mathcal{F}})$ are the cylinder intervals, and we use the common
shorthand notation $f_{i_{1}\dots i_{n}}:=f_{i_{1}}\circ\cdots\circ f_{i_{n}}$
for an $(i_{1},\dots,i_{n})\in[m]^{n}$.

The following three IFS families, each defined on $\mathbb{R}$, appear on
Figure 1 .

  1. (a)

Self-similar IFS:
$\mathcal{F}=\left\\{f_{i}(x)=\rho_{i}x+t_{i}\right\\}_{k=1}^{m}$, where
$\rho_{i}\in(-1,1)\setminus\left\\{0\right\\}$ and $t_{i}\in\mathbb{R}$.

  2. (b)

Hyperbolic IFS: $\mathcal{F}=\left\\{f_{1},\dots,f_{m}\right\\}$, where each
$f_{k}:J\to J$ is a $C^{1+\varepsilon}(J)$ contracting self-mappings of a non-
empty open interval $J\subset\mathbb{R}$.

  3. (c)

We introduce the Continuous Piecewise Linear Iterated Function Systems
(CPLIFS): These are IFSs of the form
$\mathcal{F}=\left\\{f_{1},\dots,f_{m}\right\\}$, where
$f_{k}:\mathbb{R}\to\mathbb{R}$ are continuous, piececewise linear
contractions (not necesserily injective) with all slopes different from zero.

Figure 1.

We note that the interval $I^{\mathcal{F}}$ intruduced in (1.2) is spanned by
the attractor in the first two cases, but not necesserily for non-injective
CPLIFSs.

The $t$-dimensional Hausdorff measure of an open Borel set
$A\subset\mathbb{R}$ is

(1.4)
$\mathcal{H}^{t}(A):=\sup\limits_{\delta>0}\left\\{\inf\left\\{\sum_{i=1}^{\infty}|A_{i}|^{t}:A\subset\bigcup\limits_{i=1}^{\infty}A_{i},\
|A_{i}|<\delta\right\\}\right\\}.$

Let $\mathcal{F}$ be an arbitrary IFS on the line. We can give a natural upper
bound on $\mathcal{H}^{t}(\Lambda^{\mathcal{F}})$ if we substitute the
covering system made of the cylinder intervals
$\left\\{I^{\mathcal{F}}_{i_{1}\dots i_{n}}\right\\}_{(i_{1}\dots
i_{n})\in[m]^{n}}$ in the place of the most efficient cover in (1.4), just
like in (1.3). For this particular system the right hand side of (1.4) is
related to the sequence of sums $\left\\{S_{n}^{t}\right\\}_{n=1}^{\infty}$

(1.5) $S_{n}^{t}:=\sum_{(i_{1}\dots
i_{n})\in[m]^{n}}|I^{\mathcal{F}}_{i_{1}\dots i_{n}}|^{t}.$

Namely, if the exponetial growth rate of
$\left\\{S_{n}^{t}\right\\}_{n=1}^{\infty}$ is negative (positive), then it
suggests that $\mathcal{H}^{t}(\Lambda^{\mathcal{F}})$ is equal to zero
(infinity). Therefore the best guess for the Hausdorff dimension of
$\Lambda^{\mathcal{F}}$ is the value of the exponent $t$ for which this
exponential growth rate is zero. This motivates the introduction of the
exponential growth rate

(1.6)
$\Phi^{\mathcal{F}}(s):=\limsup_{n\rightarrow\infty}\frac{1}{n}\log\sum_{i_{1}\dots
i_{n}}|I^{\mathcal{F}}_{i_{1}\dots i_{n}}|^{s}.$

It is easy to see that we can obtain $\Phi^{\mathcal{F}}(s)$ above as a
special case of the non-additive upper capacity topological pressure
introduced by Barreira in [2, p. 5]. According to [2, Theorem 1.9], the zero
of $\Phi^{\mathcal{F}}(s)$ is well defined

(1.7) $s_{\mathcal{F}}:=(\Phi^{\mathcal{F}})^{-1}(0).$

Barreira considered generalized Moran constructions [2, Section 2.1.2].
Condition (b) of such a construction requires that the cylinder intervals
$\left\\{I^{\mathcal{F}}_{i_{1}\dots i_{n}}\right\\}$ are sufficiently well
separated. This separation assumption does not hold for the constructions
considered in this paper, thus not all of his results apply. In the last
inequality of [2, Theorem 2.1 part (a)] Barreira proves that the upper box
dimension of $\Lambda^{\mathcal{F}}$ is less than or equal to the root of the
non-additive upper capacity topological pressure. It is easy to see that its
proof does not require any separation conditions on the cylinder intervals.
That is

######  Corollary 1.1 (Barreira).

For any IFS $\mathcal{F}$ on the line

(1.8) $\overline{\dim}_{\rm B}\Lambda^{\mathcal{F}}\leq s_{\mathcal{F}.}$

For a given IFS $\mathcal{F}$ on the line we name $s_{\mathcal{F}}$ the
natural dimension of the system. It is easy to see that in the self-simiar
case $s_{\mathcal{F}}$ is the solution of the self-similar equation

(1.9) $\sum_{k=1}^{m}|\rho_{k}|^{s_{\mathcal{F}}}=1$

and called similarity dimension. In the hyperbolic case $s_{\mathcal{F}}$ is
the root of the so-called pressure formula (see [13]). In both cases the Open
Set Condition (OSC) implies that

(1.10) $\dim_{\rm
H}\Lambda^{\mathcal{F}}=\min\left\\{s_{\mathcal{F}},1\right\\}.$

OSC is a separation condition which is to express that the cylinder intervals
are well separated (see [5, p. 35]). However, less strict separation
conditions might lead to the same result. In the self-similar case the
celebrated Hochman Theorem [6] yields that (1.10) also follows from the
Exponential Searation Condition (ESC).

###  1.2. Introducing Continuous Piecewise Linear IFSs

The IFSs we consider in this paper are consisting of piecewise linear
functions. Thus their derivatives might change at some points, but they are
linear over given intervals of $\mathbb{R}$. We always assume that the
functions are continuous, piecewise linear, strongly contracting with non-zero
slopes, and that the slopes can only change at finitely many points. This
setup enables us to investigate the case of non-injective functions as well,
which is not achievable with neither hyperbolic nor self-similar systems.

A special case of CPLIFSs can be handled using the theory of F. Hofbauer [8]
and P. Raith [14]. In particular, a family of CPLIFSs satisfying the one
dimensional version of the rectangular open set condition.

######  Definition 1.2.

We say that a CPLIFS $\mathcal{F}=\\{f_{k}\\}_{k=1}^{m}$ satisfies the
Interval Open Set Condition (IOSC) if the first cylinder intervals
$\\{I^{\mathcal{F}}_{k}\\}_{k=1}^{m}$ are pairwise disjoint.

Take a CPLIFS $\mathcal{F}=\\{f_{k}\\}_{k=1}^{m}$ of injective functions that
satisfies the IOSC. Each $f_{k}$ function can be considered as the local
inverse over $I^{\mathcal{F}}_{k}$ of a strictly expanding map on
$\mathbb{R}$. Therefore, the Hausdorff dimension of the attractor of such
systems equal to the root of a topological pressure function [14]. Further,
according to [8], for this special CPLIFS family the Hausdorff and the box
dimensions of the attractor are also equal.

In general, when we have no information about the nature of the overlapping of
the cylinders, we can claim only that (1.10) holds in some sense typically.
Instead of a particular IFS, it is natural to consider its so-called
translation family.

######  Definition 1.3.

For every $\boldsymbol{\tau}=(\tau_{1},\dots,\tau_{m})\in\mathbb{R}^{m}$ we
define the translation family of $\mathcal{F}$ by

(1.11)
$\left\\{\mathcal{F}^{\boldsymbol{\tau}}\right\\}_{\boldsymbol{\tau}\in\mathbb{R}^{m}},\mbox{
where
}\mathcal{F}^{\tau}=\\{f_{1}^{\boldsymbol{\tau}},\dots,f_{m}^{\boldsymbol{\tau}}\\}$

and $f_{k}^{\boldsymbol{\tau}}(x):=f_{k}(x)+\tau_{k}$ for a $k\in[m]$.
Moreover, for an $\mathbf{i}=(i_{1},\dots,i_{n})\in\Sigma^{*}$ we define the
function

(1.12)
$f_{\mathbf{i}}^{\boldsymbol{\tau}}(x):=f_{i_{1}}^{\boldsymbol{\tau}}\circ\cdots\circ
f_{i_{n}}^{\boldsymbol{\tau}}(x).$

The classical results [3], [4] and [15] about the translation families read
like this: For typical (in some sense) translations $\boldsymbol{\tau}$, the
Hausdorff dimension of the translated attractor $\Lambda^{\boldsymbol{\tau}}$
is equal to its natural dimension. Here the sense of typicality depends on the
family of the IFS considered. For example, for hyperbolic IFS on the line the
typicality above means "typical with respect to the $m$-dimensional Lebesgue
measure" [15] (at least in the case when all contractions are stronger than
$\frac{1}{2}$). In this case we say that the Hausdorff dimension and the
natural dimension coincide for Lebesgue typical translations. On the other
hand, Hochman [7] proved a much stronger theorem for the translation family of
self-similar IFSs that we call Multi Parameter Hochman Theorem (see Theorem
3.5). This theorem implies that the exceptional set of translations
$\boldsymbol{\tau}$ for which the ESC does not hold for
$\mathcal{F}^{\boldsymbol{\tau}}$ (and consequently $\dim_{\rm
H}\Lambda^{\boldsymbol{\tau}}$ is different from the similarity dimension),
has packing dimension (and in this way also Hausdorff dimension) less than or
equal to $m-1$. Remember that the dimension of the parameter space was equal
to $m$. Motivated by this, we introduce the following terminology related to
more general IFS families on the line:

Terminology: Let
$\left\\{\mathcal{F}^{\boldsymbol{\lambda}}\right\\}_{\boldsymbol{\lambda}\in
U}$ be a family of IFSs on the line where the parameter set $U$ is an open
subset of $\mathbb{R}^{d}$ for some $d\geq 1$. We say that a property
$\mathfrak{P}$, which makes sense for all elements of this family, holds
$\dim_{\rm P}$-typically if the exceptional set $E$ of those parameters
$\boldsymbol{\lambda}\in U$ for which $\mathfrak{P}$ does not hold satisfies
$\dim_{\rm P}E<d$. That is the packing dimension of the exceptional set
$E\subset U$ is smaller than the dimension of the parameter space $U$.

Our main result states that $\dim_{\rm P}$-typically the Hausdorff dimensision
of the attractor of a CPLIFS that satisfies certain regularity conditions is
equal to its natural dimension. More precisely, a CPLIFS $\mathcal{F}$ is
called regular if it has sufficiently small contraction ratios and there
exists an $N$ such that for all $(i_{1},\dots,i_{N})\in[m]^{N}$ the cylinder
interval $I_{i_{1}\dots i_{N}}^{\mathcal{F}}$ does not contain any point of
non-differentiability of any of the functions of $\mathcal{F}$. We will show
the following:

  1. (a)

For a regular CPLIFS $\mathcal{F}$ we consider its generated self-similar IFS
$\mathcal{S}_{\mathcal{F}}$ (see Section 4.1). If it satisfies the so-called
Exponential Separation Condition (ESC), then its $N$-th iterate system
$\mathcal{S}_{\mathcal{F}}^{N}$ and any subsystem of
$\mathcal{S}_{\mathcal{F}}^{N}$ will also satisfy the ESC. We point out that
we can select a suitable subsystem
$\mathscr{S}_{\mathcal{F}}\subset\mathcal{S}_{\mathcal{F}}^{N}$ and form a
graph-directed self-similar IFS from the functions of
$\mathscr{S}_{\mathcal{F}}$ such that the attractor of this graph-directed
system and $\Lambda^{\mathcal{F}}$ coincide. Then we use the Jordan Rapaport
Theorem [10] to compute the dimension of this graph-directed attractor which,
as we just mentioned, is the same as $\Lambda^{\mathcal{F}}$. In this way we
obtain that

(1.13) $\dim_{\rm H}\Lambda^{\mathcal{F}}=s_{\mathcal{F}}.$

  2. (b)

We verify in our Main Proposition that for a $\dim_{\rm P}$-typical set of
parameters (see our Terminology) the above mentioned regularity property
holds. Then we apply the multi-parameter Hochman Theorem [7, Theorem 1.10] to
conclude that the ESC is valid also for a $\dim_{\rm P}$-typical set of
parameters. That is by part (a) (1.13) holds $\dim_{\rm P}$-typically.

To state these more precisely we need some notations. See Figure 2 for
guidance.

Figure 2. A general CPLIFS with the related notations.

###  1.3. Notations

Let $\mathcal{F}=\left\\{f_{k}\right\\}_{k=1}^{m}$ be a CPLIFS and
$I^{\mathcal{F}}\subset\mathbb{R}$ be the compact interval defined in (1.2).
For any $k\in[m]$ let $l(k)$ be the number of breaking points
$\left\\{b_{k,i}\right\\}_{i=1}^{l(k)}$ of $f_{k}$. They determine the
$l(k)+1$ open intervals of linearity $\left\\{J_{k,i}\right\\}_{i=1}^{l(k)+1}$
(see Figure 2). We write $S_{k,i}$ for the contracting similarity on
$\mathbb{R}$ that satisfies $S_{k,i}|_{J_{k,i}}\equiv f_{k}|_{J_{k,i}}$. We
define $\left\\{\rho_{k,i}\right\\}_{k\in[m],i\in[l(k)+1]}$ and
$\left\\{t_{k,i}\right\\}_{k\in[m],i\in[l(k)+1]}$ such that

(1.14) $S_{k,i}(x)=\rho_{k,i}x+t_{k,i}.$

We say that
$\mathcal{S}_{\mathcal{F}}:=\left\\{S_{k,i}\right\\}_{k\in[m],i\in[l(k)+1]}$
is the self-similar IFS generated by the CPLIFS $\mathcal{F}$. In order to
prove a certain transversality like condition we need to require that the
system is strongly contracting. Namely, let

(1.15)
$\rho_{k}:=\max_{i\in[l(k)]}{|\rho_{k,i}|},\quad\rho_{\max}:=\max_{k}\rho_{k},\mbox{
and }\rho_{\min}:=\min_{k\in[m]}\min_{i\in[l(k)]}|\rho_{k,i}|$

######  Definition 1.4.

We say that $\mathcal{F}$ is small if both of the following two requirements
hold:

  1. (a)

$\sum\limits_{k=1}^{m}\rho_{k}<1$.

  2. (b)

Our second requirement depends on the injectivity of $f_{k}$:

    1. (i)

If $f_{k}$ is injective then we require that $\rho_{k}<\frac{1}{2}$.

    2. (ii)

If $f_{k}$ is not injective then we require that
$\rho_{k}<\frac{1-\rho_{\max}}{2}$, which always holds if
$\rho_{\max}<\frac{1}{3}$.

The first assumption is required to know a priori that $\dim_{\rm H}\Lambda<1$
in a certain uniform manner, and the later assumptions are required for a kind
of transversality argument.

######  Definition 1.5.

We say that a small CPLIFS $\mathcal{F}$ is regular if its attractor
$\Lambda_{\mathcal{F}}$ does not contain any of the breaking points
$\left\\{b_{k,i}\right\\}_{k\in[m],i\in[l(k)]}$.

Remember, our aim is to prove that _$\dim_{\rm P}$ -typically_

(1.16) $\mbox{ $\mathcal{F}$ is small}\Longrightarrow\mbox{$\mathcal{F}$ is
regular,}\quad\&\quad\mbox{$\mathcal{F}$ is regular}\Longrightarrow\dim_{\rm
H}\Lambda=s_{\mathcal{F}}.$

Here _$\dim_{\rm P}$ -typically_ means that if we fix all the contraction
ratios satisfying the condition in the left side but perturb all other
parameters, then for all parameters other then an exceptional set of less than
full packing dimension the statement on the right in (1.16) holds, for each
implication respectively.

The translation family of CPLIFSs does not have enough parameters to claim a
$\dim_{\rm P}$-typical result for the Hausdorff dimension of the attractor.
Thus besides the vertical translations, as in the translation families, in the
case of CPLIFSs we also consider the breaking points as parameters (but we fix
all of the contraction ratios). In this way besides the $m$ vertical
translation parameters $\boldsymbol{\tau}=(\tau_{1},\dots,\tau_{m})$ we have
$L:=\sum\limits_{k=1}^{m}l(k)$ parameters which correspond to all the breaking
points. Let us denote temporarily this $L+m$-dimensional parameter space by
$U$. For fixed contraction ratios
$\left\\{\rho_{k,i}\right\\}_{k\in[m],i\in[l(k)+1]}$ the elements of this
parameter space $U$ determine a CPLIFS.he

###  1.4. The steps of our argument

We fix the set of contractions
$\left\\{\rho_{k,i}\right\\}_{k\in[m],i\in[l(k)+1]}$ which are small in the
sense introduced above.

