# Local Geometry of Self-similar Sets: Typical Balls, Tangent Measures and
Asymptotic Spectra

Manuel Morán 1,2, Marta LLorente3 and María Eugenia Mera1

###### Abstract

We analyse the local geometric structure of self-similar sets with open set
condition through the study of the properties of a distinguished family of
spherical neighbourhoods, the typical balls. We quantify the complexity of the
local geometry of self-similar sets, showing that there are uncountably many
classes of spherical neighbourhoods that are not equivalent under similitudes.
We show that, at a tangent level, the uniformity of the Euclidean space is
recuperated in the sense that any typical ball is a tangent measure of the
measure $\nu$ at $\nu$-a.e. point, where $\nu$ is any self-similar measure. We
characterise the spectrum of asymptotic densities of metric measures in terms
of the packing and centred Hausdorff measures. As an example, we compute the
spectrum of asymptotic densities of the Sierpinski gasket.

1 Departamento de Análisis Económico y Economía Cuantitativa. Universidad
Complutense de Madrid. Campus de Somosaguas, 28223 Madrid, Spain.
2 IMI-Institute of Interdisciplinary Mathematics. Universidad Complutense de
Madrid. Plaza de Ciencias 3, 28040 Madrid, Spain.
3 Departamento de Análisis Económico: Economía Cuantitativa. Universidad
Autónoma de Madrid, Campus de Cantoblanco, 28049 Madrid, Spain.

Emails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Short Title: Local Geometry of Self-similar Sets

Keywords: Self-Similar Sets, Hausdorff Measures, Tangent Measures, Density of
Measures, Computability of Fractal Measures, Complexity of Topological Spaces,
Sierpinski Gasket.
[2020] MSC: 28A78, 28A80, 28A75, 54A05, 54A25

##  1 Introduction and main results

In order to gauge the vastness of the set of spherical neighbourhoods of a
metric space $X,$ it is useful to consider the quotient spaces
$Sph_{X}/\simeq_{\mathcal{F}},$ where $Sph_{X}$ is the set of spherical
neighbourhoods of $X$ and $\simeq_{\mathcal{F}}$ is the equivalence class
associated with some group $\mathcal{F}$ of self-mappings of $X:$
$B\simeq_{\mathcal{F}}B^{\prime}\Leftrightarrow$ $B=f(B^{\prime})$ for $B,$
$B^{\prime}\in$ $Sph_{X}$ and some $f\in\mathcal{F}.$ The regularity of the
Euclidean space $\mathbb{R}^{n}$ is made clear by the fact that if
$\mathcal{S}_{n}$ is the set of similarities of $\mathbb{R}^{n},$ then
$Sph_{\mathbb{R}^{n}}/\simeq_{\mathcal{S}_{n}}$ consists of a unique
equivalence class.

In this paper, we study the local geometry of a self-similar set
$E\subset\mathbb{R}^{n}$ satisfying the open set condition (OSC), geometry
which is described by the spherical neighbourhoods of $E$ as a metric subspace
of $\mathbb{R}^{n},$ i.e. by restricted balls of the form $B\cap E,$ where $B$
is a Euclidean ball. For general points $x,$ $y\in E,$ if $B(x,d)$ denotes the
closed Euclidean ball centred at $x$ and with radius $d,$ then $B(x,d)\cap E$
and $B(y,d)\cap E$ are not equivalent by translation, and $B(x,d)\cap E$ and
$B(x,d^{\prime})\cap E$ with $d\neq d^{\prime}$ are not homothetic-equivalent.
Using classical tools of fractal geometry, namely, the $s$-densities of metric
measures on balls (see Definitions 21 and 22), and Marstrand’s Theorem [1],
together with the results in Sec. 3.2, we are able to prove that, for general
self-similar sets with OSC, there are uncountably many equivalence classes in
the quotient spaces $Sph_{E}/\simeq_{\mathcal{S}_{n}}.$ This gives account of
the complexity of the purely deterministic self-similar geometry.

In spite of these facts, the literature has established the existence of a
strong kind of regularity, on a tangent level and on average, in the
neighbourhoods of a self-similar set.

Recall that a self-similar set is defined as the unique compact set
$E\subset\mathbb{R}^{n}$ that satisfies the basic equation of self-similarity

$E=\cup_{i=0}^{m-1}f_{i}(E).$ (1)

for a given system $\Psi=\left\\{f_{i}\right\\}_{i\in M},\
M:=\left\\{0,1,\dots,m-1\right\\}$ of contractive similitudes in
$\mathbb{R}^{n}$. We shall assume that the system $\Psi$ satisfies the OSC,
meaning that there is an open set $\mathcal{O\subset}\mathbb{R}^{n}$ such that
$f_{i}\mathcal{(O)\subset O}$ for all $i\in M$ and
$f_{i}(\mathcal{O)\cap}f_{j}(\mathcal{O)=\varnothing}$ for $i,$ $j\in M,\
i\neq j.$ We shall refer to such a set $\mathcal{O}$ as a feasible open set
for $\Psi.$ We can assume, without loss of generality, as we shall from now
on, that $\mathcal{\ O\cap}E\neq\varnothing$ holds, also called strong open
set condition (SOSC) (cf. [2] and [3], see also [4]). If $f_{i}(E)\cap
f_{j}(E)=\varnothing$ for $i,$ $j\in M,$ $i\neq j,$ it is said that the strong
separation condition (SSC) holds, in which case the OSC is also fulfilled.

We want to understand the local geometry of $E$ through the study of the local
behaviour of the metric $s$-measures,

$\mathcal{M}^{s}\lfloor_{E}:=\left\\{\mu,\text{
}\mathcal{H}^{s}\lfloor_{E},\text{ }\mathcal{H}_{Sph}^{s}\lfloor_{E},\text{
}C^{s}\lfloor_{E},\text{ }P^{s}\lfloor_{E}\right\\}$ (2)

where $s$ is the similarity dimension of $E,$ $\dim E,$ that is, the unique
real number $s$ that satisfies $\sum_{i\in M}r_{i}^{s}=1,$ $r_{i}$ being the
contraction constant of the similarity $f_{i},\ i\in M.$ Here
$\beta\lfloor_{E}$ stands for a measure $\beta$ restricted to the set $E.$ The
measures

$\mathcal{M}^{s}:=\left\\{\mathcal{H}^{s},\text{ }\mathcal{H}_{Sph}^{s},\text{
}C^{s},\text{ }P^{s}\right\\}$ (3)

are the $s$-dimensional Hausdorff measure, spherical Hausdorff measure,
centred Hausdorff measure and packing measure, respectively. Any two measures
in $\mathcal{M}^{s}\lfloor_{E}$ are multiple of each other, moreover, in the
case that $s$ takes the integer value $n,$ they are also multiple of the
$n$-dimensional Lebesgue measure. Each measure in $\mathcal{M}^{s}\lfloor_{E}$
highlights different basic geometric properties of subsets of
$\mathbb{R}^{n}.$ For $\alpha\in\mathcal{M}^{s}\lfloor_{E},$
$0<\alpha(E)<\infty$ holds and $E$ is called an $s$-set (see [5] for further
details and Sec. 2.2 for the definitions of the measures in
$\mathcal{M}^{s}$). We shall present in Sec. 2.1 below the natural probability
measure $\mu$. For the time being, we can see it as the normalised measure,
$\frac{\alpha}{\alpha(E)}$ of any other $\alpha\in\mathcal{M}^{s}\lfloor_{E}.$

The results in this paper about the regularity of the metric measures are also
shared by the wider class of self-similar measures,
$\mathcal{M}_{\mathcal{S}}(E)$ (see [6] and Sec. 2.1 for a definition).
Whereas the metric measures, $\mathcal{M}^{s},$ convey a strong geometric
meaning, self-similar measures are an essential tool in multifractal analysis
of logarithmic densities, a topic that has generated a vast amount of
literature for the past 30 years.

###  1.1 Scenery flow, tangent distribution and tangent measures

Let $\nu$ be a Radon measure on $\mathbb{R}^{n}$ and let $x$ be a point in the
support of $\nu.$ We can access the local geometry of $\nu\lfloor_{E}$ around
$x$ through the following zooming process: let $T_{x,t}(y)=t(y-x),\ t>0,$ be
the homothety that maps the ball $B(x,t^{-1})$ onto the unit ball $D:=B(0,1).$
Let $\nu_{x,t}$ be the probability measure on $D$ obtained from the
normalisation of the restriction to $D$ of the image measure of
$\nu\lfloor_{E}$ under the homothety $T_{x,t}.$ If $\mathcal{M(}D)$ denotes
the set of Radon measures on $D,$ then the mapping $t\rightarrow\nu_{x,t}$ can
be considered as a measure-valued time series that takes values in the metric
space $\mathcal{M(}D)$ endowed with the weak topology. This time series is
called scenery flow of $\nu$ around $x$ (cf. [7]). The empirical distributions
$\Phi_{x,t}(\nu),$ $t>0,$ associated with such “time” series, are probability
measures on $\mathcal{M(}D)$ (so they belong to the set
$\mathcal{M}(\mathcal{M(}D))$ of Radon measures on $\mathcal{M(}D)).$ The
empirical distribution $\Phi_{x,t}(\nu)$ gives weight to a set
$A\subset\mathcal{M(}D)$ according to the rate of the time interval $[0,t]$
that the “empirical” data $\delta_{\nu_{x,t}}$ (unit mass at $\nu_{x,t})$ stay
in $A.$ If the empirical distribution $\Phi_{x,t}(\nu)$ converges to a limit
$\Phi_{x}(\nu)$ as $t$ tends to infinity, then the limiting distribution
$\Phi_{x}(\nu)$ is called the tangent distribution of $\nu$ at $x$ (see [8]).

S. Graf [9] proved that if $E$ is a self-similar set with OSC and
$\nu\in\mathcal{M}_{\mathcal{S}}(E),$ then the limit $\Phi_{x}(\nu)$ exists
$\nu$-a.e. $x,$ and it does not depend on $x.$ Moreover, he constructed an
explicit formula for the tangent distribution. This author gave credit for the
first of these results to C. Bandt in [8], and Bandt in turn gives credit for
the same result to S. Graf [10] (indeed a most refreshing case). M. Arbeiter
[11], C. Bandt [10] and A. Pyörälä [12] extended these results in different
ways. The uniqueness and independence of the limit $\Phi_{x}(\nu)$ from $x$ is
what M. Gavish, [13], calls, when displayed by a measure, the uniform scaling
scenery property of such a measure. This means that, at a tangent level and in
this sense, the flow scenery recovers the uniformity of the Euclidean space.

###### Remark 1

There is another way to pass to the limit at the tangent level that leads to
_tangent measures,_ a concept prior to tangent distributions introduced by D.
Preiss [14]. There, starting from a measure $\nu$ in the set
$\mathcal{M(}\mathbb{R}^{n})$ of Radon measures on $\mathbb{R}^{n},$ he
considers unrestricted zoomings $\nu_{x,t}$ of $\nu$ at $x$ by homotheties
$T_{x,t}$ as above. Instead of performing an averaging procedure, Preiss
considers non-null and locally finite limits, in the vague topology of
$\mathcal{M(}\mathbb{R}^{n}),$ of sequences

$\left\\{c_{n}\nu_{x,t_{n}}\right\\}\quad\text{with}\quad
t_{n}\overset{n\rightarrow\infty}{\rightarrow}\infty\quad\text{and }\quad
c_{n}>0.$

Such limit points are called tangent measures of $\nu$ at $x,$ and
$Tan(\nu,x)$ denotes the set of all such limits.
In our approach, following C. Bandt [8], the measures $\nu_{x,t_{n}}$ are
restricted and normalised zoomings, but the zoomings are through general
expanding similitudes, rather than only homotheties.

