# New bounds for the Hausdorff dimension
of a dynamically defined Cantor set

Fernando Sánchez-Salas  Departamento de Matemáticas, Facultad Experimental de
Ciencias, Universidad del Zulia, Avenida Universidad, Edificio Grano de Oro,
Maracaibo, Venezuela<EMAIL_ADDRESS>
(Date: April 13, 2021)

###### Abstract.

In this paper we use the additive thermodynamic formalism to obtain new bounds
of the Hausdorff and box-counting dimension of certain non conformal
hyperbolic repellers defined by piecewise smooth expanding maps on a
$d$-dimensional smooth manifold $M$.

##  1\. Introduction

In many important situations the study of the time evolution of a system can
be reduced to the iterations of a map $f$ over a phase space $M$. These
iterations produce recursive sequences, called orbits, which are mathematical
representions of the time evolution of the system. If we are interested in a
qualitative description of the dynamics we may direct our inquiries to the
geometry of these recursively defined sets. This approach motivate the study
of topological properties of dynamical systems. The idea goes back to the work
of Poincaré and is the subject of the modern theory of dynamical systems. See
[7]

In this note we study some geometric properties of a class of sets originated
from the iteration of piecewise differentiable maps $f$ over smooth manifolds
$M$. Concretely we are interested in the fractal geometry of Cantor sets
defined by iterations of a piecewise expanding maps $f:\bigcup_{i}U_{i}\to M$.
For instance, the classical ternary Cantor set can be seen as the set of
points whose orbits do not escape from the intervals $[0,1/3]\cup[2/3,1]$
under the iterations of the expanding map $f(x)=3x\mod 1$ acting on the circle
$M=\mathbb{S}^{1}$.

The fractal geometry of sets and measures studies irregular sets such as non
rectifiable curves and Cantor sets. To do this, important dimensional
characteristics as the Hausdorff and box-counting dimensions are introduced.
Theses quantities capture some important geometric properties of fractal sets.
It is also important to determine whether or not the Hausdorff measure, used
to calculate the Hausdorff dimension, it is finite and positive as well as the
continuity of these quantities under small perturbations, among other things.
We refer to [4] and [12] for a complete exposition of the theory and some of
its applications.

In this paper we consider the case of non conformal hyperbolic repellers
defined by piecewise $C^{r}$, $r>1$ differentiable expanding maps. We refer to
[1] and [16] for an overview of problems and results on the subject.
Concretely we use the additive thermodynamic formalism to obtain some useful
bounds for the Hausdorff and box-counting dimension of dynamically defined
Cantor sets. The problem had been discussed previously in [6], [9], [10],
[11], [13], [14], [18]. Our approach will rely upon some ideas in [20], where
classical and well known ideas from the (additive) thermodynamic formalism
were used to bound the Hausdorff dimension of a nonuniformly expanding
repeller. In next section we recall how thermodynamic formalism appears
naturally in the dimension theory of dynamical systems, to motivate the
statement of results.

##  2\. Thermodynamics and dimension: the setting

Let us start recalling the notion of Hausdorff dimension of a set. Let $(X,d)$
a complete metric space and $Z\subset X$ a subset. We say that a countable
collection of open sets $\mathcal{U}=\\{U_{i}\\}$ of $Z$ is a
$\delta$-covering if $Z\subset\bigcup_{i}U_{i}$ and
$\mbox{diam}(U_{i})<\delta$ for every $i$. We define an outer measure:

$\mathcal{H}_{a,\delta}(Z)=\inf_{\mathcal{U}}\sum_{i=1}^{+\infty}\mbox{diam}(U_{i})^{a}$

where infimum is taken over all the $\delta$-coverings of $Z$. The outer
measure $\mathcal{H}_{a,\delta}(Z)$ is a non decreasing function of $\delta$
so we define the $a$-Hausdorff measure of $Z$ as

$\mathcal{H}_{a}(Z)=\lim_{\delta\to
0^{+}}\mathcal{H}_{a,\delta}(Z)=\sup_{\delta>0}\mathcal{H}_{a,\delta}(Z)$

since $\mathcal{H}_{a,\delta}(Z)$ is non decreasing in $\delta$. This is the
well known Carathéodory’s method. $\mathcal{H}_{a}$ is a Borel regular measure
and can be equally defined using coverings by closed or convex sets. Moreover,
it holds that, if $\mathcal{H}_{a}(Z)>0$ the $\mathcal{H}_{s}(Z)=+\infty$ for
every $s<a$. Similarly so if $\mathcal{H}_{a}(Z)<+\infty$ then
$\mathcal{H}_{s}(Z)=0$ for every $s>a$. See [12, Theorem 4.7, Chapter 4]. This
property permits to define the Hausdorff dimension of a subset $Z\subset X$
as:

$\dim_{\mathcal{H}}\,(Z)=\inf\\{a>0:\mathcal{H}_{a}(Z)=0\\}=\sup\\{a>0:\mathcal{H}_{a}(Z)=+\infty\\}$

This is a dimensional characteristic of the set. It satisfies the following
well-known properties:

  1. (1)

$\dim_{\mathcal{H}}\,(Z)\leq\dim_{\mathcal{H}}\,(Y)$ if $Z\subseteq Y$;

  2. (2)

$\dim_{\mathcal{H}}\,(\\{point\\})=0$ and

  3. (3)

$\dim_{\mathcal{H}}\,\left(\bigcup_{i=1}^{+\infty}Z_{i}\right)=\sup_{i>0}\dim_{\mathcal{H}}\,(Z_{i})$.

The Hausdorff dimension is invariant under Lipschitz continuous map. See for
instance [4] and [12]. Another important dimensional characteristic of a set
is the box countig dimension or limit capacity: let $Z\subset X$ be a compact
set and define:

$\mathcal{N}(X,\rho)=\inf\\{\ \\#\,\mathcal{U}:\
\mathcal{U}=\\{B(x_{i},\rho)\\},\ \text{finite covering of $Z$ by
$\rho$-balls}\ B(x,\rho)\ \\}$

Then,

$\overline{\dim_{B}}(Z)=\limsup_{\rho\to
0^{+}}\dfrac{\log\,\mathcal{N}(X,\rho)}{\log(1/\rho)}\quad\text{and}\quad\underline{\dim_{B}}(Z)=\liminf_{\rho\to
0^{+}}\dfrac{\log\,\mathcal{N}(X,\rho)}{\log(1/\rho)}$

are respectively the upper box-counting dimension and lower box-counting
dimension of a set $Z$. The box-counting dimension

$\dim_{B}(Z)=\lim_{\rho\to
0^{+}}\dfrac{\log\,\mathcal{N}(X,\rho)}{\log(1/\rho)}$

is defined if the limit exists. Box-counting dimension can also be defined
using the Carathéodory method: let $Z\subset X$ and $a>0$ we define the
measure

$B_{a,\rho}(Z)=\inf_{\mathcal{U}}\sum_{i}\mbox{diam}(B(x_{i},\rho))^{a},$

where infimum is taken over the family of finite open coverings by
$\rho$-balls $\mathcal{U}=\\{B(x_{i},\delta)\\}$. Then, up to a constant,
$B_{a,\rho}(Z)=\mathcal{N}(Z,\rho)\rho^{a}$ and we then define

$\overline{B}_{a}(Z)=\limsup_{\rho\to 0^{+}}B_{a,\rho}(Z).$

If $\overline{B}_{a}(Z)=0$ then $\overline{\dim_{B}}(Z)\leq a$ for, in this
case, given $\epsilon>0$ there exists $\rho_{0}>0$ such that

$\mathcal{N}(Z,\delta)\rho^{a}<\epsilon,\quad\text{for every}\quad
0<\rho<\rho_{0},$

which implies

$\dfrac{\log(\mathcal{N}(X,\rho))}{\log(1/\rho)}<a+\dfrac{\epsilon}{\log(1/\rho)},$

for every sufficiently small $\rho>0$. Therefore,

$\limsup_{\rho\to
0^{+}}\dfrac{\log(\mathcal{N}(X,\rho))}{\log(1/\rho)}\quad\leq\quad a.$

Hence, $\overline{\dim_{B}}(Z)\leq\inf\\{a>0:\overline{B}_{a}(Z)=0\\}$.
Similarly for the lower dimension:

$\sup\\{a>0:\quad\underline{B}_{a}(Z)=+\infty\\}\quad\leq\quad\underline{\dim_{B}}(Z)$

where, $\underline{B}_{a}(Z)=\liminf_{\rho\to 0^{+}}B_{a,\rho}(Z)$. Finally,
if $\dim_{B}(\Lambda)=\underline{\dim_{B}}(Z)=\overline{\dim_{B}}(Z)$ then

$\dim_{B}(\Lambda)=\inf\\{a>0:B_{a}(Z)=0\\}=\sup\\{a>0:B_{a}(Z)=+\infty\\},$

where $B_{a}(Z)=\lim_{\rho\to 0^{+}}B_{a,\rho}(Z)$. It follows from the
definitions that,

$\dim_{\mathcal{H}}\,(Z)\leq\underline{\dim_{B}}(Z)\leq\overline{\dim_{B}}(Z),$

since every covering by $\rho$-balls is a $\rho$-covering and therefore,
$\mathcal{H}_{a,\rho}(Z)\leq\underline{B}_{a,\rho}(Z)\leq\overline{B}_{a,\rho}(Z)$
for every $a>0$ and $\rho>0$.

We are concerned with the Hausdorff dimension of the limit set of certain
geometric constructions. Let us describe our basic model. Let $N\subset M$ be
a compact domain with non empty interior. What we have in mind is basically a
closed $d$-cube $[0,1]^{d}$ embedded in $M$ with piecewise smooth border
$\partial{N}$, where $d=\dim(M)$. We also need finitely many one-to-one maps
$g_{i}:N\to N$, $i=1,\cdots,s$ satisfying the following conditions:

  * •

the maps $g_{i}$, $i=1,\cdots s$ are $C^{r}$ ($r>1$) contractive:

$\|Dg\|:=\max_{1\leq i\leq s}\sup_{x\in U}\|Dg_{i}(x)\|<1;$

  * •

open condition: there exists an open neighborhood $N\subset U$ such that
$g_{i}(U)\subset int(N)$ are pairwise disjoint: $g_{i}(U)\cap
g_{j}(U)=\emptyset$ for $i\neq j$;

  * •

border condition: we will suppose in addition that
$\partial{N_{i}}\cap\partial{N}=\emptyset$.

It is not usual to ask for a border condition as above, however we need it in
some parts of our arguments. It is not too restrictive as long as we can
approximate a generic geometric construction by a configuration satisfying
this property. See below.

By a well known theorem due to Hutchinson, there exists a limit set

$\Lambda=\bigcap_{n=1}^{+\infty}\bigcup_{(i_{1},\dots,i_{n})\in
s^{n}}g_{i_{1}}\circ\cdots\circ g_{i_{n}}(N),$

which is an attractor of the IFS ${\bf g}=\\{g_{i}\\}$ acting, as a set map,
upon the (complete) metric space of compact subsets of $N$ with the Hausdorff
distance:

${\bf g}(A)=\bigcup_{i}g_{i}(A)\quad\text{with a composition law}\quad{\bf
g}\circ{\bf g}(A)=\bigcup_{i,j}g_{i}\circ g_{j}(A).$

The family of sets $\wp_{n}=\\{g_{i_{1}}\circ\cdots\circ
g_{i_{n}}(N):(i_{1},\dots,i_{n})\in s^{n}\\}$, defined at the $n$-stage of the
construction, will be called atoms of generation $n$:

${\bf g}^{n}(N)=\bigcup_{P\in\wp_{n}}P.$

The family of sets of first generation will be called the template of the
construction. Topologically, the attractor of a geometric construction
$\Lambda$ satisfying the open and border condition is a Cantor set, that is, a
compact, totally disconnected, perfect set. In particular $\Lambda$ has zero
topological dimension. Furthermore, our method requires that there exists a
well defined bounded relation between diameter and volume in the following
sense.

######  Definition 2.1.

We say that a subset $X\subset M$ is volume reducible if there exist some
$k<d$ such that $X$ can be covered with at most countably many smooth subsets
$B_{i}$ of positive $k$-volume, where $\mbox{Vol}\,_{k}(B_{i})\asymp
radius(B)^{k}$ uniformly. Here $d$ is the dimension of $M$, the ambient
manifold. By smooth we mean that $B_{i}\subset M$ is an smoothly embedded
$k$-disc. A subset is volume irreducible if it is not volume reducible.

A cartesian product $X\times Y$ of Cantor sets $X,Y\subset\mathbb{R}$ is
irreducible since it can not be covered by countably many intervals. Similarly
so a Cantor set $\Lambda\subset\mathbb{R}^{d}$ whose projections onto the
every $k$ dimensional coordinate plane is a Cantor set. Moreover, if some of
these projections have positive Lebesgue measure, the set is still volume
irreducible, whenever there exits at least one Cantorian projection. A
sufficient condition for a limit set $\Lambda$ to be volume reducible is that
there exists an embedded smooth submanifold $Z\subset M$ of codimension $\geq
1$ such that $\Lambda\subset Z$. This situation appears in some examples of
self-affine subsets, depending of the initial configuration of the template.
Volume reducible configurations are exceptional in that this condition can be
removed by an small perturbation of the initial template of the IFS.

If the construction converges to a non rectifiable connected set, as in the
case of Weierstrass-like graphs, the limit set it is not volume reducible in
the sense of our defintion since it can not be covered by countably many
smooth $k$-discs, with $k<d$. However, in some sense it is reducible, in that
it can be certainly covered by countably many topological $k$-discs with
$k<d$. Problem is that, being not rectifiable, there is not a well defined
notion of $k$-dimensional volume for the sets in these coverings. In
particular, this type of examples are not directly tractable with our methods.
However, in some cases, we can approximate these graphs-like curves or
connected sets by irreducible Cantor sets, as we will see in brief. Then, we
will suppose, for simplicity, that the limit of our constructions are volume
irreducible Cantor sets in the $d$ dimensional ambient manifold $M$.

######  Definition 2.2.

We say that a contractive Iterated Function System (IFS) $\\{g_{i}\\}$
satisfying the open and border condition, whose limit set is volume
irreducible, is a regular geometric construction.

In principle, our definition exclude some interesting examples: let $n>m$ be
two positive integers and divide the unit square ${\bf\sf R}=[0,1]^{2}$ into
rectangles of width $1/m$ and heigth $1/n$. Then choose finitely many
rectangles ${\bf\sf R}_{ij}$, $(i,j)\in
S\subset\\{1,\cdots,m\\}\times\\{1,\cdots,n\\}$, from this collection of $nm$
rectangles and introduce affine contractions

$g_{ij}(x,y)=\left[\begin{array}[]{cc}m^{-1}&0\\\
0&n^{-1}\end{array}\right]\left[\begin{array}[]{c}x\\\
y\end{array}\right]+(x_{i},y_{j})\quad\text{such that}\quad{\bf\sf
R}_{ij}=g_{ij}({\bf\sf R}),\ (i,j)\in S.$

The limit $\Lambda$ of this IFS is known as a Bedford-McMullen carpet and it
basic example of a non conformal self-affine subset of the plane. We refer to
the survey [5] for a review of results on fractal geometry of these
interesting self-affine subsets. It might happen however that, depending on
the choice of the rectangles, this geometric construction may violate our
hypothesis of regularity. This may happens if the initial template have
rectangles with overlapping border. Moreover, the limit set can be reducible.
It might also happen that $\Lambda$ is contained in a vertical or horizontal
interval, depending on the choice of an initial template. We will call these
cases degenerate. Moreover, in a number of interesting cases, the limit may
develop connected components and even converge to a Weierstrass-like graph.
However, we can slightly perturb the IFS, assuming that it cannot be reduced
one of the degenerate cases outlined above, to obtain a regular construction
converging to a Cantor set contained inside $\Lambda$. For this we choose a
small $\epsilon>0$ and introduce the contraction:

$g^{\epsilon}_{ij}(x,y)=\left[\begin{array}[]{cc}(m+\epsilon)^{-1}&0\\\
0&(n+\epsilon)^{-1}\end{array}\right]\left[\begin{array}[]{c}x\\\
y\end{array}\right]+(x^{\epsilon}_{i},y^{\epsilon}_{j})\quad\text{such
that}\quad{\bf\sf R}^{\epsilon}_{ij}=g^{\epsilon}_{ij}({\bf\sf R}),\ (i,j)\in
S.$

A suitable choice of the traslations $(x^{\epsilon}_{i},y^{\epsilon}_{j})$
makes the perturbed IFS $\\{g^{\epsilon}_{ij}\\}$ regular, that is, the
resulting rectangles ${\bf\sf R}_{ij}^{\epsilon}$ are pairwise disjoint,
satisfy the open and border condition and do not degenerate into a volume
reducible set. We use these perturbations to prove that $\Lambda$ is the limit
of a sequence of regular geometric constructions. For this we let
$\epsilon_{n}\downarrow 0$ be a decreasing sequence of positive numbers and
define $\Lambda_{n}$ as the limit of the corresponding perturbed regular
geometric construction given by the IFS $\\{g^{\epsilon_{n}}_{i}\\}$ such that
$R^{\epsilon_{n}}_{ij}\subset{\bf\sf R}_{ij}$ for every $(i,j)\in S$ and
${\bf\sf R}^{\epsilon_{n}}_{ij}\uparrow{\bf\sf R}_{ij}$ when
$\epsilon_{n}\downarrow 0^{+}$. Then

$\Lambda=\overline{\bigcup_{n>0}\Lambda_{n}},\quad\quad\text{where}\quad\quad\Lambda_{n}\subset\Lambda_{n+1},\quad\text{for
every}\quad n>0.$

In other words, generically, Bedford-McMullen carpets are limits of increasing
sequences of Cantor sets defined by suitable regular geometric constructions,
in the sense of our definition. This cover as well the case of the non
rectifiable Weierstrass-like graphs mentioned before.

Of course it might happen that there are not available exact formulas for the
Hausdorff dimension of these perturbed approximations. However, as we will see
below, our methods produce robust bounds for the Hausdorff and box-counting
dimension which are still valid for these slightly perturbed geometric
constructions. This could be sufficient for the applications that we have in
mind. Observe that we are not telling that, in general, the Hausdorff
dimension varies continuously, even for this regular geometric constructions.