  1. (a)

First we observe that there exists a non-singular affine correspondence
between the elements of $U$ and
$\left\\{t_{k,i}\right\\}_{k\in[m],i\in[l(k)+1]}$, the set of translations of
the generated self-similar system.

  2. (b)

We may apply the Multi Parameter Hochman Theorem to conclude that $\dim_{\rm
P}$-typically the ESC holds for the generated self-similar IFS.

  3. (c)

We prove in our Main Proposition that $\dim_{\rm p}$-typically there are no
breaking points on the attractors.

  4. (d)

It implies, that $\dim_{\rm P}$-typically we can construct a self-similar
graph-directed IFS whose attractor coincides with the attractor of our CPLIFS.

  5. (e)

When we are in both of the $\dim_{\rm P}$-typical situations described by the
previous points, we introduce an ergodic (Markov) measure $\nu$ on the
symbolic space determined by the previously mentioned graph-directed IFS such
that $\dim_{\rm H}\nu$ is equal to the Hausdorff dimension of the attractor of
our CPLIFS.

  6. (f)

To show it, we use a recent theorem of Jordan-Rapaport [10, Theorem 1.1] to
compute $\dim_{\rm H}\nu$. Then we conclude that the Hausdorff dimension and
the natural dimension of the attractor coincide.

###  1.5. The organization of the paper

The first two sections are dedicated to introduce and motivate the continuous
piecewise linear iterated function systems, and the pressure function we use
to calculate the natural dimension of the attractor. We introduce here all the
important definitions and notations, and the typicality we use in our theorems
as well. At the end of the chapter we state our main theorems.

In Section 3 we present some recent results from the field of self-similar
iterated function systems, that will be useful in our proofs. Following [6] we
define the exponential Separation Condition (ESC) and state some important
theorems of M. Hochman that utilizes this assumption. We also state here a
recent result of Jordan and Rapaport [10] that will help us give a lower bound
on the Hausdorff dimension of the attractor of a regular CPLIFS. At the second
half of this section we introduce the self-similar graph-directed iterated
function systems and their dimension theory based on [12] and [5].

To draw our results, we need to associate regular CPLIFSs to graph-directed
iterated function systems. Section 4 contains in steps how we relate these two
IFS families to each other. We also introduce here an ergodic invariant
measure supported on the attractor of a regular CPLIFS, which let us calculate
the Hausdorff dimension of the attractor of the corresponding CPLIFS with the
help of the Jordan-Rapaport theorem.

Section 5 contains the proof of our main theorems assuming the main
proposition, and some useful lemmas that are not just helpful in the proofs,
but they also give us more details about CPLIFSs. The proof of our main
proposition is contained in section 6.

In the Appendix we present how our method of associating a regular CPLIFS to a
graph-directed function system can be used to extend one of Hochman’s theorems
(Namely part (b) of Theorem 3.2 in this paper) from self-similar sets to graph
directed ones.

##  2\. Preliminaries and main results

###  2.1. Parametrizing CPLIFSs

Let $\mathcal{F}=\left\\{f_{k}\right\\}_{k=1}^{m}$ be a CPLIFS. We write
$l(k)$ for the number of breaking points of $f_{k}$ for $k\in[m]$, and we say
that the type of the CPLIFS is the vector

(2.1) $\boldsymbol{\ell}=(l(1),\dots,l(m)).$

For example the type of the CPLIFS on Figure 2 is $\boldsymbol{\ell}=(2,1)$.
If $\mathcal{F}$ is a CPLIFS of type $\boldsymbol{\ell}$, then we write

$\mathcal{F}\in\mathrm{CPLIFS}_{\boldsymbol{\ell}}.$

The breaking points of $f_{k}$ are denoted by $b_{k,1}<\cdots<b_{k,l(k)}$. Let
$L:=\sum\limits_{k=1}^{m}l(k)$ be the total number of breaking points of the
functions of $\mathcal{F}$ with multiplicity if some of the breaking points of
two different elements of $\mathcal{F}$ coincide. We arrange all the breaking
points in an $L$ dimensional vector $\mathfrak{b}\in\mathbb{R}^{L}$ in a way
described below. First we partition $[L]=\left\\{1,\dots,L\right\\}$ into
blocks of length $l(k)$ for $k\in[m]$. The $k$-th block is

(2.2) $L^{k}:=\left\\{p\in\mathbb{N}:1+\sum\limits_{j=1}^{k-1}l(j)\leq
p\leq\sum\limits_{j=1}^{k}l(j)\right\\}$

where $\sum\limits_{j=1}^{k-1}$ is meant to be $0$ when $k=1$. We use this
convention without further mentioning it throughout the paper. The breaking
points of $f_{k}$ occupy the components belonging to the block $L(k)$ in
increasing order. That is

(2.3)
$\mathfrak{b}=(\underbrace{b_{1,1},\dots,b_{1,l(1)}}_{L^{1}},\underbrace{b_{2,1},\dots,b_{2,l(2)}}_{L^{2}},\dots,\underbrace{b_{m,1},\dots,b_{m,l(m)}}_{L^{m}}).$

The set of breaking points vectors $\mathfrak{b}$ for a type
$\boldsymbol{\ell}$ CPLIFS is

(2.4)
$\mathfrak{B}^{\boldsymbol{\ell}}:=\left\\{\mathbf{x}\in\mathbb{R}^{L}:x_{i}<x_{j}\mbox{
if }i<j\mbox{ and }\exists k\in[m]\mbox{ with }i,j\in L^{k}\right\\}.$

The $l(k)$ breaking points of the piecewise linear continuous function $f_{k}$
determines the $l(k)+1$ intervals of linearity $J_{k,i}^{\mathfrak{b}}$, among
which the first and the last are actually half lines:

(2.5)
$J_{k,i}:=J_{k,i}^{\mathfrak{b}}:=\left\\{\begin{array}[]{ll}(-\infty,b_{k,i}),&\hbox{if
$i=1$;}\\\ (b_{k,i-1},b_{k,i}),&\hbox{if $2\leq i\leq l(k)$;}\\\
(b_{k,l(k)},\infty),&\hbox{if $i=l(k)+1$.}\end{array}\right.$

The derivative of $f_{k}$ exists on $J_{k,i}$ and is equal to the constant

(2.6) $\rho_{k,i}:\equiv f^{\prime}_{k}|_{J_{k,i}}.$

We arrange the contraction ratios $\rho_{k,i}\in(-1,1)\setminus\\{0\\}$ into a
vector $\boldsymbol{\rho}$ in an analogous way as we arranged the breaking
points into a vector in (2.3), but taking into account that there is one more
contraction ratio for each $f_{k}$ than breaking points:

(2.7)
$\boldsymbol{\rho}:=\boldsymbol{\rho}_{\mathcal{F}}:=(\underbrace{\rho_{1,1},\dots,\rho_{1,l(1)+1}}_{\widetilde{L}^{1}},\dots,\underbrace{\rho_{m,1},\dots,\rho_{m,l(m)+1}}_{\widetilde{L}^{m}})\in\left((-1,1)\setminus\\{0\\}\right)^{L+m},$

where

(2.8)
$\widetilde{L}^{k}:=\left\\{p\in\mathbb{N}:1+\sum\limits_{j=1}^{k-1}\left(1+l(j)\right)\leq
p\leq\sum\limits_{j=1}^{k}\left(1+l(j)\right)\right\\}.$

We call $\boldsymbol{\rho}$ the vector of contractions. The set of all
possible values of $\boldsymbol{\rho}$ for an
$\mathcal{F}\in\mathrm{CPLIFS}_{\boldsymbol{\ell}}$ is

(2.9)
$\mathfrak{R}^{\boldsymbol{\ell}}:=\left\\{\boldsymbol{\rho}\in\left((-1,1)\setminus\\{0\\}\right)^{L+m}:\forall
k\in[m],\ \\!\forall
i,i+1\in\widetilde{L}^{k},\\!\rho_{i}\neq\rho_{i+1}\right\\},$

where $\boldsymbol{\rho}=(\rho_{1},\dots,\rho_{L+m})$. Recall the definitions
of $\rho_{k}$, $\rho_{\max}$ and $\rho_{\min}$ from (1.15). Moreover, let
$\rho_{k_{1}\dots k_{n}}:=\rho_{k_{1}}\cdots\rho_{k_{n}}$. Clearly,

(2.10) $|f^{\prime}_{k_{1}\dots k_{n}}(x)|\leq\rho_{k_{1}\dots k_{n}},\mbox{
for all }x.$

Finally, we write

(2.11) $\tau_{k}:=f_{k}(0),\mbox{ and
}\boldsymbol{\tau}:=(\tau_{1},\dots,\tau_{m})\in\mathbb{R}^{m}.$

So, the parameters that uniquely determine an
$\mathcal{F}\in\mathrm{CPLIFS}_{\boldsymbol{\ell}}$ can be organized into a
vector

(2.12)
$\boldsymbol{\lambda}=(\mathfrak{b},\boldsymbol{\tau},\boldsymbol{\rho})\in\boldsymbol{\Gamma}^{\boldsymbol{\ell}}:=\mathfrak{B}^{\boldsymbol{\ell}}\times\mathbb{R}^{m}\times\mathfrak{R}^{\boldsymbol{\ell}}\subset\mathbb{R}^{L}\times\mathbb{R}^{m}\times\mathbb{R}^{L+m}=\mathbb{R}^{2L+2m}.$

For a $\boldsymbol{\lambda}\in\boldsymbol{\Gamma}^{\boldsymbol{\ell}}$ we
write $\mathcal{F}^{\boldsymbol{\lambda}}$ for the corresponding CPLIFS,
$\Lambda^{\boldsymbol{\lambda}}$ for its attractor, and
$s_{\boldsymbol{\lambda}}$ for its natural dimension. Similarly, for an
$\mathcal{F}\in\mathrm{CPLIFS}_{\boldsymbol{\ell}}$ we write
$\boldsymbol{\lambda}(\mathcal{F})$ for the corresponding element of
$\boldsymbol{\Gamma}^{\boldsymbol{\ell}}$.

We can handle only those situations when the contraction ratios are
sufficiently small. Along the lines of Definition 1.4, we define the set of
small contraction vectors for a given type $\boldsymbol{\ell}$ as

(2.13)
$\mathfrak{R}^{\boldsymbol{\ell}}_{\mathrm{small}}:=\left\\{\boldsymbol{\rho}\in\mathfrak{R}^{\boldsymbol{\ell}}:\mathcal{F}^{(\mathfrak{b},\boldsymbol{\tau},\boldsymbol{\rho})}\mbox{
is small}\right\\},$

where $\mathfrak{b}\in\mathfrak{B}^{\boldsymbol{\ell}}$ and
$\boldsymbol{\tau}\in\mathbb{R}^{m}$ are arbitrary.

With the help of these notations we can taylor the terminology of $\dim_{\rm
P}$-typicality for CPLIFSs.

Terminology: Let $\mathfrak{P}$ be a property that makes sense for every
CPLIFS. For a contraction vector
$\boldsymbol{\rho}\in\mathfrak{R}_{\mathrm{small}}^{\boldsymbol{\boldsymbol{\ell}}}$
we consider the (exceptional) set

(2.14)
$E_{\mathfrak{P},\boldsymbol{\ell}}^{\boldsymbol{\rho}}=:\left\\{(\mathfrak{b},\boldsymbol{\tau})\in\mathfrak{B}^{\boldsymbol{\ell}}\times\mathbb{R}^{m}:\mathcal{F}^{(\mathfrak{b},\boldsymbol{\tau},\boldsymbol{\rho})}\mbox{
does not have property }\mathfrak{P}\right\\}.$

We say that property $\mathfrak{P}$ holds $\dim_{\rm P}$-typically if for all
type $\boldsymbol{\ell}$ and for all
$\boldsymbol{\rho}\in\mathfrak{R}_{\mathrm{small}}^{\boldsymbol{\ell}}$ we
have

(2.15) $\dim_{\rm
P}E_{\mathfrak{P},\boldsymbol{\ell}}^{\boldsymbol{\rho}}<L+m,$

where $\boldsymbol{\ell}=(l(1),\dots,l(m))$ and $L=\sum\limits_{k=1}^{m}l(k)$
as above.

###  2.2. Our main results

With the help of all these definitions and notations we can state now the main
results of this paper.

######  Theorem 2.1 (Main Theorem).

For a $\dim_{\rm P}$-typical small CPLIFS $\mathcal{F}$ we have

(2.16) $\dim_{\rm H}\Lambda^{\mathcal{F}}=\dim_{\rm
B}\Lambda^{\mathcal{F}}=s_{\mathcal{F}}.$

To prove our main result we verify the following theorem and proposition:

######  Theorem 2.2.

Let $\mathcal{F}$ be a regular CPLIFS for which the generated self-similar IFS
(defined in Section 4.1) satisfies the Exponential Separation Condition (ESC)
(see Definition 3.1). Then

(2.17) $\dim_{\rm H}\Lambda^{\mathcal{F}}=\dim_{\rm
B}\Lambda^{\mathcal{F}}=s_{\mathcal{F}}.$

To prove this, we use the multi-parameter Hochman Theorem [7, Theorem 1.10]
and a very recent result of Jordan and Rapaport [10, Theorem 1.1].

######  Proposition 2.3 (Main Proposition).

A $\dim_{\rm P}$-typical small CPLIFS is regular.

Together with [5, Theorem 3.2], this proposition implies the following
corollary.

######  Corollary 2.4.

For a $\dim_{\rm P}$-typical small CPLIFS $\mathcal{F}$ let $s:=\dim_{\rm
H}\Lambda^{\mathcal{F}}$ be the Hausdorff dimension of its attractor. Further,
assume that all functions in $\mathcal{F}$ are injective. Then we have

(2.18) $s=\dim_{\rm H}\Lambda^{\mathcal{F}}=\dim_{\rm
B}\Lambda^{\mathcal{F}}.$

Further, $\mathcal{H}^{s}(\Lambda^{\mathcal{F}})<\infty$, where
$\mathcal{H}^{s}$ denotes the $s$-dimensional Hausdorff measure.

##  3\. Self-similar and self-similar graph-directed IFSs on $\mathbb{R}$

Here we summarize the relevant results from the theory of self-similar IFSs.
Let $\mathcal{F}=\left\\{f_{k}\right\\}_{k=1}^{M}$ be an IFS on $\mathbb{R}$
and $\Lambda$ be the attractor of $\mathcal{F}$. We denote the symbolic space
by $\Sigma=\\{1,\ldots,M\\}^{\mathbb{N}}$, and its elements by
$\mathbf{i}=(i_{1},i_{2},\ldots)\in\Sigma$. We write $\Sigma^{\ast}$ for the
set of all finite length words. Elements of $\Sigma^{\ast}$ are denoted by
$\overline{\imath}=i_{1}\ldots i_{n}$, and sometimes we obtain them as
truncations of infinite length words $\mathbf{i}|n=i_{1}\ldots i_{n}$. As
usual, the natural projection $\Pi:\Sigma\to\Lambda$ is defined by

(3.1) $\Pi(\mathbf{i})=\lim\limits_{n\to\infty}f_{\mathbf{i}|n}(0).$

We write $\sigma$ for the left-shift on $\Sigma$. For the definition of
invariance, ergodicity, and entropy (denoted by $h(\mu)$) of the measure $\mu$
we refer the reader to Walters’ book [16]. If the measure $\mu$ is ergodic and
the derivatives of all of the mappings $f_{i}$ are continuous, we define the
Lyapunov exponent of $\mu$ by

(3.2)
$\chi(\mu):=\int\log|f^{\prime}_{i_{i}}(\Pi(\sigma\mathbf{i}))|d\mu(\mathbf{i}).$

Let $\nu$ be a Borel probability measure on $\mathbb{R}$. The Hausdorff
dimension of $\nu$ is defined as

(3.3) $\dim_{\rm H}\nu:=\inf\left\\{\dim_{\rm H}E:\nu(E)>0\right\\}.$

###  3.1. Self-similar IFSs on the line

In the special case when all $f_{i}$ are similarities (the slopes are
constants) $\mathcal{F}$ is called self-similar IFS. In this case for all
$i\in[M]:=\left\\{1,\dots,M\right\\}$ the mappings $f_{i}$, can be presented
in the form

(3.4) $f_{i}(x)=r_{i}x+t_{i},\mbox{ where
}r_{i}\in(-1,1)\setminus\left\\{0\right\\}\mbox{ and }t_{i}\in\mathbb{R}.$

The simplest guess for the dimension of the attractor $\Lambda$ can be
expressed in terms of the similarity dimension $\dim_{\rm S}\Lambda$, which is
defined as the solution $s$ of the equation
$\sum\limits_{K=1}^{M}|r_{k}|^{s}=1$. That is