Let $\mathcal{I}_{n}$ be the group of isometries of $\mathbb{R}^{n}.$ We may
define, in the set $\mathcal{M}(\mathbb{R}^{n}),$ the equivalence relationship

$\alpha\cong\beta\Leftrightarrow\text{there is a }g\in\mathcal{I}_{n}\text{
and a }\lambda>0\text{ such that
}\beta=\lambda\left(g_{\sharp}(\alpha\right)),$ (4)

where $g_{\sharp}(\alpha)$ is the image measure of $\alpha$ under $g,$ i.e.
$g_{\sharp}(\alpha)(A)=\alpha(g^{-1}(A))$ for $\alpha$-measurable
$A\subset\mathbb{R}^{n}.$ Thus, we identify two measures if they are equal up
to an isometry (see, for instance, [10], where equivalent measures up to
isometries are identified in the construction of tangent measures), and we
also identify all measures in the half-straight line
$\left\\{\lambda\alpha:\lambda>0,\text{
}\alpha\in\mathcal{M}(\mathbb{R}^{n})\right\\}.$ For
$\alpha\in\mathcal{M(}\mathbb{R}^{n}),\,$ let $\widetilde{\alpha}$ denote the
equivalence class in $\mathcal{M(}\mathbb{R}^{n})/\cong$ to which $\alpha$
belongs, i.e.

$\widetilde{\alpha}=\left\\{\beta\in\mathcal{M(}\mathbb{R}^{n}):\beta\cong\alpha\right\\}$
(5)

Given a measure $\nu\in\mathcal{M(}\mathbb{R}^{n}),$ we now consider the
zoomings $\nu_{x,t_{n}}$ be of the form $\left(g_{n}\right)_{\sharp}$
$\nu\lfloor_{B(x,dt_{n}^{-1})}$ where $g_{n}$ is a similitude of contraction
ratio $t_{n},$ $d\leq 1,$ and $x\in spt(\nu)$ (see (23)). We define the
quotient space $\widetilde{\mathcal{M}}(\mathbb{R}^{n})$ and the set of
tangent equivalence classes of measures, $\widetilde{Tan}(\nu,x),$ by

$\displaystyle\widetilde{\mathcal{M}}(\mathbb{R}^{n})$ $\displaystyle=$
$\displaystyle\left\\{\widetilde{\alpha}:\alpha\in\mathcal{M(}\mathbb{R}^{n})\right\\}$
(6) $\displaystyle\widetilde{Tan}(\nu,x)$ $\displaystyle=$
$\displaystyle\left\\{\widetilde{\alpha}:\text{there is a sequence
}c_{n}\nu_{x,t_{n}}\xrightarrow[n\to\infty]{w}\alpha,\text{with
}t_{n}\rightarrow\infty,\text{ }\alpha\neq 0\text{ and
}\alpha\in\mathcal{M(}\mathbb{R}^{n})\right\\},$ (7)

where $\overset{w}{\rightarrow}$ denotes the weak convergence of measures on
$\mathcal{M}(\mathbb{R}^{n}).$

It turns out that, in the course of our research, the case in which the
convergence of the magnifications occurs in the strong topology of measures in
$\mathcal{M}(\mathbb{R}^{n})$ is relevant (see Sec. 1.2 below for a discussion
of this result). We shall write $\widetilde{Tan}^{st}(\nu,x)$ for the set of
equivalence classes, w.r.t. $\cong,$ of such strong limits.

###### Remark 2

In our definition (7) any two zoomings, $\beta=\left(g_{n}\right)_{\sharp}$
$\nu\lfloor_{B(x,dt_{n}^{-1})}$ and
$\beta^{\prime}=\left(h_{n}\right)_{\sharp}\nu\lfloor_{B(x,dt_{n}^{-1})}$ of a
given spherical neighbourhood $B(x,dt_{n}^{-1})$ are considered as valid steps
in the construction of a tangent limiting measure $\alpha,$ where
$g_{n},h_{n}$ are different similitudes. This can be considered as the
identification of $\beta$ and $\beta^{\prime}$ as equivalent zoomings. Notice
that $\beta^{\prime}=\left(g_{n}^{-1}\circ h_{n}\right)_{\sharp}\beta$ and
that $g_{n}^{-1}\circ h_{n}$ is an isometry. Thus, the equivalence
relationship (4) and the definition in (7) are consistent.

In contrast to the enlightening results obtained in [9], [10] and [11] on the
uniform scaling scenery property of self-similar measures, to the best of our
knowledge, the members of $Tan(\nu,x)$ for
$\nu\in\mathcal{M}_{\mathcal{S}}(E)$ remain unknown. Several natural issues
arise here: What is the relationship between $\Phi_{x}(\nu)$ and $Tan(\nu,x)?$
What do the measures in $Tan(\nu,x)$ look like? Do they display some uniform
property? As for the first question, see Proposition 1 in [15]. Below, we give
a partial answer to the second and third questions for measures in
$\mathcal{M}_{\mathcal{S}}(E)$ (see (10) and Theorem 12).

###  1.2 Typical balls

A distinguished class of neighbourhoods of $E$, in terms of which our results
are expressed, is the class of typical balls.

###### Definition 3

A ball $B(x,d)$ is said to be typical if $x\in E$ and
$B(x,d)\subset\mathcal{O},$ where $\mathcal{O}$ is some feasible open set. We
shall write $\mathcal{B}$ for the set of typical balls.

The family of typical balls is invariant under the semigroup $G$ generated by
$\Psi$ (see Sec. 2), since, for $f\in G,$ it follows from
$f(\mathcal{O)\subset O}$ that $f\mathcal{(B)\subset B}$ holds. Consider now
the set of typical spherical $\mathcal{B}$-measures,

$\mathcal{M}_{\mathcal{S}}\mathcal{(B)}:=\left\\{\alpha\lfloor
B:B\in\mathcal{B},\ \alpha\in\mathcal{M}_{S}(E)\right\\}.$ (8)

It is well known [6] that, for any $x\in E,$ the set $\left\\{f(x):f\in
G\right\\}$ is dense in $E,$ so the balls in $\mathcal{B}$ are typical in the
sense that, if $B\in\mathcal{B},$ then similar copies of $B$ are densely
spread over $E$ at small scales by the action of $G.$ These copies are a
countable set of balls. As Theorem 12 shows, the measures in
$\mathcal{M}_{\mathcal{S}}\mathcal{(B)}$ are also typical in a deeper sense
since, for any $f\in G,$ $B\in\mathcal{B}$ and
$\alpha\in\mathcal{M}_{\mathcal{S}}\mathcal{(B)},$ the equality
$\alpha\lfloor_{f(B)}=p_{f}f_{{}_{\sharp}}(\alpha\lfloor_{B})$ holds for a
certain constant $p_{f}<1$ associated with $f.$ This means that the images of
typical balls are identical copies, up to the constant $p_{f},$ to the
original ones not only as subsets, but also from the point of view of any
property expressible in terms of self-similar measures. Moreover, in Theorem
12 it is shown that, for any typical ball $B(x,d),$ for any measure
$\alpha\in\mathcal{M}_{\mathcal{S}}\mathcal{(}E\mathcal{)}$ and for all points
$y$ in a set $\widehat{E}$ with full $\alpha$-measure, there is a sequence of
balls $\left\\{B(y,d_{k})\right\\}$ with $d_{k}\rightarrow 0,$ a sequence
$\left\\{f_{k}\right\\}$ of similitudes in $G$ and constants
$p_{f_{k}}^{-1}\rightarrow\infty,$ such that

$p_{f_{k}}^{-1}\left(f_{k}^{-1}\right)_{\sharp}\left(\alpha\lfloor_{(B(y,d_{k})}\right)\xrightarrow[k\to\infty]{st}\alpha\lfloor_{B(x,d)},$
(9)

where the convergence in (9) is in the sense of the strong topology of Radon
measures.

Theorem 12 also states that, for all $x\in\widehat{E}$ and
$\alpha\in\mathcal{M}_{\mathcal{S}}\mathcal{(B)},$

$\widetilde{\mathcal{M}}_{\mathcal{S}}\mathcal{(B)}\subset\widetilde{Tan}^{st}(\alpha,x)$
(10)

holds, where

$\widetilde{\mathcal{M}}_{\mathcal{S}}\mathcal{(B)=}\left\\{\widetilde{\alpha}:\alpha\in\mathcal{M}_{\mathcal{S}}\mathcal{(B)}\right\\},$
(11)

(see (5) for the notation $\widetilde{\alpha}).$

The results above imply that the use of general zooming similitudes, grants
the strong convergence of the zoomings to the tangent measures, whereas in the
ordinary spaces of tangent measures, where only homotheties are allowed,
convergence can only be ensured in a weak topology sense. See Sec. 3.1.1 below
for further details on identifications and topologies of measures.

###### Remark 4

Putting the results in Sec. 3.3, described in the first paragraph of this
section, together with (10), we see that the self-similar scenery at $x\in E$
depends on $x$ on large scales, meaning that there is a broad variety of balls
$B(x,d)$ for varying $x$ that, moreover, also vary with $d$ for fixed $x.$
Additionally, on a tangent scale, for each
$\alpha\in\mathcal{M}_{\mathcal{S}}\mathcal{(B)}$ and each $x\in\widehat{E},$
each typical class of measures in
$\widetilde{\mathcal{M}}_{\mathcal{S}}\mathcal{(B)}$ is a feasible outcome of
the zooming process of $\alpha$ at $x,$ so there is a wide variety of limiting
measures in $\widetilde{Tan}^{st}(\alpha,x),$ $x\in\widehat{E}.$ The
uniformity of the self-similar setting emerges here in the fact that the
inclusion
$\widetilde{\mathcal{M}}_{\mathcal{S}}\mathcal{(B)}\subset\widetilde{Tan}^{st}(\alpha,x)$
stands true for any $x\in\widehat{E},$ so all the points in $\widehat{E}$
share the set $\widetilde{\mathcal{M}}_{\mathcal{S}}\mathcal{(B)}$ of tangent
measures.

###  1.3 Spectrum of local densities of a self-similar set: the Sierpinski
gasket case

The relevance of the typical balls is stressed by the connection between
typical balls and the spectrum of densities, which in turn determines some
basic geometric features of $E.$

Let $\alpha\in\mathcal{M(}\mathbb{R}^{n}),$ $0\leq s\leq\infty$ and
$x\in\mathbb{R}^{n}.$ The upper and lower spherical $s$-densities of $\alpha$
at $x$ are defined, respectively, by

$\displaystyle\overline{\theta}_{\alpha}^{s}(x)$
$\displaystyle=\limsup_{d\rightarrow 0}\text{ }\theta_{\alpha}^{s}(x,d),$ (12)
$\displaystyle\underline{\theta}_{\alpha}^{s}(x)$
$\displaystyle=\liminf_{d\rightarrow 0}\text{ }\theta_{\alpha}^{s}(x,d),$ (13)

where the $s$-density of the ball $B(x,d),$ $\theta_{\alpha}^{s}(x,d),$ is
given by

$\theta_{\alpha}^{s}(x,d)=\frac{\alpha(B(x,d))}{(2d)^{s}}.$

Here the zooming process is summarised in only two scalars, $\eqref{upper-
density}$ and $\eqref{lower-density}$. If
$\overline{\theta}_{\alpha}^{s}(x)=\underline{\theta}_{\alpha}^{s}(x),$ then
we write $\theta_{\alpha}^{s}(x)$ for the common value and call it $s$-density
of $\alpha$ at $x$. Densities and their connections to their underlying
measures have been studied extensively in the context of geometric measure
theory. A major contribution from Marstrand (Marstrand’s theorem, [1]) asserts
that, in the Euclidean setting, if the $s$-density $\theta_{\alpha}^{s}(x)$
exists in a set with a finite and positive $\alpha$-measure with
$\alpha\in\mathcal{M(}\mathbb{R}^{n}),$ then $s$ is an integer.

The widest class of subsets of Euclidean spaces that are $s$-sets (i.e. sets
with a finite and positive $\alpha$-measure) is either the class of self-
similar sets that satisfy the OSC, with $s$ being their similarity dimension
(see (2)), or some variations of it, like the Mauldin and Williams graph-
directed constructions, cf. [16], and controlled Moran constructions, cf.
[17]. Here, we are interested in the case in which the similarity dimension
$s$ is not an integer and, by Marstrand’s theorem described above,
$\underline{\theta}_{\alpha}^{s}(x)$ and $\overline{\theta}_{\alpha}^{s}(x)$
do not coincide in subsets with a positive $\alpha$-measure. This leads to the
following definition of asymptotic spectrum of densities of a given measure
$\alpha$ at a point and, more in general, in a subset of points.