Let us see now how the dynamics generates dimensional indicators. Dynamically,
our limits sets are hyperbolic repellers of a suitable piecewise expanding
map. Let $f:\bigcup_{i}U_{i}\to U$ be defined a piecewise $C^{r}$, $r>1$
differentiable map defined as $f|U_{i}=g^{-1}_{i}$, where $U_{i}=g_{i}(U)$,
$i=1,\cdots,s$ and $N\subset U$ is the open neighborhood provided by the open
condition. Then,

$\Lambda=\bigcap_{n=0}^{+\infty}f^{-n}N$

is the locally maximal $f$-invariant subset of $f$ in $\bigcup_{i}N_{i}$.
$\Lambda$ is the set of point which never escapes from the initial template
$\\{N_{i}\\}$. This is the simplest model of a non linear hyperbolic repeller
topologically conjugated to a full-shift on $s$-symbols: there are constants
$C>0$ and $\lambda>1$ such that:

$\|Df^{n}(x)v\|\geq C\lambda^{n}\|v\|\quad\forall v\in T_{x}M,\ x\in\Lambda$

and there exists a homeomorphism $h:B^{+}(s)\to\Lambda$ intertwining $f$ and
the shift $\sigma$: $f\circ h=h\circ\sigma$, where
$\sigma(\omega)(n)=\omega(n+1)$, $\omega\in
B^{+}(s)=\\{1,\cdots,s\\}^{\mathbb{Z}^{+}}$ with the product topology. In
contrast with other approaches we do not use directly this symbolic dynamics.

There exists a deep connection between the dimension theory of dynamical
systems and thermodynamic formalism. Let us recall some crucial notions of the
theory.

Let $(X,d)$ the a compact metric space and $f:X\to X$ a continuous self map.
We say that a subset $E\subset X$ is $(\epsilon,n)$-separated if for every
pair of points $x,y\in E$ there exists $0\leq k<n$ such that
$d(f^{k}(x),f^{k}(y))>\epsilon$. The cardinal of a maximal
$(\epsilon,n)$-separated subset is essentially the number of dynamically
different orbits up to time $n$ with precision $\epsilon>0$. The topological
entropy, a quantitative indicator of the complexity of the dynamics, can be
computed as the rate of growing of dynamically different orbits up to finite
arbitrarily small precision. The topological pressure of a continuous
potential $\phi$ is a generalization of the topological entropy and can be
defined as a the rate of growing of weighted dynamically different orbits up
to finite time and small precision,

$P(\phi)=\lim_{\epsilon\to
0^{+}}\limsup_{n\to+\infty}\dfrac{1}{n}\log\left(\sup_{E}\sum_{x\in
E}e^{S_{n}\phi(x)}\right),$

where supremum is taken over the family of $(\epsilon,n)$-separated subsets
and $S_{n}\phi(x)=\sum_{k=0}^{n-1}\phi(f^{k}(x))$. See [3] and [21]. This
important quantity is invariant under topological conjugation and can be used
as a dynamical indicator of the statistical properties of the system. The
following variational principle is a cornestone of the thermodynamic
formalism:

$P(\phi)=\sup_{\mu\in\mathcal{M}_{f}}\left\\{h(\mu)+\int\phi{d\mu}\right\\}$

where $\mathcal{M}_{f}$ is the set of $f$-invariant Borel probabilites,
$h(\mu)$ the Kolmogorov-Sinai metric entropy and

$P_{\mu}(\phi)=h(\mu)+\int\phi{d\mu}$

is the free energy, also called the measure-theoretical pressure. We say that
$\mu\in\mathcal{M}_{f}$ is an equilibrium state for the potential $\phi$ if it
maximizes the free energy, that is:

$P(\phi)=h(\mu)+\int\phi{d\mu}.$

An important problem of the thermodynamic formalism is to give sufficient
conditions for a potential $\phi$ to have a unique or at most finitely many
equilibrium states. These ideas came from the statistical physics on infinite
one-dimensional gas whose phase state is modeled on the (bilateral) shift. We
refer to [3] and [8] for a comprehensive introduction to the subject.

It is the case that the thermodynamic formalism of $C^{r}$ ($r>1$) hyperbolic
repellers is well understood. It can be proved that, for any continuous
function $\phi:\Lambda\to\mathbb{R}$, the topological pressure is:

$P(f|\Lambda,\phi)=\limsup_{n\to+\infty}\dfrac{1}{n}\log\left(\sum_{P\in\wp_{n}}\sup_{x\in
P}\exp{S_{n}\phi(x)}\right).$

Moreover, is $\phi$ is Hölder continuous, then there exists a unique
equilibrium state $\mu_{\phi}$ which is a Gibbs measure: there exists a
constant $G>1$ such that:

(1) $G^{-1}\leq\dfrac{\mu_{\phi}(\wp_{n}(x))}{exp(-P(\phi)+S_{n}\phi(x)}\leq
G$

for every $x\in\Lambda$ and for every $n>0$. See [3] and [8].

As pointed out in [15], several pressure-like dynamical indicators constructed
in several settings of the termodynamic formalism are dimensional indicators
associated to a suitable Carathéodory’s structure. This remark had originated
a number of useful generalizations of the above thermodynamic formalism to
include more general classes of potentials, as subadditive and almost
subadditive sequences giving rise to a rich theory for non conformal sets. See
for instance [1], [2], [15] and [16]. Let us show how thermodynamic formalism
can be used to get dimensional indicators from the dynamics. Let

$\wp=\bigcup_{n\geq 1}\wp_{n}.$

be the generating net for $\Lambda$: for every $x\in\Lambda$ there exists a
decreasing sequence $\wp_{n+1}(x)\subset\wp_{n}(x)$, $n\geq 1$, such that:

$\\{x\\}=\bigcap_{n=1}^{+\infty}\wp_{n}(x)$

where $\wp_{n}(x)\in\wp_{n}$ is the atom of generation $n$ containing $x$. We
use the generating familiy $\wp$ and the volume as a set function to define a
suitable Carathéodory’s structure with a dimensional indicator as follows. We
say that a countable family of subsets $\mathcal{U}=\\{P_{i}\\}$, with
$P_{i}\in\wp$, is a $(\wp,\delta)$-covering of $X$ if a)
$X\subset\bigcup_{i}P_{i}$ and $\mbox{diam}(P_{i})<\delta$. Then we define an
outer measure:

$\mathcal{D}_{a,\delta}(X)=\inf_{\mathcal{U}}\sum_{i=1}^{+\infty}\mbox{Vol}\,(P_{i})^{a}$

where infimum is taken over the family of $(\wp,\delta)$-coverings. As usual,
$\mathcal{D}_{a,\delta}(X)$ is non decreasing in $\delta$ so we define the
following dynamical $a$-measure:

$\mathcal{D}_{a}(X)=\sup_{\delta>0}\mathcal{D}_{a,\delta}(X).$

This is a Borel regular measure and its dimensional indicator is the dynamical
dimension

$\dim_{\mathcal{D}}(X)=\inf\\{a>0:\mathcal{D}_{a}(X)=0\\}=\sup\\{a>0:\mathcal{D}_{a}(X)=+\infty\\}$

It holds out that $\alpha=\dim_{\mathcal{D}}(\Lambda)$ is precisely the
solution to the Bowen equation

(2) $P(f|\Lambda,-\alpha\log(Jf))=0,$

where $Jf=|\det(Df)|$ is the Jacobian $f$ with respect to the Riemannian
volume. Indeed, it is well known that, by the bounded distortion property of
the volume under $f$ (see below), there exists a constant $A>1$, uniform, such
that

$A{-1}\leq\mbox{Vol}\,(P)Jf^{n}(x)\leq A\quad\forall\ x\in P,\ P\in\wp_{n},\
n\geq 1.$

This is also known as the volume lemma. Therefore,
$\mbox{Vol}\,(P)^{a}\asymp\sup_{x\in P}\exp{S_{n}\phi(x)}$, meaning that the
ratios of these quantities are uniformly bounded, independent of $P\in\wp_{n}$
and $n$, by suitable constants. Here $\phi(x)=-a\log{Jf}(x)$ is the geometric
potential. This is a Hölder continuous function, by our assumptions on the
differentiability of the map. By the thermodynamic formalism there exists a
unique ergodic Borel probability $\mu_{\alpha}$ supported on $\Lambda$ which
is an equilibrium state for the geometric potential with parameter $\alpha$.
Using the volume lemma and the Gibbs property we find a constant $B>1$ such
that

$B^{-1}\leq\dfrac{\mu(P)}{\mbox{Vol}\,(P)^{\alpha}}\leq B\quad\forall\
P\in\wp.$

Then, we can prove that $0<\mathcal{D}_{\alpha}(\Lambda)<+\infty$. In
particular,

$\dim_{\mathcal{D}}\,(\Lambda)=\dfrac{h(\mu_{\alpha})}{\chi^{+}(\mu_{\alpha})},$

where $\chi^{+}(\mu)=\int\log|Jf|d\mu$ is the Lyapunov exponent of the Borel
probability $\mu$.

There are two main settings under which the above approach can be successfully
used to obtain exact formulas for the dimension of a hyperbolic repeller:

  * •

piecewise expanding interval maps and

  * •

conformal piecewise expanding maps $f:M\to M$ of a $d$-dimensional compact
Riemannian manifold,

both in the $C^{r}$, ($r>1$) category. In these cases, the following facts are
well known:

  1. (1)

$\dim_{\mathcal{H}}\,(\Lambda)=\dim_{B}(\Lambda)$;

  2. (2)

$\dim_{\mathcal{H}}\,(\Lambda)=d\alpha$, where $0<\alpha\leq 1$ is the unique
solution to the Bowen equation $P(f|\Lambda,-\alpha\log|f^{\prime}|)=0$ for
interval maps $(d=1)$ and $P(f|\Lambda,-d\alpha\log|a|)=0$, for a
multidimensional conformal map, where $Df(x)=a(x)id_{T_{x}M}$ is the
derivative at $x$;

  3. (3)

the Hausdorff measure is finite and positive
$0<\mathcal{H}_{d\alpha}(\Lambda)<+\infty$;

  4. (4)

there exists a unique Borel probability of maximal dimension $\mu_{\Lambda}$,
such that:

$\dim_{\mathcal{H}}\,(\Lambda)=\dfrac{h(\mu_{\Lambda})}{\chi^{+}(\mu_{\Lambda})}=\sup_{\mu\in\mathcal{M}_{f}}\dfrac{h(\mu)}{\chi^{+}(\mu)};$

  5. (5)

the Hausdorff dimension $\dim_{\mathcal{H}}\,(\Lambda_{f})$ varies smoothly
with $f$.

Why the dynamical dimension and the geometric dimension coincides in these
settings? Well, tt occurs that, in these cases, the atoms of the construction
$P\in\wp$ have bounded geometric distortion. Indeed, let

$r_{inner}(P)=\sup\\{r>0:\exists\ x\in P,\ B(x,r)\subset P\\}$
$r_{outer}(P)=\inf\\{r>0:\exists\ x\in P,\ P\subset B(x,r)\\}$

be the inner and outer diameters of the set $P$, respectively. It can be
proved that, if $f$ is conformal, then $r_{inner}(P)\asymp r_{outer}(P)$ are
uniformly comparable for every $P\in\wp$ and therefore
$\mbox{Vol}\,(P)\asymp\mbox{diam}(P)^{d}$, uniformly in $P\in\wp$. Therefore,
in these cases the dynamical measure $\mathcal{D}_{a}$ is equivalent to the
Hausdorff measure $\mathcal{H}_{ad}$, for every $a>0$ and thus the dynamical
and the Hausdorff dimension are equal, up to the constant $d$, the dimension
of the ambient space. Moreover, the Hausdorff measure is non trivial, there
exists a unique measure of maximal dimension which is an equilibrium state of
the geometric potential and the dimensional characteristic varies smoothly
with the map $f$, by differentiability results of the thermodynamic formalism.

If $f$ is non conformal this approach breaks down, originating a number of
interesting and difficult problems. The point is that, for non conformal maps,
the geometric distortion of the generating sets $P\in\wp$ is unbounded and
there is no longer a clear connection between the dynamical and Hausdorff
measures. A large body of literature had been produced trying to dilucidate
how the dynamics interacts with the geometry of a non conformal repeller,
letting a wide field of open problems on the subject. One direction of
approach had been to look for a generalized Bowen equation in the setting of a
non additive thermodynamic formalism, using subadditive or almost additive
potentials and a corresponding generalized pressure-like indicator which are,
after all, a dimensional indicator of the dynamics. We refer to the book of
Barreira [2] for a comprehensive exposition of these methods. Another authors
had directed their attention to the study of self-affine subsets, that is,
attractors of finite subsets of affine non conformal contractions of the plane
or multidimensional euclidean spaces. These examples serve as simplified
geometric models of the problem and are interesting on their own. One way to
look at self-affine subsets is decomposing the atoms $P\in\wp_{n}$ of the
$n$-th stage in the construction, into pieces with uniformly bounded geometric
distortion, using a characteristic scale of the construction, given in terms
of the different rates of constraction. This approach had generated a number
of interesting contributions including exact formulas for the Hausdorff
dimension of examples as Bedford-McMullen and Gatzouras-Lalley carpets among
others. We refer to the survey of Falconer [5] for results, examples and a
large list of references on the subject. We also refer to [1] and [16] for a
review, from the dynamical systems point of view, of this type of problems.

It is the goal of this paper to show that, nevertheless, the standard additive
thermodynamic formalism and the Bowen equation can still be used, not only to
get good upper and lower bounds which vary smoothly with the construction, but
to give exact formulas for the Hausdorff dimension of a non confomal repeller
in terms of a the dynamical dimensions.

##  3\. Statement of results

Before to state our main result we need to introduce the following quantities:

  * •

fix small positive numbers $\rho,\delta>0$;

  * •

define $0<\epsilon<1$ as:

$\epsilon=(1-\delta)\left(\dfrac{\|\underline{Jg}\|}{\quad\quad\|Dg\|^{d-1+\delta}}\right)^{\frac{1}{1-\delta}},$

where:

$\|\underline{Jg}\|=\min_{i=1,\cdots,s}\inf_{x\in
N_{i}}Jg_{i}(x)\quad\text{and}\quad\|Dg\|=\max_{i=1,\cdots,s}\sup_{x\in
N_{i}}\|Dg_{i}(x)\|$

where $Jg_{i}(x)=|\det(Dg_{i}(x))|$ is the Jacobian and $Dg_{i}(x)$ the
derivative at $x$ and $\delta>0$ a small positive number;

  * •

let $\lambda_{0}$ and $\lambda_{1}$ be given by the volume expansion of $f$:

$\lambda_{0}=\log(\inf_{x\in\Lambda}Jf(x)-\rho)\quad\text{and}\quad\lambda_{1}=\log(\sup_{x\in\Lambda}Jf(x)+\rho),$

where we suppose that $0<\rho<\inf_{x\in\Lambda}Jf(x)$ and
$Jf(x)=|\det(Df(x)|$ is the Jacobian of $f$ with respect to the Riemannian
volume.

We will prove the following

######  Theorem 1.

Let $g_{i}:N\to N$ be a SIF generating a regular geometric construction,
$\Lambda\subset N$ its attractor and $f:\bigcup_{i}U_{i}\to U$ be the non
conformal $C^{r}$ ($r>1$) piecewise differentiable expanding map generated by
the inverse branches $\\{g_{i}\\}$.

  1. (1)

Let $s=\alpha$ be the solution to the Bowen equation
$P(f|\Lambda,-s\log{Jf})=0$. Then

$d\underline{\beta}\leq\underline{\dim}_{B}(\Lambda)\leq\overline{\dim}_{B}(\Lambda)\leq
d\overline{\beta};$

where $\overline{\beta}$ and $\underline{\beta}$ are the solutions to the
equation

$e^{\lambda_{0}(1-\alpha)}\epsilon^{d(1-\overline{\beta})}=1\quad\text{and}\quad
e^{\lambda_{2}(1-\alpha)}\epsilon^{d(1-\underline{\beta})}=1,$

where

$\lambda_{2}=\dfrac{\lambda_{1}\log\epsilon}{\log\|Dg\|}.$

  2. (2)

Let us suppose in addition that:

$\dim_{B}(\Lambda)=\overline{\dim_{B}}(\Lambda)=\underline{\dim_{B}}(\Lambda).$

Then, there exist positive constants $\underline{\Sigma}>\overline{\Sigma}>0$
such that, if $s=\underline{\alpha}$ (resp. $s=\overline{\alpha}$) is the
solution to the modified Bowen equation

$P\left(f|\Lambda,-s\log(e^{\underline{\Sigma}}Jf)\right)=0\quad\left(\text{resp.}\quad
P\left(f|\Lambda,-s\log(e^{\overline{\Sigma}}Jf)\right)=0\right)$

then,

$d\underline{\beta}^{*}\leq\dim_{\mathcal{H}}\,(\Lambda)\leq
d\overline{\beta}^{*},$

where $\underline{\beta}^{*}$ and ${\overline{\beta}}^{*}$ are the solutions
to the equations

$e^{\lambda_{1}(1-\underline{\alpha})}\epsilon^{d(1-{\underline{\beta}}^{*})}=1\quad\text{and}\quad
e^{\lambda_{0}(1-\overline{\alpha})}\epsilon^{d(1-{\overline{\beta}}^{*})}=1,\quad\text{respectively.}$

Explicitly, the upper $U=d\overline{\beta}$ and lower $L=d\underline{\beta}$
bounds for the box-counting dimension

$U=d+\dfrac{\lambda_{0}(1-\alpha)}{\log\epsilon}\quad\text{and}\quad
L=d+\dfrac{\lambda_{2}(1-\alpha)}{\log\epsilon};$

Similarly so, the upper and lower bounds for $\dim_{\mathcal{H}}\,(\Lambda)$
are:

$U=d+\dfrac{\lambda_{0}(1-\overline{\alpha})}{\log\epsilon}\quad\text{and}\quad
L=d+\dfrac{\lambda_{1}(1-\underline{\alpha})}{\log\epsilon}.$

Let us first underline the role of the additive thermodynamic formalism in our
formulas. The solutions to the Bowen equation and our so-called modified
modified Bowen equation are dimensional indicators of the dynamics and they
provide upper and lower bounds of the geometric dimensions. Certainly they are
not equal to the Hausdorff dimension in the nonconformal case, but suggest the
existence of a functional relation between them. Also, as a consequence of the
thermodynamic formalism, the bounds are robust and vary smoothly under small
perturbations of the construction. We will use this to provide a new exact
formula for the Hausdorff dimension of a dynamically defined Cantor set
depending on a dynamical dimension.