(3.5) $\dim_{\rm H}\Lambda\leq\min\left\\{1,\dim_{\rm S}\Lambda\right\\},$

It follows from a theorem of Hochman’s (see in the next subsection) that in
some sense typically we have equality above. For an
$\overline{\imath}:=(i_{1},\dots,i_{n})$ we define

(3.6) $r_{\overline{\imath}|_{0}}:=1,\quad
r_{\overline{\imath}}=r_{i_{1}\ldots i_{n}}=r_{i_{1}}\cdot\ldots\cdot
r_{i_{n}}\mbox{ and
}t_{\overline{\imath}}:=\sum\limits_{k=1}^{n}t_{i_{k}}r_{\overline{\imath}|_{k}}.$

Clearly, we have
$f_{\overline{\imath}}(x)=r_{\overline{\imath}}x+t_{\overline{\imath}}$ and

(3.7) $\Pi(\mathbf{i})=\sum\limits_{n=1}^{\infty}r_{i_{1}\dots
i_{n-1}}t_{i_{n}}=\lim\limits_{n\to\infty}t_{\mathbf{i}|_{n}}.$

If $\mu$ is a measure on $\Sigma$ then we write $\Pi_{*}\mu$ for the push
forward measure of $\mu$. That is $\Pi_{*}\mu(E)=\mu(\Pi^{-1}E)$. Let $\mu$ be
a $\sigma$-invariant ergodic measure on $\Sigma$. Then the Lyapunov exponent
is

(3.8) $\chi(\mu)=\sum\limits_{k=1}^{m}\mu[k]\log r_{k}.$

For a probability vector $\mathbf{p}:=(p_{1},\dots,p_{M})$ we define the
measures:

(3.9) $\mu_{\mathbf{p}}([i_{1},\dots,i_{n}]):=p_{i_{1}}\cdots p_{i_{n}},\mbox{
and }\nu_{\mathbf{p}}:=\Pi_{*}\mu_{\mathbf{p}}.$

We say that $\nu_{\mathbf{p}}$ is a self-similar measure. The simplest guess
for the Hausdorff dimension of a self-similar measure $\nu_{\mathbf{p}}$ is

(3.10) $\dim_{\rm
S}\nu_{\mathbf{p}}:=\frac{h(\nu_{\mathbf{p}})}{\chi(\nu_{\mathbf{p}})}=\frac{\sum\limits_{k=1}^{M}p_{k}\log
p_{k}}{\sum\limits_{k=1}^{M}p_{k}\log r_{k}}$

Namely, $\dim_{\rm H}\nu_{\mathbf{p}}\leq\dim_{\rm S}\nu_{\mathbf{p}}$ and a
theorem of Hochman states that typically we have equality. We say that
$\dim_{\rm S}\nu_{\mathbf{p}}$ is the similarity dimension of the measure
$\nu_{\mathbf{p}}$.

Hochman [6] introduced the notion of exponential separation for self-similar
IFSs. To state it, first we need to define the distance of two similarity
mappings $g_{1}(x)=\varrho_{1}x+\tau_{1}$ and
$g_{2}(x)=\varrho_{2}x+\tau_{2}$,
$\varrho_{1},\varrho_{2}\in(-1,1)\setminus\left\\{0\right\\}$, on
$\mathbb{R}$. Namely,

(3.11)
$\mathrm{dist}\left(g_{1},g_{2}\right):=\left\\{\begin{array}[]{ll}|\tau_{1}-\tau_{2}|,&\hbox{if
$\varrho_{1}=\varrho_{2}$;}\\\ \infty,&\hbox{otherwise.}\end{array}\right.$

######  Definition 3.1.

Given a self-similar IFS $\mathcal{F}=\left\\{f_{k}(x)\right\\}_{k=1}^{m}$ on
$\mathbb{R}$. We say that $\mathcal{F}$ satisfies the Exponential Separation
Condition (ESC) if there exists a $c>0$ and a strictly increasing sequence of
natural numbers $\left\\{n_{\ell}\right\\}_{\ell=1}^{\infty}$ such that

(3.12)
$\mathrm{dist}\left(f_{\overline{\imath}},f_{\overline{\jmath}}\right)\geq
c^{n_{\ell}}\mbox{ for all }\ell\mbox{ and for all
}\overline{\imath},\overline{\jmath}\in\left\\{1,\dots,M\right\\}^{n_{\ell}},\
\overline{\imath}\neq\overline{\jmath}.$

####  3.1.1. Some important results about self-similar IFSs

Here we present three very important theorems about the dimension theory of
self-similar IFSs. In this subsection
$\mathcal{F}=\left\\{f_{k}(x)=r_{i}x+t_{i}\right\\}_{i=1}^{M}$, and
$r_{k}\in(-1,1)\setminus\left\\{0\right\\}$ as above, and we use the notation
of the previous subsections. First we recall two theorems of Hochman.

######  Theorem 3.2 (Hochman [6]).

Let $\mathcal{F}$ be a self-similar IFS on $\mathbb{R}$ which satisfies the
Exponential Separation Condition (ESC). Then

  1. (a)

For every self-similar measure $\nu$ we have

(3.13) $\dim_{\rm H}\nu=\min\\{1,\dim_{\rm S}\Lambda\\}.$

  2. (b)

Consequently,

(3.14) $\dim_{\rm H}\Lambda=\min\left\\{1,\dim_{\rm S}\Lambda\right\\}$

The following theorem of Hochman is about the translation family of a self
similar IFS. To state it we need some further notation.

######  Definition 3.3.

We consider the translation family
$\left\\{\mathcal{F}^{\boldsymbol{\tau}}\right\\}_{\boldsymbol{\tau}\in\mathbb{R}^{M}}$
(defined in Definition 1.3) of a self-similar IFS $\mathcal{F}$. We denote the
attractor of $\mathcal{F}^{\tau}$ by $\Lambda^{\boldsymbol{\tau}}$. For
$\overline{\imath},\overline{\jmath}\in[M]^{n}$ we write

(3.15)
$\Delta_{\overline{\imath},\overline{\jmath}}(\boldsymbol{\tau}):=f_{\overline{\imath}}^{\boldsymbol{\tau}}(0)-f_{\overline{\jmath}}^{\boldsymbol{\tau}}(0).$

Let us define the exceptional set

(3.16)
$E:=\bigcap\limits_{\varepsilon>0}\left(\bigcup\limits_{N=1}^{\infty}\bigcap\limits_{n>N}\left(\bigcup\limits_{\overline{\imath},\overline{\jmath}\in[M]^{n}\atop\overline{\imath}\neq\overline{\jmath}}\Delta^{-1}_{\overline{\imath},\overline{\jmath}}(-\varepsilon^{n},\varepsilon^{n})\right)\right)$

One can easily check that the following simple fact holds:

######  Fact 3.4.

Using the notation of Definition 3.3, we have

(3.17)
$\left\\{\boldsymbol{\tau}\in\mathbb{R}^{M}:\mathcal{F}^{\boldsymbol{\tau}}\mbox{
does not satisfy the ESC }\right\\}\subset E.$

The following theorem is a Corollary of [7, Theorem 1.10]

######  Theorem 3.5 (Multi parameter Hochman Theorem [7]).

Using the notation of Definition 3.3, we have

(3.18) $\dim_{\rm P}E\leq M-1.$

Consequently,

(3.19) $\dim_{\rm
P}\left\\{\boldsymbol{\tau}\in\mathbb{R}^{M}:\mathcal{F}^{\boldsymbol{\tau}}\mbox{
does not satisfy the ESC }\right\\}\leq M-1.$

The consequences of the following theorem will be very important in this
paper.

######  Theorem 3.6 (Jordan, Rapaport[10]).

Let $\mathcal{F}=\left\\{f_{k}\right\\}_{k=1}^{m}$ be a self-similar IFS on
$\mathbb{R}$ which satisfies the ESC. Moreover, let $\mu$ be an invariant
ergodic probability measure on $\Sigma=[M]^{\mathbb{N}}$. Then

(3.20) $\dim_{\rm
H}\Pi_{*}\mu=\min\left\\{1,\frac{h(\mu)}{\chi(\mu)}\right\\}.$

In the Appendix in Corollary 7.2 we prove that with the help of this result we
can extend part (b) of Theorem 3.2 to self-similar graph-directed iterated
function systems.

###  3.2. Self-similar graph-directed IFSs on the line

We present here the most important notations and results related to self-
similar Graph-Directed Iterated function Systems (GDIFS). In this subsection
we follow the book [5] and the papers [12] and [11]. The major difference is
that in the first two references the authors assume separation in between the
graph-directed sets. In [11] no separation is assumed, and we follow that
line.

To define the graph-directed iterated function systems we need a directed
graph $\mathcal{G}=\left(\mathcal{V,E}\right)$. We label the vertices of this
graph with the numbers $\\{1,2,...,q\\}$, where $|\mathcal{V}|=q$. This
$\mathcal{G}$ graph is not assumed to be simple, it might have multiple edges
between the same vertices, or even loops. For an edge $e=(i,j)\in\mathcal{E}$
we write $s(e):=i$ for the source and $t(e):=j$ for the target of $e$. Denote
with $\mathcal{E}_{i,j}$ the set of directed nodes from vertex $i$ to vertex
$j$, and write $\mathcal{E}_{i,j}^{k}$ for the set of length $k$ directed
paths between $i$ and $j$. Similarly, we write $\mathcal{E}^{n}$ for the set
of all paths of length $n$ in the graph. We assume that $\mathcal{G}$ is
strongly connected. That is for every $i,j\in\mathcal{V}$ there is a directed
path in $\mathcal{G}$ from $i$ to $j$.

For all edge $e\in\mathcal{E}$ given a contracting similarity mapping
$F_{e}:\mathbb{R}\rightarrow\mathbb{R}$. The contraction ratio is denoted by
$r_{e}\in(-1,1)\setminus\\{0\\}$. Let $e_{1}\dots e_{n}$ be a path in
$\mathcal{G}$. Then we write $F_{e_{1}\dots e_{n}}:=F_{e_{1}}\circ\cdots\circ
F_{e_{n}}$. It follows from the proof of [12, Theorem 1.1] that there exists a
unique family of non-empty compact sets $\Lambda_{1},...,\Lambda_{q}$ labeled
by the elements of $\mathcal{V}$, for which

(3.21)
$\Lambda_{i}=\bigcup\limits_{j=1}^{q}\bigcup\limits_{e\in\mathcal{E}_{i,j}}F_{e}(\Lambda_{j}),\quad
i=1,\dots,q.$

We call the sets $\\{\Lambda_{1},...,\Lambda_{q}\\}$ graph-directed sets, and
we say that $\Lambda:=\bigcup\limits_{i=1}^{q}\Lambda_{i}$ is the attractor of
the self-similar graph-directed IFS
$\mathcal{F}=\\{F_{e}\\}_{e\in\mathcal{E}}$. We abbreviate it self-similar
GDIFS.

By iterating (3.21) we get

(3.22)
$\Lambda_{i}=\bigcup_{j=1}^{q}\bigcup_{(e_{1},...e_{k})\in\mathcal{E}_{i,j}^{k}}F_{e_{1}\dots
e_{k}}(\Lambda_{j}).$

To get the most natural guess for the dimension of $\Lambda$, for every $s\geq
0$, we define a $q\times q$ matrix with the following entries

(3.23) $C^{(s)}=(c^{(s)}(i,j))_{i,j=1}^{q}\mbox{ and
}c^{(s)}(i,j)=\left\\{\begin{array}[]{ll}0,&\hbox{if
$\mathcal{E}_{i,j}=\emptyset$;}\\\
\sum\limits_{e\in\mathcal{E}_{i,j}}|r_{e}|^{s},&\hbox{otherwise.}\end{array}\right.$

The spectral radius of $C^{(s)}$ is denoted by $\varrho(C^{(s)})$. Mauldin and
Williams [12, Theorem 2] proved that the function $s\mapsto\varrho(C^{(s)})$
is strictly decreasing, continuous, greater than $1$ at $s=0$, and less than
$1$ if $s$ is large enough.

######  Definition 3.7.

For the self-similar GDIFS
$\mathcal{F}=\left\\{f_{e}\right\\}_{e\in\mathcal{E}}$ there exists a unique
$\alpha=\alpha(\mathcal{F})$ satisfying

(3.24) $\varrho(C^{(\alpha)})=1.$

The relation of $\alpha$ to the dimension of the attractor is given by the
following theorem. Both parts appeared in [12] apart from the box dimension
part, which is from [5].

######  Theorem 3.8.

Let $\mathcal{F}=\left\\{F_{e}\right\\}_{e\in E}$ be a self-similar GDIFS as
above. In particular, the graph $\mathcal{G}=(\mathcal{V},E)$ is strongly
connected and let $\Lambda$ be the attractor.

  1. (a)

$\dim_{\rm H}\Lambda\leq\alpha$.

  2. (b)

Let $I_{k}$ be the interval spanned by $\Lambda_{k}$ for all
$k\in\mathcal{V}$. If the intervals $\left\\{I_{k}\right\\}_{k=1}^{q}$ are
pairwise disjoint then $\dim_{\rm H}\Lambda=\dim_{\rm B}\Lambda=\alpha$.
Moreover, $0<\mathcal{H}^{\alpha}(\Lambda)<\infty$.

##  4\. The preparation for the proof of the Main Theorem

In this rather technical section we will complete the following steps:

  1. (1)

First we recall the definition of the generated self-similar IFS
$\mathcal{S}_{\mathcal{F}}$ for a CPLIFS $\mathcal{F}$.

  2. (2)

We assume that $\mathcal{F}$ is regular and we consider a suitable sub-system
$\mathscr{S}_{\mathcal{F}}$ (called associated IFS) of the $N$-th iterate
$\mathcal{S}_{\mathcal{F}}^{N}$ of the generated self-similar IFS
$\mathcal{S}_{\mathcal{F}}$ for the appropriate $N$.

  3. (3)

We represent $\Lambda^{\mathcal{F}}$ as the attractor of a graph directed
self-similar IFS whose functions are elements of $\mathscr{S}_{\mathcal{F}}$.

  4. (4)

Then we define an appropriate invariant and ergodic measure (actually a Markov
measure) for the self-similar IFS $\mathscr{S}_{\mathcal{F}}$, whose support
is the attractor of the previously mentioned grapgh directed self-similar IFS
which coincides with $\Lambda_{\mathcal{F}}$ as we mentioned above.

Later we will apply the Jordan Rapaport theorem (Theorem 3.6) for this measure
to get the dimension of $\Lambda^{\mathcal{F}}$.

###  4.1. The generated self-similar IFS

Let
$\mathcal{F}=\left\\{f_{k}\right\\}_{k=1}^{m}\in\mathrm{CPLIFS}_{\boldsymbol{\ell}}$
for a type $\boldsymbol{\ell}=(l(1),\dots,l(m))$ with
$L=\sum\limits_{k=1}^{m}l(k)$. As we already said in Subsection 1.3, the
generated self-similar IFS $S_{\mathcal{F}}$ consists of those similarity
mappings on $\mathbb{R}$ whose graph coincide with the graph of $f_{k}$ for
some $k\in[m]$ on some interval of linearity $J_{k,i}$, $i\in l(k)+1$ of
$f_{k}$. Hence, it is natural to parameterize the generated self-similar IFS
by the $l(k)+1$ intervals of linearity $J_{k,1},\dots,J_{k,l(k)+1}$ of the
mapping $f_{k}$ for all $k\in[m]$. That is

(4.1)
$\mathcal{S}:=\mathcal{S}_{\mathcal{F}}=\left\\{S_{k,i}(x)=\rho_{k,i}\cdot
x+t_{k,i}\right\\}_{k\in[m],i\in[l(k)+1]},\quad S_{k,i}|_{J_{k,i}}\equiv
f_{k}|_{J_{k,i}}.$

We organize the translation parts
$\left\\{t_{k,i}\right\\}_{k\in[m],i\in[l(k)+1]}$ of the mappings of
$\mathcal{S}_{\mathcal{F}}$ into a vector

(4.2)
$\mathfrak{t}=\mathfrak{t}(\mathfrak{b},\boldsymbol{\tau},\boldsymbol{\rho}):=(t_{1,1},\dots,t_{1,l(1)+1},\dots,t_{m,1},\dots,t_{m,l(m)+1})\in\mathbb{R}^{L+m}.$

If $b_{k,i}>0$ for all $i\in[l(k)]$ then the formula for $t_{k}$ is simple:

(4.3) $t_{k,1}=\tau_{k}\mbox{ and
}t_{k,i}=\tau_{k}+\sum\limits_{p=1}^{i}b_{k,i}\left(\rho_{k,i}-\rho_{k,i}\right).$

Using this and an appropriate conjugation in the general case when some of
$b_{k,i}\leq 0$ we obtain by simple calculation that the following Fact holds:

######  Fact 4.1.

For any fixed $\boldsymbol{\rho}\in\mathfrak{R}^{\boldsymbol{\ell}}$, consider
the mapping
$\Phi_{\boldsymbol{\rho}}:\mathfrak{B}^{\boldsymbol{\ell}}\times\mathbb{R}^{m}\to\mathbb{R}^{L+m}$
defined by

(4.4)
$\Phi_{\boldsymbol{\rho}}(\mathfrak{b},\boldsymbol{\tau}):=\mathfrak{t},$

where $\mathfrak{t}\in\mathbb{R}^{L+m}$ was defined in (4.2). Then
$\Phi_{\boldsymbol{\rho}}$ is a non-singular affine transformation. Hence
$\Phi_{\boldsymbol{\rho}}$ and its inverse $\Phi_{\boldsymbol{\rho}}^{-1}$
preserve Hausdorff and packing dimensions, and also preserve the sets of zero
measure with respect to $\mathcal{L}^{L+m}$.