###### Definition 5

Given a subset $A\subset\mathbb{R}^{n},$ we define the _asymptotic spectrum of
(non-logarithmic) spherical $s$-densities,_ $Spec(\alpha,A),$ for a locally
finite measure $\alpha$ by

$Spec(\alpha,A)=\left\\{\lim_{k\rightarrow\infty}\theta_{\alpha}^{s}(x,d_{k}):x\in
A\text{ and }\lim_{k\rightarrow\infty}d_{k}=0\right\\}.$ (14)

We insert the non-logarithmic epithet above because there is a ample
literature on the so-called multifractal spectrum of logarithmic spherical
densities. This literature also focuses on the limiting behaviour of $\alpha$
on small balls, but the interest is in the upper and lower limits of the
quotients $\frac{\log\alpha(B(x,d))}{\log d}$ when $d\rightarrow 0$ (for $x\in
E)$ and, in particular, in the fractal dimension of both the ($\alpha$-null)
sets where these limits exist and take particular values [18] and the sets of
divergence points (see [19], [20], [21]) where the limits do not coincide.
Much less is known about the behaviour of non-logarithmic densities, and the
research in this paper can be considered a preliminary step in that direction.

In particular, in Sec. 3, Theorem 14, we present the knowledge to date about
the spectrum of non-logarithmic $\alpha$-densities,
$\alpha\in\mathcal{M}^{s}\lfloor_{E},$ of self-similar sets $E$ that satisfy
the OSC. In particular, we show that $Spec(\alpha,x)$ is contained in the
closed interval
$\left[\frac{\alpha(E)}{P^{s}(E)},\frac{\alpha(E)}{C^{s}(E)}\right]$ for all
$x$ in a subset $\widehat{E}$ of $E$ with a full $\alpha$-measure. There
arises a natural class of self-similar sets with nice properties, the
$\alpha$-exact self-similar sets (see notation in 15), which are sets for
which the endpoints of such interval belong to $Spec(\alpha,x),$
$x\in\widehat{E}.$ Whereas the results for general self-similar sets with OSC
presented in Sec. 3 are of a qualitative nature, in Sec. 4 we shall focus on
our prime example of $\alpha$-exact self-similar set, the Sierpinski gasket
$S,$ and exploit its regularity to accurately approximate the range of values
taken by its spectrum, which is the content of Theorem 26. Moreover, we give a
full characterisation of the spectrum of all the points in $S,$ which is given
by the union of two closed intervals of positive length, namely,

$Spec(\alpha,S)=\left[\alpha(S)\underline{\theta}_{\mu}^{s}(z_{0}),\alpha(S)\overline{\theta}_{\mu}^{s}(z_{0})\right]\cup\left[\frac{\alpha(S)}{P^{s}(S)},\frac{\alpha(S)}{C^{s}(S)}\right],\text{
}\alpha\in\mathcal{M}^{s}\lfloor_{S},$

where $z_{0}:=(0,0).$ Using the numerical approximations of
$\underline{\theta}_{\mu}^{s}(z_{0}),$ $\overline{\theta}_{\mu}^{s}(z_{0})$
obtained in Sec. 4 and of $P^{s}(S)$ and $C^{s}(S)$ obtained in [22] and [23],
we can also show that these two intervals are disjointed. In the case that
$\alpha\in\\{\mu,P^{{}^{s}}\lfloor_{S},C^{s}\lfloor_{S}\\},$ we have numerical
estimations of these two disjointed intervals. The Sierpinski gasket is, as
far as we know, the first connected self-similar with non-integer dimension
for which the entire spectrum has been computed.

##  2 Notation and preliminaries

The self-similar set $E$ given in (1) can be parametrised as
$E=\left\\{\pi(i):i\in\Sigma\right\\}$ with parameter space
$\Sigma:=M^{\infty}$ and geometric projection mapping $\pi:\Sigma\rightarrow
E$ given by $\pi(i)=\cap_{k=1}^{\infty}f_{i(k)}E,\ $where $i(k)$ denotes the
curtailment $i_{1}\dots i_{k}\in M^{k}$ of $i=i_{1}i_{2}\dots\in\Sigma$ and
$f_{i_{1}\dots i_{k}}=f_{i_{1}}\circ f_{i_{2}}\circ
f_{i_{3}}\circ...f_{i_{k}}$. We adopt the convention $M^{0}=\varnothing$ and
write $M^{\ast}=\cup_{k=0}^{\infty}M^{k}$ for the set of words of finite
length. Expressed in this notation, the semigroup generated by $\Psi$ can be
written as $G=\left\\{f_{i}:i\in M^{\ast}\right\\}.$

For any $i\in M^{\ast},$ we denote by $E_{i}$ the cylinder sets $f_{i}(E),$
and if $i\in M^{0},$ then $f_{i}(E):=E.$ The sets $E_{i}$ are called
$k$-cylinders if $i\in M^{k}.$ We also shorten the notation $f_{i}(A)$ to
$A_{i}$ for a general set $A\subset\mathbb{R}^{n}.$ We write
$r_{i}:=r_{i_{1}}r_{i_{2}}\dots r_{i_{k}}$ for the contraction ratio of the
similitude $f_{i}.$

Moreover, $\sigma\ :\Sigma\rightarrow\Sigma\ $ shall stand for the shift map
given by $\sigma(i_{1}i_{2}i_{3}\dots)=i_{2}i_{3}i_{4}\dots$ The code shift
can be projected (as a correspondence) onto $E,$ yielding the geometric shift

$\mathcal{T}(x):=\pi\circ\sigma\circ\pi^{-1}(x),$ (15)

$x\in E.$ The shift orbit of $x\in E$ is given by
$\left\\{\mathcal{T}^{k}(x):k\in\mathbb{N}\right\\}.$

###### Remark 6

Observe that $x\in\mathcal{T}^{k}(A)$ if and only if $f_{i}(x)\in A$ for some
$i\in M^{k}.$

###  2.1 Self-similar measures

Let $\mathcal{P}(\mathbb{R}^{n})$ be the space of compactly supported
probability Borel measures on $\mathbb{R}^{n},$ let
$\mathbf{p}=(p_{0},...,p_{m-1})\in\mathbb{R}^{m}$ be a probability vector and
let $\mathbf{M}_{\mathbf{p}}\mathbf{:}$
$\mathcal{P}(\mathbb{R}^{n})\rightarrow\mathcal{P}(\mathbb{R}^{n})$ be the
Markov operator defined by

$\mathbf{M}_{\mathbf{p}}(\alpha)=\sum_{i=0}^{m-1}p_{i}\alpha\circ
f_{i}^{-1},\text{ }\alpha\in\mathcal{P}(\mathbb{R}^{n}).$

The unique fixed point of the contractive operator $\mathbf{M}_{\mathbf{p}}$
is called the self-similar measure $\mu_{\mathbf{p}};$ that is,

$\mu_{\mathbf{p}}=\sum_{i\in M}p_{i}\mu_{\mathbf{p}}\circ f_{i}^{-1}.$ (16)

Moreover,

$\mathbf{M}_{\mathbf{p}}^{k}(\alpha)=\sum_{i\in M^{k}}p_{i}\alpha\circ
f_{i}^{-1}\underset{k\rightarrow\infty}{\xrightarrow{w}}\mu_{\mathbf{p}}$ (17)

for any $\alpha\in\mathcal{P}(\mathbb{R}^{n}),$ where, for $i\in M^{k},$
$p_{i}:=$ $p_{i_{1}}\cdot\cdot\cdot p_{i_{k}}.$ Here
$\mathbf{M}_{\mathbf{p}}^{k}$ is the $k$-th iterate of
$\mathbf{M}_{\mathbf{p}}$ (see [6] and [24] for further details). Set

$\mathcal{M}_{\mathcal{S}}(E):=\left\\{\mu_{\mathbf{p}}:\text{
}\sum_{i=0}^{m-1}p_{i}=1,\ p_{i}>0,\text{ }i=0,...,m-1\right\\}.$ (18)

For $\mathbf{p}_{s}:=(r_{0}^{s},...,r_{m-1}^{s}),$ where $s$ is the similarity
dimension of $E$ (recall that $r_{i}$ is the contraction constant of the
similarity $f_{i},$ $i\in M),$ the measure $\mu_{\mathbf{p}_{s}}$ is called
the natural probability measure on $E.$ Furthermore, if
$\alpha\in\mathcal{M}^{s}\lfloor_{E}$ (see (2) for notation), then

$\mu:=\mu_{\mathbf{p}_{s}}=\frac{\alpha}{\alpha(E)}$ (19)

(see [25]).

Notice that, whereas the measures in $\mathcal{M}^{s}$ (see (3) for notation)
convey an strong geometrical meaning, the measures $\mu_{\mathbf{p}}$ in
$\mathcal{M}_{\mathcal{S}}(E)$ do not. They are concentrated in dense subsets
of $E,$ $E_{\mathbf{p}},$ whose dimension is given by $\dim(E_{\mathbf{p}})=$
$s_{\mathbf{p}}:=\frac{\sum_{i=0}^{m-1}p_{i}\log
p_{i}}{\sum_{i=0}^{m-1}p_{i}\log r_{i}},$ but the measure $\mu_{\mathbf{p}}$
is singular w.r.t. the measures $\mathcal{H}^{s_{\mathbf{p}}}$ and
$P^{s_{\mathbf{p}}}$ (see [26] and [27]).

###  2.2 Metric measures

We now briefly recall metric measures. They are the classical tools for
analysing the geometric properties of subsets of $\mathbb{R}^{n}.$

The Hausdorff centred measure, $C^{s}(A),$ of a subset
$A\subset\mathbb{R}^{n},$ was defined by Saint Raymond and Tricot [28] in a
two-step process. First, the premeasure $C_{0}^{s}(A)$ is defined for any
$s>0$ by

$C_{0}^{s}(A)=\lim_{\delta\rightarrow
0}\inf\left\\{\sum\limits_{i=1}^{\infty}(2d_{i})^{s}\ :\
2d_{i}\leq\delta,\text{ }i=1,2,\dots\right\\},$ (20)

where the infimum is taken over all coverings,
$\left\\{B(x_{i},d_{i})\right\\}_{i\in\mathbb{N}^{+}},$ of $A$ by closed balls
$B(x_{i},d_{i})$ centred at points $x_{i}\in A.$ Then, the centred Hausdorff
$s$-dimensional measure is defined by

$C^{s}(A)=\sup\left\\{C_{0}^{s}(F):F\subset A,\ F\text{ closed}\right\\}.$

The second step in the definition of $C^{s}(A)$ is due to the lack of
monotonicity of $C_{0}^{s}$ (see [29] and [30, Example 4]). However, in [30],
it was shown that the second step can be omitted when restricting oneself to
self-similar sets with OSC.

With regard to metric measures based on packings, the standard packing measure
$P^{s}$ (see [28] and [31]) is also defined in a two-step process,

$P_{0}^{s}(A)=\lim_{\delta\rightarrow
0}\sup\left\\{\sum\limits_{i=1}^{\infty}(2d_{i})^{s}:2d_{i}\leq\delta,\text{
}i=1,2,\dots\right\\},$

where the supremum is taken over all packings
$\left\\{B(x_{i},d_{i})\right\\}_{i\in\mathbb{N}^{+}},$ with $x_{i}\in A$ for
all $i,$ and $B(x_{i},d_{i})\cap B(x_{j},d_{j})=\varnothing$ for $i\neq j.$
Then,

$P^{s}(A)=\inf\left\\{\sum\limits_{i=1}^{\infty}P_{0}^{s}(F_{i})\right\\},$

where the infimum is taken over all coverings
$\left\\{F_{i}\right\\}_{i\in\mathbb{N}^{+}}$ of $A$ by closed sets $F_{i}$
(cf.[32]). In [33], it was proved that if $A$ is a compact set with
$P_{0}^{s}(A)<\infty,$ then $P^{s}(A)=P_{0}^{s}(A),$ so this simplification
applies to any compact subset of a self-similar set with OSC.