######  Corollary 3.1.

Let $\Lambda$ be the limit of regular geometric construction given by a
$C^{r}$ ($r>1$) IFS $\mathcal{G}=\\{g_{i}\\}$. For every $\epsilon>0$, there
exists $\delta>0$ such that,

$|L-L(\delta)|<\epsilon\quad\text{and}\quad|U-U(\delta)|<\epsilon,$

for every $\delta$-$C^{1}$ near regular geometric construction
$\mathcal{G}_{\delta}=\\{g_{\delta,j}\\}$, where $L=L(\underline{\alpha})$
(resp. $U=U(\overline{\alpha})$) are the lower bound (resp. upper bound) of
the Hausdorff dimension of the attractor $\Lambda$ and $L(\delta)$ and
$U(\delta)$ the corresponding upper and lower bounds for $\Lambda(\delta)$,
the limit of the perturbed IFS.

This is a plain consequence of the smoothness of the additive pressure
$P(\phi)$ with respect to the potential $\phi$ and the robustness and
structural stability of hyperbolic repellers

The quantities involved in the definition of our upper and lower bounds are
continuous functions of $\rho$, $\delta$, $\epsilon$ and the solutions to
Bowen’s equations, both ordinary and modified, and reflects a delicate
interplay between the volume and distance contraction of the inverse branches
$g_{i}:N\to N$ and the geometry of $\Lambda$. The choice of constants $\rho$
and $\delta$ is quite arbitrary, as long as they are sufficiently small
positive numbers. For $\rho$ we have the upper bound
$0<\rho<\|\underline{Jf}\|$ and for $\delta$ we will find a small
$\delta_{1}>0$ such that $0<\delta<\delta_{1}$ is required for the argument to
hold on. The number $\epsilon>0$ is a sort of characteristic scale at which
the dynamical measure and the box-counting measure generated by suitable
$2\epsilon^{n}$-coverings are uniformly comparable over atoms $P\in\wp_{n}$ at
the $n$-th stage of the construction, independently of $n>0$. The constants
$\underline{\Sigma}$ and $\overline{\Sigma}$ involved in the modified Bowen
equation depends on $\Sigma(x)$, the rate of exponential decay, as
$n\to+\infty$, of the volume fraction of an atom $\wp_{n}(x)$ which is covered
by a suitable efficient covering of $\Lambda\cap\wp_{n}(x)$ by
$2\epsilon^{n}$-balls. Moreover, we will prove that, for
$0<\delta<\delta_{1}$,

$0<\log\left(\dfrac{\inf_{x\in\Lambda}Jf(x)}{\epsilon^{-(d(\Lambda)+\delta-d)}}\right)\leq\Sigma(x)\leq\log\left(\dfrac{\sup_{x\in\Lambda}Jf(x)}{(\|Df\|\epsilon)^{-(d_{B}(\Lambda)-\delta)}\epsilon^{d}}\right)<+\infty,$

where $d_{B}(\Lambda)=\dim_{B}(\Lambda)$ is the box-counting dimension of
$\Lambda$. Then,

$\underline{\Sigma}=\sup_{x\in\Lambda}\Sigma(x)\quad\text{and}\quad\overline{\Sigma}=\inf_{x\in\Lambda}\Sigma(x)$

Moreover, we get a new exact formula for the Hausdorff dimension of his type
of constructions.

######  Theorem 2.

Let $\Lambda$ be the limit of regular geometric construction given by a
$C^{r}$ ($r>1$) IFS $\mathcal{G}=\\{g_{i}\\}$. Suppose that

$d\underline{\beta}^{*}<\dim_{\mathcal{H}}\,(\Lambda)<d\beta_{0}$

where $\beta_{0}$ is the unique solution to the equation:

$e^{\lambda_{1}(1-\alpha)}\epsilon^{d(1-\beta_{0})}=1$

and $s=\alpha$ the solution to the Bowen equation $P(f|\Lambda,-s\log{Jf})=0$
Then, there exists a unique parameter
$0<\Sigma=\Sigma(\Lambda)<\underline{\Sigma}$ such that

$\dim_{\mathcal{H}}\,(\Lambda)=d+\dfrac{\lambda_{1}(1-\alpha(\Sigma))}{\log\epsilon}$

where $\alpha(\Sigma)$ is the solution to the modified Bowen equation of
parameter $\Sigma$:

$P(f|\Lambda,-s\log(e^{\Sigma}Jf))=0.$

###### Proof.

The solution $\alpha(t)$ to the modified Bowen equation

$P(s,t)=P(f|\Lambda,-s\log(e^{t}Jf))=0$

is a smooth function of $t$, by differentiability properties of the pressure
of a uniformly hyperbolic repeller and the implicit function theorem.
Moreover,

$\partial_{s}P(s,t)=-\int\log(e^{t}Jf)d\mu_{t}\quad\text{and}\quad\partial_{t}P(s,t)=-s,$

where, $\mu_{t}$ is the equilibrium state of $\phi_{t}=-\log(e^{t}Jf))$. Then,

$\alpha^{\prime}(t)=-\left.\dfrac{\partial_{t}P(s,t)}{\partial_{s}P(s,t)}\right|_{(\alpha(t),t)}=-\dfrac{\alpha(t)}{\int\log(e^{t}Jf)d\mu_{t}}<0,$

for every $t>0$. Thus, $\alpha(t)$ is strictly decreasing and then

$D(t)=d+\dfrac{\lambda_{1}(1-\alpha(t))}{\log\epsilon}$

will be strictly decreasing too, since $1-\alpha(t)$ is increasing and
$\lambda_{1}/\log\epsilon<0$. By hypothesis
$D(\underline{\Sigma})<\dim_{\mathcal{H}}\,(\Lambda)<D(0)$ so there exists a
unique $0<\Sigma<\underline{\Sigma}$ such that
$D(\Sigma)=\dim_{\mathcal{H}}\,(\Lambda)$, by Bolzano’s property, i.e.

$\dim_{\mathcal{H}}\,(\Lambda)=d+\dfrac{\lambda_{1}(1-\alpha(\Sigma))}{\log\epsilon},$

concluing the proof. ∎

######  Remark 3.1.

The functions $\epsilon=\epsilon(f)$ and $\lambda_{1}=\lambda_{1}(f)$ are
Hölder continuous in the $C^{1}$-topology of $f$. On the other side,
$\alpha(f)=\alpha(\Sigma(\Lambda_{f}))$ is determined by the regularity of
$\dim_{\mathcal{H}}\,(\Lambda_{f})$ as a function of $f$ and reciprocally.
This opens the door to look at the regularity properties of
$\dim_{\mathcal{H}}\,(\Lambda_{f})$, using the smoothness properties of the
pressure and the thermodynamic formalism, eventhough our formula is explicit
and not implicit as in the conformal case. We left this and related problems
for future indagations on the subject.

##  4\. Outline of the proof

The main idea of the method was borrow from the work of Vanderlei Horita and
Marcelo Viana [20], where it is proved that certain nonuniformly expanding
repeller $\Lambda$ has Hausdorff dimension strictly less that the dimension of
ambient manifold. The starting step is the simple remark that
$\sum_{i}\mbox{Vol}\,(B_{i})^{\beta}$ are uniformly comparable with
$\sum_{i}\mbox{diam}(B_{i})^{d\beta}$ for covering by metric balls $B_{i}$.
Then a characteristic scale $0<\epsilon<1$ is chosen such that, for every atom
$P\in\wp_{n}$ in the $n$-th stage of the construction, there exists a covering
$\mathcal{B}_{n}(P)$ by $2\epsilon^{n}$-balls such that
$\sum_{B\in\mathcal{B}_{n}(P)}\mbox{Vol}\,(B)$ is uniformly comparable with
$\mbox{Vol}\,(P)$. These converings serve as a sort of decompositions of $P$
into pieces with uniformly bounded geometric distortion, taking care of the
non conformality of the IFS. One can prove that the sums
$\sum_{B\in\mathcal{B}_{n}}\mbox{Vol}\,(B)^{\beta}$ are comparable with
$\sum_{P\in\wp_{n}}\mbox{Vol}\,(P)^{\alpha}$, bounded by a function of
$\lambda_{0}$, $\epsilon$, $\beta$ and $\alpha$, where $\alpha$ is the
dynamical dimension. By thermodynamic formalism
$\sum_{P\in\wp_{n}}\mbox{Vol}\,(P)^{\alpha}$ is uniformly bounded. Then, one
deduce that $\sum_{B\in\mathcal{B}_{n}}\mbox{Vol}\,(B)^{\beta}$ is bounded by
a suitable geometric progression with ratio $\theta=\theta(\beta)$:
$\sum_{B\in\mathcal{B}_{n}}\mbox{Vol}\,(B)^{\beta}\leq
C[\overline{\theta}]^{n}$, where $\overline{\theta}=\overline{\theta}(\beta)$
is a stricly decreasing function of $\beta$. If $0<\beta<1$ is chosen
conveniently then $0<\overline{\theta}<1$ and
$\sum_{B\in\mathcal{B}_{n}}\mbox{Vol}\,(B)^{\beta}\to 0^{+}$ as $n\to+\infty$,
so proving that $\dim_{\mathcal{H}}\,(\Lambda)\leq d\beta$, strictly less that
the dimension of the ambient space. In fact, Horita-Viana’s argument proves
that $\underline{\dim}_{B}(\Lambda)\leq\overline{\dim}_{B}(\Lambda)\leq
d\overline{\beta}$, where $\overline{\beta}$ is the solution to the equation
$\overline{\theta}(\beta)=1$.

Eventhough [20] was mainly focused on to obtain an upper bound for the
Hausdorff dimension, the method can be used for lower bounds too. Actually,
one can prove that $\sum_{B\in\mathcal{B}_{n}}\mbox{Vol}\,(B)^{\beta}$ is
bounded from below by a suitable geometric progression with ratio $>1$ if
$\beta$ is lower than a certain critical parameter. However, to bound from
below the box-counting dimension, one need to work out
$\mathcal{N}(\Lambda,\rho)\rho^{a}$ which is lower than or equal to
$\sum_{B\in\mathcal{B}_{n}}\mbox{Vol}\,(B)^{\beta}$, for $\rho=2\epsilon^{n}$.
This originates a correction term in the ratio $\theta$ so that, in the end,
one can prove that
$\mathcal{N}(\Lambda,2\epsilon^{n})(2\epsilon^{n})^{d\beta}\geq
C\underline{\theta}^{n}$ for a suitable decreasing function
$\underline{\theta}=\underline{\theta}(\beta)$ of $\beta$, depending on the
parameters $\lambda_{2}$, $\epsilon$ and $\alpha$, the dynamical dimension.
Therefore, if $\underline{\beta}$ is a solution to the equation
$\underline{\theta}(\beta)=1$ and $\beta<\underline{\beta}$ then
$\underline{\theta}(\beta)>1$ so that
$\mathcal{N}(\Lambda,2\epsilon^{n})(2\epsilon^{n})^{d\beta}$ diverges to
infinite exponentially when $n\to+\infty$. This proves that
$d\underline{\beta}\leq\underline{\dim}_{B}(\Lambda)\leq\overline{\dim}_{B}(\Lambda)$.

A similar argument will be used for the Hausdorff dimension bounds. However,
this case is more delicate since we need to bound the corresponding
$\beta$-sums defined over nonuniform $\zeta$-coverings by balls with variable
radius. For this we start with an arbitrary $\zeta$-covering $\mathcal{B}_{0}$
showing that the sums $\sum_{B\in\mathcal{B}_{0}}\mbox{Vol}\,(B)^{\beta}$ are
bounded below, for sufficiently large $n$, by
$\sum_{B\in\mathcal{B}_{n}}\mbox{Vol}\,(B)^{\beta}$, up to a function of
$\beta$, $n$, $\epsilon$ and $\zeta$. On the other hand, as the coverings
$\mathcal{B}_{n}$ are too large to estimate the Hausdorff measure, we choose a
suitable efficient covering $\mathcal{B}_{n}(\Lambda)$ of $\Lambda$. We then
define $\rho(P)$ the fraction of $\mbox{Vol}\,(P)$ which is covered by this
minimal covering and prove that the asymptotic rate of decay of
$\rho(\wp_{n}(x))$ is a bounded measurable function $\Sigma(x)$. This function
is bounded from above and from below so we fix the supremum
$\underline{\Sigma}$ and infimum $\overline{\Sigma}$. By thermodynamic
formalism, the sums $\sum_{P\in\wp_{n}}(\rho(P)\mbox{Vol}\,(P))^{\alpha}$ are
uniformly bounded if $\alpha$ is the solution to a modified Bowen equation
with parameters $\underline{\Sigma}$ and $\overline{\Sigma}$. This allows to
prove that the sums
$\sum_{B\in\mathcal{B}_{n}(\Lambda)}\mbox{Vol}\,(B)^{\beta}$ are bounded from
above and from below by suitable geometric progressions with ratios
$\overline{\theta}^{*}$ and $\underline{\theta}^{*}$, respectively, which are
decreasing functions of $\beta$, involving the constants $\lambda_{1}$,
$\epsilon$, and the solutions to the modified Bowen equation,
$\overline{\alpha}$ and $\underline{\alpha}$, respectively. Then,
$\sum_{B\in\mathcal{B}_{n}(\Lambda)}\mbox{Vol}\,(B)^{\beta}\leq
C(\overline{\theta}^{*})^{n}$, where $\overline{\theta}^{*}<1$ if
$\beta>\overline{\beta}^{*}$, the solution to the equation,
$\overline{\theta}^{*}(\beta)=1$ so proving that
$\dim_{\mathcal{H}}\,(\Lambda)\leq d\overline{\beta}^{*}$.

For the lower bound we combine our first estimate for
$\sum_{B\in\mathcal{B}_{0}}\mbox{Vol}\,(B)^{\beta}$ and a lower bound for
$\sum_{B\in\mathcal{B}_{n}(\Lambda)}\mbox{Vol}\,(B)^{\beta}$ to prove that
$\sum_{B\in\mathcal{B}_{0}}\mbox{Vol}\,(B)^{\beta}$ is bounded from below, for
sufficiently large $n$, by a function of $\beta$, $n$, $\zeta$, $\lambda_{1}$,
$\epsilon$ and $\underline{\alpha}$, the solution to a modified Bowen
equation. It holds that if $\underline{\beta}^{*}$ is the solution to a
suitable equation and $n$ is chosen conveniently, then the factor $\zeta$ can
be cancelled in the lower bound, proving that the sum
$\sum_{B\in\mathcal{B}_{0}}\mbox{Vol}\,(B)^{\underline{\beta}^{*}}$ is bounded
from below by a constant $C>0$ which depends neither on $\zeta$ nor on the
$\zeta$-covering $\mathcal{B}_{0}$. As $\mathcal{B}_{0}$ is arbitrary, this
proves that $\mathcal{H}_{d\underline{\beta}^{*},\zeta}(\Lambda)\geq C>0$ for
every sufficiently small $\zeta$. Thus
$\mathcal{H}_{d\underline{\beta}^{*}}(\Lambda)>0$ and, therefore,
$\mathcal{H}_{d\beta}(\Lambda)=+\infty$ for every
$\beta<\underline{\beta}^{*}$ finishing the proof.

The plan of the exposition is as follows. In section 5 we prove the bounds for
the box-counting dimension introducing the characteristic scale $\epsilon>0$
and the special coverings $\mathcal{B}_{n}$. In section 6 we introduce an
effcient covering $\mathcal{B}_{n}(\Lambda)$ and prove the existence of the
rate of decaying $\Sigma(x)$ to get bounds for the Hausdorff dimension. In
section 7 we apply our estimates to a well known example of plane self-affine
non conformal set. We rely upon the existing well known formulae for this
example as a test for our methods. Finally, we conclude in section 8 with a
discussion of the potential applications of these ideas.

##  5\. Proof of theorem 1: lower and upper bounds for the box-counting
dimension

We first prove our upper bound. Let us recall the set up: let
$g=\\{g_{i}:i=1,\cdots,s\\}$ be finitely many contractions $g_{i}:N\to N$ from
an embedded $d$-cube $N\subset M$ with positive volume into $N$. We suppose in
addition that the boundary is piecewise smooth formed by finitely many
codimension one embedded submanifolds. In particular,
$\dim_{B}(\partial{N})=d-1$. We also use the following volume estimates: there
are two constants $0<A_{0}<A_{1}$, only depending on the Riemannian structure,
such that

(3) $A_{0}r^{d}\leq\mbox{Vol}\,(B(x,r))\leq A_{1}r^{d}\quad\text{for
every}\quad x\in N\ r>0$

Moreover, we get from (3), that, for every $C>0$,

(4)
$\dfrac{A_{0}C^{d}}{A_{1}}\mbox{Vol}\,(B(x,r))\leq\mbox{Vol}\,(B(x,Cr))\leq\dfrac{A_{1}C^{d}}{A_{0}}\mbox{Vol}\,(B(x,r))$

The following lemma introduces the characteristic scale $0<\epsilon<1$.

######  Lemma 5.1.

There exists $0<\epsilon<1$, $N_{0}=N_{0}(\epsilon)>0$ a postive integer and a
constant $C_{0}>0$, such that for every $n$-generation atom $P\in\wp_{n}$,
there exists a covering ${\mathcal{B}}_{n}(P)$ by $2\epsilon^{n}$-balls such
that,

(5)
$\mbox{Vol}\,(P)\leq\sum\limits_{B\in{\mathcal{B}}_{n}(P)}\mbox{Vol}\,(B)\leq
C_{0}\mbox{Vol}\,(P)\quad\forall\ n\geq N_{0}.$

The constant $C_{0}$ only depends on $\dim_{B}(\partial{N})$, the box-counting
dimension of the border $\partial{N}$, the volume $\mbox{Vol}\,(N)$ and the
geometry of the ambient manifold.