To get a more effective method of labeling the $f_{k}\in\mathcal{F}$
functions’ intervals of linearity, and in this way the functions in
$S_{\mathcal{F}}$, we introduce

(4.5) $\mathcal{A}:=\left\\{(k,i):k\in[m]\mbox{ and }i\in[l(k)+1]\right\\}.$

Moreover, we write
$\mathcal{A}^{N}:=\underbrace{\mathcal{A}\times\cdots\times\mathcal{A}}_{N}$
for $N\in\mathbb{N}\cup\left\\{\infty\right\\}$. That is
$\mathcal{S}_{\mathcal{F}}=\left\\{S_{a}\right\\}_{a\in\mathcal{A}}$. Note
that $\\#\mathcal{A}=L+m$. Recall that the slope $\rho_{a}$ of $S_{a}$ was
defined in (2.6), and we define $t_{a}\in\mathbb{R}$ such that

(4.6) $S_{a}(x)=\rho_{a}x+t_{a}.$

###  4.2. Associated self-similar and graph-directed self-similar IFS for a
regular CPLIFS

Let $\mathcal{F}=\left\\{f_{k}\right\\}_{k=1}^{m}$ be a regular CPLIFS and let
$N=N(\mathcal{F})\in\mathbb{N}$ be the smallest $n$ such that for all
$\mathbf{u}=(u_{1},\dots,u_{n})\in[m]^{n}$ the cylinder interval
$I_{\mathbf{u}}^{\mathcal{F}}$ does not contain any breaking points. Then we
say that $\mathcal{F}$ is regular of order $N$. The collection of regular
CPLIFS of type $\boldsymbol{\ell}$ and order $N$ is denoted by
$\mathrm{CPLIFS}_{\boldsymbol{\ell},N}$.

Given an $\mathcal{F}\in\mathrm{CPLIFS}_{\boldsymbol{\ell},N}$. Let
$\mathcal{G}_{\mathcal{F}}$ be the following directed full graph

(4.7) $\mathcal{G}_{\mathcal{F}}:=(\mathcal{V},\mathcal{E})\mbox{ with
}\mathcal{V}:=[m]^{N}\mbox{ and
}\mathcal{E}:=\left\\{(\mathbf{v},\mathbf{u}):\mathbf{v},\mathbf{u}\in\mathcal{V}\right\\}$

Recall from Section 3.2 that we denote the source and the target of an edge
$e\in\mathcal{E}$ by $s(e)$ and $t(e)$ respectively. Further, for a
$k\in\mathbb{N}$ and $\mathbf{v},\mathbf{u}\in\mathcal{V}$ we write
$\mathcal{E}_{\mathbf{v},\mathbf{u}}^{k}$ for the set of length $k$ directed
paths $(e_{1},\dots,e_{k})$ such that $s(e_{1})=\mathbf{v}$,
$t(e_{k})=\mathbf{u}$ and $t(e_{i})=s(e_{i+1})$ for $i\in[k-1]$.

This section is organized in two steps:

  1. (a)

First we associate a self-similar IFS $\mathscr{S}_{\mathcal{F}}$ to every
$\mathcal{F}\in\mathrm{CPLIFS}_{\boldsymbol{\ell},N}$ which is the relevant
subsystem of $\mathcal{S}_{\mathcal{F}}^{N}$.

  2. (b)

Then, we use the directed graph
$\mathcal{G}_{\mathcal{F}}=\left(\mathcal{V},\mathcal{E}\right)$ to construct
a graph-directed self-similar IFS $\left\\{F_{e}\right\\}_{e\in\mathcal{E}}$
which consits of the functions of $\mathscr{S}_{\mathcal{F}}$. Moreover, the
attractor $\Lambda^{\mathcal{G}_{\mathcal{F}}}$ of this self-similar GDIFS
coincides with $\Lambda^{\mathcal{F}}$.

The construction of $\mathscr{S}_{\mathcal{F}}$. Let
$e=(\mathbf{v},\mathbf{u})\in\mathcal{E}$. For a $p\in[N]$ we consider
$f_{\sigma^{p}\mathbf{v}}I_{\mathbf{u}}$, where $\sigma$ is the left shift.
That is

$f_{\sigma^{p}\mathbf{v}}I_{\mathbf{u}}=\left\\{\begin{array}[]{ll}f_{v_{p+1}\dots
v_{N}}I_{\mathbf{u}},&\hbox{if $p\in[N-1]$;}\\\ I_{\mathbf{u}},&\hbox{ if
$p=N$.}\end{array}\right.$

Clearly, $f_{\sigma^{p}\mathbf{v}}I_{\mathbf{u}}$ is contained in the
$N$-cylinder $I_{v_{p+1}\dots v_{N}u_{1}\dots u_{p}}$. That is
$f_{\sigma^{p}\mathbf{v}}I_{\mathbf{u}}$ contains no breaking points of any
functions from $\mathcal{F}$. Namely, $f_{\sigma^{p}\mathbf{v}}I_{\mathbf{u}}$
is the subset of a linearity interval of $f_{v_{p}}$ for every $p\in[N]$.

In particular, there exists a unique $i(e,p)\in[l(v_{p})+1]$ such that

(4.8) $f_{\sigma^{p}\mathbf{v}}I_{\mathbf{u}}\subset J_{v_{p},i(e,p)},\mbox{
for any }p\in[N]$

Now we define a mapping
$\psi=\psi_{\mathcal{F}}:\mathcal{E}\to\mathcal{A}^{N}$

(4.9) $\psi(e)=\mathbf{a}=(a_{1},\dots,a_{N})\>\mbox{, where
}a_{p}:=\left(v_{p},i(e,p)\right).$

Let
$e=(\mathbf{v},\mathbf{u}),e^{\prime}=(\mathbf{v}^{\prime},\mathbf{u}^{\prime})\in\mathcal{E}$.
It is immediate from the construction that

(4.10)
$\psi(e)=\psi(e^{\prime})\Longrightarrow\mathbf{v}=s(e)=s(e^{\prime})=\mathbf{v}^{\prime}.$

Hence, for an $\mathbf{a}\in\mathscr{A}$ it makes sense to write

(4.11) $\psi_{1}^{-1}(\mathbf{a}):=\mathbf{v}\quad\mbox{ and
}\quad\mathbf{u}\in\psi_{2}^{-1}(\mathbf{a}),\quad\mbox{ if
}\psi(e)=\mathbf{a}\mbox{ and }e=(\mathbf{v},\mathbf{u}).$

Put

(4.12) $S_{\mathbf{a}}:=S_{a_{1}}\circ\cdots\circ S_{a_{N}}\mbox{ for
}\psi(e)=\mathbf{a}=(a_{1},\dots,a_{N}).$

By (4.1) for any $p\in[N]$ and $a_{p}=(v_{p},i(e,p))$

(4.13) $\displaystyle S_{a_{p}}|_{J_{v_{p},i(e,p)}}\equiv
S_{v_{p},i(e,p)}|_{J_{v_{p},i(e,p)}}\equiv f_{v_{p}}|_{J_{v_{p},i(e,p)}}.$

Then, using (4.13) and (4.8) together it is clear that

(4.14) $S_{\mathbf{a}}|_{I_{\mathbf{u}}}\equiv
f_{\mathbf{v}}|_{I_{\mathbf{u}}},\mbox{ for }\mathbf{a}=\psi(e)\mbox{ with
}e=(\mathbf{v},\mathbf{u})\in\mathcal{E}.$

With the help of a new alphabet

$\mathscr{A}:=\mathscr{A}_{\mathcal{F}}:=\left\\{\mathbf{a}\in\mathcal{A}^{N}:\exists
e\in\mathcal{E},\mathbf{a}=\psi(e)\right\\},$

we define the associated self-similar IFS

(4.15)
$\mathscr{S}:=\mathscr{S}_{\mathcal{F}}:=\left\\{S_{\mathbf{a}}\right\\}_{\mathbf{a}\in\mathscr{A}}.$

The mapping $\psi:\mathcal{E}\to\mathscr{A}$, what we use to associate the
elements of $\mathscr{A}$ to edges, is onto but not necessarily $1-1$.

The construction of the associated graph-directed self-similar IFS For an
$e=(\mathbf{v},\mathbf{u})\in\mathcal{E}$ we consider $S_{\mathbf{a}}$ for
$\mathbf{a}=(a_{1},\dots,a_{N})=\psi(e)$ and define the mapping
$F_{e}:I_{\mathbf{u}}\to I_{\mathbf{v}}$ by

(4.16)
$F_{e}(x):=S_{\mathbf{a}}|_{I_{\mathbf{u}}}=f_{\mathbf{v}}|_{I_{\mathbf{u}}}.$

Moreover, for
$\mathbf{e}=(e_{1},\dots,e_{k})\in\mathcal{E}^{k}_{\mathbf{v}^{1},\mathbf{v}^{k+1}}$
with $e_{i}=(\mathbf{v}^{i},\mathbf{v}^{i+1}),\>i\in[k]$ we have

(4.17) $\forall x\in I_{\mathbf{v}^{k+1}}:\quad
F_{\mathbf{e}}(x):=F_{e_{1}}\circ\dots\circ
F_{e_{k}}(x)=f_{\mathbf{v}^{1}}\circ\dots\circ f_{\mathbf{v}^{k}}(x).$

Obviously, $F_{e}$ is a similarity mapping restricted to $I_{\mathbf{u}}$ with
contraction ratio

(4.18)
$\rho_{\mathbf{v},\mathbf{u}}:=\rho_{e}:=\rho_{\mathbf{a}}:=S^{\prime}_{\mathbf{a}}\equiv\rho_{a_{1}}\cdots\rho_{a_{N}}.$

Recall that we defined the directed full graph in (4.7). We call
$\mathcal{F}^{\mathcal{G}}:=\left\\{F_{e}\right\\}_{e\in\mathcal{E}}$ the
associated self-similar graph-directed IFS. The symbolic space of
$\mathcal{F}^{\mathcal{G}}$ is the set of infinite paths in the full graph
$\mathcal{G}_{\mathcal{F}}$ that we denote by $\mathcal{E}_{\infty}$

(4.19) $\displaystyle\mathcal{E}_{\infty}$ $\displaystyle:=$
$\displaystyle\left\\{\mathfrak{p}:=(e_{1},e_{2},\dots):t(e_{i})=s(e_{i+1}),\
e_{i}\in\mathcal{E}\mbox{ for all }i\in\mathbb{N}\right\\}$ $\displaystyle=$
$\displaystyle\left\\{\mathfrak{p}=(\underbrace{(\mathbf{v}^{1},\mathbf{v}^{2})}_{e_{1}},\underbrace{(\mathbf{v}^{2},\mathbf{v}^{3})}_{e_{2}},\underbrace{(\mathbf{v}^{3},\mathbf{v}^{4})}_{e_{3}},\dots):\mathbf{v}^{k}\in\mathcal{V},\quad\forall
k\in\mathbb{N}\right\\}.$

The attractor of $\mathcal{F}^{\mathcal{G}}$ is

(4.20)
$\Lambda^{\mathcal{F}^{\mathcal{G}}}:=\bigcup_{\mathbf{v}\in\mathcal{V}}\bigcap_{k=1}^{\infty}\bigcup_{\mathbf{u}\in\mathcal{V}}\bigcup_{(e_{1}\dots
e_{k})\in\mathcal{E}_{\mathbf{v},\mathbf{u}}^{k}}F_{e_{1}\dots
e_{k}}I_{\mathbf{u}}.$

Then by (4.17) we have

(4.21) $\Lambda^{\mathcal{F}}=\Lambda^{\mathcal{F}^{\mathcal{G}}}.$

Thus $\Lambda^{\mathcal{F}}$ can be represented as a graph-directed attractor.
In the next section we show how $\Lambda^{\mathcal{F}}$ relates to the
associated self-similar IFS $\mathscr{S}_{\mathcal{F}}$.

###  4.3. Constructing a subshift in $\mathscr{A}^{\mathbb{N}}$

Now we show that we can consider $\Lambda^{\mathcal{F}}$ also as the
projection of a subshift $\mathscr{A}^{\mathbb{N}}_{\rm
Good}\subset\mathscr{A}^{\mathbb{N}}$ (defined below) by
$\Pi_{\mathscr{S}_{\mathcal{F}}}$, the natural projection corresponding to
$\mathscr{S}_{\mathcal{F}}$. In order to construct
$\mathscr{A}^{\mathbb{N}}_{\rm Good}$ we define two bijections:
$\vartheta:\mathcal{V}^{\mathbb{N}}\to\mathcal{E}_{\infty}$ and
$\Psi:\mathcal{E}_{\infty}\to\mathscr{A}^{\mathbb{N}}_{\rm Good}$.

Our first bijection $\vartheta$ is a very simple one:

(4.22)
$\vartheta(\mathbf{v}^{1},\mathbf{v}^{2},\mathbf{v}^{3},\mathbf{v}^{4},\dots):=\left((\mathbf{v}^{1},\mathbf{v}^{2}),(\mathbf{v}^{2},\mathbf{v}^{3}),(\mathbf{v}^{3},\mathbf{v}^{4}),\dots\right).$

To define our second bijection, first recall the definition of
$\psi:\mathcal{E}\to\mathscr{A}$ from (4.9). Now we apply this componentwise
to define the mapping $\Psi$ on $\mathcal{E}_{\infty}$ by

(4.23)
$\Psi(e_{1},e_{2},\dots)=\left(\mathbf{a}^{1},\mathbf{a}^{2},\dots\right)\in\mathscr{A}^{\mathbb{N}},\quad\mathbf{a}^{k}:=\psi(e_{k}),\quad
k\in\mathbb{N}.$

Let

$\mathscr{A}_{\mathrm{Good}}^{\mathbb{N}}:=\Psi(\mathcal{E}_{\infty}).$

We claim that $\Psi:\mathcal{E}_{\infty}\to\mathscr{A}^{\mathbb{N}}_{\rm
Good}$ is a bijection. By definition $\psi(e_{i})=\mathbf{a}^{i}$ for
$i=1,\dots,k$. By (4.10), this determines $e_{1},\dots,e_{k-1}$ uniquely.
Since this holds for all $k\in\mathbb{N}$ we obtain that $\Psi$ is $1-1$.
Clearly $\mathscr{A}_{\mathrm{Good}}^{\mathbb{N}}$ is compact and forward
invariant. That is

(4.24)
$\sigma(\mathscr{A}_{\mathrm{Good}}^{\mathbb{N}})\subset\mathscr{A}_{\mathrm{Good}}^{\mathbb{N}}\subset\mathscr{A}^{\mathbb{N}}.$

That is $\mathscr{A}_{\mathrm{Good}}^{\mathbb{N}}$ is a subshift in
$\mathcal{A}^{\mathbb{N}}$.

As we defined earlier, let
$\Pi_{\mathcal{F}^{N}}:\mathcal{V}^{\mathbb{N}}\to\mathbb{R}$ and
$\Pi_{\mathscr{S}_{\mathcal{F}}}:\mathscr{A}^{\mathbb{N}}\to\mathbb{R}$ be the
natural projections corresponding to the $N$-th iterate system of the CPLIFS
$\mathcal{F}$ and the self-similar IFS $\mathscr{S}_{\mathcal{F}}$
respectively. (For the definition of the natural projection of a general IFS
see Section3.)

Consider the following two diagrams:

(4.25)
$\textstyle{\mathcal{V}^{\mathbb{N}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{\vartheta}$$\textstyle{\mathcal{V}^{\mathbb{N}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vartheta}$$\textstyle{\mathcal{E}_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{\Psi}$$\textstyle{\mathcal{E}_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi}$$\textstyle{\mathscr{A}_{\mathrm{Good}}^{\mathbb{N}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{\mathscr{A}_{\mathrm{Good}}^{\mathbb{N}}}$
$\textstyle{\mathcal{V}^{\mathbb{N}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi\circ\vartheta}$$\scriptstyle{\Pi_{\mathcal{F}^{N}}}$$\textstyle{\mathscr{A}_{\mathrm{Good}}^{\mathbb{N}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Pi_{\mathscr{S}_{\mathcal{F}}}}$$\textstyle{\Lambda^{\mathcal{F}}}$

It is obvious that the first diagram is commutative.