The spherical $s$-dimensional Hausdorff measure, $\mathcal{H}_{Sph}^{s}(A),$
is obtained by removing in (20) the requirement that the balls are centred at
points of $A$. The classical $s$-dimensional Hausdorff measure,
$\mathcal{H}^{s}(A),$ results if coverings of $A$ by arbitrary subsets,
$\left\\{U_{i}\right\\}_{i\in\mathbb{\ N}^{+}},$ are considered and $2d_{i}$
is replaced in (20) with the diameter of $U_{i},$ $\left|U_{i}\right|$ (see
[34] and [5]). No second step is required for these last two measures.

The packing and the centred Hausdorff measures have a much simpler expression
when dealing with self-similar sets $E$ that satisfy the OSC as the browse for
optimal packings or coverings can be reduced to the search for optimal density
balls within the class of typical balls, $\mathcal{B}$ (see Definition 3). In
particular, for any self-similar $E$ that satisfies the OSC and with
similarity dimension $s,$ it is known (see [36]) that

$P^{s}(E)=\left(\inf\left\\{\theta_{\mu}^{s}(x,d):B(x,d)\in\mathcal{B}\right\\}\right)^{-1},$
(21)

and, Lemma (13) of Sec. 3.2 implies that

$C^{s}(E)=\left(\sup\left\\{\theta_{\mu}^{s}(x,d):B(x,d)\in\mathcal{B}\right\\}\right)^{-1}.$
(22)

##  3 Local structure and typical balls

Now we shall study the local structure of a self-similar set $E$ that
satisfies the OSC for a feasible open set $\mathcal{O}$ through the study of
the scenery flow of $\alpha\in\mathcal{M}_{\mathcal{S}}(E)$ at a.e. $x\in E,$
and through the characterisation of the spectrum of the spherical
$s$-densities of measures in $\mathcal{M}^{s}\lfloor_{E}$ (Sec. 3.2), a
limiting set that helps to summarise the structure in the neighbourhood of a
point (Sec. 1.3).

###  3.1 Scenery flow and tangent measures

We start by giving details on the construction of $\widetilde{Tan}(\nu,x)$ for
$\nu\in\mathcal{M}_{\mathcal{S}}(E)$ and $x\in E$ (see 18 for notation).

####  3.1.1 Tangent measures, identifications and topologies.

Recall that the construction of the sets $\widetilde{Tan}(\nu,x)$ and
$\widetilde{Tan}^{st}(\nu,x)$ employs the identification, in the set
$\mathcal{M}(\mathbb{R}^{n})$, of those measures that are equal up to
isometries or mutual multiples (see (4), (5), (6) and (7) for notation). We
now examine the construction of the spaces of equivalence classes of tangent
measures above in more detail.

For $\nu\in\mathcal{M}(\mathbb{R}^{n})$ and $x\in spt(\nu),$ we first consider
sequences $\\{c_{n}\nu_{x,t_{n}}\\}_{n=0}^{\infty},$ where for every
$n\in\mathbb{N},$ $c_{n}>0,$

$\nu_{x,t_{n}}:=\frac{1}{\nu(B(x,dt_{n}^{-1}))}\left(g_{x,t_{n}}\right)_{\sharp}\nu\lfloor_{B(x,dt_{n}^{-1})},$
(23)

$d\leq 1$ and $g_{x,t_{n}}$ is some similarity with expanding ratio $t_{n}$
that maps the ball $B(x,t_{n}^{-1})$ onto the ball $B(z_{n},1),$ with
$z_{n}=g_{x,t_{n}}(x),$ so each $\nu_{x,t_{n}}$ is a probability measure
supported on $B(z_{n},d).$ Then, $\widetilde{Tan}(\nu,x)$ and
$\widetilde{Tan}^{st}(\nu,x)$ consist of the equivalence classes of non-null
weak and strong limits, respectively, as $t_{n}\rightarrow\infty,$ of such
sequences $\\{c_{n}\nu_{x,t_{n}}\\}_{n=0}^{\infty}$ (see (7)). Lemma 8 shows
that the elements in $\widetilde{Tan}(\nu,x)$ and
$\widetilde{Tan}^{st}(\nu,x)$ do not depend on either the sequence of
constants $c_{n}$ or the particular elements chosen in the equivalence classes
$\widetilde{\nu_{x,t_{n}}}$ as long as the convergence of these elements is
guaranteed.

###### Remark 7

The unit ball $D$ does not play any essential role in our definition of
tangent measures in the quotient space
$\widetilde{\mathcal{M}}(\mathbb{R}^{n}).$ In the opposite direction (second
approach) we may, in a more akin way to the classical approach, require the
similarities $g_{x,t_{n}}$ to map $B(x,t_{n}^{-1})$ onto $B(0,1),$ and then
define $Tan_{D}(\nu,x)$ and $Tan_{D}^{st}(\nu,x)$ as weak and strong limits in
$\mathcal{M}(D),$ respectively, of sequences of such measures $\nu_{x,t_{n}},$
and $\widetilde{Tan}(\nu,x),$ $\widetilde{Tan}^{st}(\nu,x)$ as the sets of
equivalence classes of measures in $Tan_{D}(\nu,x)$ and $Tan_{D}^{st}(\nu,x),$
respectively.
This second method gives spaces of tangent equivalence classes which are
particular cases of these in our primary approach. Are these equivalent
methods? In order to answer this question, let a sequence
$\\{c_{n}\nu_{x,t_{n}}\\}_{n=0}^{\infty},$ as in (23), converge to a non-null
Radon measure $\alpha.$ By Lemma 8 we may assume $c_{n}=1$ for all
$n\in\mathbb{N}^{+}.$ Since the measures $\nu_{x,t_{n}}$ are supported on
balls $B(z_{n},d)$ with $d\leq 1$ (see Theorem 12 (i)), the limiting measure
$\alpha$ must also be supported on a ball $B(z,d)$ with
$z_{n}\underset{n\rightarrow\infty}{\longrightarrow}z.$ Each measure
$\nu_{x,t_{n}}^{\prime}=\left(\tau_{z_{n}}\right)_{\sharp}\nu_{x,t_{n}},$
where $\tau_{z_{n}}(y)=y-z_{n},$ is equivalent by translation to
$\nu_{x,t_{n}},$ and $\nu_{x,t_{n}}^{\prime}$ is supported on $D.$ It is easy
to see that $\nu_{x,t_{n}}\xrightarrow[n\to\infty]{w}\alpha$ implies that
$\nu_{x,t_{n}}^{\prime}\xrightarrow[n\to\infty]{w}\alpha^{\prime}=\left(\tau_{z}\right)_{\sharp}\alpha,$
so $\alpha^{\prime}$ is equivalent to $\alpha$ and supported on $D.$ Thus, the
second method gives the same space $\widetilde{Tan}(\nu,x)$ than our primary
method. But $\nu_{x,t_{n}}\xrightarrow[n\to\infty]{st}\alpha$ does not imply
that $\nu_{x,t_{n}}^{\prime}\xrightarrow[n\to\infty]{st}\alpha^{\prime},$ so
the second method does not produce the same space
$\widetilde{Tan}^{st}(\nu,x)$ than our method.
Observe that, if we let
$\nu_{x,t_{n}}^{\prime}=\left(\tau_{z}\right)_{\sharp}\nu_{x,t_{n}},$ then
$\nu_{x,t_{n}}\xrightarrow[n\to\infty]{st}\alpha$ does imply
$\nu_{x,t_{n}}^{\prime}\xrightarrow[n\to\infty]{st}\alpha^{\prime}=\left(\tau_{z}\right)_{\sharp}\alpha.$
But now the measure $\nu_{x,t_{n}}^{\prime}$ is supported on the ball
$B(z_{n}-z,d)$ rather than on $D.$ This observation is useful because $D$ and
all the balls $B(z_{n}-z,d)$ are contained in some ball $B(0,R)$ for $R$ large
enough (notice that $z_{n}$ is a convergent sequence of points), so the
convergence
$\nu_{x,t_{n}}\underset{n\rightarrow\infty}{\longrightarrow}\alpha$ (weak or
strong) occurs in $\mathcal{M(}B(0,R)),$ and we can see that, if we consider
vague convergence of measures, we do not obtain anything new, since in the
Polish space $B(0,R)$ both convergences are equivalent ([15], Appendix).

###### Lemma 8

(i) The sequences $\\{c_{n}\\}_{n=0}^{\infty}$ in the construction of $\
\widetilde{Tan}(\nu,x)$ and $\widetilde{Tan}^{st}(\nu,x)$ can be taken to be
$c_{n}=1,$ $n=0,1,2,...$
(ii) Let $\nu\in\mathcal{M}(\mathbb{R}^{n}),$ $x\in spt(\nu)$ and $\alpha\in
Tan(\nu,x).$ Let $\\{t_{n}\\}_{n=0}^{\infty}\uparrow\infty$ be such that
$\\{\nu_{x,t_{n}}\\}_{n=0}^{\infty}$ $\xrightarrow[n\to\infty]{w}\alpha.$
Assume also that there is a sequence $\\{f_{n}\\}_{n=0}^{\infty}$ in the set
of isometries $\mathcal{I}_{n}$ such that
$\\{\left(f_{n}\right)_{\sharp}\nu_{x,t_{n}}\\}_{n=0}^{\infty}\xrightarrow[n\to\infty]{w}\alpha^{\prime}.$
Then, there is $f$ $\in\mathcal{I}_{n}$ such that
$(f)_{\sharp}\alpha=\alpha^{\prime}.$ The same is true if the convergence
holds in the topology of the strong convergence in
$\mathcal{M}(\mathbb{R}^{n}).$

Proof. (i) By definition, a weak limiting measure $\alpha$ as in (7) is a non-
null measure in $\mathcal{M}(\mathbb{R}^{n}).$ Therefore, the sequence of
constants $\left\\{c_{n}\right\\}$ must be bounded above and below by two
positive and finite constants. We can choose a subsequence
$\left\\{c_{n_{k}}\right\\}_{k=0}^{\infty}$ that converges to a constant $c,$
and then the sequence $c\nu_{x,t_{n}}$ must converge to the weak limit
$\alpha.$ This gives $\nu_{x,t_{n}}\xrightarrow[n\to\infty]{w}c^{-1}\alpha,$
which belongs to the same equivalence class in $\widetilde{Tan}(\nu,x)$ as
$\alpha.$ On the other hand, the non-null weak limits in
$\mathcal{M}(\mathbb{R}^{n})$ of sequences
$\\{\nu_{x,t_{n}}\\}_{n=0}^{\infty}$ are particular cases of those of
sequences $\left\\{c_{n}\nu_{x,t_{n}}\right\\}.$ This completes the proof of
part (i) for weak limits. The argument also holds true for strong limits.
(ii) For any $n\in\mathbb{N}^{+},$ we can write
$f_{n}(\cdot)=g_{n}(\cdot)+a_{n},$ where $g_{n}$ is an orthogonal map and
$a_{n}\in\mathbb{R}^{n}.$ Recall that $\nu_{x,t_{n}}$ is supported on
$B(z_{n},d),$ so $\left(g_{n}+a_{n}\right)_{\sharp}(\nu_{x,t_{n}})$ is
supported on $a_{n}+B(z_{n},d)$ ([5], Theorem 1.18), with
$z_{n}\underset{n\rightarrow\infty}{\longrightarrow}z.$ This means that, if
$\nu_{x,t_{n}}^{\prime}:=\left(f_{n}\right)_{\sharp}\nu_{x,t_{n}}$ converges,
in the weak topology of $\mathcal{M}(\mathbb{R}^{n}),$ to some non-null
measure $\alpha^{\prime}$ in $\mathcal{M}(\mathbb{R}^{n}),$ the sequence
$a_{n}$ must be bounded, and then the sequence $\\{f_{n}\\}_{n=0}^{\infty}$ is
also bounded in the supremum norm. Therefore, there is a convergent
subsequence, $\\{f_{n_{k}}\\}_{k=0}^{\infty},$ of
$\\{f_{n}\\}_{n=0}^{\infty}.$ Let $f:=\lim_{k\rightarrow\infty}f_{n_{k}}.$
Since the sequence
$\\{\left(f_{n_{k}}\right)_{\sharp}(\nu_{x,t_{n_{k}}})\\}_{k=0}^{\infty}$
converges to $\alpha^{\prime},$ we have that