###### Proof.

We will choose $0<\epsilon<1$ after establishing some conditions necessary to
satisfy the required property in the statement of the lemma. We first recall
that

$\dim_{B}(\partial{N})=\lim_{\rho\to
0^{+}}\dfrac{\log\mathcal{N}(\partial{N},\rho)}{\log(1/\rho)}=d-1$

is the box-counting dimension of the border and choose $\delta>0$ such that
$d_{0}:=d-1+\delta<d$, $\rho_{0}>0$ and $D_{0}>0$ such that:

$\forall\ 0<\rho<\rho_{0}:\quad\partial{N}\quad\text{{can be covered by at
most}}\quad D_{0}\rho^{-d_{0}}\quad\text{{balls of radius}}\quad\rho.$

Moreover, we choose $0<\rho_{0}<1$ sufficiently small such that $\Lambda$ is
contained in the complement of every covering of the border $\partial{N}$ by
$\rho_{0}$-balls

$\Lambda\subset N-\bigcup_{i}B(z_{i},\rho_{0}).$

This is possible by the border condition on the construction. Our first
condition to define $\epsilon$ is the following:

$\|Dg\|^{-n}2\epsilon^{n}<\rho_{0}\quad\text{for sufficiently large $n$.}$

This can be achieved if

$0<\|Dg\|^{-1}\epsilon<1\quad\text{and}\quad n\geq
N_{0}=\dfrac{\log(\rho_{0}/2)}{\log(\|Dg\|^{-1}\epsilon)}.$

As we can seen the definition of $N_{0}=N(\epsilon)$ depends on the choice of
$\epsilon$ whenever $0<\|Dg\|^{-1}\epsilon<1$. We will fix $\epsilon>0$ and
then $N_{0}$ and the end of the proof.

For every $n\geq N_{0}$, the border $\partial{N}$ can be covered by no more
than $D_{0}(\|Dg\|^{-n}2\epsilon^{n})^{-d_{0}}$ balls of radius
$r=\|Dg\|^{-n}2\epsilon^{n}$. We will choose such a covering as follows: let
$\\{B(x_{i},\|Dg\|^{-n}\epsilon^{n})\\}$ be maximal family of pairwise
disjoint open $\|Dg\|^{-n}\epsilon^{n}$-balls covering the border. Then the
open $\|Dg\|^{-n}2\epsilon^{n}$-balls covers $\partial{N}$ and we can
therefore extract a minimal subcovering which will be denotated
$\mathcal{B}_{n}(\partial{N})=\\{B(x_{i},\|Dg\|^{-n}2\epsilon^{n})\\}$. For
every $n>0$, the open set

$W_{n}=\bigcup_{i}B(g^{n}_{P}(x_{i}),2\epsilon^{n})\quad\text{covers
$\partial{P}$,}$

where $g^{n}_{P}$ is the branch of the IFS generating the $n$-atom
$P\in\wp_{n}$. Moreover, for $n\geq N_{0}$ it has volume:

$\mbox{Vol}\,(W_{n})\leq
D_{0}A_{1}(\|Dg\|^{-n}2\epsilon^{n})^{-d_{0}}(2\epsilon^{n})^{d}$

due to (3) and our bound on the cardinality of $\mathcal{B}_{n}(\partial{N})$.
We can even suppose that,

$\Lambda\cap P\subset P-W_{n}(P)\quad\text{for every}\quad n\geq N_{0},$

since $\Lambda\subset N-\bigcup_{i}B(x_{i},\rho_{0})$ and
$B_{i}\subset\bigcup_{i}B(x_{i},\rho_{0})$ for every
$B_{i}\in\mathcal{B}_{n}(\partial{N})$ by our choice of $\epsilon$ and
$N_{0}$.

Now we choose a maximal family of closed balls
$\overline{B}(z_{j},\epsilon^{n})$ contained in $P-W_{n}$ with pairwise
disjoint interiors. Then, $\\{B(z_{j},2\epsilon^{n})\\}$ covers $P-W_{n}$. Let

${\mathcal{B}}_{n}(P)=\\{B(g^{n}_{P}(x_{i}),2\epsilon^{n})\\}\cup\\{B(z_{j},2\epsilon^{n})\\}.$

be the covering so constructed.

By (4),

$\sum_{j}\mbox{Vol}\,(B(z_{j},2\epsilon^{n}))\leq\dfrac{D_{0}A_{1}2^{d}}{A_{0}}\sum_{j}\mbox{Vol}\,(B(z_{j},\epsilon^{n}))\leq\dfrac{A_{1}2^{d}}{A_{0}}\mbox{Vol}\,(P),$

since $\overline{B}(z_{j},\epsilon^{n})\subset P-W_{n}$ have pairwise disjoint
interiors. Then,

$\displaystyle\sum\limits_{B\in{\mathcal{B}}_{n}(P)}\mbox{Vol}\,(B)$
$\displaystyle\leq$ $\displaystyle
D_{0}A_{1}(\|Dg\|^{-n}2\epsilon^{n})^{-d_{0}}(2\epsilon^{n})^{d}+\dfrac{A_{1}2^{d}}{A_{0}}\mbox{Vol}\,(g^{n}(R))$
$\displaystyle=$ $\displaystyle
D_{0}A_{1}2^{d-d_{0}}(\|Dg\|^{d_{0}}\epsilon^{d-d_{0}})^{n}+\dfrac{A_{1}2^{d}}{A_{0}}\mbox{Vol}\,(P).$

On the other hand,

$\|\underline{Jg}\|^{n}\leq\dfrac{\mbox{Vol}\,(P)}{\mbox{Vol}\,(N)}.$

Indeed, $\mbox{Vol}\,(P)\geq\inf_{x\in
N}|Jg^{n}_{P}(x)|\mbox{Vol}\,(N)\geq\|\underline{Jg}\|^{n}\mbox{Vol}\,(N)$. If
$\epsilon>0$ is chosen, such that:

(6) $\|Dg\|^{d_{0}}\epsilon^{d-d_{0}}\leq\|\underline{Jg}\|$

then, one would have,

$\sum\limits_{B\in{\mathcal{B}}_{n}(P)}\mbox{Vol}\,(B)\leq\left(\dfrac{A_{1}2^{d-d_{0}}}{\mbox{Vol}\,(N)}+\dfrac{A_{1}2^{d}}{A_{0}}\right)\mbox{Vol}\,(P),$

and therefore,

$C_{0}=\dfrac{D_{0}A_{1}2^{d-d_{0}}}{\mbox{Vol}\,(N)}+\dfrac{A_{1}2^{d}}{A_{0}}$

would define a constant as required in (5). The solution to the inequality (6)
is:

$0<\epsilon\leq\left(\dfrac{\|\underline{Jg}\|}{\ \
\|Dg\|^{d_{0}}}\right)^{\frac{1}{d-d_{0}}}$

Now, notice that $\|\underline{Jg}\|\leq\|Dg\|^{d}$. Therefore:

$\left(\dfrac{\|\underline{Jg}\|}{\ \
\|Dg\|^{d_{0}}}\right)^{\frac{1}{d-d_{0}}}=\left(\dfrac{\|\underline{Jg}\|}{\quad\
\
\|Dg\|^{d-1+\delta}}\right)^{\frac{1}{1-\delta}}\leq(\|Dg\|^{1-\delta})^{\frac{1}{1-\delta}}=\|Dg\|.$

Thus, given $0<\delta<1$ we choose $\epsilon=\epsilon(\delta)$ as:

$\epsilon=(1-\delta)\left(\dfrac{\|\underline{Jg}\|}{\ \
\|Dg\|^{d-1+\delta}}\right)^{\frac{1}{1-\delta}}.$

With this choice of $\epsilon$ we have that $0<\|Dg\|^{-1}\epsilon<1$. ∎

Let us collect some facts concerning the collection of coverings
${\mathcal{B}}_{n}(P)$ so constructed which will be used later:

  1. (1)

first:

(7) $\Lambda\subset P-W_{n}(P),\quad\text{where}\quad
W_{n}(P)=\bigcup_{B\in{\mathcal{B}}_{n}(P)}B;$

  2. (2)

the members of the reduced family

(8)
${\mathcal{B}}^{*}_{n}(P)=\\{B(g^{n}_{P}(x_{i}),\epsilon^{n}/2)\\}\cup\\{B(z_{j},\epsilon^{n}/2)\\}$

are pairwise disjoint:

     * •

$B(z_{j},\epsilon^{n})\cap B(z_{j^{\prime}},\epsilon^{n})=\emptyset$ for
$j\neq j^{\prime}$, by construction;

     * •

$B(g^{n}_{P}(x_{i}),\epsilon^{n}/2)\cap B(z_{j},\epsilon^{n})=\emptyset$ since
$B(g^{n}_{P}(x_{i}),\epsilon^{n}/2)\subset W_{n}$ and
$B(z_{j},\epsilon^{n})\subset P-W_{n}$ and

     * •

if $x\in B(g^{n}_{P}(x_{i}),\epsilon^{n}/2)\cap
B(g^{n}_{P}(x_{i^{\prime}}),\epsilon^{n}/2)$ then
$d(g^{n}_{P}(x_{i}),g^{n}_{P}(x_{i^{\prime}}))<\epsilon^{n}$ and then
$d(x_{i},x_{i^{\prime}})<\|Dg\|^{-n}\epsilon_{n}$, but
$B(x_{i},\|Dg\|^{-n}\epsilon_{n})\cap
B(x_{i^{\prime}},\|Dg\|^{-n}\epsilon_{n})=\emptyset$, by construction.

Our next lemma uses the well known Bounded distortion property:

There exists a constant $C>0$ such that, for every $n>0$ and for every $x,y\in
P$ and every $P\in\wp_{n}$,

(9) $\left|\dfrac{Jf^{n}(x)}{Jf^{n}(y)}-1\right|\leq Cd(f^{n}(x),f^{n}(y))$

######  Lemma 5.2.

There exist constants $C_{1}>1$ and $0<\lambda_{0}<\lambda_{1}$ such that, for
every $x\in\Lambda$:

(10) $C_{1}^{-1}e^{-n\lambda_{1}}\leq\mbox{Vol}\,(\wp_{n}(x))\leq
C_{1}e^{-n\lambda_{0}}\quad\forall\ n>0.$

###### Proof.

As $f^{n}(P)=N$, for every $P\in\wp_{n}$ then, by the bounded distortion
property, there exists a constant $C_{1}>1$, only depending on the volume
distortion of $f$ and the volume of $N$, such that

$C_{1}^{-1}\leq\mbox{Vol}\,(\wp_{n}(x))Jf^{n}(x)\leq C_{1},$

for every $x\in\Lambda$ and for every $n>0$. Then,

$\sup_{x\in\Lambda}[Jf^{n}(x)]^{-1}\leq\left[\inf_{x\in\Lambda}Jf^{n}(x)\right]^{-1}\leq\left[\inf_{x\in\Lambda}Jf(x)\right]^{-n}$

Similarly so,

$\inf_{x\in\Lambda}[Jf^{n}(x)]^{-1}\geq\left[\sup_{x\in\Lambda}Jf^{n}(x)\right]^{-1}\geq\left[\sup_{x\in\Lambda}Jf(x)\right]^{-n}$

Therefore,for any given $0<\rho<\inf_{x\in\Lambda}Jf(x)$ the constants:

(11) $\displaystyle\lambda_{0}$ $\displaystyle=$
$\displaystyle\log\left(\inf_{x\in\Lambda}Jf(x)-\rho\right)$ (12)
$\displaystyle\lambda_{1}$ $\displaystyle=$
$\displaystyle\log\left(\sup_{x\in\Lambda}Jf(x)+\rho\right).$

still satisfy (10). ∎

######  Lemma 5.3.

Let $s=\alpha$ be the solution to the Bowen equation
$P(f|\Lambda,-s\log(Jf))=0$. Then,

(13)
$0<\inf\limits_{n>0}\sum\limits_{P\in\wp_{n}}\mbox{Vol}\,(P)^{\alpha}\leq\sup\limits_{n>0}\sum\limits_{P\in\wp_{n}}\mbox{Vol}\,(P)^{\alpha}<+\infty.$

###### Proof.

Let $\mu_{\alpha}$ be the equilibrium state for the Hölder continuous
potential $\phi=-\alpha\log(Jf)$. This is a Gibbs measure. Then by the bounded
distortion property and Gibbs property, we can find a constant $C_{2}>1$ such
that:

$C_{2}^{-1}\leq\dfrac{\mu_{\alpha}(\wp_{n}(x))}{\mbox{Vol}\,(\wp_{n}(x))^{\alpha}}\leq
C_{2}$

for every $x\in\Lambda$ and every $n>0$. Then (13) follows immediately by an
standard argument, since $\wp_{n}$ is a decomposition into disjoint pieces at
the $n$-stage of the construction. ∎

######  Lemma 5.4.

Let $0<\alpha\leq 1$ be the solution to the Bowen equation and
$0<\overline{\beta}<1$ be the unique solution to the equation

(14) $e^{\lambda_{0}(1-\alpha)}\epsilon^{d(1-\overline{\beta})}=1$

Then,

(15) $\underline{\dim_{B}}(\Lambda)\leq\overline{\dim_{B}}(\Lambda)\leq
d\overline{\beta}.$

###### Proof.

By lemma 5.1 there exists a uniform constant $C_{0}>0$ and a positive
$\epsilon>0$ such that for every $n>0$ and for every atom of generation $n$,
$P\in\wp_{n}$, there exits a finite covering ${\mathcal{B}}_{n}(P)$ by
$2\epsilon^{n}$-balls $B_{i}=B(x_{i},2\epsilon^{n}))$ such that
$\sum_{B\in{\mathcal{B}}_{n}(P)}\mbox{Vol}\,(B)\leq C_{0}\mbox{Vol}\,(P)$. We
define

${\mathcal{B}}_{n}=\\{B:B\in{\mathcal{B}}_{n}(P),\ P\in\wp_{n}\\}$

the open covering by these $2\epsilon^{n}$-balls of $\bigcup_{P\in\wp_{n}}P$,
the $n$-th stage in the construction of the limit $\Lambda$.

We will show that if $\beta>\overline{\beta}$ then

(16) $\sum\limits_{B\in{\mathcal{B}}_{n}}\mbox{Vol}\,(B)^{\beta}\to
0^{+}\quad\text{exponentially, as}\quad n\to+\infty.$

Actually, if $\beta>\overline{\beta}$ then there exists $C>0$ and
$0<\theta=\theta(\beta)<1$ such that

$\sum\limits_{B\in{\mathcal{B}}_{n}}\mbox{Vol}\,(B)^{\beta}\leq
C\theta^{n}\quad\text{for every}\quad n\geq 1.$

This would prove our assertion since
$A_{0}\mbox{diam}(B)^{d}\leq\mbox{Vol}\,(B)$ for every $B\in{\mathcal{B}}_{n}$
and therefore,

$A_{0}\mathcal{N}(\Lambda,2\epsilon^{n})(2\epsilon^{n})^{d\beta}\leq
A_{0}\\#{\mathcal{B}}_{n}(2\epsilon^{n})^{d\beta}\leq\sum\limits_{B\in{\mathcal{B}}_{n}}\mbox{Vol}\,(B)^{\beta}\leq
C\theta^{n},$

which would imply that

$\liminf_{\rho\to
0^{+}}\dfrac{\log\mathcal{N}(\Lambda,\rho)}{\log(1/\rho)}\leq\limsup_{\rho\to
0^{+}}\dfrac{\log\mathcal{N}(\Lambda,\rho)}{\log(1/\rho)}<d\beta$

for every $\beta>\overline{\beta}$ so proving (15). So, let $\beta>0$. Then:

$\sum\limits_{B\in{\mathcal{B}}_{n}(P)}\mbox{Vol}\,(B)^{\beta}\leq\sum\limits_{B\in{\mathcal{B}}_{n}(P)}\mbox{Vol}\,(B)\max_{B\in{\mathcal{B}}(P)}\mbox{Vol}\,(B)^{\beta-1}.$

Hence,

$\sum\limits_{B\in{\mathcal{B}}_{n}(P)}\mbox{Vol}\,(B)^{\beta}\leq
C_{0}\mbox{Vol}\,(P)[A_{1}2^{d}\epsilon^{nd}]^{\beta-1}.$

Therefore,

$\sum\limits_{B\in{\mathcal{B}}_{n}}\mbox{Vol}\,(B)^{\beta}\leq
C_{0}[A_{1}2^{d}]^{\beta-1}[\epsilon^{d(\beta-1)}]^{n}\sum\limits_{P\in\wp_{n}}\mbox{Vol}\,(P).$

On the other hand, by (5) in lemma 5.1,

$\begin{array}[]{ccc}\sum\limits_{P\in\wp_{n}}\mbox{Vol}\,(P)&\leq&\sum\limits_{P\in\wp_{n}}\mbox{Vol}\,(P)^{\alpha}\max_{P\in\wp_{n}}\mbox{Vol}\,(P)^{1-\alpha}\\\
&\leq&C_{1}^{1-\alpha}e^{-n\lambda_{0}(1-\alpha)}\sum\limits_{P\in\wp_{n}}\mbox{Vol}\,(P)^{\alpha}.\end{array}$

for every $n>0$, by (10). Thus

(17) $\sum\limits_{B\in{\mathcal{B}}_{n}}\mbox{Vol}\,(B)^{\beta}\leq
C_{3}(\beta)[e^{\lambda_{0}(\alpha-1)}\epsilon^{d(\beta-1)}]^{n},$

where

$C_{3}(\beta)=C_{0}C_{1}^{1-\alpha}[A_{1}2^{d}]^{\beta-1}\sup_{n>0}\sum\limits_{P\in\wp_{n}}\mbox{Vol}\,(P)^{\alpha}.$

does not depend on $n>0$. Now,
${\overline{\theta}}(\beta)=e^{\lambda_{0}(\alpha-1)}\epsilon^{d(\beta-1)}$ is
strictly decreasing in $\beta$, therefore
$e^{\lambda_{0}(\alpha-1)}\epsilon^{d(\beta-1)}<1$ for
$\beta>\overline{\beta}$, where $\overline{\beta}$ is the unique solution to
the equation

$e^{\lambda_{0}(\alpha-1)}\epsilon^{d(\overline{\beta}-1)}=e^{\lambda_{0}(1-\alpha)}\epsilon^{d(1-\overline{\beta})}=1,$

so proving that $\sum_{B\in{\mathcal{B}}_{n}}\mbox{Vol}\,(B)^{\beta}\to 0^{+}$
exponentially as $n\to+\infty$, bounded by a geometric progression with ratio

$\overline{\theta}(\beta)=e^{\lambda_{0}(\alpha-1)}\epsilon^{d(\beta-1)}<1.$

∎

######  Lemma 5.5.

Let $\Lambda_{n}=\bigcup_{P\in\wp_{n}}P$ be the $n$-th stage in the
construction of $\Lambda$: $\Lambda=\bigcap_{n\geq 1}\Lambda_{n}$. Then, for
every $\rho>0$:

$\mathcal{N}(\Lambda,\rho)=\inf_{n>0}\mathcal{N}(\Lambda_{n},\rho)$

Moreover, there exists $N_{1}>0$ such that

$\mathcal{N}(\Lambda,\rho)=\mathcal{N}(\Lambda_{n},\rho)\quad\text{for
every}\quad n>N_{1}.$