######  Fact 4.2.

The second diagram in (4.25) is commutative.

###### Proof.

Write $\Pi_{\mathcal{F}^{\mathcal{G}}}$ for the natural projection defined by
the associated self-similar GDIFS $\mathcal{F}^{\mathcal{G}}$. Since
$\Lambda^{\mathcal{F}}=\Lambda^{\mathcal{F}^{\mathcal{G}}}$ we may dissect the
diagram in the following way:

(4.26)
$\textstyle{\mathcal{V}^{\mathbb{N}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vartheta}$$\scriptstyle{\Pi_{\mathcal{F}^{N}}}$$\textstyle{\mathcal{E}_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Pi_{\mathcal{F}^{\mathcal{G}}}}$$\textstyle{\Lambda^{\mathcal{F}}}$
---
$\textstyle{\mathcal{E}_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi}$$\scriptstyle{\Pi_{\mathcal{F}^{\mathcal{G}}}}$$\textstyle{\mathscr{A}_{\mathrm{Good}}^{\mathbb{N}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Pi_{\mathscr{S}_{\mathcal{F}}}}$$\textstyle{\Lambda^{\mathcal{F}^{\mathcal{G}}}}$

We already showed in (4.21) that the left diagram in (4.26) is commutative.
Using (4.16) and the definition of $\mathscr{A}_{\mathrm{Good}}^{\mathbb{N}}$
it is easy to see that the right diagram is also commutative.

∎

In particular we obtained that

(4.27)
$\Lambda^{\mathcal{F}}=\Lambda^{\mathcal{F}^{N}}=\Pi_{\mathcal{F}^{N}}(\mathcal{V}^{\mathbb{N}})=\Pi_{\mathscr{S}_{\mathcal{F}}}\left(\mathscr{A}_{\mathrm{Good}}^{\mathbb{N}}\right).$

###  4.4. An ergodic measure on $\Lambda^{\mathscr{S}_{\mathcal{F}}}$
supported by $\Lambda^{\mathcal{F}}$

Our aim is to define an appropriate invariant ergodic measure $\mathfrak{m}$
on $\mathscr{A}^{\mathbb{N}}$ which is supported by
$\mathscr{A}^{\mathbb{N}}_{\mathrm{Good}}$. Then we will take its push-forward
measure by $\Pi_{\mathscr{S}_{\mathcal{F}}}$. This way, according to (4.27),
the push-forward measure $\Pi_{\mathscr{S}_{\mathcal{F}}*}\mathfrak{m}$ is
supported by $\Lambda^{\mathcal{F}}$.

For every $\beta\geq 0$ we define the $m^{N}\times m^{N}$ matrix $C^{(\beta)}$
just like we did in Subsection 3.2. First we order the elements of
$\mathcal{V}$ according to lexicographical order. This will be the order of
the rows and columns of $C^{(\beta)}$. Then set

(4.28)
$C^{(\beta)}:=(|\rho_{\mathbf{v},\mathbf{u}}|^{\beta})_{(\mathbf{v},\mathbf{u})\in\mathcal{V}\times\mathcal{V}},$

where $\rho_{\mathbf{v},\mathbf{u}}$ was defined in (4.18). According to
Definition 3.7, the number $\alpha=\alpha(\mathcal{F})$ is uniquely defined by
$\varrho\left(C^{(\alpha)}\right)=1$.

Since $C^{(\alpha)}$ is an irreducible matrix, both the left and the right
eigenvectors
$\underline{\boldsymbol{\mathfrak{u}}}=(\mathfrak{u}_{\mathbf{v}})_{\mathbf{v}\in\mathcal{V}}$,
$\underline{\boldsymbol{\upnu}}=(\upnu_{\mathbf{v}})_{\mathbf{v}\in\mathcal{V}}$
corresponding to eigenvalue $1$ can be choosen to have all positive
components. That is

(4.29)
$\sum\limits_{\mathbf{v}\in\mathcal{V}}\mathfrak{u}_{\mathbf{v}}\cdot|\rho_{\mathbf{v},\mathbf{u}}|^{\alpha}=\mathfrak{u}_{\mathbf{u}},\quad\sum\limits_{\mathbf{u}\in\mathcal{V}}|\rho_{\mathbf{v},\mathbf{u}}|^{\alpha}\upnu_{\mathbf{u}}=\upnu_{\mathbf{v}},\quad\mathfrak{u}_{\mathbf{v}},\upnu_{\mathbf{v}}>0\mbox{
for all }\mathbf{v}\in\mathcal{V}.$

We normalize them in such a way that

(4.30)
$\sum\limits_{\mathbf{v}\in\mathcal{V}}\upnu_{\mathbf{v}}=1,\quad\sum\limits_{\mathbf{v}\in\mathcal{V}}\mathfrak{u}_{\mathbf{v}}\cdot\upnu_{\mathbf{v}}=1.$

Now we define the stochastic matrix
$P=(p_{\mathbf{v},\mathbf{u}})_{\mathbf{v},\mathbf{u}\in\mathcal{V}}$ and its
stationary distribution
$\mathbf{p}=(p_{\mathbf{v}})_{\mathbf{v}\in\mathcal{V}}$, which corresponds to
the matrix $C^{(\alpha)}$. That is

(4.31)
$p_{\mathbf{v},\mathbf{u}}:=\frac{|\rho_{\mathbf{v},\mathbf{u}}|^{\alpha}\upnu_{\mathbf{u}}}{\upnu_{\mathbf{v}}},\qquad
p_{\mathbf{v}}:=(\upnu_{\mathbf{v}}\cdot\mathfrak{u}_{\mathbf{v}}),\quad\mathbf{v},\mathbf{u}\in\mathcal{V}.$

Clearly,

(4.32) $\mathbf{p}^{T}\cdot P=\mathbf{p}^{T}\mbox{ that is
}\sum\limits_{\mathbf{v}}\upnu_{\mathbf{v}}\mathfrak{u}_{\mathbf{v}}\cdot\frac{|\rho_{\mathbf{v},\mathbf{u}}|^{\alpha}\upnu_{\mathbf{u}}}{\upnu_{\mathbf{v}}}=\upnu_{\mathbf{u}}\mathfrak{u}_{\mathbf{u}}.$

####  4.4.1. Auxiliary measures

Now we consider the one-sided Markov shift on $\mathcal{V}^{\mathbb{N}}$ (see
[16, p. 22]) corresponding to $(\mathbf{p},P)$. This gives us the ergodic
Borel measure $\mu=\mu_{\mathcal{F}}$ on $\mathcal{V}^{\mathbb{N}}$ defined on
the $n$-cylinders
$\left[\mathbf{v}^{1},\dots,\mathbf{v}^{n}\right]\subset\mathcal{V}^{n}$ by

(4.33)
$\mu\left(\left[\mathbf{v}^{1},\dots,\mathbf{v}^{n}\right]\right):=p_{\mathbf{v}^{1}}\cdot
p_{\mathbf{v}^{1},\mathbf{v}^{2}}p_{\mathbf{v}^{2},\mathbf{v}^{3}}\cdots
p_{\mathbf{v}^{n-1},\mathbf{v}^{n}}.$

Then this extends to an ergodic measure on $\mathcal{V}^{\mathbb{N}}$ (see
[16, Theorem 1.19]). Using that
$\Psi\circ\vartheta:\mathcal{V}^{\mathbb{N}}\to\mathscr{A}^{\mathbb{N}}_{\mathrm{Good}}$
is a homeomorphism, and the first diagram in (4.25) is commutative, we get
that the measure

(4.34) $\nu:=\left(\Psi\circ\vartheta\right)_{*}(\mu)$

is an invariant and ergodic measure on
$(\mathscr{A}^{\mathbb{N}}_{\mathrm{Good}},\sigma)$. We have pointed out that
$\mathscr{A}^{\mathbb{N}}_{\mathrm{Good}}$ is subshift in
$\mathscr{A}^{\mathbb{N}}$. So we can extend $\nu$ from
$\mathscr{A}^{\mathbb{N}}_{\mathrm{Good}}$ to $\mathscr{A}^{\mathbb{N}}$ in a
obvious way, such that after extension we still have an ergodic invariant
measure. Namely, we define the measure $\mathfrak{m}$ on
$\mathscr{A}^{\mathbb{N}}$ such that for a Borel set
$H\subset\mathscr{A}^{\mathbb{N}}$

(4.35) $\mathfrak{m}(H):=\nu(H\cap\mathscr{A}^{\mathbb{N}}_{\mathrm{Good}}).$

Using that $\mathscr{A}^{\mathbb{N}}_{\mathrm{Good}}$ is compact we obtain
that the support
$\mathrm{spt}(\mathfrak{m})=\mathscr{A}^{\mathbb{N}}_{\mathrm{Good}}$.
Moreover, it follows from (4.24) and [16, Theorem 1.6] that $\mathfrak{m}$ is
an ergodic measure. The invariance of $\mathfrak{m}$ is obvious from the
definition.

####  4.4.2. The entropy and Lyapunov exponent of $\mathfrak{m}$

First we estimate the measure of an $n$-cylinder for an arbitrary
$n\in\mathbb{N}$. For an
$\mathfrak{a}:=(\mathbf{a}^{1},\mathbf{a}^{2},\dots,\mathbf{a}^{n},\dots)\in\mathscr{A}^{\mathbb{N}}$
we write
$\mathfrak{a}|_{n}:=(\mathbf{a}^{1},\mathbf{a}^{2},\dots,\mathbf{a}^{n})$.
Then

$[\mathfrak{a}|_{n}]=\left\\{\widehat{\mathfrak{a}}=(\widehat{\mathbf{a}}^{1},\widehat{\mathbf{a}}^{2},\dots)\in\mathscr{A}^{\mathbb{N}}:\widehat{\mathbf{a}}^{1}=\mathbf{a}^{1},\dots,\widehat{\mathbf{a}}^{n}=\mathbf{a}^{n}\right\\}.$

For every $n$ and $\mathbf{v}^{n+1}\in\Psi_{2}^{-1}(\mathbf{a}^{n})$ we have

(4.36)
$\left[\mathbf{v}^{1},\dots,\mathbf{v}^{n},\mathbf{v}^{n+1}\right]\subset\left(\Psi\circ\vartheta\right)^{-1}\left([\mathfrak{a}|_{n}]\right)\subset\left[\mathbf{v}^{1},\dots,\mathbf{v}^{n}\right]$

Hence,

(4.37)
$\mu\left(\left[\mathbf{v}^{1},\dots,\mathbf{v}^{n},\mathbf{v}^{n+1}\right]\right)\leq\nu\left(\left[\mathbf{a}^{1},\dots,\mathbf{a}^{n}\right]\right)\leq\mu\left(\left[\mathbf{v}^{1},\dots,\mathbf{v}^{n}\right]\right)$

By substituting the formulas given in (4.31) for $p_{\mathbf{v},\mathbf{u}}$
$p_{\mathbf{v}}$ into the formula (4.33) we obtain that there exist
$C_{1},C_{2}>0$ such that

(4.38)
$C_{1}\leq\frac{\nu\left(\left[\mathbf{a}^{1},\dots,\mathbf{a}^{n}\right]\right)}{|\rho_{\mathbf{a}^{1},\dots,\mathbf{a}^{n}}|^{\alpha}}\leq
C_{2}.$

By definition (see (3.2)) the Lyapunov exponent of $\mathfrak{m}$ is

(4.39)
$\chi_{\mathfrak{m}}:=-\sum\limits_{\mathbf{a}\in\mathscr{A}}\mathfrak{m}([\mathbf{a}])\cdot\log|\rho_{\mathbf{a}}|,$

where we defined $\rho_{\mathbf{a}}$ in (4.18). Then by the ergodicity of
$\mathfrak{m}$ and by the Shannon-McMillan-Breiman Theorem we have

(4.40)
$h(\mathfrak{m})=-\lim\limits_{n\to\infty}\frac{1}{n}\log\mathfrak{m}([\mathfrak{a}|_{n}])\mbox{
for $\mathfrak{m}$-a.a. }\mathfrak{a}\in\mathscr{A}^{\mathbb{N}}.$

Moreover,

(4.41)
$\chi_{\mathfrak{m}}:=-\lim\limits_{n\to\infty}\frac{1}{n}\log|S^{\prime}_{\mathfrak{a}|_{n}}|=-\lim\limits_{n\to\infty}\frac{1}{n}\log|\rho_{\mathfrak{a}|_{n}}|\mbox{
for $\mathfrak{m}$-a.a. }\mathfrak{a}\in\mathscr{A}^{\mathbb{N}}.$

Putting together (4.41),(4.40) and (4.37) we get

(4.42) $\frac{h(\mathfrak{m})}{\chi_{\mathfrak{m}}}=\alpha.$

##  5\. The proof of Theorems 2.2, 2.1 assuming the Main Proposition

This section contains the proofs of our main results, under the assumption
that the Main Proposition holds. First we prove some facts and a lemma.

######  Lemma 5.1.

Let $\mathcal{F}$ be a regular CPLIFS. Then we have

(5.1) $\alpha(\mathcal{F})=s_{\mathcal{F}}.$

To prove this Lemma we will use the following facts.

######  Definition 5.2.

Let $\mathcal{F}$ be a CPLIFS. We say that $\mathcal{F}$ satisfies the Bounded
Distortion Property (BDP) if there exist $0<C_{1}<C_{2}$ such that

(5.2) $\forall n,\;\forall|\mathbf{j}|=n,\;\forall x,y\in I:\quad
C_{1}\leq\frac{|f_{\mathbf{j}}^{\prime}(x)|}{|f_{\mathbf{j}}^{{}^{\prime}}(y)|}\leq
C_{2}.$

It is not true in general that a CPLIFS satisfies the BDP. For example, if a
function’s fixed point coincides with one of its breaking points, then this
condition trivially fails. However, in the regular case the BDP does hold.

######  Fact 5.3.

Let $\mathcal{F}$ be a regular CPLIFS of order $N$. Then $\mathcal{F}$
satisfies the bounded distortion property.

###### Proof.

As there are finitely many words of length at most $N$, the statement
trivially holds for $n<N$. Let $\mathbf{j}=j_{1}\dots j_{n}$ with $n>N$. Let
$x,y\in I$ be arbitrary numbers. First we investigate the derivative of
$f_{\mathbf{j}}$ at a given $x\in I$.

(5.3)
$f_{j_{1},\dots,j_{n}}^{{}^{\prime}}(x)=f_{j_{1}}^{{}^{\prime}}(f_{\sigma\mathbf{j}}(x))\cdots
f_{j_{n-N}}^{{}^{\prime}}(f_{\sigma^{n-N}\mathbf{j}}(x))f_{\sigma^{n-N}\mathbf{j}}^{{}^{\prime}}(x).$

We said that $N$ is the order of our CPLIFS. It means that the functions of
$\mathcal{F}$ have constant slopes over all the $N$ cylinders. Hence
$f_{k}^{{}^{\prime}}(\widehat{x})=f_{k}^{{}^{\prime}}(\widehat{y})$ if
$\widehat{x},\widehat{y}$ are elements of the same cylinder interval of level
$N$.

For $1\leq l\leq n-N$ the words $\sigma^{l}\mathbf{j}$ has length at least
$N$. It implies that $f_{\sigma^{l}\mathbf{j}}(I)$ is a subset of a cylinder
interval of level $N$. Thus $f_{\sigma^{l}\mathbf{j}}(x)$ and
$f_{\sigma^{l}\mathbf{j}}(y)$ are contained in the same cylinder interval of
level $N$ for $1\leq l\leq n-N$.

We conclude that for $1\leq l\leq n-N$

(5.4)
$f_{k}^{{}^{\prime}}(f_{\sigma^{l}\mathbf{j}}(x))=f_{k}^{{}^{\prime}}(f_{\sigma^{l}\mathbf{j}}(y)),$

for any $k\in[m]$.

After substituting (5.4) into (5.3) we obtain that

(5.5)
$C_{1}:=\frac{\rho_{\min}^{N}}{\rho_{\max}}\leq\frac{|f_{j_{1},\dots,j_{n}}^{{}^{\prime}}(x)|}{|f_{j_{1},\dots,j_{n}}^{{}^{\prime}}(y)|}=\frac{|f_{\sigma^{n-N}\mathbf{j}}^{{}^{\prime}}(x)|}{|f_{\sigma^{n-N}\mathbf{j}}^{{}^{\prime}}(x)|}\leq\frac{\rho_{\max}}{\rho_{\min}^{N}}=:C_{2},$

where $\rho_{min}$ and $\rho_{max}$ were defined in (1.15). ∎

If we assume Proposition2.3, we can easily proof Corollary 2.18 with the help
of this Fact.