$\alpha^{\prime}=\lim_{k\rightarrow\infty}f_{n_{k}\sharp}(\nu_{x,t_{n_{k}}})=f_{\sharp}\alpha,$
(24)

which proves that $\alpha^{\prime}\cong\alpha.$ The second equality in (24)
holds true because, for any $\varphi$ in the space $C_{0}(\mathbb{R}^{n})$ of
continuous, compactly supported functions on $\mathbb{R}^{n}$ and for any
$\varepsilon>0,$ there is $k_{0}>0$ such that for $k\geq k_{0},$ we have

$\left\|\varphi\circ f_{n_{k}}-\varphi\circ
f\right\|\leq\frac{\varepsilon}{2},$ $\left\|\int\varphi\circ f\text{
}d(\nu_{x,t_{n_{k}}})-\int\varphi\circ f\text{
}d\alpha\right\|\leq\frac{\varepsilon}{2},$

and then

$\displaystyle\left\|\int\varphi\text{
}d\left(f_{n_{k}}{}_{\sharp}(\nu_{x,t_{n_{k}}})\right)-\int\varphi\text{
}d(f_{\sharp}\alpha)\right\|$ $\displaystyle\leq\int\left\|\varphi\circ
f_{n_{k}}-\varphi\circ f\text{
}\right\|d(\nu_{x,t_{n_{k}}})+\left\|\int\varphi\circ f\text{
}d(\nu_{x,t_{n_{k}}})-\int\varphi\circ f\text{
}d\alpha\right\|\leq\varepsilon.$

If $\\{\nu_{x,t_{n}}^{\prime}\\}_{n=0}^{\infty}$ converges to
$\alpha^{\prime}$ in the strong topology of $\mathcal{M}(D),$ then it also
converges in the weak topology and the argument above applies.

####  3.1.2 Scaling properties of typical balls and scenery flow

We need some preliminary lemma and the following definition.

###### Definition 9

Given a measure $\alpha\in\mathcal{M}^{s}\lfloor_{E},$ two Euclidean balls
$B(x,d)$ and $B(x^{\prime},d^{\prime})$ are said to be $\alpha$-density
equivalent if
$\theta_{\alpha}^{s}(x,d)=\theta_{\alpha}^{s}(x^{\prime},d^{\prime}).$

We start with two elementary scaling properties of typical balls for measures
in $\mathcal{M}_{\mathcal{S}}(E)$ and in $\mathcal{M}^{s}\lfloor_{E}.$

###### Lemma 10

Let $E$ be a self-similar set generated by the system $\
\Psi=\left\\{f_{i}\right\\}_{i\in M}$ of similarities of $\ \mathbb{R}^{n},$
with $M=\left\\{0,1,\dots,m-1\right\\},$ and similarity dimension $s.$ Let
$\mathcal{O}$ be a feasible open set (for $\Psi)$ and let $i\in M^{\ast}.$
Then
(i)

$\mu_{\mathbf{p}}(f_{i}(A))=p_{i}\mu_{\mathbf{p}}(A),\text{ for
}\mu_{\mathbf{p}}\in\mathcal{M}_{\mathcal{S}}(E)\quad\text{and
}\mu_{\mathbf{p}}\text{-measurable }A\subset\mathcal{O},$ (25)

(ii)

$\mu_{\mathbf{p}}(f_{i}^{-1}(C))=p_{i}^{-1}\mu_{\mathbf{p}}(C)\text{ for
}\mu_{\mathbf{p}}\in\mathcal{M}_{\mathcal{S}}(E)\text{ and
}\mu_{\mathbf{p}}\text{-measurable }C\subset\mathcal{O}_{i},$ (26)

(iii)

$B(f_{i}(x),r_{i}d)\text{ is}\ \alpha\text{-density equivalent to}\
B(x,d)\text{ for }\alpha\in\mathcal{M}^{s}\lfloor_{E}\ \text{and
}B(x,d)\subset\mathcal{O},$ (27)

(iv)

$f_{i}^{-1}(B(f_{i}(x),r_{i}d))\text{ is}\ \alpha\text{-density equivalent
to}\ B(x,d)\ \text{for }\alpha\ \in\mathcal{M}^{s}\lfloor_{E}\ \text{and }\
B(x,d)\subset\mathcal{O}_{i}.$ (28)

Proof. The proof of (25) is trivial from (16) if $E$ satisfies SSC. If SSC
does not hold, then

$\mu_{\mathbf{p}}(f_{j}^{-1}(f_{i}(A)))\leq\mu_{\mathbf{p}}(\partial\mathcal{O)}=0\text{
for }j\neq i,$

because $A\subset\mathcal{O}$ and, hence, $f_{j}^{-1}(f_{i}(A))\cap
E\subset\partial\mathcal{O},$ which is known to be a $\mu_{\mathbf{p}}$\- null
set (cf. [27]), so (25) also follows from (16). If we set $A=f_{i}^{-1}(C)$ in
(25), we obtain (26) (see also [10]). By (19), we can apply (25) and (26) to
any measure $\alpha\in\mathcal{M}^{s}\lfloor_{E},$ which easily gives (27) and
(28).

Before stating the main theorem of this section, we will see the following
lemma.

###### Lemma 11

(i) Let $g,$ $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n},$
$\alpha\in\mathcal{M}(\mathbb{R}^{n}),$ $\lambda>0$ and
$A\subset\mathbb{R}^{n}$ be an $\alpha$-measurable subset. Then, the following
equalities hold true:
$\mathbf{\bullet}$
$\lambda\left(g\right)_{\sharp}(\alpha)=\left(g\right)_{\sharp}(\lambda\alpha),\newline
\mathbf{\bullet}$ $(f\circ g)_{\sharp}\alpha=f_{\sharp}(g)_{\sharp}(\alpha),$
and
$\mathbf{\bullet}$
$\left(g_{\sharp}\alpha\right)\lfloor_{A}=g_{\sharp}(\alpha\lfloor_{g^{-1}(A)})$
(ii) Let $\alpha$ be a measure on $\mathcal{M}(\mathbb{R}^{n}),$
$g:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ a bijective mapping and
$\beta:=g_{\sharp}\left(\alpha\right).$ Then, $\alpha=(g^{-1})_{\sharp}\beta.$
(iii) If $\left\\{\alpha_{k}\right\\}_{k\in\mathbb{N}}$ is a sequence of
measures on $\mathcal{M}(\mathbb{R}^{n})$ and
$(g)_{\sharp}\left(\alpha_{k}\right)\xrightarrow[k\to\infty]{st}\beta,$ then
$\alpha_{k}\xrightarrow[k\to\infty]{st}\left(g^{-1}\right)_{\sharp}\left(\beta\right).$
(iv) Let $B(x_{n},d):=B_{n}$ be a sequence of closed balls that converges in
the Hausdorff metric to a closed ball $B(x,d):=B,$ and let
$\alpha\in\mathcal{M}(B)$ with $\alpha(\partial B)=0.$ Then
$\alpha\lfloor_{B_{n}}:=\alpha_{n}$ $\xrightarrow[n\to\infty]{st}\alpha.$

Proof.
Parts (i)-(iii) easily follows from the definitions.
Recall that $\alpha_{n}\xrightarrow[n\to\infty]{st}$ $\alpha$ means that
$\alpha_{n}(A)\underset{n\rightarrow\infty}{\longrightarrow}\alpha(A)$ for any
Borel set $A\subset\mathbb{R}^{n}.$ Let $\alpha\in\mathcal{M(}B)$ and let $K$
be any compact set contained in the interior $U$ of $B.$ The distance
$d(K,\partial B)=\min\left\\{\left\|x-y\right\|:x\in K,\text{ }y\in\partial
B\right\\}$ must be a quantity $\varepsilon>0$ and then $K\subset
B(x,d-\varepsilon).$ The convergence of $B_{n}$ to $B$ implies that there is
an $n_{0}\in\mathbb{N}^{+}$ such that, for $n>n_{0},$
$\left\|x-x_{n}\right\|\leq\varepsilon.$ Then, if $z\in$ $K,$

$\left\|z-x_{n}\right\|\leq\left\|z-x\right\|+\left\|x-x_{n}\right\|\leq d,$

which shows that $K\subset B\cap B_{n}$ for $n>n_{0}.$ Then, for such values
of $n,$ we have

$\alpha_{n}\left(K\right)=\alpha(B_{n}\cap K)=\alpha(K)$

We now prove that $\alpha_{n}\xrightarrow[n\to\infty]{st}\alpha$ also holds in
the $\sigma$-field $\mathfrak{B}(B)$ of Borel subsets of $B.$ Let

$\mathcal{A}:=\left\\{A\subset B:A\text{ is }\alpha\text{-measurable and
}\lim_{n\rightarrow\infty}\alpha_{n}(A)=\alpha(A)\right\\}.$

(Notice that any $\alpha$-measurable set is also $\alpha_{n}$-measurable for
all $n\in\mathbb{N}^{+}).$ It is easy to check that $B\in\mathcal{A},$ that
$B-A:=A^{c}\in\mathcal{A}$ if $A\in\mathcal{A},$ and that $\mathcal{A}$ is
closed under a finite union of its members or, in short, that $\mathcal{A}$ is
a field. Let $F_{k}$ be a sequence of members of $\mathcal{A}.$ In order to
show that $\cup_{k\in\mathbb{N}^{+}}F_{k}\in\mathcal{A},$ we first write
$\cup_{k\in\mathbb{N}^{+}}F_{k}=$ $\cup_{k\in\mathbb{N}^{+}}G_{k},$ where
$G_{k}=\cup_{i=1}^{k}F_{i}.$ This shows that $\cup_{k\in\mathbb{N}^{+}}F_{k}$
can be expressed as a countable union of the increasing sequence $G_{k}$ of
members of $\mathcal{A}.$ Furthermore, $\cup_{k\in\mathbb{N}^{+}}F_{k}=$
$\cup_{k\in\mathbb{N}^{+}}H_{k},$ where $H_{k}=(G_{k}-G_{k-1})$ with
$G_{0}=\varnothing.$ Now, each $H_{k}\in\mathcal{A}$ and $H_{k}\cap
H_{k^{\prime}}=\varnothing$ for $k\neq k^{\prime}.$ Then, using that each
$\alpha_{n}$ is a measure, we have

$\lim_{n\rightarrow\infty}\alpha_{n}\left({\displaystyle\bigcup\limits_{k\in\mathbb{N}^{+}}}H_{k}\right)={\displaystyle\sum\limits_{k\in\mathbb{N}^{+}}}\lim_{n\rightarrow\infty}\alpha_{n}\left(H_{k}\right)={\displaystyle\sum\limits_{k\in\mathbb{N}^{+}}}\alpha\left(H_{k}\right)=\alpha\left({\displaystyle\bigcup\limits_{k\in\mathbb{N}^{+}}}H_{k}\right).$

This completes the proof that $\mathcal{A}$ is a $\sigma$-field. Notice that
any closed set $K\subset B$ can be written as the union of the $\alpha$ and
$\alpha_{n}$-null set $K\cap\partial B$ and of the set $K-\partial B,$ which
belongs to $\mathcal{A}$ as a countable union of compact sets $K\cap
B(x,d-n^{-1})\subset U.$ Thus, the class $\mathcal{K}$ of closed subsets of
$B$ is contained in $\mathcal{A}.$ We know that the $\sigma$-fields generated
by $\mathcal{K}$ and by $\mathcal{A}$ satisfy
$\mathfrak{B}\mathcal{(}B)=\sigma(\mathcal{K)\subset\sigma(A)=A}.$ This gives
the strong convergence of $\alpha_{n}$ to $\alpha$ on
$\mathfrak{B}\mathcal{(}B).$