###### Proof.

In fact, if $\Lambda_{m}\subset\Lambda_{n}$ then
$\mathcal{N}(\Lambda_{m},\rho)\leq\mathcal{N}(\Lambda_{n},\rho)$ since every
$\rho$-covering of $\Lambda_{n}$ is a $\rho$-covering of $\Lambda_{m}$, that
is, if $\mathcal{U}(\Lambda_{n},\rho)$ is the family of finite
$\rho$-coverings of $\Lambda_{n}$, then $\Lambda_{m}\subset\Lambda_{n}$
implies $\mathcal{U}(\Lambda_{n},\rho)\subset\mathcal{U}(\Lambda_{m},\rho)$
and, therefore,

$\mathcal{N}(\Lambda_{m},\rho)\quad=\quad\inf_{\mathcal{B}\in\mathcal{U}(\Lambda_{m},\rho)}\\#\mathcal{B}\quad\leq\quad\inf_{\mathcal{B}\in\mathcal{U}(\Lambda_{n},\rho)}\\#\mathcal{B}\quad=\quad\mathcal{N}(\Lambda_{n},\rho).$

Similarly, $\mathcal{N}(\Lambda,\rho)\leq\mathcal{N}(\Lambda_{n},\rho)$ for
every $n>0$ and thus,
$\mathcal{N}(\Lambda,\rho)\leq\inf_{n>0}\mathcal{N}(\Lambda_{n},\rho)$. On the
other hand, if $\mbox{diam}(P)<\rho$ for every $P\in\wp_{n}$, then
$\wp_{n}(x)\subset B(x,\rho)$ for every $x\in\Lambda$, hence, every
$\rho$-cover of $\Lambda$ is a $\rho$-cover of $\Lambda_{n}$, that is:

$\mathcal{U}(\Lambda,\rho)\subset\mathcal{U}(\Lambda_{n},\rho)\quad\text{and
then}\quad\mathcal{N}(\Lambda_{n},\rho)\leq\mathcal{N}(\Lambda,\rho).$

This is because $\mbox{diam}(\wp_{n}(x))\to 0$ as $n\to+\infty$, uniformly so,
for sufficiently large $n>0$, $\mbox{diam}(\wp_{n}(x))<\rho$ for every
$X\in\Lambda$, so proving that

$\mathcal{N}(\Lambda,\rho)=\inf_{n>0}\mathcal{N}(\Lambda_{n},\rho).$

In fact, as $P=g^{n}_{P}(N)$ for every $P\in\wp_{n}$, then
$\mbox{diam}(P)\leq\|Dg\|^{n}\mbox{diam}(N)$ and then, $\mbox{diam}(P)<\rho$
if $\|Dg\|^{n}\mbox{diam}(N)<\rho$. Therefore,

$\mathcal{N}(\Lambda,\rho)=\mathcal{N}(\Lambda_{n},\rho)\quad\text{whenever}\quad
n>N_{1}(\rho):=\dfrac{\log(\rho/\mbox{diam}(N))}{\log\|Dg\|}.$

∎

######  Remark 5.1.

Notice that we are not saying that the box-counting dimension is continuous
and that $\dim_{B}(\Lambda_{n})\to\dim_{B}(\Lambda)$. What can be proved from
the above lemma is,

$\liminf_{n\to+\infty}\dim_{B}(\Lambda_{n})\geq\overline{\dim}_{B}(\Lambda),$

but we are not going to use this fact in the present discussion.

######  Lemma 5.6.

Let $0<\alpha\leq 1$ be the solution to the Bowen equation and
$0<\underline{\beta}<1$ be the unique solution to the equation

(18) $e^{\lambda_{2}(1-\alpha)}\epsilon^{d(1-\underline{\beta})}=1,$

where

(19) $\lambda_{2}=\dfrac{\lambda_{1}\log\epsilon}{\log\|Dg\|}.$

Then,

(20)
$d\underline{\beta}\leq\underline{\dim_{B}}(\Lambda)\leq\overline{\dim_{B}}(\Lambda)$

###### Proof.

Let $n,m$ be positive integers, $\beta>0$, a positive real number and
$\mathcal{B}_{n}$ a finite $2\epsilon^{n}$-cover of $\Lambda_{m}$. Then, for
every $P\in\wp_{m}$:

$\sum\limits_{B\in{\mathcal{B}}_{n}(P)}\mbox{Vol}\,(B)^{\beta}\geq\sum\limits_{B\in{\mathcal{B}}_{n}(P)}\mbox{Vol}\,(B)\min_{B\in{\mathcal{B}}(P)}\mbox{Vol}\,(B)^{\beta-1}.$

Hence,

$\sum\limits_{B\in{\mathcal{B}}_{n}(P)}\mbox{Vol}\,(B)^{\beta}\geq\mbox{Vol}\,(P)[A_{0}2^{d}\epsilon^{nd}]^{\beta-1}$

and then:

$\sum\limits_{B\in{\mathcal{B}}_{n}}\mbox{Vol}\,(B)^{\beta}\geq[A_{0}2^{d}]^{\beta-1}[\epsilon^{d(\beta-1)}]^{n}\sum\limits_{P\in\wp_{m}}\mbox{Vol}\,(P).$

On the other hand,

$\begin{array}[]{ccc}\sum\limits_{P\in\wp_{m}}\mbox{Vol}\,(P)&\geq&\sum\limits_{P\in\wp_{m}}\mbox{Vol}\,(P)^{\alpha}\min_{P\in\wp_{m}}\mbox{Vol}\,(P)^{1-\alpha}\\\
&\geq&C_{1}^{\alpha-1}e^{-m\lambda_{1}(1-\alpha)}\sum\limits_{P\in\wp_{m}}\mbox{Vol}\,(P)^{\alpha}.\end{array}$

Then:

$\sum\limits_{B\in{\mathcal{B}}_{n}}\mbox{Vol}\,(B)^{\beta}\geq
C_{4}(\beta)[e^{m\lambda_{1}(\alpha-1)}\epsilon^{nd(\beta-1)}]\quad\text{for
every}\quad n,m>0,$

where

$C_{4}(\beta)=C_{1}^{\alpha-1}[A_{0}2^{d}]^{\beta-1}\inf_{n>0}\sum\limits_{P\in\wp_{n}}\mbox{Vol}\,(P)^{\alpha}.$

Now,

$\sum\limits_{B\in{\mathcal{B}}_{n}}\mbox{Vol}\,(B)^{\beta}=(2\epsilon^{n})^{d\beta}\\#{\mathcal{B}}_{n}$

and then

$\\#{\mathcal{B}}_{n}\geq(2\epsilon^{n})^{-d\beta}C_{4}(\beta)[e^{m\lambda_{1}(\alpha-1)}\epsilon^{nd(\beta-1)}]$

as ${\mathcal{B}}_{n}$ is an arbitrary finite $2\epsilon^{n}$-covering of
$\Lambda_{m}$ and the right side do not depend on ${\mathcal{B}}_{n}$, then:

$\mathcal{N}(\Lambda_{m},2\epsilon^{n})(2\epsilon^{n})^{d\beta}\geq
C_{4}(\beta)[e^{m\lambda_{1}(\alpha-1)}\epsilon^{nd(\beta-1)}]$

for every $n,m>0$. Hence, if

$N_{1}(2\epsilon^{n}))=\dfrac{\log(2\epsilon^{n}/\mbox{diam}\,N)}{\log\|Dg\|}$

then,

$\mathcal{N}(\Lambda,2\epsilon^{n})=\mathcal{N}(\Lambda_{m},2\epsilon^{n})\quad\text{for
every}\quad m>N_{1}.$

Let us choose

$m=\left[\dfrac{\log(2\epsilon^{n}/\mbox{diam}(N))}{\log\|Dg\|}\right]+1$

where $[x]$ is the largest integer less than or equal to $x$, i.e. the integer
part of a real number $x$, then

$\dfrac{\log(2\epsilon^{n}/\mbox{diam}(N))}{\log\|Dg\|}<m<\dfrac{\log(2\epsilon^{n}/\mbox{diam}(N))}{\log\|Dg\|}+1.$

Therefore,

(21) $\displaystyle m\lambda_{1}(\alpha-1)$ $\displaystyle>$
$\displaystyle\left(\dfrac{\log(2\epsilon^{n}/\mbox{diam}(N))}{\log\|Dg\|}+1\right)\lambda_{1}(\alpha-1)$
(22) $\displaystyle=$
$\displaystyle\left(\dfrac{\log(2/\mbox{diam}(N))}{\log\|Dg\|}+\dfrac{n\log\epsilon}{\log\|Dg\|}+1\right)\lambda_{1}(\alpha-1)$
(23) $\displaystyle=$ $\displaystyle n\lambda_{2}(\alpha-1)+C_{5}$

where,

$\lambda_{2}=\dfrac{\lambda_{1}\log\epsilon}{\log\|Dg\|}\quad\text{and}\quad
C_{5}=\left(\dfrac{\log(2/\mbox{diam}(N))}{\log\|Dg\|}+1\right)\lambda_{1}(\alpha-1)$

which do not depend on $n$. Hence,

$\mathcal{N}(\Lambda,2\epsilon^{n})(2\epsilon^{n})^{d\beta}\geq
C_{6}(\beta)[e^{\lambda_{2}(\alpha-1)}\epsilon^{d(\beta-1)}]^{n}$

where,

$C_{6}(\beta)=e^{C_{5}}C_{4}(\beta)>0,$

is a uniform constant not depending on $n>0$. If $\underline{\beta}$ is the
solution to the equation (18) and $\beta<\underline{\beta}$ then

$e^{\lambda_{2}(\alpha-1)}\epsilon^{d(\beta-1)}>1,$

so that

$\mathcal{N}(\Lambda,2\epsilon^{n})(2\epsilon^{n})^{d\beta}\to+\infty,\quad\text{exponentially,}\quad\text{as}\quad
n\to+\infty,$

so proving that

$\overline{\dim}_{B}(\Lambda)\geq\underline{\dim}_{B}(\Lambda)\geq d\beta$

for every $\underline{\beta}>\beta$, concluding that

$\overline{\dim}_{B}(\Lambda)\geq\underline{\dim}_{B}(\Lambda)\geq
d\underline{\beta}.$

∎

##  6\. Proof of theorem 1: upper and lower bound for the Hausdorff dimension

To bound the Hausdorff dimension we need to use an arbitrary $\zeta$-covering
${\mathcal{B}}_{0}=\\{B^{0}_{k}\\}$ by open balls of variable radius, where we
will suppose, without loss of generality, that ${\mathcal{B}}_{0}$ is finite.
By lemma 5.1, for every $P\in\wp_{n}$ there exists a finite open cover

${\mathcal{B}}_{n}(P)=\\{B(x_{i}(P),2\epsilon^{n}),B(y_{j}(P),2\epsilon^{n})\\}$

such that $\sum_{B\in{\mathcal{B}}_{n}(P)}\mbox{Vol}\,(B)\leq
C_{0}\mbox{Vol}\,(P)$ for a suitable constant $C_{0}>0$. Let us choose $n>0$
large enough such that $2\epsilon^{n}<\lambda({\mathcal{B}}_{0})<\zeta$, the
Lebesgue number of ${\mathcal{B}}_{0}$.

We can suppose that the elements of the reduced family

${\mathcal{B}}^{*}_{n}(P)=\\{B(x_{i}(P),\epsilon^{n}/2),B(y_{j}(P),\epsilon^{n}/2)\\}$

are pairwise disjoint, as we proved before. Let

${\mathcal{B}}_{n}=\\{B\in{\mathcal{B}}_{n}(P):P\in\wp_{n}\\}$

be the $2\epsilon^{n}$-covering of $\Lambda_{n}$, the $n$-stage of the
geometric construction. Using the Lebesgue number property we can arrange the
open balls $B\in{\mathcal{B}}_{n}$ as follows:

$\begin{array}[]{ccc}{\mathcal{B}}_{n}(B^{0}_{1})&:=&\\{B\in{\mathcal{B}}_{n}:B\subset
B^{0}_{1}\\}\\\
{\mathcal{B}}_{n}(B^{0}_{2})&:=&\\{B\in{\mathcal{B}}_{n}-{\mathcal{B}}_{n}(B^{0}_{1}):B\subset
B^{0}_{2}\\}\\\ &\vdots&\\\
{\mathcal{B}}_{n}(B^{0}_{l})&:=&\\{B\in{\mathcal{B}}_{n}-\bigcup_{l^{\prime}<l}{\mathcal{B}}_{n}(B^{0}_{l^{\prime}}):B\subset
B^{0}_{l}\\}\\\ \end{array}$

For each $B^{0}_{k}\in{\mathcal{B}}_{0}$ the family
${\mathcal{B}}_{n}(B^{0}_{k})$ is made of collections of open balls in
${\mathcal{B}}_{n}(P)$, where $P\cap B^{0}_{k}\neq\emptyset$. However,
eventhough for every $P$ the $\epsilon^{n}$-balls of the reduced covering
${\mathcal{B}}^{*}_{n}(P)$ are disjoint it can occur that, for different atoms
$P,Q\subset B^{0}_{k}$, there are overlapping balls $B\cap
B^{\prime}\neq\emptyset$ with $B\in{\mathcal{B}}^{*}_{n}(P)$ and
$B^{\prime}\in{\mathcal{B}}^{*}_{n}(Q)$. In order to control those
overlappings we argue as follows.

######  Lemma 6.1.

There exists a constant $D>1$ with the following property: for every
$B^{0}_{k}\in{\mathcal{B}}_{0}$ there exists a collection of disjoint
$\epsilon^{n}$-balls $B_{i}\in{\mathcal{B}}^{*}_{n}(B^{0}_{k})$ such that

(24)
$\sum_{B\in{\mathcal{B}}_{n}(B^{0}_{k})}\mbox{Vol}\,(B)\leq\dfrac{A_{1}D}{A_{0}}\sum_{i}\mbox{Vol}\,(B_{i}).$

######  Remark 6.1.

The constant $D>1$ only depends on the topological dimension of the border of
the atoms generating $\Lambda$ and is, therefore, independent of the
$\zeta$-covering $\mathcal{B}_{0}$.

###### Proof.

For any given $P\in\wp_{n}$, the $\epsilon^{n}/2$-balls
$B(y_{j}(P),\epsilon^{n}/2)$ are disjoint, by definition. Moreover, by the
open condition, for every $Q\neq P$, the corresponding interior balls
$B(y_{j}(P),\epsilon^{n}/2)$ and $B(y_{k}(Q),\epsilon^{n}/2)$ are pairwise
disjoint, so the unique overlappings between elements in
${\mathcal{B}}^{*}_{n}(P)$ for different atoms $P\in\wp_{n}$ can occur near
the border. As
$\partial\mathcal{B}_{n}=\\{B(x_{i}(P),2\epsilon^{n}):P\in\wp_{n},i>0\\}$ is a
minimal $2\epsilon^{n}$-covering of the border of the $n$-th stage in the
construction, $\partial{\wp_{n}}=\bigcup_{P\in\wp_{n}}\partial{P}$, we can
find a constant $D>1$, only depending on the topological dimension of
$\partial{\wp_{n}}$ such that there are at most $D$-overlappings: for every
$i$,

$\\#\\{(j,Q):B(x_{i}(P),\epsilon^{n}/2)\cap
B(x_{j}(Q),\epsilon^{n}/2)\neq\emptyset\\}\leq D.$

To rpove this we begin with some
$B_{1}=B(x_{i}(P),\epsilon^{n}/2)\in\mathcal{B}^{*}_{n}(B^{0}_{k})$. It is
straightforward to prove that, for all the overlapping balls:

$\bigcup\limits_{(j,Q)}{B(x_{j}(Q),\epsilon^{n}/2)}\subset 4B_{1},$

where $tB(x,r):=B(x,tr)$, and

$\sum\limits_{(j,Q)}\mbox{Vol}\,(B(x_{j}(Q),\epsilon^{n}/2))\leq\dfrac{A_{1}D}{A_{0}}\mbox{Vol}\,(B_{1}).$

Then we remove all those overlappings balls $B(x_{j}(Q),\epsilon^{n}/2)$ for
$Q\neq P$ and choose $B_{2}\in\partial\mathcal{B}_{n}$ out of the set of
overlappings associated to $B_{1}$:

$B_{2}\in\partial\mathcal{B}_{n}-\\{B(x_{j}(Q),2\epsilon^{n}):B_{1}\cap
B(x_{j}(Q),\epsilon^{n}/2)\neq\emptyset\\}.$

Then we remove the $B_{2}$ overlappings and proceed inductively until we get a
disjoint collection $\\{B_{i}\\}$ with the required properties. ∎

Now, we are ready to prove the following lemma which says how to compare the
sums
$\sum\limits_{B^{0}_{k}\in{\mathcal{B}}_{0}}\mbox{Vol}\,(B^{0}_{k})^{\beta}$
and $\sum\limits_{B\in{\mathcal{B}}_{n}}\mbox{Vol}\,(B)^{\beta}$, for an
arbitrary $\zeta$-covering ${\mathcal{B}}_{0}$.