###### Proof of Corollary 2.18 assuming Proposition 2.3.

According to Proposition 2.3, it is enough to prove our statement for a small
regular CPLIFS $\mathcal{F}=\\{f_{i}\\}_{i=1}^{m}$.

We proceed by applying [5, Theorem 3.2] to $\Lambda^{\mathcal{F}}$. Without
loss of generality we assume that $|\Lambda^{\mathcal{F}}|=1$. Let $r_{0}:=1$
and $a=\frac{\rho_{min}}{2C_{2}}$, where
$C_{2}=\frac{\rho_{\max}}{\rho^{N}_{\min}}$. Fix an arbitrary $0<r<r_{0}$ and
$x\in\Lambda^{\mathcal{F}}$, and consider the interval $D=[x-r,x+r]$. We
choose $(i_{1},\dots,i_{n})\in[m]^{n}$ such that
$x\in\Lambda_{i_{1},\dots,i_{n}}\subset D$ but
$\Lambda_{i_{1},\dots,i_{n-1}}\not\subset D$. Then
$\frac{r}{2}<|\Lambda_{i_{1},\dots,i_{n-1}}|$.

Let $I^{\mathcal{F}}$ be the interval we defined in (1.2). There exists
$x_{1}\in I^{\mathcal{F}}$ for which

(5.6)
$\frac{r}{2}<|\Lambda_{i_{1},\dots,i_{n-1}}|\leq|f_{i_{1},\dots,i_{n-1}}^{{}^{\prime}}(x_{1})|.$

According to Fact 5.3 $\mathcal{F}$ satisfies the bounded distortion property.
Namely, (5.5) implies that for every $x_{2}\in I^{\mathcal{F}}$ we have

(5.7) $|f_{i_{1},\dots,i_{n-1}}^{{}^{\prime}}(x_{1})|\leq
C_{2}|f_{i_{1},\dots,i_{n-1}}^{{}^{\prime}}(f_{n}(x_{2}))|\leq
C_{2}\underbrace{|f_{i_{1},\dots,i_{n-1}}^{{}^{\prime}}(f_{i_{n}}(x_{2}))|\cdot|f_{i_{n}}^{{}^{\prime}}(x_{2})|}_{|f^{{}^{\prime}}_{i_{1},\dots,i_{n}}(x_{2})|}\frac{1}{\rho_{\min}},$

where in the last inequality we used that $\rho_{\min}$ is the smallest slope
in the system in absolute value. Together (5.6) and (5.7) gives that for all
$x_{2}\in I^{\mathcal{F}}$

(5.8)
$|f^{{}^{\prime}}_{i_{1},\dots,i_{n}}(x_{2})|>\frac{\rho_{\min}}{2C_{2}}r=a\cdot
r.$

All the functions in $\mathcal{F}$ are continuous and piecewise linear, hence
for any $y,z\in\Lambda^{\mathcal{F}},\>y<z$ there exists a $\xi\in(y,z)$ such
that

(5.9)
$|f_{i_{1},\dots,i_{n}}(z)-f_{i_{1},\dots,i_{n}}(y)|\geq|f_{i_{1},\dots,i_{n}}^{{}^{\prime}}(\xi)|\cdot(z-y)\geq
a\cdot r(z-y),$

where we used that (5.8) applies for any element of $I^{\mathcal{F}}$.

Since
$f_{i_{1},\dots,i_{n}}\Lambda^{\mathcal{F}}\subset\Lambda^{\mathcal{F}}\cap
D$, all conditions of [5, Theorem 3.2] hold which completes the proof of our
theorem.

∎

######  Fact 5.4.

Let $\mathcal{F}$ be a regular CPLIFS. Then

(5.10)
$\Phi(s)=\lim\limits_{n\to\infty}\frac{1}{n}\log\sum\limits_{i_{1},\dots,i_{n}}|I_{i_{1}\dots
i_{n}}|^{s}.$

That is we can write $\lim$ instead of $\limsup$ in (1.6).

###### Proof.

Since $\mathcal{F}$ is regular, it satisfies the BDP according to Fact 5.3. It
follows that for every $n$ there is a suitably big $k\geq n-N$ and constants
$D_{1},D_{2}$ such that

(5.11)
$D_{1}\left|I_{i_{1},\dots,i_{kN}}\right|\leq\left|I_{i_{1},\dots,i_{n}}\right|\leq
D_{2}\left|I_{i_{1},\dots,i_{kN}}\right|.$

Using this we have that

(5.12) $\displaystyle\Phi(s)$
$\displaystyle=\limsup_{n\rightarrow\infty}\frac{1}{n}\log\sum_{i_{1},\dots,i_{n}}\left|I_{i_{1},\dots,i_{n}}\right|^{s}$
$\displaystyle=\limsup_{k\rightarrow\infty}\frac{1}{kN}\log\sum_{i_{1},\dots,i_{kN}}\left|I_{i_{1},\dots,i_{kN}}\right|^{s}$
$\displaystyle=\limsup_{k\rightarrow\infty}\frac{1}{kN}\log\sum_{i_{1},\dots,i_{kN}}\left|f_{i_{1},\dots
i_{(k-1)N}}\left(I_{i_{(k-1)N+1},\dots,i_{kN}}\right)\right|^{s}$
$\displaystyle=\limsup_{k\rightarrow\infty}\frac{1}{kN}\log\sum_{\mathbf{v}^{1},\dots,\mathbf{v}^{k}\in\mathcal{V}}\left|F_{\mathbf{v}^{1},\mathbf{v}^{2}}\circ\cdots\circ
F_{\mathbf{v}^{k-1},\mathbf{v}^{k}}\left(I_{\mathbf{v}^{k}}\right)\right|^{s}$
$\displaystyle=\limsup_{k\rightarrow\infty}\frac{1}{kN}\log\sum_{\mathbf{v}^{1},\dots,\mathbf{v}^{k}\in\mathcal{V}}|\rho_{\mathbf{v}^{1},\mathbf{v}^{2}}|^{s}\cdots|\rho_{\mathbf{v}^{k-1},\mathbf{v}^{k}}|^{s}\left|\left(I_{\mathbf{v}^{k}}\right)\right|^{s}$
$\displaystyle=\limsup_{k\rightarrow\infty}\frac{1}{kN}\log\sum_{\mathbf{v}^{1},\dots,\mathbf{v}^{k}\in\mathcal{V}}|\rho_{\mathbf{v}^{1},\mathbf{v}^{2}}|^{s}\cdots|\rho_{\mathbf{v}^{k-1},\mathbf{v}^{k}}|^{s},$

where in the last equality we used that the length of any cylinder interval of
level $N$ can be easily bounded by suitable constants.

The series
$c_{k}:=\log\sum_{\mathbf{v}^{1},\dots,\mathbf{v}^{k}\in,\mathcal{V}}|\rho_{\mathbf{v}^{1},\mathbf{v}^{2}}|^{s}\cdots|\rho_{\mathbf{v}^{k-1},\mathbf{v}^{k}}|^{s}$
is subadditive, and hence by Fekete’s lemma the limit
$\lim_{k\rightarrow\infty}\frac{c_{k}}{k}$ exists. Applying this fact to
(5.12) yields

(5.13)
$\Phi(s)=\lim_{k\rightarrow\infty}\frac{1}{kN}\log\sum_{\mathbf{v}^{1},\dots,\mathbf{v}^{k}\in\mathcal{V}}|\rho_{\mathbf{v}^{1},\mathbf{v}^{2}}|^{s}\cdots|\rho_{\mathbf{v}^{k-1},\mathbf{v}^{k}}|^{s}.$

Together (5.13) and (5.12) imply that

(5.14)
$\Phi(s)=\lim_{n\rightarrow\infty}\frac{1}{n}\log\sum_{i_{1},\dots,i_{n}}\left|I_{i_{1},\dots,i_{n}}\right|^{s}.$

∎

###### The proof of Lemma 5.1.

We define

(5.15) $\displaystyle\overline{Q}$
$\displaystyle:=\max_{i_{1},\dots,i_{N}}\left|I_{i_{1},\dots,i_{N}}\right|,\quad\underline{Q}:=\min_{i_{1},\dots,i_{N}}\left|I_{i_{1},\dots,i_{N}}\right|,$
(5.16) $\displaystyle\overline{q}$
$\displaystyle:=\max_{\mathbf{u},\mathbf{v}\in[m]^{N}}\mathfrak{u}_{\mathbf{u}}\upnu_{\mathbf{v}},\quad\underline{q}:=\min_{\mathbf{u},\mathbf{v}\in[m]^{N}}\mathfrak{u}_{\mathbf{u}}\upnu_{\mathbf{v}}.$

By Fact 5.4 and its proof we have

(5.17)
$\Phi(s)=\lim_{k\rightarrow\infty}\frac{1}{kN}\log\sum_{\mathbf{v}^{1},\dots,\mathbf{v}^{k}\in\mathcal{V}}|\rho_{\mathbf{v}^{1},\mathbf{v}^{2}}|^{s}\cdots|\rho_{\mathbf{v}^{k-1},\mathbf{v}^{k}}|^{s}\left|\left(I_{\mathbf{v}^{k}}\right)\right|^{s}.$

Observe that by the definition of measure $\mu$ we have

(5.18)
$\underline{q}|\rho_{\mathbf{v}^{1},\mathbf{v}^{2}}\cdots\rho_{\mathbf{v}^{k-1},\mathbf{v}^{k}}|^{\alpha}\leq\mu([\mathbf{v}^{1},\dots,\mathbf{v}^{k}])\leq\overline{q}|\rho_{\mathbf{v}^{1},\mathbf{v}^{2}}\cdots\rho_{\mathbf{v}^{k-1},\mathbf{v}^{k}}|^{\alpha},$

Now we substitute $s=\alpha(=\alpha_{\mathcal{F}})$ into (5.12), and use the
bounds of (5.18) and the constants introduced in (5.15) to obtain

(5.19) $\displaystyle\Phi(\alpha)$
$\displaystyle\leq\lim_{k\rightarrow\infty}\frac{1}{kN}\log\sum_{\mathbf{v}^{1},\dots,\mathbf{v}^{k}\in\mathcal{V}}|\rho_{\mathbf{v}^{1},\mathbf{v}^{2}}\cdots\rho_{\mathbf{v}^{k-1},\mathbf{v}^{k}}|^{\alpha}\overline{Q}^{\alpha}$
$\displaystyle\leq\lim_{k\rightarrow\infty}\frac{1}{kN}\frac{\overline{Q}^{\alpha}}{\underline{q}}\log\underbrace{\sum_{\mathbf{v}^{1},\dots\mathbf{v}^{k}\in\mathcal{V}}\mu([\mathbf{v}^{1},\dots\mathbf{v}^{k}])}_{=1}=0,$

since $\mu$ is a probability measure. Note that it only holds for
$s=\alpha(\mathcal{F})$, otherwise $\mu$ would fail to be a probability
measure. Similar calculations show that $0$ is also a lower bound.

(5.20)
$\Phi(\alpha)\geq\lim_{k\rightarrow\infty}\frac{1}{kN}\frac{\underline{Q}^{\alpha}}{\overline{q}}\log\underbrace{\sum_{\mathbf{v}^{1},\dots\mathbf{v}^{k}\in\mathcal{V}}\mu([\mathbf{v}^{1},\dots\mathbf{v}^{k}])}_{=1}=0.$

As $s_{\mathcal{F}}$ is the unique zero of $\Phi(s)$, we just proved that
$s_{\mathcal{F}}=\alpha$. ∎

###### Proof of Theorem 2.2.

Now we fix an arbitrary regular CPLIFS $\mathcal{F}$ of order $N$, for which
the generated self-similar IFS $\mathcal{S}_{\mathcal{F}}$ satisfies ESC. Let
$\mathscr{S}:=\mathscr{S}_{\mathcal{F}}:=\\{S_{\mathbf{a}}\\}_{\mathbf{a}\in\mathcal{A}}$
be its associated self-similar system as defined in Section 4.2. Since the ESC
holds for $\mathcal{S}_{\mathcal{F}}$, it also holds for $\mathscr{S}$, as the
later one is a subsystem of $\mathcal{S}_{\mathcal{F}}^{N}$ (see [1]).

Recall that the measure $\mathfrak{m}$ defined in (4.35) satisfies

(5.21)
$\mbox{spt}(\Pi_{\mathscr{S}_{\mathcal{F}}\ast}\mathfrak{m})=\Lambda^{\mathcal{F}}.$

We also know that $h(\mathfrak{m})/\chi(\mathfrak{m})=\alpha$ by (4.42). Now
we can apply the Jordan Rapaport Theorem (Theorem 3.6) to obtain

(5.22) $\min\\{1,\alpha\\}\leq\dim_{\rm H}\Lambda^{\mathcal{F}}.$

Applying Lemma 5.1 yields

(5.23) $\min\\{1,s_{\mathcal{F}}\\}\leq\dim_{\rm
H}\Lambda^{\mathcal{F}}\leq\dim_{\rm
B}\Lambda^{\mathcal{F}}\leq\min\\{1,s_{\mathcal{F}}\\},$

where the last inequality comes from Corollary 1.1. ∎

Now we can prove our Main Theorem.

###### Proof of Theorem 2.1 assuming Proposition 2.3.

Fix an arbitrary type
$\boldsymbol{\ell}=(l(1),\dots,l(m))$ and
$\boldsymbol{\rho}\in\mathfrak{R}_{\mathrm{small}}^{\boldsymbol{\ell}}$. Let
us define the exceptional sets

(5.24)
$E_{\boldsymbol{\rho}}:=\left\\{(\mathfrak{b},\boldsymbol{\tau})\in\mathfrak{B}^{\boldsymbol{\ell}}\times\mathbb{R}^{m}:\dim_{\rm
H}\Lambda^{(\mathfrak{b},\boldsymbol{\tau},\boldsymbol{\rho})}\neq
s_{(\mathfrak{b},\boldsymbol{\tau},\boldsymbol{\rho})}\right\\},$ (5.25)
$E_{\boldsymbol{\rho},\mathrm{ESC}}:=\left\\{(\mathfrak{b},\boldsymbol{\tau})\in\mathfrak{B}^{\boldsymbol{\ell}}\times\mathbb{R}^{m}:\mathscr{S}_{\mathcal{F}^{(\mathfrak{b},\boldsymbol{\tau},\boldsymbol{\rho})}}\mbox{
does not satisfy ESC}\right\\},$ (5.26)
$E_{\boldsymbol{\rho},\mathrm{regular}}:=\left\\{(\mathfrak{b},\boldsymbol{\tau})\in\mathfrak{B}^{\boldsymbol{\ell}}\times\mathbb{R}^{m}:\mathcal{F}^{(\mathfrak{b},\boldsymbol{\tau},\boldsymbol{\rho})}\mbox{
is not regular }\right\\}.$

Using this terminology Theorem 2.2 asserts that

(5.27) $E_{\boldsymbol{\rho}}\subset E_{\boldsymbol{\rho},\mathrm{ESC}}\cup
E_{\boldsymbol{\rho},\mathrm{regular}}.$

Recall that the mapping $\Phi_{\boldsymbol{\rho}}$ (defined in (4.4))
associates each
$(\mathfrak{b},\boldsymbol{\tau})\in\mathfrak{B}^{\boldsymbol{\ell}}\times\mathbb{R}^{m}$
to a vector $\mathbf{t}\in\mathbb{R}^{L+m}$ uniquely.
$\Phi_{\boldsymbol{\rho}}(\mathfrak{b},\boldsymbol{\tau})$ is the translation
vector of
$\mathscr{S}_{\mathcal{F}^{(\mathfrak{b},\boldsymbol{\tau},\boldsymbol{\rho})}}$.
Hence, by Hochmann’s theorem (Theorem 3.5)

(5.28) $\dim_{\rm
P}\Phi_{\boldsymbol{\rho}}(E_{\boldsymbol{\rho},\mathrm{ESC}})<L+m.$

According to Fact 4.1, $\Phi_{\boldsymbol{\rho}}$ preserves Packing dimension,
and thus

(5.29) $\dim_{\rm P}E_{\boldsymbol{\rho},\mathrm{ESC}}<L+m.$

It follows from Proposition 2.3 that

(5.30) $\dim_{\rm H}E_{\boldsymbol{\rho},\mathrm{regular}}<L+m.$

Combining (5.27) with the last two inequalities completes the proof of the
Theorem. ∎

##  6\. The proof of the Main Proposition

###### Proof of Proposition 2.3.

Our aim in this section is to verify Proposition 2.3. That is we will prove
that the attractor $\Lambda^{\mathcal{F}}$ of a $\dim_{\rm P}$-typical small
CPLIFS $\mathcal{F}$ does not contain any breaking points. More precisely, we
prove that for every
$\boldsymbol{\rho}\in\mathfrak{R}_{\mathrm{small}}^{\boldsymbol{\ell}}$

(6.1) $\dim_{\rm
P}\left(E_{\boldsymbol{\rho},\mathrm{regular}}\right)<L+m-1+s_{*},$

where $E_{\boldsymbol{\rho},\mathrm{regular}}$ was defined in (5.26) and
$0<s_{*}<1$ is defined by

(6.2) $\sum\limits_{k=1}^{m}\rho_{k}^{s_{*}}=1.$

By symmetry, it is enough to prove that

(6.3) $b_{1,1}\not\in\Lambda^{\mathcal{F}},\quad\mbox{ for a $\dim_{\rm
P}$-typical small CPLIFS }\mathcal{F}.$

Fix an arbitrary type $\boldsymbol{\ell}=(l(1),\dots,l(m))$ and a
$\boldsymbol{\rho}\in\mathfrak{R}_{\mathrm{small}}^{\boldsymbol{\ell}}$. We
always assume that
$(\mathfrak{b},\boldsymbol{\tau})\in\mathfrak{B}^{\boldsymbol{\ell}}\times\mathbb{R}^{m}$,
and write $\Lambda^{(\mathfrak{b},\boldsymbol{\tau},\boldsymbol{\rho})}$ for
the attractor of the CPLIFS
$\mathcal{F}^{(\mathfrak{b},\boldsymbol{\tau},\boldsymbol{\rho})}$. As usual,
we write $L:=\sum\limits_{k=1}^{m}l(k)$.