We can now go to the scenery flow of measures in
$\mathcal{M}_{\mathcal{S}}(E).$

###### Theorem 12

Let $E$ be a self-similar set generated by the system
$\Psi=\left\\{f_{i}\right\\}_{i\in M}$ of similarities on $\mathbb{R}^{n},$
with $M=\left\\{0,1,\dots,m-1\right\\}$ and similarity dimension $s.$ Let
$\mathcal{O}$ be a feasible open set (for $\Psi$) and
$\mu_{\mathbf{p}}\in\mathcal{M}_{\mathcal{S}}(E).$ Then, for any
$\mu_{\mathbf{p}}$-measurable set $B\subset\mathcal{O}$ and $i\in M^{\ast},$
the following statements hold true.
(i)

$\mu_{\mathbf{p}}\lfloor_{B_{i}}=p_{i}\left(f_{i}\right)_{\sharp}\left(\mu_{\mathbf{p}}\lfloor_{B}\right).$

(ii)

$\mu_{\mathbf{p}}\lfloor_{B}=p_{i}^{-1}\left(f_{i}^{-1}\right)_{\sharp}\left(\mu_{\mathbf{p}}\lfloor_{B_{i}}\right).$
(29)

(iii) There is a subset $\widehat{E}\subset E$ with
$\mu_{\mathbf{p}}(\widehat{E})=1$ such that, if $x\in E$ and
$B(x,d)\subset\mathcal{O},$ then for any $y\in\widehat{E},$ there is a
sequence $\left\\{i_{j}\right\\}_{j\in\mathbb{N}^{+}}$ with $i_{j}\in
M^{\ast}$ and a sequence of balls
$\left\\{B(y,dr_{i_{j}})\right\\}_{j\in\mathbb{N}^{+}}$ such that

$p_{i_{j}}^{-1}\left(f_{i_{j}}^{-1}\right)_{\sharp}\left(\mu_{\mathbf{p}}\lfloor_{B(y,dr_{i_{j}})}\right)\
\xrightarrow[j\to\infty]{st}\mu_{\mathbf{p}}\lfloor_{B(x,d)}$

(iv) For any $x\in\widehat{E},$

$\widetilde{\mathcal{M}}_{\mathcal{S}}\mathcal{(B)\subset}\widetilde{Tan}^{st}(\mu_{\mathbf{p}},x),$

where $\mathcal{M}_{\mathcal{S}}\mathcal{(B)}$ is defined in (8).

Proof. In order to show (i), let
$\mu_{\mathbf{p}}\in\mathcal{M}_{\mathcal{S}}(E),$ $i\in M^{\ast}$ and let
$B\subset\mathcal{O}$ and $A\subset\mathbb{R}^{n}$ be
$\mu_{\mathbf{p}}$-measurable sets. Then,

$\displaystyle\left(p_{i}({f_{i}})_{\sharp}\left(\mu_{\mathbf{p}}\lfloor_{B}\right)\right)(A)$
$\displaystyle=p_{i}\left(\mu_{\mathbf{p}}\lfloor_{B}\right)\left(f_{i}^{-1}(A\right))$
$\displaystyle=p_{i}\mu_{\mathbf{p}}(f_{i}^{-1}(A\cap
B_{i}))=\mu_{\mathbf{p}}\lfloor_{B_{i}}(A),$

where the third equality follows from (26) and (i) is proved. Analogously,
(ii) follows from (25).
Now, let

$\widehat{E}=\left\\{y\in
E:\left\\{\mathcal{T}^{k}(y):k\in\mathbb{N^{+}}\right\\}\text{ is dense in
}E\right\\}$ (30)

(see (15) in Sec. 2 the definition of $\mathcal{T}).$ It is well known (cf.
[37]) that the set $\widehat{E}$ has a full $\mu_{\mathbf{p}}$-measure. Let
$x\in E,$ $B(x,d)\subset\mathcal{O},$ $y\in\widehat{E}$ and
$\left\\{x_{j}\right\\}_{j\in\mathbb{N}^{+}}$ such that
$\lim_{j\rightarrow\infty}x_{j}=x$ (in the Euclidean metric) with
$x_{j}\in\mathcal{T}^{k_{j}}(y)$ for every ${j}\in\mathbb{N^{+}}.$ We may also
assume that $B(x_{j},d)\subset\mathcal{O}$ for every $j\in\mathbb{N^{+}}.$ We
shorten $B(x_{j},d)$ to $B^{j}$ and $B(x,d)$ to $B.$ Since
$\lim_{j\rightarrow\infty}x_{j}=x,$ it follows that
$\left\\{B^{j}\right\\}_{j\in\mathbb{N}}$ converges to $B$ in the Hausdorff
metric. Also, $\mu_{\mathbf{p}}(\partial B)=0$ because
$\mu_{\mathbf{p}}\in\mathcal{M}_{\mathcal{S}}(E)$ (cf. [35]). Then, Lemma 11
(iv) implies that

$\mu_{\mathbf{p}}\lfloor_{B^{j}}\xrightarrow[j\to\infty]{st}\mu_{\mathbf{p}}\lfloor_{B}.$
(31)

Now, notice that, since $x_{j}\in\mathcal{T}^{k_{j}}(y)$ for each
$j\in\mathbb{N},$ there is $i_{j}$ $\in M^{k_{j}}$ such that
$f_{i_{j}}(x_{j})=y$ (see Remark 6). Then,
$f_{i_{j}}^{-1}(B(y,dr_{i_{j}}))=B^{j}.$ By (29) applied to $B^{j}$ and
$i_{j}\in M^{\ast},$ we see that

$\mu_{\mathbf{p}}\lfloor_{B^{j}}=p_{i_{j}}^{-1}\left(f_{i_{j}}^{-1}\right)_{\sharp}\left(\mu_{\mathbf{p}}\lfloor_{{B(y,dr_{i_{j}})}}\right),$
(32)

which concludes the proof of (iii).
Observe that, in the terminology of Sec. 3.1.1, the right hand term in (32)
is, $c_{t_{j}}\nu_{y,t_{j}}$ for $\nu=\mu_{\mathbf{p}},$
$t_{j}=r_{i_{j}}^{-1}$ and
$c_{t_{j}}=p_{i_{j}}^{-1}\mu_{\mathbf{p}}(B(y,dr_{i_{j}}))$ (recall that
$\nu_{y,t_{j}}$ was a normalised blowup and notice also that we may assume,
rescaling $E$ if necessary, that all typical balls have a radius $d\leq 1).$
So, (31) and (7) give
$\widetilde{\mu_{\mathbf{p}}\lfloor_{B}}\in\widetilde{Tan}^{st}(\mu_{\mathbf{p}},x)$
and part (iv) is proved.

###  3.2 Asymptotic spectra and measure-exact self-similar sets

We shall write $\mathit{Im}(\theta_{\alpha}^{s},\mathcal{B})$ to designate the
set

$\mathit{Im}(\theta_{\alpha}^{s},\mathcal{B}):=\left\\{\theta_{\alpha}^{s}(x,d):B(x,d)\in\mathcal{B}\right\\}$

(see notation in Definition 3), which plays a relevant role in the geometric
analysis of $E$ (see (21) and the lemma below).

###### Lemma 13

Let $E$ be a self-similar set generated by the system of similarities of
$\mathbb{R}^{n},\ \Psi=\left\\{f_{i}\right\\}_{i\in M},$ with
$M=\left\\{0,1,\dots,m-1\right\\},$ and similarity dimension $s.$ If $E$
satisfies the OSC, then
(i)

$C^{s}(E)=\left(\sup\left\\{\theta_{\mu}^{s}(x,d):B(x,d)\in\mathcal{B}_{\mathcal{O}}\right\\}\right)^{-1},$

where $\mathcal{B}_{\mathcal{O}}:=\left\\{B(x,d)\in\mathcal{B}:\text{
}B(x,d)\subset\mathcal{O}\right\\}$ and $\mathcal{O}$ is any feasible open set
for $\Psi.$
(ii)

$C^{s}(E)=\left(\sup\mathit{Im}(\theta_{\mu}^{s},\mathcal{B})\right)^{-1}.$

Proof. It is known that for a general self-similar set that satisfies the OSC
(see [36] and [30]),

$C^{s}(E)=\left(\sup\\{\theta_{\mu}^{s}(x,d):x\in E\text{ and
}d>0\\}\right)^{-1}$ (33)

holds. Let $\mathcal{O}$ be any feasible open set. Then, it is enough to show
that

$\sup_{(x,d)\in
E\times\mathbb{R}^{+}}\theta_{\mu}^{s}(x,d)\leq\sup_{B(x,d)\in\mathcal{B}_{\mathcal{O}}}\theta_{\mu}^{s}(x,d).$

Should this not be the case, there would exist $(x_{0},d_{0})\in
E\times\mathbb{R}^{+}$ such that
$B(x_{0},d_{0})\notin\mathcal{B}_{\mathcal{O}}$ and

$\theta_{\mu}^{s}(x_{0},d_{0})>\sup_{B(x,d)\in\mathcal{B}_{\mathcal{O}}}\theta_{\mu}^{s}(x,d).$

In order to show that this contradicts (33), take $x^{\ast}\in
E\cap\mathcal{O}$ such that there is $i\in M^{\ast}$ with
$f_{i}(x^{\ast})=x^{\ast}.$ Let
$\rho_{1}:=\min\left\\{\left\|x^{\ast}-z\right\|:z\in\partial\mathcal{O}\right\\}.$
Observe that, if we take $\rho_{2}>0$ so that $B(x_{0},d_{0})\subset
B(x^{\ast},\rho_{2})$ and $k\in\mathbb{N}^{+},$ satisfying that
$r_{i}^{k}\rho_{2}<\rho_{1},$ then

$f_{i}^{k}(B(x_{0},d_{0}))\subset
f_{i}^{k}(B(x^{\ast},\rho_{2}))=B(x^{\ast},r_{i}^{k}\rho_{2})\subset\mathcal{O},$

which, using that $f_{i}^{k}(B(x_{0},d_{0})\cap S)\subset
f_{i}^{k}(B(x_{0},d_{0}))\cap S,$ raises the contradiction

$\theta_{\mu}^{s}(x_{0},d_{0})\leq\frac{r_{i}^{-ks}\mu(f_{i}^{k}(B(x_{0},d_{0})))}{(2d_{0})^{s}}=\frac{\mu(B(f_{i}^{k}(x_{0}),r_{i}^{k}d_{0}))}{(2d_{0}r_{i}^{k})^{s}}\leq\sup_{B(x,d)\in\mathcal{B}_{\mathcal{O}}}\theta_{\mu}^{s}(x,d).$

Part (ii) is trivial from (i).