######  Lemma 6.2.

(25) $\sum\limits_{B\in{\mathcal{B}}_{n}}\mbox{Vol}\,(B)^{\beta}\leq
C_{4}(\beta)[\epsilon^{nd(\beta-1)}\zeta^{d(1-\beta)}]\sum\limits_{B^{0}_{k}\in{\mathcal{B}}_{0}}\mbox{Vol}\,(B^{0}_{k})^{\beta},$

where

$C_{4}(\beta)=\dfrac{2^{d(\beta-1)}A_{1}D}{A_{0}}.$

###### Proof.

By lemma 6.1

$\sum\limits_{B\in{\mathcal{B}}_{n}(B^{0}_{k})}\mbox{Vol}\,(B)\leq\dfrac{A_{1}D}{A_{0}}\sum_{i}\mbox{Vol}\,(B_{i})\leq\dfrac{A_{1}D}{A_{0}}\mbox{Vol}\,(B^{0}_{k}).$

Hence,

$\displaystyle\sum\limits_{B\in{\mathcal{B}}_{n}}\mbox{Vol}\,(B)^{\beta}$
$\displaystyle\leq$
$\displaystyle\max_{B\in{\mathcal{B}}_{n}}\mbox{Vol}\,(B)^{\beta-1}\sum\limits_{B\in{\mathcal{B}}_{n}}\mbox{Vol}\,(B)$
$\displaystyle\leq$
$\displaystyle[A_{1}2^{d}\epsilon^{nd}]^{\beta-1}\dfrac{A_{1}D}{A_{0}}\sum\limits_{B^{0}_{k}\in{\mathcal{B}}_{0}}\mbox{Vol}\,(B^{0}_{k})$
$\displaystyle\leq$
$\displaystyle[A_{1}2^{d}\epsilon^{nd}]^{\beta-1}\dfrac{A_{1}D}{A_{0}}\max_{B^{0}_{k}\in{\mathcal{B}}_{0}}\mbox{Vol}\,(B^{0}_{k})^{1-\beta}\sum\limits_{B^{0}_{k}\in{\mathcal{B}}_{0}}\mbox{Vol}\,(B^{0}_{k})^{\beta}$
$\displaystyle\leq$
$\displaystyle[A_{1}2^{d}\epsilon^{nd}]^{\beta-1}\dfrac{A_{1}D}{A_{0}}[A_{1}\zeta^{d}]^{1-\beta}\sum\limits_{B^{0}_{k}\in{\mathcal{B}}_{0}}\mbox{Vol}\,(B^{0}_{k})^{\beta}$
$\displaystyle\leq$ $\displaystyle
C_{4}(\beta)[\epsilon^{nd(\beta-1)}\zeta^{d(1-\beta)}]\sum\limits_{B^{0}_{k}\in{\mathcal{B}}_{0}}\mbox{Vol}\,(B^{0}_{k})^{\beta}.$

∎

Now, we construct an efficient covering of $\Lambda$ from ${\mathcal{B}}_{n}$
as follows. For $n\geq N_{0}$ we choose a $\|Dg\|^{-n}2\epsilon^{n}$-covering
of the border $\mathcal{B}_{n}(\partial{N})$ and generate a
$2\epsilon^{n}$-covering of $\partial{P}$, for every $P\in\wp_{n}$ defining an
open neighborhood $W_{n}(P)$. Then we choose a $2\epsilon^{n}$-covering of
$P-W_{n}(P)$ as follows: fix a maximal family of non-overlapping open
$\epsilon^{n}$-balls $\\{B(z_{j},\epsilon^{n})\\}$ inside the closed set
$P-W_{n}(P)$ so that $\\{B(z_{j},2\epsilon^{n})\\}$ is on open covering of
$P-W_{n}(P)$. This choice was completely arbitrary whenever the non-
overlapping condition is fullfiled. So let us choose now this collection as
follows: let us start, for every $P\in\wp_{n}$, with a maximal collection of
non-overlapping open $\epsilon^{n}$-balls $\\{B(y_{k},\epsilon^{n})\\}$ with
$y_{k}\in\Lambda\cap P$ contained in $P-W_{n}(P)$. This is possible since
$\Lambda\cap P\subset P-W_{n}(P)$, by our border condition and the choice of
$n\geq N_{0}$. Then, we extend to a maximal collection
$\\{B(z_{j},\epsilon^{n})\\}$ of non-overlapping open $\epsilon^{n}$-balls in
$P-W_{n}(P)$. We call

$\mathcal{B}_{n}(\Lambda\cap
P)=\\{B(y_{k},2\epsilon^{n})\\}\quad\text{and}\quad\mathcal{B}_{n}(P)=\\{B(y_{k},2\epsilon^{n})\\}\cup\\{B(z_{j},2\epsilon^{n})\\}\cup\\{B(x_{i},2\epsilon^{n})\\},$

where $y_{k}\in\Lambda$, $z_{j}\in
P-\left(W_{n}(P)\cup\bigcup_{k}B(y_{k},\epsilon^{n})\right)$ and
$x_{i}\in\partial{P}$. We then define

${\mathcal{B}}_{n}(\Lambda)=\bigcup_{P\in\wp_{n}}{\mathcal{B}}_{n}(\Lambda\cap
P).$

and

(26) $\rho(P)=\dfrac{\sum_{B\in{\mathcal{B}}_{n}(\Lambda\cap
P)}\mbox{Vol}\,(B)}{\mbox{Vol}\,(P)}.$

$\rho(P)$ is the fraction of $P$ which is covered by the $2\epsilon^{n}$-balls
in ${\mathcal{B}}_{n}(\Lambda\cap P)$.

######  Proposition 6.3.

There exists a bounded measurable non negative function,

$\Sigma(x)=\limsup_{n\to+\infty}\,-\dfrac{\log\rho(\wp_{n}(x))}{n},\quad\forall\
x\in\Lambda.$

Moreover, for sufficiently small $\delta>0$,

$-\infty<\log\left(\dfrac{\inf_{x\in\Lambda}Jf(x)}{\epsilon^{-(d(\Lambda)+\delta-d)}}\right)\leq\Sigma(x)\leq\log\left(\dfrac{\sup_{x\in\Lambda}Jf(x)}{(\|Df\|\epsilon)^{-(d_{B}(\Lambda)-\delta)}\epsilon^{d}}\right)<+\infty$

for every $x\in\Lambda$

To prove proposition 6.3 we need first a couple of lemmas.

######  Lemma 6.4.

Let $\|Df\|=\sup_{x\in\bigcup_{i}N_{i}}\|Df(x)\|$. There exists $\delta_{0}>0$
such that $\|Df\|\epsilon<1$ for every $0<\delta<\delta_{0}$.

By definition,

$\epsilon=(1-\delta)\left(\dfrac{\|\underline{Jg}\|}{\quad\quad\|Dg\|^{d-1+\delta}}\right)^{\frac{1}{1-\delta}},$

where

$\|\underline{Jg}\|=\min_{i}\inf_{x\in
N}Jg_{i}(x)\quad\text{and}\quad\|Dg\|=\max_{i}\sup_{x\in N}Dg_{i}(x).$

We will use the singular values of $Dg_{i}(x)$, $0<s^{d}_{i}(x)\leq\cdots\leq
s^{1}_{i}(x)<1$ to estimate the product $\|Df\|\epsilon$. Indeed, as

$Jg_{i}(x)=s^{1}_{i}(x)\cdots
s^{d}_{i}(x)\quad\quad\|Dg_{i}(x)\|=s^{1}_{i}(x)\quad\quad
Df(g_{i}(x))=(s^{d}_{i}(x))^{-1}$

we are led to estimate:

$\Delta(x,\delta)=(1-\delta)\left(\dfrac{s^{1}_{i}(x)\cdots
s^{d}_{i}(x)}{(s^{1}_{i}(x))^{d-1+\delta}}\right)^{\frac{1}{1-\delta}}(s^{d}_{i}(x))^{-1}.$

This is a continuous function of $x\in N$ and $\delta\geq 0$. Then, provided
that at least one singular value $s^{k}_{i}(x)<s^{1}_{i}(x)$, which is
certainly the case because we are dealing with a non conformal map, then:

$\Delta(x,0)=\dfrac{s^{1}_{i}(x)\cdots
s^{d}_{i}(x)}{(s^{1}_{i}(x))^{d-1}}(s^{d}_{i}(x))^{-1}=\dfrac{s^{2}(x)}{s^{1}(x)}\cdots\dfrac{s^{d-1}(x)}{s^{1}(x)}<1.$

Thus we can find, by continuity and a compacity argument, $\delta_{0}>0$, a
small positive number, such that:

$\Delta(x,\delta)<1\quad\text{for every}\quad
0<\delta<\delta_{0},\quad\forall\ x\in\Lambda.$

Then, taking supremum and infimum properly, we conclude that
$\|Df\|\epsilon<1$, as we claimed.

######  Lemma 6.5.

There exists $D_{1}>1$ such that, for every $0<\delta<\delta_{0}$,

(27)
$D^{-1}_{1}(\|Df\|^{n}2\epsilon^{n})^{-(d_{B}(\Lambda)-\delta)}\leq\\#\mathcal{B}_{n}(\Lambda\cap\wp_{n}(x))),$

for every $x\in\Lambda$ and $n>0$, where $\epsilon=\epsilon(\delta)$.

###### Proof.

As $\|Df\|\epsilon<1$ then $(\|Df\|\epsilon)^{n}\to 0$ as $n\to+\infty$ and
therefore, we can find a constant $D_{1}>1$ such that:

$D^{-1}_{1}(\|Df\|^{n}2\epsilon^{n})^{-(d_{B}(\Lambda)-\delta)}\leq\mathcal{N}(\Lambda,\|Df\|^{n}2\epsilon^{n})\leq
D_{1}(\|Df\|^{n}2\epsilon^{n})^{-(d_{B}(\Lambda)+\delta)}\quad\forall\ n>0,$

Now, notice that

$\mathcal{N}(\Lambda,\|Df\|^{n}2\epsilon^{n})\leq\\#\mathcal{B}_{n}(\Lambda\cap\wp_{n}(x))),$

since

$f^{n}(\Lambda\cap P)=\Lambda\quad\text{and}\quad
f^{n}(B(x_{i},2\epsilon^{n}))\subset B(f(x_{i}),\|Df\|^{n}2\epsilon^{n}),$

defining an open $\|Df\|^{n}2\epsilon^{n}$-covering of $\Lambda$ with the same
cardinality of $\\#\mathcal{B}_{n}(\Lambda\cap\wp_{n}(x))$. Here we use the
fact that $f:\Lambda\to\Lambda$ es topologically exact, that is, for every
open set $U\subset\Lambda$ there exists $n>0$ such that $f^{n}(U)=\Lambda$,
since $f|\Lambda$ es topologically conjugated to the full unilateral shift.
Hence:

$D^{-1}_{1}(\|Df\|^{n}2\epsilon^{n})^{-(d_{B}(\Lambda)-\delta)}\leq\\#\mathcal{B}_{n}(\Lambda\cap\wp_{n}(x))),$

as we wanted to prove. ∎

###### Proof of proposition 6.3.

We are looking for upper and lower bounds for

$\rho(\wp_{n}(x))=\dfrac{\sum_{B\in{\mathcal{B}}_{n}(\Lambda\cap\wp_{n}(x))}\mbox{Vol}\,(B)}{\mbox{Vol}\,(\wp_{n}(x))}$

For this we start recalling that there exists $C_{2}>1$ such that, for every
$x\in\Lambda$,

(28) $C_{2}^{-1}\leq\mbox{Vol}\,(\wp_{n}(x))Jf^{n}(x)\leq C_{2}\quad\forall\
x\in\Lambda,\ n>0.$

This, together with (27) provides a lower bound:

$\rho(\wp_{n}(x))\geq\dfrac{D^{-1}_{1}(\|Df\|^{n}2\epsilon^{n})^{-(d_{B}(\Lambda)-\delta)}(2\epsilon^{n})^{d}}{C_{2}[\sup_{x\in\Lambda}Jf(x)]^{n}}=C_{3}D_{3}^{n}$

where

$C_{3}=\dfrac{D^{-1}_{1}2^{-(d_{B}(\Lambda)-\delta-d)}}{C_{2}}\quad\text{and}\quad
D_{3}=\dfrac{(\|Df\|\epsilon)^{-(d_{B}(\Lambda)-\delta)}\epsilon^{d}}{\sup_{x\in\Lambda}Jf(x)}.$

Now, notice that, as
${\mathcal{B}}_{n}(\Lambda\cap\wp_{n}(x))=\\{B(y_{k},2\epsilon^{n})\\}$ and
$\\{B(y_{k},\epsilon^{n})\\}$ is a maximal family of non-overlapping open
$\epsilon^{n}$-balls with $y_{k}\in\Lambda\cap\wp_{n}(x)$, then it does not
cover totally $\Lambda\cap\wp_{n}(x)$. Therefore,

$\\#{\mathcal{B}}_{n}(\Lambda\cap\wp_{n}(x))\leq\mathcal{N}(\Lambda\cap\wp_{n}(x),\epsilon^{n})\leq\mathcal{N}(\Lambda,\epsilon^{n}).$

Let $0<\delta<\delta_{0}$, where $\delta_{0}>0$ was defined in lemma 6.5.
Then, there exists a constant $D_{2}>1$ such that

$D_{2}^{-1}(\epsilon^{n})^{-(d(\Lambda)-\delta}\leq\mathcal{N}(\Lambda,\epsilon^{n})\leq
D_{2}(\epsilon^{n})^{-(d(\Lambda)+\delta)}\quad\forall\ n>0.$

Thus,

$\rho(\wp_{n}(x))\leq\dfrac{D_{2}(\epsilon^{n})^{-(d(\Lambda)+\delta)}(2\epsilon^{n})^{d}}{C^{-1}_{2}[\inf_{x\in\Lambda}Jf(x)]^{n}}=\dfrac{C_{2}D_{2}2^{d}(\epsilon^{n})^{-(d(\Lambda)+\delta-d)}}{[\inf_{x\in\Lambda}Jf(x)]^{n}}=C_{4}D^{n}_{4}$

where

$C_{4}=C_{2}D_{2}2^{d}\quad\text{and}\quad
D_{4}=\dfrac{\epsilon^{-(d(\Lambda)+\delta-d)}}{\inf_{x\in\Lambda}Jf(x)}$

Hence,

$-C_{4}-n\log{D_{4}}\leq-\log(\rho(\wp_{n}(x))\leq-
C_{3}-n\log{D_{3}},\quad\quad x\in\Lambda,\ n>0,$

and then

$\overline{\Sigma}(x)=\limsup_{n\to+\infty}\,-\dfrac{\log(\rho(\wp_{n}(x))}{n}$

is a bounded Borel measurable non negative function defined over $\Lambda$.
Moreover,

$\log\left(\dfrac{\inf_{x\in\Lambda}Jf(x)}{\epsilon^{-(d(\Lambda)+\delta-d)}}\right)\leq\overline{\Sigma}(x)\leq\log\left(\dfrac{\sup_{x\in\Lambda}Jf(x)}{(\|Df\|\epsilon)^{-(d_{B}(\Lambda)-\delta)}\epsilon^{d}}\right)$

∎

We notice that the lower and upper bounds for $\Sigma=\Sigma(x)$ depends on
several parameters as $\delta$, $\epsilon$, the largest volume and distance
expansion produced by iteration of $f$. The next lemma shows that $\delta$ can
be chosen small enough in such way that the lower bound of $\Sigma$ is
positive.

######  Lemma 6.6.

There exists $0<\delta_{1}<\delta_{0}$ such that:

$\log\left(\dfrac{\inf_{x\in\Lambda}Jf(x)}{\epsilon^{-(d(\Lambda)+\delta-d)}}\right)>0\quad\text{for
every}\quad 0<\delta<\delta_{1}$

###### Proof.

We want to bound

$\dfrac{\inf_{x\in\Lambda}Jf(x)}{\epsilon^{-(d(\Lambda)+\delta-d)}}=\inf_{x\in\Lambda}Jf(x)\epsilon^{d(\Lambda)+\delta-d}.$

Let $0<s^{d}_{i}(x)\leq\cdots\leq s^{1}_{i}(x)<1$ be the singular values of
$Dg_{i}(x)$. As

$\epsilon=(1-\delta)\left(\dfrac{s^{1}_{i}(x)\cdots
s^{d}_{i}(x)}{(s^{1}_{i}(x))^{d-1+\delta}}\right)^{\frac{1}{1-\delta}}$

and

$Jf(x)=(s^{1}_{i}(x)\cdots s^{d}_{i}(x))^{-1}\quad\text{for every}\quad x\in
N_{i},$

then:

$Jf(x)\epsilon^{d(\Lambda)+\delta-d}=(1-\delta)(s^{1}_{i}(x)\cdots
s^{d}_{i}(x))^{\tau_{1}(\delta)}(s^{1}_{i}(x))^{\tau_{2}(\delta)}=\Delta(x,\delta)$

where:

$\tau_{1}(\delta)=\dfrac{d(\Lambda)+\delta-d}{1-\delta}=\dfrac{d(\Lambda)+2\delta-d-1}{1-\delta}$

and

$\tau_{2}(\delta)=\,-\,\dfrac{(d-1+\delta)(d_{B}(\Lambda)+\delta-d)}{1-\delta}.$

For every $x\in\Lambda$, $\Delta(x,\delta)$ is a continuous function of
$\delta\geq 0$. Evaluating at $\delta=0$ we get:

$\Delta(x,0)=(s^{1}_{i}(x)\cdots
s^{d}_{i}(x))^{d(\Lambda)-d-1}(s^{1}_{i}(x))^{-(d-1)(d_{B}(\Lambda)-d)}$

as $d_{B}(\Lambda)<d$ and $d\geq 2$, $d(\Lambda)-d-1<0$ so that:

$(s^{1}_{i}(x)\cdots
s^{d}_{i}(x))^{d(\Lambda)-d-1}\geq(s^{1}_{i}(x))^{d(d(\Lambda)-d-1)},$

thus

$\Delta(x,0)\geq(s^{1}_{i}(x))^{d(d(\Lambda)-d-1)}(s^{1}_{i}(x))^{(d-1)(d-d_{B}(\Lambda))}.$

Now,

$d(d(\Lambda)-d-1)+(d-1)(d-d_{B}(\Lambda))=-2d,$

so we get that

$\Delta(x,0)\geq(s^{1}_{i}(x))^{-2d}>1.$

Hence, by continuity and a compacity argument, we find
$0<\delta_{1}<\delta_{0}$ such that:

$\Delta(x,\delta)>1\quad\text{for every}\quad
0<\delta<\delta_{1},\quad\forall\ x\in\Lambda.$

so proving that,

$\log\left(\dfrac{\inf_{x\in\Lambda}Jf(x)}{\epsilon^{-(d(\Lambda)+\delta-d)}}\right)>0,$

for every $0<\delta<\delta_{1}$ ∎

######  Corollary 6.1.

For every $0<\delta<\delta_{1}$:

$0<\inf_{x\in\Lambda}\Sigma(x)\leq\sup_{x\in\Lambda}\Sigma(x)<+\infty$

We fix $0<\delta<\delta_{1}$ once for all. This fixes
$\epsilon=\epsilon(\delta)$

######  Lemma 6.7.

Let $\overline{\Sigma}=\sup_{x\in\Lambda}\Sigma(x)>0$ be the supremum of
$\Sigma=\Sigma(x)$ and $s=\underline{\alpha}$ be the solution to the modified
Bowen equation

(29) $P(f|\Lambda,-s\log(e^{\overline{\Sigma}}Jf))=0.$

Then:

(30)
$\inf\limits_{n>0}\sum\limits_{P\in\wp_{n}}(\rho(P)\mbox{Vol}\,(P))^{\underline{\alpha}}\
>\ 0.$

Similarly, let $\underline{\Sigma}=\sup_{x\in\Lambda}\Sigma(x)>0$ be the
infimum and $s=\overline{\alpha}$ be the solution to the modified Bowen
equation

(31) $P(f|\Lambda,-s\log(e^{\underline{\Sigma}}Jf))=0.$

Then

(32)
$\sup\limits_{n>0}\sum\limits_{P\in\wp_{n}}(\rho(P)\mbox{Vol}\,(P))^{\overline{\alpha}}\
<\ +\infty.$

###### Proof.

By proposition 6.3,

$\rho(\wp_{n}(x))\geq e^{-n\overline{\Sigma}}\quad\forall\ x\in\Lambda,\ n\geq
1,$

where $\overline{\Sigma}\geq 0$ is the supremum of $\Sigma(x)$, the asymptotic
fraction of the volume of $P$ which is covered by $2\epsilon^{n}$-balls of the
minimal covering ${\mathcal{B}}_{n}(\Lambda\cap P)$. Let
$\mu_{\underline{\alpha}}$ be the equilibrium state for the Hölder continuous
potential $\underline{\phi}=-\underline{\alpha}\log(e^{\overline{\Sigma}}Jf)$.
Then, for $\mu_{\underline{\alpha}}$-a.e. $x\in\Lambda$:

$\displaystyle(\rho(\wp_{n}(x))\mbox{Vol}\,(\wp_{n}(x)))^{\underline{\alpha}}$
$\displaystyle\geq$
$\displaystyle(e^{-n\overline{\Sigma}}\mbox{Vol}\,(\wp_{n}(x)))^{\underline{\alpha}}$
$\displaystyle\geq$ $\displaystyle
C_{1}^{-\underline{\alpha}}(e^{-n\overline{\Sigma}}[Jf^{n}(x)]^{-1})^{\underline{\alpha}}$
$\displaystyle=$ $\displaystyle
C_{1}^{-\underline{\alpha}}\left(e^{n\overline{\Sigma}}\prod_{k=0}^{n-1}Jf(f^{k}(x))\right)^{-\underline{\alpha}}$
$\displaystyle=$ $\displaystyle
C_{1}^{-\underline{\alpha}}\exp\left(\sum_{k=0}^{n-1}\underline{\phi}(f^{k}(x))\right)$
$\displaystyle\geq$
$\displaystyle(C_{1}^{\underline{\alpha}}G)^{-1}\mu_{\underline{\alpha}}(\wp_{n}(x)),$

by the Gibbs property (1). Thus,

$\sum\limits_{P\in\wp_{n}}(\rho(P)\mbox{Vol}\,(P))^{\underline{\alpha}}\geq
C_{1}^{-\underline{\alpha}}G_{1}^{-1}>0$

for every $n>0$, so proving (30).