For arbitrary $U<V$ we write

(6.4)
$E_{U,V}:=\left\\{(\mathfrak{b},\boldsymbol{\tau})\in\left(\mathfrak{B}^{\boldsymbol{\ell}}\cap(U,V)^{L}\right)\times(U,V)^{m}:b_{1,1}\in\Lambda^{(\mathfrak{b},\boldsymbol{\tau},\boldsymbol{\rho})}\subset(U,V)\right\\}.$

To show that (6.1) holds, it is enough to prove that for all $U<V$ we have

(6.5) $\dim_{\rm P}E_{U,V}<L+m-1+s_{*}.$

From now on we fix $U<V$, and we write

(6.6)
$\Delta:=\left\\{(\mathfrak{b},\boldsymbol{\tau})\in\left(\mathfrak{B}^{\boldsymbol{\ell}}\cap(U,V)^{L}\right)\times(U,V)^{m}:\Lambda^{(\mathfrak{b},\boldsymbol{\tau},\boldsymbol{\rho})}\subset(U,V)\right\\}.$

Let $D$ be an arbitrary closed cube contained in $\Delta$:

(6.7) $D=D_{1}\times D_{2},\mbox{ where }D_{1}\subset\mathbb{R}\mbox{ and
}D_{2}\subset\mathbb{R}^{L+m-1}.$

To verify (6.5) we need to prove that

(6.8) $\dim_{\rm P}(E_{U,V}\cap D)<L+m-1+s_{*}.$

For an $r>0$ we define a kind of symbolic Moran cover

(6.9)
$\mathcal{M}_{r}:=\left\\{(k_{1},\dots,k_{n})\in\Sigma^{*}:|\rho_{k_{1}\dots
k_{n}}|\leq r<|\rho_{k_{1}\dots k_{n-1}|}\right\\}.$

By a standard argument it is easy to see that

(6.10)
$\frac{1}{r^{s_{*}}}\leq\\#\mathcal{M}_{r}<\frac{1}{\rho_{\mathrm{min}}^{s_{*}}r^{s_{*}}}.$

Clearly, for all $r>0$ we have

(6.11) $E_{U,V}\cap D=\bigcup_{(k_{1}\dots
k_{n})\in\mathcal{M}_{r}}\widetilde{K}_{k_{1},\dots,k_{n}}.$

where
$\widetilde{K}_{k_{1},\dots,k_{n}}:=\left\\{(\mathfrak{b},\boldsymbol{\tau})\in
D:b_{1,1}\in\Lambda_{k_{1}\dots
k_{n}}^{(\mathfrak{b},\boldsymbol{\tau},\boldsymbol{\rho})}\right\\}.$ Let
$\widetilde{N}_{k_{1}\dots k_{n}}(r)$ be the minimal number of $r$-mesh cubes
required to cover $\widetilde{K}_{k_{1},\dots,k_{n}}$. Using (6.10), it is
enough to prove that there exists a constant $C_{1}$ which may depend on $D$
but independent of $r$ and $(k_{1},\dots,k_{n})\in\mathcal{M}_{r}$ such that

(6.12) $\widetilde{N}_{k_{1}\dots
k_{n}}(r)\leq\frac{C_{1}}{r^{L+m-1}},\quad\forall r>0,\
(k_{1},\dots,k_{n})\in\mathcal{M}_{r}.$

Namely, putting together this and (6.9) we get from (6.11) that

$\overline{\dim}_{\rm B}\left(E_{U,V}\cap D\right)\leq L+m-1+s_{*}$

for all cubes $D\subset\Delta$, which implies (6.8). Using that $\Delta$ is
the countable union of such cubes we obtain (6.5).

Without loss of generality we may assume that $U=1$. Now we fix an $r>0$ which
is so small that for every $(\mathfrak{b},\boldsymbol{\tau})\in D$ and
$(k_{1},\dots,k_{n})\in\mathcal{M}_{r}$ we have

(6.13) $f_{k_{1}\dots
k_{n}}^{\mathfrak{b},\boldsymbol{\tau},\boldsymbol{\rho}}([0,V+1])\subset\left(\frac{1}{2},V+\frac{1}{2}\right)\mbox{
and }|\rho_{k_{1}\dots k_{n}}|<\min\left\\{|D_{1}|,\frac{1}{6(V-1)}\right\\}.$

We also fix an arbitrary $(k_{1},\dots,k_{n})\in\mathcal{M}_{r}$. Now we
verify (6.12). For the duration of this proof we introduce the following
notation

(6.14)
$\boldsymbol{\eta}:=(b_{1,2},\dots,b_{1,l(1)},\dots,b_{m,1},\dots,b_{m,l(m)},\tau_{1},\dots,\tau_{m})\in
D_{2}.$

That is, instead of $(\mathfrak{b},\boldsymbol{\rho})$ from now on we write
$(b_{1,1},\boldsymbol{\eta})\in D=D_{1}\times D_{2}.$ Let $B_{k}$ be the set
of breaking points of the function
$f_{k}^{\mathfrak{b},\boldsymbol{\tau},\boldsymbol{\rho}}(\cdot)$ and we write
$B^{(b_{1,1},\boldsymbol{\eta})}$ for the set of breaking points of
$f(b_{1,1},\boldsymbol{\eta},\cdot)$. Then

(6.15)
$B^{(b_{1,1},\boldsymbol{\eta})}=B_{k_{n}}\cup\bigcup_{p=2}^{n}f_{k_{n}}^{-1}\circ\cdots\circ
f_{k_{p}}^{-1}B_{k_{p-1}}.$

The complement $\left(B^{(b_{1,1},\boldsymbol{\eta})}\right)^{c}$ is the union
for all $(i_{1},\dots,i_{n})\in[l(k_{1})+1]\times\cdots\times[l(k_{n})+1]$ of
the open intervals

(6.16) $\displaystyle J_{(k_{1},i_{1})\dots(k_{n},i_{n})}\\!\\!\\!\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!J_{k_{n},i_{n}}\cap
S_{k_{n},i_{n}}^{-1}(J_{k_{n-1},i_{n-1}})\cap\cdots\cap
S_{k_{n},i_{n}}^{-1}\circ\cdots\circ S_{k_{2},i_{2}}^{-1}(J_{k_{1},i_{1}})$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!\left\\{x\in
J_{k_{n},i_{n}}\\!\\!:\\!\\!S_{k_{n},i_{n}}(x)\in
J_{k_{n-1},i_{n-1}},\dots,S_{(k_{2},i_{2})\cdots(S_{k_{n},i_{n}})}(x)\in
J_{k_{1},i_{1}}\right\\}.$

Let $\overline{J}_{(k_{1},i_{1})\dots(k_{n},i_{n})}$ be the closure of
$J_{(k_{1},i_{1})\dots(k_{n},i_{n})}$, then by the formulas (4.1) and (4.3) we
obtain that for all $x\in\overline{J}_{(k_{1},i_{1})\dots(k_{n},i_{n})}$ we
have

(6.17) $f_{k_{1}\dots
k_{n}}^{\mathfrak{b},\boldsymbol{\tau},\boldsymbol{\rho}}(x)=\widetilde{\rho}_{n}\cdot
x\\\
+\sum\limits_{j=1}^{n}\widetilde{\rho}_{j-1}\underbrace{\left(\tau_{k_{j}}+b_{k_{j},1}(\rho_{k_{j},1}-\rho_{k_{j},2})+\cdots+b_{k_{j},i_{j}}(\rho_{k_{j},i_{j}}-\rho_{k_{j},i_{j}+1})\right)}_{t_{k_{j},i_{j}}},$

where $\widetilde{\rho}_{q}:=\prod\limits_{z=1}^{q}\rho_{k_{z},i_{z}}$. Using
this, a simple calculation shows that the following Fact holds:

######  Fact 6.1.

For a fixed $x$ and $k_{1},\dots,k_{n}$ the following hold

  1. (a)

the function $f_{k_{1}\dots k_{n}}(x,\mathfrak{b},\boldsymbol{\tau})$ is
piecewise linear in all of its variables $\tau_{k}$ and $b_{p,q}$. So, if we
fix all variables but one then the partial derivative, against the variable
which is not fixed, exists at all but finitely many points. The next two
assertions are meant in the sense that the estimates hold where the partial
derivatives exist.

  2. (b)

For every $p\in[m]$ and $q\in[l(p)]$ we have

(6.18) $\left|\frac{\partial f_{k_{1}\dots
k_{n}}(x,\mathfrak{b},\boldsymbol{\tau})}{\partial
b_{p,q}}\right|<\gamma:=\sum\limits_{k=1}^{\infty}\rho_{\max}^{k}<1.$

  3. (c)

For every $k\in[m]$

(6.19) $\left|\frac{\partial f_{k_{1}\dots
k_{n}}(x,\mathfrak{b},\boldsymbol{\tau})}{\partial\tau_{k}}\right|<2.$

Clearly,

(6.20) $\Lambda^{(b_{1,1},\boldsymbol{\eta})}_{k_{1}\dots k_{n}}\subset
I^{(b_{1,1},\boldsymbol{\eta})}:=\left(f^{(b_{1,1},\boldsymbol{\eta})}_{k_{1}\dots
k_{n}}(1)-(V-1)\rho_{k_{1}\dots
k_{n}},f^{(b_{1,1},\boldsymbol{\eta})}_{k_{1}\dots
k_{n}}(1)+(V-1)\rho_{k_{1}\dots k_{n}}\right),$

and we also have

(6.21) $\widetilde{K}_{k_{1}\dots k_{n}}\subset K_{k_{1}\dots
k_{n}}:=\left\\{(b_{1,1},\boldsymbol{\eta})\in D:b_{1,1}\in
I^{(b_{1,1},\boldsymbol{\eta})}\right\\}.$

Like above, we write $N_{k_{1}\dots k_{n}}(r)$ for the minimal number of
$r$-mesh cubes required to cover the set $K_{k_{1}\dots k_{n}}$. By (6.21)
$\widetilde{N}_{k_{1}\dots k_{n}}(r)\leq N_{k_{1}\dots k_{n}}(r)$. Hence to
verify (6.12) we only need to prove that

######  Lemma 6.2.

There is a constant $C_{1}$ which is independent of $r$ and
$(k_{1},\dots,k_{n})\in\mathcal{M}_{r}$ such that

(6.22) $N_{k_{1}\dots k_{n}}(r)\leq\frac{C_{1}}{r^{L+m-1}},\quad\forall r>0,\
(k_{1},\dots,k_{n})\in\mathcal{M}_{r}.$

###### Proof of Lemma 6.2.

For our arbitrary but fixed $r>0$ and $(k_{1},\dots,k_{n})\in\mathcal{M}_{r}$
we define the mapping $G:D\to\left(\frac{1}{3},V+\frac{2}{3}\right)$,

(6.23)
$G(b_{1,1},\boldsymbol{\eta}):=f^{(b_{1,1},\boldsymbol{\eta})}_{k_{1}\dots
k_{n}}(1)-(V-1)\rho_{k_{1}\dots k_{n}}.$

Observe that (6.18) gives

(6.24) $\left|\frac{\partial}{\partial
b_{1,1}}G(b_{1,1},\boldsymbol{\eta})\right|<\gamma\quad\mbox{ for all
}\boldsymbol{\eta}\in D_{2}.$

Then for $w:=2(V-1)\rho_{k_{1}\dots k_{n}}$ we have

(6.25) $K_{k_{1}\dots k_{n}}=\left\\{(b_{1,1},\boldsymbol{\eta})\in
D:b_{1,1}-w<G(b_{1,1},\boldsymbol{\eta})<b_{1,1}\right\\}.$

We introduce the stripe $Z:=\left\\{(x,y)\in
D_{1}\times\mathbb{R}:x-w<y<x\right\\}$ and for an $\boldsymbol{\eta}\in
D_{2}$ we write

(6.26) $K_{k_{1}\dots k_{n}}^{\boldsymbol{\eta}}:=\left\\{b_{1,1}\in
D_{1}:(b_{1,1},G(b_{1,1},\boldsymbol{\eta}))\in Z\right\\}.$

It follows from (6.18) that for all $\boldsymbol{\eta}\in D_{2}$ we have

(6.27)
$|G(b_{1,1},\boldsymbol{\eta})-G(b_{1,1}+u,\boldsymbol{\eta})|<\gamma|u|,\quad\
b_{1,1},b_{1,1}+u\in D_{1},\boldsymbol{\eta}\in D_{2}.$

This immediately implies that

######  Fact 6.3.

For every $x$ we have

  1. (a)

$\bigg{(}G(b_{1,1},\boldsymbol{\eta})\geq
b_{1,1}-x\bigg{)}\Longrightarrow\bigg{(}G(b_{1,1}-u,\boldsymbol{\eta})>b_{1,1}-u-x\bigg{)},$
if $b_{1,1},b_{1,1}-u\in D_{1}$,

  2. (b)

$\bigg{(}G(b_{1,1},\boldsymbol{\eta})\leq
b_{1,1}-x\bigg{)}\Longrightarrow\bigg{(}G(b_{1,1}+u,\boldsymbol{\eta})<b_{1,1}+u-x\bigg{)},$
if $b_{1,1},b_{1,1}+u\in D_{1}$.

Let us write $d_{l}$ and $d_{r}$ for the left and right endpoint of $D_{1}$
respectively. For any set $A\subset\mathbb{R}^{L+m}$ we define a projective
mapping

(6.28) $\mathrm{proj}_{2}(A):=\left\\{\boldsymbol{\eta}:\>\exists
b_{1,1}\mbox{ such that }(b_{1,1},\boldsymbol{\eta})\in A\right\\}.$

Fact 6.3 implies that

(6.29) $\boldsymbol{\eta}\in\mathrm{Proj}_{2}(K_{k_{1}\dots
k_{n}})\Longleftrightarrow d_{l}-w<G(d_{l},\boldsymbol{\eta})\quad\&\quad
G(d_{r},\boldsymbol{\eta})<d_{r}.$

That is

(6.30) $\boldsymbol{\eta}\not\in\mathrm{Proj}_{2}(K_{k_{1}\dots
k_{n}})\Longleftrightarrow G(d_{l},\boldsymbol{\eta})\leq d_{l}\mbox{ or
}G(d_{r},\boldsymbol{\eta})\geq d_{r}.$

Assume that $d_{l}<G(d_{l},\boldsymbol{\eta})$ and
$G(d_{r},\boldsymbol{\eta})<d_{r}$ holds. Then there is a fixed point of the
function $G(\cdot,\boldsymbol{\eta})$. Let us denote it by
$\mathrm{Fix}(G(\cdot,\boldsymbol{\eta}))$. Now we define the function
$q:D_{2}\to[d_{l},d_{r}]$ as follows:

(6.31)
$q(\boldsymbol{\eta}):=\left\\{\begin{array}[]{ll}\mathrm{Fix}(G(\cdot,\boldsymbol{\eta})),&\hbox{
if $d_{l}<G(d_{l},\boldsymbol{\eta})$ and
$G(d_{r},\boldsymbol{\eta})<d_{r}$;}\\\ d_{l},&\hbox{if
$G(d_{l},\boldsymbol{\eta})\leq d_{l}$;}\\\ d_{r},&\hbox{if
$G(d_{r},\boldsymbol{\eta})\geq d_{r}$.}\end{array}\right.$

Figure 3. Visualization of the function $q(\boldsymbol{\eta})$ defined in
(6.31). The continuous piecewise linear map $G(\cdot,\boldsymbol{\eta})$ is
drawn for three different $\boldsymbol{\eta}$ values to show examples for all
cases.