In the next theorem, we shall establish the relationships between the
pointwise and global spectra, the set
$\mathit{Im}(\theta_{\alpha}^{s},\mathcal{B})$ and its extreme values
$\alpha(E)\left(P^{s}(E)\right)^{-1}$ and
$\alpha(E)\left(C^{s}(E)\right)^{-1}.$

###### Theorem 14

Let $E\subset\mathbb{R}^{n}$ be a self-similar set that satisfies the SOSC
with feasible open set $\mathcal{O}$ and similarity dimension $s,$ and let
$\alpha\in\mathcal{M}^{s}\lfloor_{E}.$ Then, the following statements hold
true.
(i) For $x\in E,$ it holds that

$Spec(\alpha,x)=\left[\underline{\theta}_{\alpha}^{s}(x),\overline{\theta}_{\alpha}^{s}(x)\right]$

(see (13) and (12) for notation)

$Spec(\alpha,E)\subset\left[\kappa_{1},\kappa_{2}\right]$

with $0<\kappa_{1}\leq\kappa_{2}<\infty.$
(ii) There is a subset $\widehat{E}\subset E$ with $\mu(\widehat{E})=1$ such
that, for any $y\in\widehat{E},$

$Spec(\alpha,y)=Spec(\alpha,\widehat{E})=Spec(\alpha,\mathcal{O}\cap E).$

(iii)

$\left(\frac{\alpha(E)}{P^{s}(E)},\frac{\alpha(E)}{C^{s}(E)}\right)\subset\mathit{Im}(\theta_{\alpha}^{s},\mathcal{B})\subset
Spec(\alpha,\mathcal{O\cap}E)\subset\left[\frac{\alpha(E)}{P^{s}(E)},\frac{\alpha(E)}{C^{s}(E)}\right].$

Proof. That $\underline{\theta}_{\alpha}^{s}(x)$ and
$\overline{\theta}_{\alpha}^{s}(x)$ belong to and are the extreme values of
$Spec(\alpha,x)$ follows from the definitions. That all the intermediate
values in between also belong to $Spec(\alpha,x)$ is a consequence of the
continuousness of $\theta_{\alpha}^{s}(x,d),$ with respect to $d.$ This last
property follows from the fact that the $\alpha$-measure of the boundary of
Euclidean balls is always null [35] for any measure
$\alpha\in\mathcal{M}_{\mathcal{S}}(E),$ which proves the first assertion of
(i). The second assertion is well known [6].
In order to prove (ii), let $\widehat{E}$ be the full $\mu$-measure subset of
points of $E$ that have a dense geometric shift orbit in $E$ (see (30)) and
let $y\in\widehat{E}.$ The inclusions $Spec(\alpha,y)\subset
Spec(\alpha,\widehat{E})\subset Spec(\alpha,\mathcal{O}\cap E)$ are trivial as
$\widehat{E}\subset\mathcal{O}.$ This follows from the fact that, if
$y\notin\mathcal{O},$ then $\mathcal{T}(y)\cap\mathcal{O}=\varnothing$ because
$f_{i}(\mathcal{O})\subset\mathcal{O}$ for any $i\in M,$ and repeating the
same argument, we see that $T^{k}(y)$ could not be dense in $E.$
The corresponding equalities would follow if we prove
$Spec(\alpha,\mathcal{O}\cap E)\subset Spec(\alpha,y).$ This holds true
because, if $z=\lim_{k\rightarrow\infty}\theta_{\alpha}^{s}(x,d_{k})$ for
$x\in\mathcal{O\cap}E$ and
$d_{k}\underset{k\rightarrow\infty}{\longrightarrow}0,$ since
$B(x,d_{k})\in\mathcal{B}$ for any sufficiently large $k,$ we can apply
Theorem 12 (iii) to see that, for such values of $k,$
$\theta_{\alpha}^{s}(x,d_{k})\in Spec(\alpha,y)$ and, hence,
$\lim_{k\rightarrow\infty}\theta_{\alpha}^{s}(x,d_{k})\in Spec(\alpha,y)$
easily follows from (i). This ends the proof of (ii).
Finally, the first inclusion in (iii) for $\alpha=\mu$ follows from the
continuousness of the function $\theta_{\mu}^{s}(x,d)$ on
$\mathbb{R}^{n}\times\mathbb{R}^{+}$ since

$\frac{1}{P^{s}(E)}\leq\theta_{\mu}^{s}(x,d)\leq\frac{1}{C^{s}(E)}$

holds if $B(x,d)\in\mathcal{B}$ as a straightforward consequence of (21) and
(22). The arguments given in the proof of (ii) applied to $\mu$ show that, if
$B(x,d)\in\mathcal{B},$ then $\theta_{\mu}^{s}(x,d)\in
Spec(\mu,\mathcal{O}\cap E),$ which gives the next inclusion in (iii). The
last inclusion follows from the observation that $Spec(\mu,\mathcal{O\cap}E)$
consists of limiting values of sequences with terms in
$\mathit{Im}(\theta_{\mu}^{s},\mathcal{B}),$ whose extreme values are
$\frac{1}{P^{s}(E)}$ and $\frac{1}{C^{s}(E)}.$ Using (19), we get that
$\theta_{\alpha}^{s}(x,d)=\alpha(E)\theta_{\mu}^{s}(x,d),$ and (iii) follows
for any $\alpha\in\mathcal{M}^{s}\lfloor_{E}.$

Of note is the case in which the extreme values of $\theta_{\alpha}^{s}(x,d)$
are attained on $\mathcal{B}.$ In this case, we have the following result.

###### Corollary 15

Let $\alpha\in\\{\mu,$ $P^{s}{\lfloor_{E},}$ $C^{s}{\lfloor_{E}\\}}.$ Under
the hypotheses of Theorem 14, if there are two balls $B(x_{1},d_{1})$ and
$B(x_{2},d_{2}),$ both in $\mathcal{B},$ such that

$\theta_{\mu}^{s}(x_{1},d_{1})=\inf\left\\{\theta_{\mu}^{s}(x,d):B(x,d)\in\mathcal{B}\right\\}$
(34)

and

$\theta_{\mu}^{s}(x_{2},d_{2})=\sup\left\\{\theta_{\mu}^{s}(x,d):B(x,d)\in\mathcal{B}\right\\},$
(35)

the inclusions in Theorem 14 (iii) can be replaced with equalities.

Proof. The first inclusion in Theorem 14 (iii), together with (21), (22), (34)
and (35), implies that

$\mathit{Im}(\theta_{\mu}^{s},\mathcal{B})=\left[\frac{1}{P^{s}(E)},\frac{1}{C^{s}(E)}\right],$

which, in turn, gives that
$\mathit{Im}(\theta_{\alpha}^{s},\mathcal{B})=Spec(\alpha,\mathcal{O\cap}E).$

Corollary 15 motivates the introduction of the class of $\alpha$-exact self-
similar sets with special properties.

###### Definition 16

We say that the self-similar set $E$ is $\alpha$-_exact_ if there exists
$B\in\mathcal{C}_{\alpha}$ such that

$\frac{\mu(B)}{\left|B\right|^{s}}=\sup\left\\{\frac{\mu(B)}{\left|B\right|^{s}}:B\in\mathcal{C}_{\alpha}\right\\}$

if
$\alpha\in\left\\{C^{s}{\lfloor_{E}},\mathcal{H}^{s}{\lfloor_{E}},\mathcal{H}_{Sph}{\lfloor_{E}}\right\\},$
and

$\frac{\mu(B)}{\left|B\right|^{s}}=\inf\left\\{\frac{\mu(B)}{\left|B\right|^{s}}:B\in\mathcal{C}_{\alpha}\right\\},$

if $\alpha=P^{s}{\lfloor_{E}},$ where $\mathcal{C}_{\alpha}$ is what we call
“the relevant class of sets” for the measure $\alpha,$ which is defined as
$\bullet$ $\mathcal{C}_{\alpha}:=$ $\mathcal{B}$ if
$\alpha\in\left\\{P^{s}{\lfloor_{E}},C^{s}{\lfloor_{E}}\right\\},$
$\bullet$
$\mathcal{C}_{\mathcal{H}^{s}{\lfloor_{E}}}:=\\{B\subset\mathbb{R}^{n}:\ B\
\text{ is a convex set}\\}$ and
$\bullet$
$\mathcal{C}_{\mathcal{H}_{Sph}^{s}{\lfloor_{E}}}:=\\{B\subset\mathbb{R}^{n}:\
B\ \text{ is a closed ball}\\}.$

One nice property that $\alpha$-exact self-similar sets have is that they
possess optimal coverings or packings, that is, almost-coverings (i.e.
coverings for $\alpha$-almost all points in $E)$ or packings whose $s$-volume
gives the exact value of the corresponding $\alpha$-measure, whilst if
$\alpha$-exactness is not fulfilled, we can only hope to find coverings or
packings with $s$-volume arbitrarily close to the corresponding
$\alpha$-measure.

###### Example 17

Self-similar sets $E$ with the strong separation condition are an example of
$\alpha$-exact self-similar sets. See [38] for
$\alpha\in\left\\{P^{s}{\lfloor_{E}},\text{
}\mathcal{H}^{s}{\lfloor_{E}},\text{
}\mathcal{H}_{Sph}^{s}{\lfloor_{E}}\right\\}$ and [30] for $\alpha=C^{s}.$

###### Example 18

The Sierpinski gasket $S$ is an example of a set where the strong separation
condition does not hold, and that is a $P^{s}{\lfloor}_{S}$-exact (see [22])
and $C^{s}{\lfloor_{S}}$-exact (see [23]) set.

In [39], it is shown a class of self-similar sets $E$ with OSC in the line
whose members can be non-$\mathcal{H}^{s}{\lfloor_{E}}$-exact (and,
consequently, non-$\mathcal{H}_{Sph}^{s}{\lfloor_{E}}$-exact since these two
measures coincide in the line), and the authors find conditions under which
they are $\mathcal{H}^{s}{\lfloor_{E}}$-exact.

###### Example 19

Self-similar sets $E$ with OSC in the line, with similarity dimension $s,$ and
that admit an open interval as a feasible open set, are an example of
$P^{s}{\lfloor_{E}}$-exact self-similar sets [40].

###  3.3 Complexity of the local structure of self-similar sets

We now show how these results allow us to explore the complexity of the local
geometric structure of self-similar sets that satisfy the OSC condition.

First, we need to properly define the equivalence classes of restricted balls.
Notice that different Euclidean balls, even if they share the centre, can
produce the same restricted balls. This motivates the following definitions
that are valid for general subsets of $\mathbb{R}^{n}.$

###### Definition 20

Given a subset $A\subset\mathbb{R}^{n},$ the spherical diameter of $A$ is
defined by

$\left|A\right|_{Sph}=\inf\left\\{2d:A=A\cap B(x,d)\text{ for some }x\in
A\right\\}$

###### Definition 21

Given a subset $A\subset\mathbb{R}^{n},$ we say that the restricted ball
$B(x,d)\cap A$ is proper and write $B(x,d)\cap A\in\mathfrak{P}(A)$ if $x\in
A$ and $2d=\left|B(x,d)\cap A\right|_{Sph}.$

###### Definition 22

Given a measure $\alpha$ on $\mathbb{R}^{n}$ and an $\alpha$-measurable
$s$-set $A\subset\mathbb{R}^{n},$ we define the $\alpha$-spherical $s$-density
of $A$ by

$\theta_{Sph(\alpha)}^{s}(A)=\frac{{\alpha(A)}}{\left(\left|A\right|_{Sph}\right)^{s}}.$

###### Definition 23

Given a subset $A\subset\mathbb{R}^{n}$ and two restricted balls $B(x,d)\cap
A,$ $B(x^{\prime},d^{\prime})\cap A$ both in $\mathfrak{P}(A),$ we say that
they are similarity-equivalent and write $B(x,d)\cap A$
$\simeq_{\mathcal{S}_{n}}B(x^{\prime},d^{\prime})\cap A$ if there is an
$f\in\mathcal{S}_{n}$ such that

$B(x^{\prime},d^{\prime})\cap A=f(B(x,d)\cap A).$

###### Lemma 24

Let $A\subset\mathbb{R}^{n}$ and $B(x,d)\cap A\in\mathfrak{P}(A).$
(i) If $f\in\mathcal{S}_{n}$ has similarity constant $r_{f},$ then
$f(B(x,d))\cap f(A)\in\mathfrak{P}(f(A))$ and $\left|f(B(x,d))\cap
f(A)\right|_{p}=r_{f}d.$
(ii) Let $\alpha\in\mathcal{M}^{s}$ and let $A$ be an $\alpha$-measurable
$s$-set. If $B(x,d)\cap A\simeq_{\mathcal{S}_{n}}B(x^{\prime},d^{\prime})\cap
A,$ then

$\theta_{Sph(\alpha)}^{s}(B(x,d)\cap
A)=\theta_{Sph(\alpha)}^{s}(B(x^{\prime},d^{\prime})\cap A)$

Proof. Let $A\subset\mathbb{R}^{n},$ $B(x,d)\cap A\in$ $\mathfrak{P}(A).$ In
order to show (i), assume that $f(B(x,d))\cap f(A)$ is not proper. Then, there
is a ball $B(y,\rho)$ such that

$B(y,\rho)\cap f(A)=f(B(x,d))\cap f(A)$

with $y\in f(A)$ and $\rho<r_{f}d.$ Then $B(f^{-1}(y),r_{f}^{-1}\rho)\cap
A=B(x,d)\cap A$ with $f^{-1}(y)\in A$ and $r_{f}^{-1}\rho<d,$ in contradiction
with $\left|B(x,d)\cap A\right|_{Sph}=2d.$ Therefore, $f(B(x,d))\cap
f(A)\in\mathfrak{P(}f(A))$ and $\left|f(B(x,d))\cap
f(A)\right|_{Sph}=2r_{f}d.$
Part (ii) is now trivial since $\alpha\in\mathcal{M}^{s}$ and, hence,

$\alpha(B(x^{\prime},d^{\prime})\cap A)=\alpha(f(B(x,d)\cap
A))=r_{f}^{s}\alpha(B(x,d)\cap A)$

and, by (i),

$\left(\left|B(x^{\prime},d^{\prime})\cap
A\right|_{Sph}\right)^{s}=\left(\left|f(B(x,d)\cap
f(A))\right|_{Sph}\right)^{s}=r_{f}^{s}(2d)^{s}=r_{f}^{s}\left(\left|B(x,d)\cap
A\right|_{Sph}\right)^{s}.$

Now we can proceed to state our result for the complexity of the local
geometry of self-similar sets with OSC.