The proof of upper bound (32) follows under the same arguments. For this we
start recalling that

$\rho(\wp_{n}(x))\leq e^{-n\underline{\Sigma}}\quad\forall\ x\in\Lambda,\
n\geq 1$

and then

$(\rho(\wp_{n}(x))\mbox{Vol}\,(\wp_{n}(x)))^{\overline{\alpha}}\leq(e^{-n\underline{\Sigma}}\mbox{Vol}\,(\wp_{n}(x)))^{\overline{\alpha}}\leq
C_{1}^{\underline{\alpha}}\exp\left(\sum_{k=0}^{n-1}\overline{\phi}(f^{k}(x))\right)$

where $\overline{\phi}=-\overline{\alpha}\log(e^{\underline{\Sigma}}Jf)$ and
then we use again the Gibbs property to bound from above:

$\sum\limits_{P\in\wp_{n}}(\rho(P)\mbox{Vol}\,(P))^{\overline{\alpha}}\leq
C_{1}^{\underline{\alpha}}G_{1}<+\infty.$

∎

######  Lemma 6.8.

There exist constants $C_{5}(\beta)>0$ and $C_{6}(\beta)>0$, independent of
$n>0$, such that, for every $\beta>0$,

(33)
$C_{5}(\beta)[e^{\lambda_{1}(\underline{\alpha}-1)}\epsilon^{d(\beta-1)}]^{n}\leq\sum\limits_{B\in{\mathcal{B}}_{n}(\Lambda)}\mbox{Vol}\,(B)^{\beta}\quad\leq\quad
C_{6}(\beta)[e^{\lambda_{0}(\overline{\alpha}-1)}\epsilon^{d(\beta-1)}]^{n},$

for every $n\geq N_{0}$, where $\underline{\alpha}$ is the solution to the
lower modified Bowen equation (29) and $\overline{\alpha}$ is the solution to
the upper modified Bowen equation (31).

###### Proof.

Let ${\mathcal{B}}_{n}(\Lambda\cap P)$ be the covering of $\Lambda\cap P$
defined in proposition 6.3. Then, by definition,

$\sum\limits_{B\in{\mathcal{B}}_{n}(\Lambda\cap
P)}\mbox{Vol}\,(B)=\rho(P)\mbox{Vol}\,(P).$

Multiplying and dividing each term of the sum by $\mbox{Vol}\,(B)^{\beta}$ and
using again (4) we get that

$[A_{1}2^{d}]^{(1-\beta)}\epsilon^{nd(1-\beta)}\sum\limits_{B\in{\mathcal{B}}_{n}(\Lambda\cap
P)}\mbox{Vol}\,(B)^{\beta}\geq\rho(P)\mbox{Vol}\,(P).$

Therefore,

$[A_{1}2^{d}]^{(1-\beta)}\epsilon^{nd(1-\beta)}\sum\limits_{B\in{\mathcal{B}}_{n}(\Lambda)}\mbox{Vol}\,(B)^{\beta}\geq\sum_{P\in\wp_{n}}\rho(P)\mbox{Vol}\,(P).$

Now, multiplying and dividing each term of the sum
$\sum_{P\in\wp_{n}}\rho(P)\mbox{Vol}\,(P)$ by the factor
$(\rho(P)\mbox{Vol}\,(P))^{\underline{\alpha}}$ and using the volume bounds
(10) we get that

$\sum\limits_{P\in\wp_{n}}\rho(P)\mbox{Vol}\,(P)\geq[e^{-\lambda_{1}n(1-\underline{\alpha})}]\sum\limits_{P\in\wp_{n}}(\rho(P)\mbox{Vol}\,(P))^{\underline{\alpha}}.$

We thus get easily the lower bound in (33):

$\displaystyle\sum\limits_{B\in{\mathcal{B}}_{n}(\Lambda)}\mbox{Vol}\,(B)^{\beta}$
$\displaystyle\geq$
$\displaystyle[A_{1}2^{d}]^{(\beta-1)}\epsilon^{nd(\beta-1)}\sum\limits_{P\in\wp_{n}}\rho(P)\mbox{Vol}\,(P)$
$\displaystyle\geq$
$\displaystyle[A_{1}2^{d}]^{(\beta-1)}\epsilon^{nd(\beta-1)}[e^{\lambda_{1}n(\underline{\alpha}-1)}]\sum\limits_{P\in\wp_{n}}(\rho(P)\mbox{Vol}\,(P))^{\underline{\alpha}}$
$\displaystyle\geq$ $\displaystyle
C_{5}(\beta)[e^{\lambda_{1}(\underline{\alpha}-1)}\epsilon^{d(\beta-1)}]^{n},$

where

$C_{5}(\beta)=[A_{1}2^{d}]^{\beta-1}\inf_{n>0}\sum\limits_{P\in\wp_{n}}(\rho(P)\mbox{Vol}\,(P))^{\underline{\alpha}}.$

Similarly,

$[A_{0}2^{d}]^{(1-\beta)}\epsilon^{nd(1-\beta)}\sum\limits_{B\in{\mathcal{B}}_{n}(\Lambda\cap
P)}\mbox{Vol}\,(B)^{\beta}\quad\leq\quad\rho(P)\mbox{Vol}\,(P)$

and then

$\displaystyle[A_{0}2^{d}]^{(1-\beta)}\epsilon^{nd(1-\beta)}\sum\limits_{B\in{\mathcal{B}}_{n}(\Lambda)}\mbox{Vol}\,(B)^{\beta}$
$\displaystyle\leq$ $\displaystyle\sum_{P\in\wp_{n}}\rho(P)\mbox{Vol}\,(P)$
$\displaystyle\leq$
$\displaystyle[e^{\lambda_{0}n(\overline{\alpha}-1)}]\sum\limits_{P\in\wp_{n}}(\rho(P)\mbox{Vol}\,(P))^{\overline{\alpha}},$

concluding that

$\sum\limits_{B\in{\mathcal{B}}_{n}(\Lambda)}\mbox{Vol}\,(B)^{\beta}\leq[A_{0}2^{d}]^{(\beta-1)}[e^{\lambda_{0}(\overline{\alpha}-1)}\epsilon^{d(\beta-1)}]^{n}\sum\limits_{P\in\wp_{n}}(\rho(P)\mbox{Vol}\,(P))^{\overline{\alpha}}$

and we get he upper bound in (33) setting

$C_{6}(\beta)=[A_{0}2^{d}]^{\beta-1}\sup_{n>0}\sum\limits_{P\in\wp_{n}}(\rho(P)\mbox{Vol}\,(P))^{\overline{\alpha}}>0$

∎

######  Corollary 6.2.

Let $s=\overline{\beta}^{*}$ be the unique solution to the equation

(34)
$e^{\lambda_{0}(1-\overline{\alpha})}\epsilon^{d(1-{\overline{\beta}}^{*})}=1$

where $\overline{\alpha}$ is the solution to the upper modified Bowen equation
(31). Then,

$\dim_{\mathcal{H}}\,(\lambda)\leq d\overline{\beta}^{*}$

###### Proof.

We are going to prove that, if $\beta>\overline{\beta}^{*}$ then
$\mathcal{H}_{d\beta}(\Lambda)=0$. For this we let $\zeta>0$ be a small
positive number. If $n>\max\\{\log(\zeta/2)/\log\epsilon,N_{0}\\}$ then
$\mathcal{B}_{n}(\Lambda)$ will be a $\zeta$-covering and

$\displaystyle\mathcal{H}_{d\beta,\zeta}(\Lambda)$ $\displaystyle\leq$
$\displaystyle\sum\limits_{B\in{\mathcal{B}}_{n}(\Lambda)}\mbox{diam}(B)^{d\beta}$
$\displaystyle\leq$ $\displaystyle
A_{0}^{-\beta}\sum\limits_{B\in{\mathcal{B}}_{n}(\Lambda)}\mbox{Vol}\,(B)^{\beta}$
$\displaystyle\leq$ $\displaystyle
A_{0}^{-\beta}C_{6}(\beta)[e^{\lambda_{0}(\overline{\alpha}-1)}\epsilon^{d(\beta-1)}]^{n}$

If $\beta>\overline{\beta}^{*}$,
$e^{\lambda_{0}(\overline{\alpha}-1)}\epsilon^{d(\beta-1)}<1$ and then
$\mathcal{H}_{d\beta,\zeta}(\Lambda)=0$, taking the limit $n\to+\infty$. As
$\zeta>0$ is arbitrary, this proves that $\mathcal{H}_{d\beta}(\Lambda)=0$ for
every $\beta>\overline{\beta}^{*}$. Therefore,
$\dim_{\mathcal{H}}\,(\lambda)\leq d\overline{\beta}^{*}$, as we claimed. ∎

######  Lemma 6.9.

Let $\zeta>0$ be a small positive number and ${\mathcal{B}}_{0}$ a
$\zeta$-covering of $\Lambda$. Then, there exists a constant $C_{7}(\beta)>0$
such that,

(35)
$C_{7}(\beta)[e^{n\lambda_{1}(\underline{\alpha}-1)}\zeta^{d(\beta-1)}]\leq\sum\limits_{B^{0}_{k}\in{\mathcal{B}}_{0}}\mbox{Vol}\,(B^{0}_{k})^{\beta}$

for every $\beta>0$ and for every $n>N_{0}$ such that
$2\epsilon^{n}<\lambda({\mathcal{B}}_{0})<\zeta$, where
$\lambda({\mathcal{B}}_{0})$ is the Lebesgue number of ${\mathcal{B}}_{0}$.
The constant $C_{7}(\beta)$ depends neither on the covering
${\mathcal{B}}_{0}$ nor on $\zeta$ or $n$.

###### Proof.

By (25),

$\sum\limits_{B\in{\mathcal{B}}_{n}}\mbox{Vol}\,(B)^{\beta}\leq
C_{4}(\beta)[\epsilon^{nd(\beta-1)}\zeta^{d(1-\beta)}]\sum\limits_{B^{0}_{k}\in{\mathcal{B}}_{0}}\mbox{Vol}\,(B^{0}_{k})^{\beta},$

where

$C_{4}(\beta)=\dfrac{2^{d(\beta-1)}A_{1}D}{A_{0}}.$

and by the lower bound in (33)

$C_{5}(\beta)[e^{\lambda_{1}(\underline{\alpha}-1)}\epsilon^{d(\beta-1)}]^{n}\leq\sum\limits_{B\in{\mathcal{B}}_{n}(\Lambda)}\mbox{Vol}\,(B)^{\beta},$

where

$C_{5}(\beta)=[A_{1}2^{d}]^{\beta-1}\inf_{n>0}\sum\limits_{P\in\wp_{n}}(\rho(P)\mbox{Vol}\,(P))^{\underline{\alpha}}.$

Then:

$\begin{array}[]{ccccc}\sum\limits_{B^{0}_{k}\in{\mathcal{B}}_{0}}\mbox{Vol}\,(B^{0}_{k})^{\beta}&\geq&\dfrac{C_{5}(\beta)}{C_{4}(\beta)}\dfrac{[e^{\lambda_{1}(\underline{\alpha}-1)}\epsilon^{d(\beta-1)}]^{n}}{[\epsilon^{nd(\beta-1)}\zeta^{d(1-\beta)}]}\\\
&=&C_{7}(\beta)[e^{n\lambda_{1}(\alpha-1)}\zeta^{d(\beta-1)}].\end{array}$

where

$C_{7}(\beta)=\dfrac{A_{0}C_{5}(\beta)}{2^{d(\beta-1)}A_{1}D}.$

∎

Conclusion of the proof of theorem 1:

Let $\underline{\beta}^{*}$ be the unique solution to the equation:

(36)
$e^{\lambda_{1}(\underline{\alpha}-1)}\epsilon^{d(\underline{\beta}^{*}-1)}=1,$

equivalently,
$\lambda_{1}(\underline{\alpha}-1)+d(\underline{\beta}^{*}-1)\log\epsilon=0$
Choose $\zeta>0$ a small positive number such that
$\mathcal{H}_{d\underline{\beta}^{*},\zeta}(\Lambda)<+\infty$. We can do this
for, otherwise, $\mathcal{H}_{d\underline{\beta}^{*}}(\Lambda)=+\infty$ so
that $d\underline{\beta}^{*}\leq\dim_{\mathcal{H}}\,(\Lambda)$, concluding the
proof. Let $\mathcal{B}_{0}$ be any finite $\zeta$-covering of $\Lambda$ and
fix

$n=\left[\dfrac{\log(\lambda(\mathcal{B}_{0})/2)}{\log\epsilon}\right]+1$

where $[x]$ is the integer part of $x$, that is,

$n<\dfrac{\log(\lambda(\mathcal{B}_{0})/2)}{\log\epsilon}+1\quad\quad\text{and
then}\quad\quad
n\lambda_{1}(\underline{\alpha}-1)>\left(\dfrac{\log(\lambda(\mathcal{B}_{0})/2)}{\log\epsilon}+1\right)\lambda_{1}(\underline{\alpha}-1),$

since $\lambda_{1}(\underline{\alpha}-1)<0$. Observe that the lower bound in
(35) depends on $n=n(\mathcal{B}_{0})$ and $\zeta$. We will cancel the
dependence on $\mathcal{B}_{0}$ as follows. As
$2\epsilon^{n}<\lambda(\mathcal{B}_{0})<\zeta$ we have, by (35), that

$\displaystyle\sum\limits_{B^{0}_{k}\in{\mathcal{B}}_{0}}\mbox{Vol}\,(B^{0}_{k})^{\underline{\beta}^{*}}$
$\displaystyle\geq$ $\displaystyle
C_{7}(\underline{\beta}^{*})\exp\left[n\lambda_{1}(\underline{\alpha}-1)+d(\underline{\beta}^{*}-1)\log\zeta\right]$
$\displaystyle>$ $\displaystyle
C_{7}(\underline{\beta}^{*})\exp\left[\left(\dfrac{\log(\lambda/2)}{\log\epsilon}+1\right)\lambda_{1}(\underline{\alpha}-1)+d(\underline{\beta}^{*}-1)\log\zeta\right]$
$\displaystyle=$ $\displaystyle
C_{7}(\underline{\beta}^{*})\exp\left[\lambda_{1}(\underline{\alpha}-1)+\left(\dfrac{\log(\lambda/2)}{\log\epsilon}\right)\lambda_{1}(\underline{\alpha}-1)+d(\underline{\beta}^{*}-1)\log\zeta]\right]$
$\displaystyle=$ $\displaystyle
C_{7}(\underline{\beta}^{*})\exp\left[\lambda_{1}(\underline{\alpha}-1)+\dfrac{\log\zeta}{\log\epsilon}\left(\dfrac{\log(\lambda/2)}{\log\zeta}\lambda_{1}(\underline{\alpha}-1)+d(\underline{\beta}^{*}-1)\log\epsilon\right)\right]$
$\displaystyle=$ $\displaystyle
C_{7}(\underline{\beta}^{*})\exp\left[\lambda_{1}(\underline{\alpha}-1)+\dfrac{\log\zeta}{\log\epsilon}\left(\dfrac{\log(\lambda/2)}{\log\zeta}-1\right)\lambda_{1}(\underline{\alpha}-1)\right]$
$\displaystyle=$ $\displaystyle
C_{7}(\underline{\beta}^{*})\exp\left[\lambda_{1}(\underline{\alpha}-1)+\dfrac{\log(\lambda/2\zeta)}{\log\epsilon}\lambda_{1}(\underline{\alpha}-1)\right]$
$\displaystyle>$ $\displaystyle
C_{7}(\underline{\beta}^{*})\exp\left[\lambda_{1}(\underline{\alpha}-1)+\dfrac{\log{2}}{\log\epsilon}\lambda_{1}(1-\underline{\alpha})\right]$
$\displaystyle=$ $\displaystyle
C_{7}(\underline{\beta}^{*})\exp\left[-\lambda_{1}(1-\underline{\alpha})+\dfrac{\log{2}}{\log\epsilon}\lambda_{1}(1-\underline{\alpha})\right]$
$\displaystyle=$ $\displaystyle
C_{7}(\underline{\beta}^{*})\exp\left[\dfrac{\log(2/\epsilon)}{\log\epsilon}\lambda_{1}(1-\underline{\alpha})\right]=C_{8}(\underline{\beta}),$

cancelling the dependence on the covering $\mathcal{B}_{0}$. Here we use that,

$\displaystyle\dfrac{\log(\lambda/2)}{\log\zeta}\lambda_{1}(\underline{\alpha}-1)+d(\underline{\beta}^{*}-1)\log\epsilon$
$\displaystyle=$
$\displaystyle\left(\dfrac{\log(\lambda/2)}{\log\zeta}-1\right)\lambda_{1}(\underline{\alpha}-1)$
$\displaystyle+$
$\displaystyle[\lambda_{1}(\underline{\alpha}-1)+d(\underline{\beta}^{*}-1)\log\epsilon]$
$\displaystyle=$
$\displaystyle\left(\dfrac{\log(\lambda/2)}{\log\zeta}-1\right)\lambda_{1}(\underline{\alpha}-1),$

adding and substracting terms and recalling that
$\lambda_{1}(\underline{\alpha}-1)+d(\underline{\beta}^{*}-1)\log\epsilon=0$,
by (36). We also use that