######  Fact 6.4.

The function $q:D_{2}\to[d_{l},d_{r}]$ has the following properties

  1. (a)

$q(\boldsymbol{\eta})$ is piecewise affine.

  2. (b)

There is a constant $K_{0}$ such that all of the partial derivatives of the
function $q(\boldsymbol{\eta})$ are less than $K_{0}$ in absolute value.

Part (a) is immediate from the formula (6.17). We obtain part (b) from (6.18)
and (6.19) either by a direct calculation or by applying the Implicit Function
Theorem. Now we put together (6.24), (6.26) and Fact 6.3 to obtain

(6.32) $K_{k_{1}\dots
k_{n}}^{\boldsymbol{\eta}}\subset\left[q(\boldsymbol{\eta}),q(\boldsymbol{\eta})+\frac{2(V-1)\rho_{k_{1}\dots
k_{n}}}{1-\gamma}\right],\quad\mbox{ for all }\quad\boldsymbol{\eta}\in
D_{2}.$

In this way $K_{k_{1}\dots k_{n}}\subset W_{k_{1}\dots k_{n}}$, where

(6.33) $W_{k_{1}\dots
k_{n}}:=\left\\{(b_{1,1},\boldsymbol{\eta}):\boldsymbol{\eta}\in
D_{2},b_{1,1}\in\left[q(\boldsymbol{\eta}),q(\boldsymbol{\eta})+\frac{2(V-1)\rho_{k_{1}\dots
k_{n}}}{1-\gamma}\right]\right\\}$

Recall that $D=D_{1}\times D_{2}$ is a cube and $D_{1}=(d_{l},d_{r})$. That is
$D_{2}$ is a translate of the cube $(0,d_{r}-d_{l})^{L+m-1}$. Hence we can
cover $D_{2}$ by $\left(\left\lceil\frac{d_{r}-d_{l}}{\rho_{k_{1}\dots
k_{n}}}\right\rceil+1\right)^{L+m-1}\quad$ ($L+m-1$)-dimensional $r$-mesh
cubes, since $(k_{1},\dots,k_{n})$ $\in\mathcal{M}_{r}$ and in this way

(6.34) $\rho_{\min}r<\rho_{k_{1}\dots k_{n}}\leq r.$

Let $Q$ be one of these $(L+m-1)$-dimensional $r$-mesh cubes. Then it follows
from Fact 6.4 and (6.32) that the set $\mathrm{Proj}_{2}^{-1}(Q)\cap
W_{k_{1}\dots k_{n}}$ can be covered by $(L+m-1)\cdot
K_{0}\cdot\frac{2(V-1)}{1-\gamma}\quad$ $(L+m)$-dimensional $r$-mesh cubes.
That is we can cover $W_{k_{1}\dots k_{n}}$ (and consequently $K_{k_{1}\dots
k_{n}}$) by

$\left(\left\lceil\frac{d_{r}-d_{l}}{\rho_{k_{1}\dots
k_{n}}}\right\rceil+1\right)^{L+m-1}\cdot(L+m-1)\cdot
K_{0}\cdot\frac{2(V-1)}{1-\gamma}\leq\frac{C_{1}}{r^{L+m-1}},$

$r$-mesh cubes with a suitabe constant $C_{1}$ independent of $r$ and
$(k_{1},\dots,k_{n})\in\mathcal{M}_{r}$, where in the last step we used
(6.34). This completes the proof of the Lemma. ∎

As we already mentioned, Lemma 6.2 implies that (6.12) holds. That is, using
(6.10) and (6.11), we can cover $E_{U,V}\cap D$ with at most
$\frac{C_{2}}{r^{L+m-l+s_{*}}}\quad$ $(L+m)$-dimensional $r$-mesh cubes. As it
holds for all $D\subset\Delta$ and $\Delta$ is the countable union of such
cubes, we just obtained (6.5). Since $U$ and $V$ were arbitrary, the proof of
the Main Proposition follows.

∎

##  7\. Appendix

The purpose of the Appendix is to make explicit an important consequence of
Jordan and Rapaport’s result [10, Theorem 1.1] related to the dimension of the
attractors of graph-directed self-similar systems on $\mathbb{R}$.

Given a self-similar graph-directed IFS $\mathcal{F}$ with a strongly
connected graph. We will construct an ergodic and invariant measure $\mu$ on
the symbolic space which is a Markov measure with the following property: the
entropy of $\mu$ divided by the Lypunov exponent of $\mu$ is equal to
$\alpha(\mathcal{F})$, where $\alpha(\mathcal{F})$ was defined in Definition
3.7). That is why $\mu$ can be considered as the natural measure for the self-
similar graph-directed IFS $\mathcal{F}$.

Using this and the Jordan-Rapaport Theorem (Theorem 3.6), we obtain in
Corollary 7.2, that the Hausdorff dimension of the push-forward measure of
$\mu$ is the minimum of $1$ and $\alpha(\mathcal{F})$ if ESC holds for the
self-similar IFS $\mathcal{S}$ associated to $\mathcal{F}$. This can be
considered as a generalization of part (b) of Hochman Theorem (Theorem 3.2)
for graph-directed self-similar systems.

Throughout the Appendix we use the notation of Section 3.2. Consider a self-
similar graph-directed iterated function system
$\mathcal{F}=\\{F_{e}(x)=r_{e}x+t_{e}\\}_{e\in\mathcal{E}}$ with directed
graph $\mathcal{G}=\\{\mathcal{V},\mathcal{E}\\}$. We may identify

(7.1) $\mathcal{V}=[q]=\left\\{1,\dots,q\right\\},\mbox{ and
}\mathcal{E}=\left\\{e_{1},\dots,e_{M}\right\\}.$

We assume that $\mathcal{G}$ is strongly connected. Let $\mathcal{E}_{\infty}$
be the set of infinite length directed paths in $\mathcal{G}$. Moreover, for
$l\in[q]$ we introduce

$\mathcal{E}_{\infty}^{\ell}:=\left\\{\mathbf{e}=(e_{1},e_{2},\dots)\in\mathcal{E}_{\infty}:s(e_{1})=\ell\right\\}.$

We identify $\mathcal{E}_{\infty}$ with
$\Sigma_{Z}:=\left\\{(i_{1},i_{2},\dots)\in[M]^{\mathbb{N}}:z_{i_{k},i_{k+1}}=1,\>\forall
k\in\mathbb{N}\right\\}$, where $Z=(z_{i,j})_{i,j=1}^{m}$ is an $M\times M$
matrix such that

(7.2) $z_{i,j}=\left\\{\begin{array}[]{ll}1,&\hbox{if $t(e_{i})=s(e_{j})$;}\\\
0,&\hbox{otherwise.}\end{array}\right.$

Set $\Pi_{\mathcal{F}}(e_{1},e_{2},\dots):=\lim_{n\to\infty}F_{e_{1}\dots
e_{n}}(0)$. It is clear that the non-emty compact sets
$\left\\{\Pi_{\mathcal{F}}(\mathcal{E}^{\ell}_{\infty})\right\\}_{\ell\in[q]}$
satisfy (3.21). Hence

(7.3) $\Lambda_{i}=\Pi_{\mathcal{F}}(\mathcal{E}^{i}_{\infty}),\mbox{ and
}\Lambda=\Pi_{\mathcal{F}}(\mathcal{E}_{\infty})$

We define the natural pressure function as

(7.4)
$\Phi(s)=\lim_{n\rightarrow\infty}\frac{1}{n}\log\sum_{\mathbf{e}\in\mathcal{E}^{n}}|F_{\mathbf{e}}(\Lambda_{t(e_{n})})|^{s},$

where the existence of the limit follows from the same standard argument used
in the proof of Fact 5.4. We assumed that $\mathcal{G}$ is strongly connected,
which implies that the matrix $C^{(s)}$ we defined in (3.23) is irreducible.
It is easy to see that by the Perron Frobenius Theorem we have

(7.5) $\Phi(s)=\log\varrho(C^{(s)}),$

where $\varrho$ is the spectral radius. According to [12, Theorem 2] $\Phi(s)$
is continuous, strictly decreasing, $\Phi(0)>0$, and
$\lim_{s\rightarrow\infty}\Phi(s)=-\infty$, thus there exists a unique
$0<s_{\mathcal{F}}$ for which $\Phi(s_{\mathcal{F}})=0$. For a self-similar
GDIFS with strongly connected graph we call $s_{\mathcal{F}}$ the natural
dimension of the system. By Definition 3.7 it is clear that

(7.6) $s_{\mathcal{F}}=\alpha.$

######  Definition 7.1.

We call $\mathcal{S}=\left\\{S_{k}(x)=r_{e_{k}}x+t_{e_{k}}\right\\}_{k=1}^{M}$
the self-similar IFS associated with the self-similar GDIFS $\mathcal{F}$.
Clearly,

(7.7) $S_{k}|_{\Lambda_{t(e_{k})}}\equiv F_{e_{k}}|_{\Lambda_{t(e_{k})}}.$

Now we show that Part (a) of Theorem 3.8 combined with Jordan Rapaport Theorem
(Theorem 3.6) implies that under some conditions, part (b) of Hochman’s
theorem holds for self-similar graph directed attractors as well, with
$s_{\mathcal{F}}$ in the place of $\dim_{\rm S}\Lambda$.

######  Corollary 7.2.

Let $\mathcal{F}=\left\\{F_{e}\right\\}_{e\in\mathcal{E}}$ be a self-similar
GDIFS on $\mathbb{R}$ with directed graph
$\mathcal{G}=(\mathcal{V},\mathcal{E})$ and attractor $\Lambda$. Further,
assume that $\mathcal{G}$ is strongly connected, and that the self-similar IFS
$\mathcal{S}$ associated to $\mathcal{F}$ satisfies the ESC. Then

(7.8) $\dim_{\rm H}\Lambda=\min\\{1,s_{\mathcal{F}}\\}.$

###### Proof.

Since $\mathcal{G}$ is strongly connected, we can apply Theorem 3.8 to obtain

(7.9) $\dim_{\rm H}\Lambda\leq\min\\{\alpha,1\\}\mbox{, since
}\Lambda\subset\mathbb{R}.$

In order to prove the opposite inequality, first we define an ergodic
invariant measure $\mu$ on $\mathcal{V}^{\mathbb{N}}$. To do so, we recall the
definition of $\alpha$ from (3.24) and using that we consider the irreducible
matrix $A=(a_{i,j})_{i,j=1}^{q}:=C^{(\alpha)}$. Let
$\mathbf{u}=(u_{1},\dots,u_{q})$ and $\mathbf{v}=(v_{1},\dots v_{q})$ be the
left and right eigenvector of $A$ respectively, corresponding to the leading
eigenvalue $1$, normalized in a way that

(7.10)
$\sum\limits_{k=1}^{q}u_{k}v_{k}=1,\quad\sum\limits_{k=1}^{q}v_{k}=1,\quad
u_{k},v_{k}>0.$

We define the stochastic matrix $P=(p(i,j))_{i,j=1}^{q}$ and its stationary
distribution $\mathbf{p}=(p_{1},\dots,p_{q})$, related to the matrix $A$. That
is

(7.11) $p(i,j):=\frac{a(i,j)v_{j}}{v_{i}},\quad p_{i}:=u_{i}\cdot v_{i},\quad
i,j\in[q].$

Clearly, $\mathbf{p}$ is a probability vector and we have

(7.12) $\mathbf{p}^{T}\cdot P=\mathbf{p}^{T}.$

With the help of these we can define the $(\mathbf{p},P)$ Markov measure $\mu$
on $n$-cylinders $[v_{1},\dots,v_{n}]\in\mathcal{V}^{n}$ as follows [16, p.
22]

(7.13) $\mu([v_{1},\dots,v_{n}]):=p_{v_{1}}\cdot
p_{v_{1},v_{2}}p_{v_{2},v_{3}}\cdots p_{v_{n-1},v_{n}}.$

According to [16, Theorem 1.19] this extends to an ergodic, invariant measure
on $\mathcal{V}^{\mathbb{N}}$. There is a natural bijection between
$\mathrm{spt}(\mu)\subset\mathcal{V}^{\mathbb{N}}$ and $\mathcal{E}_{\infty}$,
where $\mathrm{spt}(\cdot)$ denotes the support of the measure and

(7.14)
$\mathrm{spt}(\mu)=\left\\{(v_{1},v_{2},\dots)\in\mathcal{V}^{\mathbb{N}}:(v_{i},v_{i+1})\in\mathcal{E},\mbox{
for all }i\in\mathbb{N}\right\\}.$

For a $(v_{1},v_{2},\dots)\in\mathrm{spt}(\mu)$ we define

$\vartheta(v_{1},v_{2},\dots):=((v_{1},v_{2}),(v_{2},v_{3}),\dots)\in\mathcal{E}_{\infty},$

where the inclusion holds since $\mu$ is only supported on vertex series
$(v_{1},v_{2},\dots)$ $\in\mathcal{V}^{\mathbb{N}}$ for which
$(v_{i},v_{i+1})\in\mathcal{E}$ for all $i\geq 1$. Thus the pushforward
measure $\nu:=\vartheta_{*}\mu$ onto $\mathcal{E}_{\infty}$ is also ergodic
and invariant with respect to the left shift $\sigma$. Observe that there is a
natural embedding
$\Psi:\left(\mathcal{E}_{\infty},\sigma\right)\to\left([M]^{\mathbb{N}},\sigma\right)$
defined as $\Psi(e_{i_{1}},e_{i_{2}},\dots):=(i_{1},i_{2},\dots)$, where we
write $\sigma$ also for left shift on $[M]^{\mathbb{N}}$. With the help of the
matrix (7.2) we already defined $\Sigma_{Z}=\Psi(\mathcal{E}_{\infty})$. The
push forward measure $\Psi_{*}\nu$ is an invariant ergodic measure supported
on $\Sigma_{Z}$.

Let $\Pi_{\mathcal{S}}:[M]^{\mathbb{N}}\to\mathbb{R}$ be the natural
projection corresponding to $\mathcal{S}$. All the symbolic spaces and their
relations we introduced so far are summarized on these diagrams:

(7.15)
$\textstyle{\mathrm{spt}(\mu)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{\vartheta}$$\textstyle{\mathrm{spt}(\mu)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vartheta}$$\textstyle{\mathcal{E}_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{\Psi}$$\textstyle{\mathcal{E}_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi}$$\textstyle{\Sigma_{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{\Sigma_{Z}}$
$\textstyle{\mathcal{E}_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi}$$\scriptstyle{\Pi_{\mathcal{F}}}$$\textstyle{\Sigma_{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Pi_{\mathcal{S}}}$$\textstyle{\Lambda}$

It is immediate that both diagrams are commutative. This and (7.3) imply that

(7.16) $\mathrm{spt}\left(\Pi_{\mathcal{S}\ast}(\Psi_{*}\nu)\right)=\Lambda.$

Moreover, easy calculations give us the entropy and the Lyapunov exponent of
$\Psi_{*}\nu$:

(7.17) $\displaystyle h(\Psi_{*}\nu)$
$\displaystyle=-\lim_{n\to\infty}\frac{1}{n}\log\nu(\mathbf{e}|_{n})=-\alpha\lim_{n\to\infty}\frac{1}{n}\log
r_{\mathbf{e}|n}$ $\displaystyle\chi_{\Psi_{*}\nu}$
$\displaystyle=-\lim_{n\to\infty}\frac{1}{n}\log r_{\mathbf{e}|n},$

for $\nu$-almost all $\mathbf{e}=(e_{1},e_{2},\dots)\in\mathcal{E}_{\infty}$,
where $r_{\mathbf{e}|n}=r_{e_{1}\dots e_{n}}$ stands for the contraction ratio
of $F_{\mathbf{e}|n}$. Hence we have

(7.18) $\frac{h(\Psi_{*}\nu)}{\chi_{\Psi_{*}\nu}}=\alpha.$

We assumed that the ESC holds for $\mathcal{S}$, and we have seen that
$\Psi_{*}\nu$ is an invariant and ergodic probability measure. Thus we can
apply the Jordan Rapaport Theorem (Theorem 3.6) to obtain that

(7.19) $\dim_{\rm H}\Lambda\geq\min\\{\alpha,1\\},$

by substituting (7.16) and (7.18) into the theorem.

Together (7.19) and (7.9) yields

(7.20) $\min\\{\alpha,1\\}\leq\dim_{\rm H}\Lambda\leq\min\\{\alpha,1\\}.$

According to (7.6) $s_{\mathcal{F}}=\alpha$, hence (7.20) implies

$\dim_{\rm H}\Lambda=\min\\{s_{\mathcal{F}},1\\}.$

∎

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