$\dfrac{\log(\lambda/2\zeta)}{\log\epsilon}\lambda_{1}(\underline{\alpha}-1)=-\dfrac{\log(\lambda/2\zeta)}{\log\epsilon}\lambda_{1}(1-\underline{\alpha})>\dfrac{\log{2}}{\log\epsilon}\lambda_{1}(1-\underline{\alpha}),$

which follows from $\log(\lambda/2\zeta)<\log(1/2)$ since
$\lambda(\mathcal{B}_{0})<\zeta$. Thus, for every finite $\zeta$-covering
${\mathcal{B}}_{0}$,

$\sum\limits_{B^{0}_{k}\in{\mathcal{B}}_{0}}\mbox{Vol}\,(B^{0}_{k})^{\underline{\beta}^{*}}\geq
C_{8}(\underline{\beta}^{*})>0.\quad\text{and
therefore}\quad\sum\limits_{B^{0}_{k}\in{\mathcal{B}}_{0}}\mbox{diam}(B^{0}_{k})^{d\underline{\beta}^{*}}\geq
C_{9}(\underline{\beta}^{*}),$

where

$C_{9}(\underline{\beta}^{*})=\dfrac{2^{d{\underline{\beta}^{*}}}C_{8}(\underline{\beta}^{*})}{A_{1}^{\underline{\beta}^{*}}}>0,\quad\text{since}\quad\mbox{Vol}\,(B^{0}_{k})^{\underline{\beta}^{*}}\leq
A_{1}^{\underline{\beta}^{*}}\left(\dfrac{\mbox{diam}(B^{0}_{k})}{2}\right)^{d{\underline{\beta}^{*}}}\quad\text{by
(\ref{volume.diameter.constants.1}).}$

The constant $C_{9}(\underline{\beta}^{*})$ only depends on the geometry of
the ambient manifold, the rates of volume expansion of $f$ and the rates of
contraction of its inverse branches and several other fixed topological data
of the construction. As $C_{9}(\underline{\beta}^{*})$ is independet of the
$\zeta$-covering $\mathcal{B}_{0}$, we conclude that:

$0<C_{9}(\underline{\beta}^{*})\leq{\mathcal{H}}_{d\underline{\beta}^{*},\zeta}(\Lambda)<+\infty$

and therefore, ${\mathcal{H}}_{d\underline{\beta}^{*}}(\Lambda)\geq
C_{9}(\underline{\beta}^{*})>0$, since
${\mathcal{H}}_{d\underline{\beta}^{*},\zeta}(\Lambda)$ is non-decreasing in
$\zeta>0$. Thus, by an standard argument for the Hausdorff measure, we
conclude that

${\mathcal{H}}_{d\beta}(\Lambda)=+\infty\quad\text{for
every}\quad\beta<\underline{\beta}^{*},$

so proving that $d\underline{\beta}^{*}\leq\dim_{\mathcal{H}}\,(\Lambda)$. QED

##  7\. Discussion

In this section we will test our approach with a popular example appearing in
the theory of self-affine subsets. Let $0<\tau<\beta$ with $\tau,\beta\neq
1/2$, be real numbers and consider the couple of affine contractions:

$g_{0}(x,y)=\left[\begin{array}[]{cc}\beta&0\\\
0&\tau\end{array}\right]\left[\begin{array}[]{c}x\\\
y\end{array}\right]\quad\text{and}\quad
g_{1}(x,y)=\left[\begin{array}[]{cc}\beta&0\\\
0&\tau\end{array}\right]\left[\begin{array}[]{c}x\\\
y\end{array}\right]+\left[\begin{array}[]{c}1-\beta\\\
1-\tau\end{array}\right]$

This generates a limit set $\Lambda$ starting with two rectangles ${\bf\sf
R}_{0}$ and ${\bf\sf R}_{1}$ of width $\beta$ and height $\tau$ attached to
the down-left and upper-right corners of unit rectangle, respectively. This is
the simplest non conformal self affine limit set. If $\tau<1/2$ the rectangles
are disjoint and the limit is a plane non conformal Cantor set. These Cantor
sets are volume-irreducible. Degenerate configurations in which rectangles
${\bf\sf R}_{i}$ are situated above a single interval in the $OX$ axis, are
excluded. In this case $\Lambda$ is contained in a single vertical line and it
is volume reducible. If $\tau=1/2$ the rectangles intersect at the border and
the construction converges to the graph of a continuous non differentiable
function, unless $\beta=1/2$, in which case the set is conformal and the limit
is the diagonal $x=y$, at least with this choice of translations. A remarkable
feature of this example is that, for almost every $\beta$ such that
$0<\tau<1/2<\beta$, the projection of $\Lambda$ onto the $OX$ axis is an
absolutely continuous Bernoulli convolution. See [5] and [17]. For this
particular choice of parameters, Hausdorff and box-counting dimension counting
dimension are equal to

$\dim_{\mathcal{H}}\,(\Lambda)=\dim_{B}(\Lambda)=1-\dfrac{\log(2\beta)}{\log\tau}=\dfrac{\log(2\beta/\tau)}{\log(1/\tau)}$

Notice also that the construction does not satisfy our border condition.
However, we can separate the rectangles from the border with the same
arguments used previously for Bedford-McMullen carpets. Certainly these
perturbations modify the number-theoretical properties which are responsible
for the remarkable geometry of these examples. However, as we are arguing, our
goal is to give bounds for the Hausdorff dimension. We are just using exact
formulas as a test case for our inequalities which are robust. In [17] an
explicit a formulae for the Hausdorff and box-counting dimension is given:

Explicit formulae

  * •

$0<\tau<\beta<1/2$:

$\dim_{B}(\Lambda)=\dim_{\mathcal{H}}\,(\lambda)=\dfrac{\log{2}}{\log(1/\beta)};$

  * •

$\beta=2^{-\frac{1}{n}},\ n\mathbb{N}$:

$\dim_{B}(\Lambda)=\dim_{\mathcal{H}}\,(\lambda)=\dfrac{\log(2\beta/\tau)}{\log(1/\tau)};$

  * •

there exists a $0<\gamma<1$ such that for almost every $\gamma<\beta<1$:

$\dim_{B}(\Lambda)=\dim_{\mathcal{H}}\,(\lambda)=\dfrac{\log(2\beta/\tau)}{\log(1/\tau)};$

  * •

$\beta>1/2$:

$\dim_{B}(\Lambda)=\dfrac{\log(2\beta/\tau)}{\log(1/\tau)};$

See [17]. We will use these formulas to compare the goodness of our bounds.
Indeed, as $Dg_{i}=Dg$ and $Jg_{i}=Jg$ for $i=0,1$ are constant, our formulas
can be easily calculated:

$\epsilon=(1-\delta)\left(\dfrac{Jg}{\|Dg\|^{d-1+\delta}}\right)^{\frac{1}{1-\delta}}=(1-\delta)\left(\dfrac{\beta\tau}{\beta^{1+\delta}}\right)^{\frac{1}{1-\delta}}=(1-\delta)(\beta^{-\delta}\tau)^{\frac{1}{1-\delta}},$

from where:

$\log\epsilon=\log(1-\delta)+\dfrac{\log(\beta^{-\delta}\tau)}{1-\delta}=\dfrac{\log((1-\delta)^{1-\delta}\beta^{-\delta}\tau)}{1-\delta}$

On the other hand,

$\lambda_{0}=\log(Jf-\rho)=\log((\beta\tau)^{-1}-\rho)\quad\text{and}\quad\lambda_{1}=\log(Jf+\rho)=\log((\beta\tau)^{-1}+\rho).$

The solution $\alpha$ to the Bowen equation is given by $2(Jg)^{\alpha}=1$.
Indeed, as the Jacobian $Jf$ is constant we have

$-s\sum_{k=0}^{n-1}\log(Jf)=\log[Jf]^{-sn}$

and thus

$\sum_{(i_{1},\cdots,i_{n})\in
2^{n}}\exp\left(-s\sum_{k=0}^{n-1}\log(Jf)\right)=\sum_{(i_{1},\cdots,i_{n})\in
2^{n}}[Jf]^{-sn}=2^{n}[Jf]^{-sn}=(2Jg^{s})^{n}.$

Therefore,

$\limsup_{n\to+\infty}\dfrac{1}{n}\log\left(\sum_{(i_{1},\cdots,i_{n})\in
2^{n}}\exp\left(-s\sum_{k=0}^{n-1}\log(Jf)\right)\right)=\log(2Jg^{s})=0$

if and only if $2Jg^{s}=1$ so giving:

$\alpha=-\dfrac{\log{2}}{\log(\beta\tau)}\quad\text{and}\quad
1-\alpha=\dfrac{\log(2\beta\tau)}{\log(\beta\tau)}.$

Hence, the upper bound $U(\delta,\rho)=2\overline{\beta}$ is

$\displaystyle U(\delta,\rho)$ $\displaystyle=$ $\displaystyle
2+\dfrac{\lambda_{0}(1-\alpha)}{\log(\epsilon)}(1-\alpha)$ $\displaystyle=$
$\displaystyle
2+(1-\delta)\dfrac{\log((\beta\tau)^{-1}-\rho)}{\log((1-\delta)^{1-\delta}\beta^{-\delta}\tau)}\dfrac{\log(2\beta\tau)}{\log(\beta\tau)}.$

which is an increasing function in $\delta$, for every fixed small $\rho>0$,
as one can verify easily. Suppose that $\rho=0$ and let $\delta\to 0^{+}$,
then we get the limit

$2-\dfrac{\log(2\beta\tau)}{\log\tau}=1-\dfrac{\log(2\beta)}{\log\tau}=-\dfrac{\log(2\beta/\tau)}{\log\tau}=\dfrac{\log(2\beta/\tau)}{\log(1/\tau)}.$

This shows that:

$\dim_{\mathcal{H}}\,(\Lambda)\leq\dim_{B}(\Lambda)\leq\dfrac{\log(2\beta/\tau)}{\log(1/\tau)}$

for every $0<\tau<\beta<1$ and $\tau,\beta\neq 1/2$, excluding degenerate
configurations leading to volume reducible sets. According to Pollicot-Weiss
formulae:

$\dim_{\mathcal{H}}\,(\Lambda)=\dim_{B}(\Lambda)=\dfrac{\log(2\beta/\tau)}{\log(1/\tau)}$

for a large set of contraction rates $\beta$ and $\tau$. Nevertheless,
$\dim_{\mathcal{H}}\,(\Lambda)<\dim_{B}(\Lambda)$ for some specific choices of
$\beta$ and $\tau$, depending on delicate number-theoretical properties. This
is the case if $\tau=1/2$ and $\beta^{-1}$ is a Pisot number, according to
[19]. This rises the question to find lower bounds for
$\dim_{\mathcal{H}}\,(\Lambda)$. According to theorem 1 a lower bound
$L(\delta,\rho,\Sigma)=2\underline{\beta}$ is given in terms of
$\underline{\alpha}$, the solution to a modified Bowen equation with parameter
$\Sigma$. Doing a similar calculation as we did before for the standard Bowen
equation we get that $\underline{\alpha}$ is the solution to

$2(e^{\Sigma}Jf)^{-\underline{\alpha}}=1.$

That is

$\underline{\alpha}=\dfrac{\log{2}}{\Sigma+\log{Jf}}=\dfrac{\log{2}}{\Sigma-\log{\beta\tau}}$

and then:

$1-\underline{\alpha}=\dfrac{\Sigma-\log(2\beta\tau)}{\Sigma-\log(\beta\tau)}.$

Then our lower bound is:

$L(\delta,\rho,\Sigma)=2+(1-\delta)\left[\dfrac{\log((\beta\tau)^{-1}+\rho)}{\log((1-\delta)^{1-\delta}\beta^{-\delta}\tau)}\right]\left[\dfrac{\Sigma-\log(2\beta\tau)}{\Sigma-\log(\beta\tau)}\right],$

for a not too large $\Sigma$. If $\delta=\rho=0$ then

$L(\Sigma)=2+\left[\dfrac{\log(\beta\tau)}{\log(\tau)}\right]\left[\dfrac{\Sigma-\log(2\beta\tau)}{\Sigma-\log(\beta\tau)}\right]$

which is a strictly decreasing function of the variable $\Sigma$. Moreover, if
$\Sigma=0$ we recover the formula

$L(0)=\dfrac{\log(2\beta/\tau)}{\log(1/\tau)},$

which is an upper bound for the Hausdorff dimension. Now, suppose that
$\beta^{-1}$ is a Pisot number and that
$\dim_{\mathcal{H}}\,(\Lambda)<\dim_{B}(\Lambda)$. If $\rho$ and $\delta$ are
positive and sufficiently small, then the function
$L(\Sigma)=L(\delta,\rho,\Sigma)$ is still strictly decreasing in $\Sigma$ and
it gives a lower bound for the Hausdorff dimension, for some
$\underline{\Sigma}$ not too large, with an upper bound which can be estimated
from the data problem, explicitly:

$L(\underline{\Sigma})<\dim_{\mathcal{H}}\,(\Lambda)<L(0),$

where

$\underline{\Sigma}\leq\log\left(\dfrac{Jf}{(\|Df\|\epsilon)^{-(d_{B}(\Lambda)-\delta)}\epsilon^{d}}\right).$

Notice that we only need an upper bound for $d_{B}(\Lambda)$ to estimate how
large $\Sigma$ can be, in order to calculate $L(\underline{\Sigma})$, the
lower bound of the Hausdorff dimension.

Lets do a numerical example. Take $\beta=0.77$ and $\tau=0.35$. It is almost
sure that $\dim_{B}(\Lambda)=\dim_{\mathcal{H}}\,(\Lambda)=1.4113$, according
to Pollicot-Weiss formulae. Let us choose $\delta=0.1$ and $\rho=0.1$. With
these numbers we get the upper bound $U=1.5131$. As for the lower bound we
first compute an upper bound for the adjustable parameter $\Sigma$, giving
$\overline{\Sigma}=1.6909$ by estimating the box-counting dimension as $1.5$.
If we plot $L=L(\Sigma)$ over the interval $[0,\overline{\Sigma}]$, with the
above choices for $\delta$ and $\rho$ we get the graphic in figure 1. A lower
bound for Hausdorff dimension can be given at $\underline{\Sigma}=0.5$ with
value $L=1.3355$, approximately. A draw back of the method is that we need an
estimation of the box-counting dimension to calculate an upper bound for
$\Sigma$, but this is not too serious as, usually, it is easier to compute
$\dim_{B}(\Lambda)$ than $\dim_{\mathcal{H}}\,(\Lambda)$. We can even
approximate its value numerically if we lack of an exact formula.

Figure 1. Plot of the lower bound $L=L(\Sigma)$ over the interval
$\Sigma\in[0,1.7]$

Another feature of the present approach is that our bounds do not depend on
the position of the initial rectangles, unless the limit degenerate into a
volume reducible subset. Many other examples can be examined with these
methods, as Bedford-McMullen, Gatzouras-Lalley, Barański and Feng-Wang
carpets. See [5]. Although we do not recover exact formulas for these
examples, as in the present case, good enough bounds can be found. This is the
subject of a work in progress and will appear elsewhere.

##  8\. Conclusions

The present results answers a question possed to me by Marcelo Viana during
the preparation of my doctoral dissertation at IMPA. He asked what kind of
relationship exists between the dynamical and Hausdorff dimension. In this
note we prove that there is indeed a precise functional relationship between
these dimensional indicators coming from robust upper and lower bounds for the
box-counting and the Hausdorff dimension. In particular, we have found a new
exact formula for the Hausdorff dimension of the attractor of a regular
geometric construction, opening the door to investigate regularity properties
of the Hausdorff dimension, using smoothness results of the thermodynamic
formalism.

A possible explanation for these functional relations between the dynamical
and geometric dimensions is the following. One can think the dynamical
dimension as a geometric dimension of an associated real line IFS defined by a
sort of “volumetric representation” of the geometric construction. That is, we
can associate the construction a nested sequence of disjoint closed intervals
in the real line, associating to each $n$-atom $P\in\wp_{n}$ an interval
$I\subset\mathbb{R}$ with length equal to the $P$-volume,
$|I|=\mbox{Vol}\,(P)$, defining a volumetric IFS $\\{G_{i}\\}$ acting upon
this collection of intervals through the Jacobian:
$G_{i}(I)=\int_{P}Jg_{i}d\mbox{Vol}\,$. The dynamical dimension is precisely
the Hausdorff dimension of the limit of this “volumetric representation” of
$\Lambda$. This explain why the dynamical dimension is insensitive to some
geometric details of the construction and why some sort of “volume
irreducibility” is necessary for our methods to work. The characteristic scale
$\epsilon$ and thermodynamics lay a bridge between the volumetric and spatial
representations, originating a whole set of equations reflecting the balance
between volume and space contractions. Possibly those equations could be used
to obtain new results on the subject, besides the upper and lower bounds for
the Hausdorff and box-counting dimensions that we have found.

The previous discussion outline the potential application of these methods to
the estimation, both from above and from below, of the most important
dimensional indicators of a class of hyperbolic repellers. While we are
dealing with a special class of dynamically defined Cantor sets, the
possibility to approximate more general attractors as limits of suitable
sequences of regular geometric constructions opens new perspectives in the
study of their geometry, due to the robustness of our estimations.

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