###### Corollary 25

Under the assumptions of Theorem 14, assume that $s$ is a non-integer real
number. Then, there is an uncountable number of equivalence classes in the
quotient space $Sph_{E}/\simeq_{\mathcal{S}_{n}}.$

Proof. By Lemma 24 (ii), we know that all restricted balls in an equivalence
class of $Sph_{E}/\simeq_{\mathcal{S}_{n}}$ share the same $\mu$-spherical
$s$-density, which allows us to naturally define a mapping $\theta_{\mu}^{s}:$
$Sph_{E}/\simeq_{\mathcal{S}_{n}}\rightarrow\mathit{Im}(\theta_{\mu}^{s},\mathcal{B}).$
This implies that the inverse
$\left(\theta_{\mu}^{s}\right)^{-1}:\mathit{Im}(\theta_{\mu}^{s},\mathcal{B})\mathbb{\rightarrow}Sph_{E}/\simeq_{\mathcal{S}_{n}}$
of such mapping is an injective correspondence. Using Marstrand’s Theorem,
parts (ii) and (iii) of Theorem 14 and that $\mu(\widehat{E})=1>0,$ it follows
that either $C^{s}(E)<P^{s}(E)$ or $s$ is an integer (notice that from the
definitions in Sec. 2.2 it is easy to see that $C^{s}(E)\leq P^{s}(E)).$ This,
together with Theorem 14 (iii), means that
$\mathit{Im}(\theta_{\mu}^{s},\mathcal{B})$ contains an interval with
uncountably many points and the proof is completed.

##  4 The spectrum of the Sierpinski gasket

In this section, we shall apply the results obtained in Theorem 14 to fully
characterise the asymptotic spectra of the Sierpinski gasket $S.$

Recall that the Sierpinski gasket or Sierpinski triangle is a special case of
a self-similar set generated by a system $\Psi=\\{f_{0,}f_{1,}f_{2}\\}$ of
three contracting similitudes of the plane, with contraction ratios
$r_{i}:=1/2,$ $i\in M,$ given by

$f_{0}(x,y)=\frac{1}{2}(x,y),\quad
f_{1}(x,y)=\frac{1}{2}(x,y)+(\frac{1}{2},0)\quad\text{and}\quad
f_{2}(x,y)=\frac{1}{2}(x,y)+\frac{1}{2}(\frac{1}{2},\frac{\sqrt{3}}{2}).$ (36)

We shall denote by $z_{i}$ the fixed point of each $f_{i},$ $i=0,1,2$ that is,
$z_{0}=(0,0),$ $z_{1}=(1,0)$ and $z_{2}=(\frac{1}{2},\frac{\sqrt{3}}{2}),$ and
by $T$ the equilateral triangle with vertexes $z_{i},$ $\,i\in M.$

It is well known that $S$ is a connected set that satisfies the OSC and has
similarity dimension $s=\frac{\log 3}{\log 2}.$

Thanks to previous work on the packing and Hausdorff centred measures of the
Sierpinski gasket (cf. [22] and [23]), we know that $S$ is both
$P^{s}{\lfloor_{S}}$ and $C^{s}{\lfloor_{S}}$-exact, and we have fairly
precise approximations of the values of $P^{s}(S)$ and $C^{s}(S).$

###  4.1 Theoretical results

###### Theorem 26

Let $S$ be the Sierpinski gasket, $\widehat{S}=\left\\{y\in
S:\left\\{\mathcal{T}^{k}(y):k\in\mathbb{N}\right\\}\text{ is dense in
}S\right\\},$ $\mathcal{B}$ be the collection of typical balls, $\mathcal{R}$
be a feasible open set for $S,$ and $\alpha\in\mathcal{M}^{s}\lfloor_{S}{.}$
Then, the following statements hold true.
(i)

$Spec(\alpha,y)=Spec(\alpha,\widehat{S})=Spec(\alpha,\mathcal{R}\cap
S)=\mathit{Im}(\theta_{\alpha}^{s},\mathcal{B})=\left[\frac{\alpha(S)}{P^{s}(S)},\frac{\alpha(S)}{C^{s}(S)}\right],\text{
}y\in\widehat{S}.$ (37)

(ii) $Spec(\alpha,S)$ is given by the union of two closed intervals of
positive length:

$Spec(\alpha,S)=\left[\underline{\theta}_{\alpha}^{s}(z_{0}),\overline{\theta}_{\alpha}^{s}(z_{0})\right]\cup\left[\frac{\alpha(S)}{P^{s}(S)},\frac{\alpha(S)}{C^{s}(S)}\right],$
(38)

where $z_{0}=(0,0).$ Furthermore,

$\underline{\theta}_{\alpha}^{s}(z_{0})=\min\left\\{\theta_{\alpha}^{s}(z_{0},d):\frac{1}{2}\leq
d\leq 1\right\\}$ (39)

and

$\overline{\theta}_{\alpha}^{s}(z_{0})=\max\left\\{\theta_{\alpha}^{s}(z_{0},d):\frac{1}{2}\leq
d\leq 1\right\\}.$ (40)

Proof. Our previous work guarantees that $S$ is a $P^{s}$-exact (see [22]) and
$C^{s}$-exact (see [23]) set. Then, the four equalities in (i) follow as a
consequence of Theorem 14 and Corollary 15.

In order to prove (38), let $\mathcal{R}_{i},$ $i\in\\{0,1,2\\}$ be the three
open rhombi composed of the topological interior of the union of the triangle
$T$ and its reflection across the edge of $T$ opposite the point $z_{i},$
$i\in M$ (see $\mathcal{R}_{2}$ in Fig. 1). Using that

$S=\left\\{z_{0},z_{1},z_{2}\right\\}\cup(S\cap\cup_{i=0}^{2}\mathcal{R}_{i}),$
(41)

we obtain

$\displaystyle Spec(\alpha,S)$
$\displaystyle=Spec(\alpha,S\cap\cup_{i=0}^{2}\mathcal{R}_{i})\cup\big{(}\cup_{i=0}^{2}Spec(\alpha,z_{i})\big{)}=$
$\displaystyle=\left[\frac{\alpha(S)}{P^{s}(S)},\frac{\alpha(S)}{C^{s}(S)}\right]\cup
Spec(\alpha,z_{0}),$

where the last equality follows from (37), (41) and the fact that, by
symmetry, $Spec(\alpha,z_{i})$ must be identical for
$i\in\left\\{0,1,2\right\\}.$

Figure 1: A feasible open set.
An open rhombus $\mathcal{R}_{2}$ that is a feasible open set for $S.$

Observe now that, if $d\leq 1/2,$ then $B(z_{0},d)\cap S=B(z_{0},d)\cap
f_{0}(S).$ Hence, using that $\alpha$ is an $s$-dimensional metric measure

$\displaystyle\theta_{\alpha}^{s}(z_{0},d)$
$\displaystyle=\frac{\alpha(B(z_{0},d)\cap
f_{0}(S))}{(2d)^{s}}=\frac{\alpha(f_{0}(B(z_{0},2d)\cap S))}{(2d)^{s}}$
$\displaystyle=\frac{\alpha(B(z_{0},2d)\cap
S))}{(4d)^{s}}=\theta_{\alpha}^{s}(z_{0},2d).$

If $2d\leq 1/2,$ we can repeat the argument $k$ times until $1/2\leq
2^{k}d\leq 1$ and
$\theta_{\alpha}^{s}(z_{0},d)=\theta_{\alpha}^{s}(z_{0},2^{k}d).$ This shows
that

$\min\left\\{\theta_{\alpha}^{s}(z_{0},d):0\leq d\leq
1\right\\}=\min\left\\{\theta_{\alpha}^{s}(z_{0},d):\frac{1}{2}\leq d\leq
1\right\\}=\underline{\theta}_{\alpha}^{s}(z_{0}),$

where the last equality can easily be checked and, analogously, (40) holds.

###### Remark 27

Notice that part (i) of Theorem 26 shows that there is a set of full
$\alpha$-measure whose points exhibit a strongly regular behaviour, whereas
part (ii) underlines the special local behaviour of the vertexes as the most
isolated points in $S.$ However, the set of exceptional points does not
consist only of the vertexes as there might be other exceptional points, all
of them belonging to the set $\cup_{i=0}^{2}\left(\mathcal{R}_{i}\cap
S\right)-\widehat{S}.$ The pointwise $\alpha$-density spectrum of such points
is contained in
$\left[\frac{\alpha(S)}{P^{s}(S)},\frac{\alpha(S)}{C^{s}(S)}\right].$ The
detection and characterisation of the behaviour of these points remains an
open issue.

###  4.2 Numerical results

Following the structure of the algorithms developed in [41, 42, 22, 23] for
the numerical estimation of the metric measures of self-similar sets, the
construction of the computational algorithm used in this work in order to
approximate the values of $\underline{\theta}_{\mu}^{s}(z_{0})$ and
$\overline{\theta}_{\mu}^{s}(z_{0})$ relies upon the discrete approximations
of both the Sierpinski gasket and its invariant measure $\mu.$ Recall that any
two measures in $\mathcal{M}^{s}\lfloor_{S}$ are mutually multiple of each
other (see (19)), so we can obtain $Spec(\alpha,S)$ from $Spec(\mu,S)$ if we
know $\alpha(S).$

The Sierpinski gasket, as the attractor of $\Psi=\\{f_{0},f_{1},f_{2}\\}$ (see
(36)), is the unique non-empty compact set that admits the self-similar
decomposition $S=F(S),$ where $F$ is the Hutchinson operator defined, for
$A\subset\mathbb{R}^{2},$ by

$F(A):=f_{0}(A)\cup f_{1}(A)\cup f_{2}(A).$

It is well-known that, for any non-empty compact subset
$A\subset\mathbb{R}^{2},$ $S$ can be built with an arbitrary level of detail
by increasing the iterations $k$ in $F^{k}(A),$ where $F^{k}=F\circ F...\circ
F$ is the $k$-th iterate of the contracting operator $F.$ This is because
$F^{k}(A)\overset{k\rightarrow\infty}{\rightarrow}S$ in the Hausdorff metric
(cf. [6]). Furthermore, if $A\subset S,$ then $F^{k}(A)\subset S$ for any
$k\in\mathbb{N^{+}}.$ In particular, if we take
$A_{1}:=\\{z_{0},z_{1},z_{2}\\}$ as the initial compact set, we obtain the set

$A_{k}:=F^{k-1}(A_{1})\subset S,\text{ }k\geq 2,$ (42)

which approximates $S$ at the iteration $k$ of our algorithm.

The relation between the Markov operator and the natural probability measure
$\mu_{\mathbf{p}_{s}}$ given in (17), with $s=\frac{\log 3}{\log 2}$ and
$p_{i}=r_{i}^{s}=3^{-k},$ and (19) leads to the following relation:

$\mathbf{M}_{\mathbf{p}_{s}}^{k}(\alpha)=\frac{1}{3^{k}}\sum_{i\in
M^{k}}\alpha\circ f_{i}^{-1}\overset{w}{\rightarrow}\mu,\text{
}\quad\alpha\in\mathcal{P}(\mathbb{R}^{2}).$ (43)

If we consider
$\mu_{1}:=\frac{1}{3}(\delta_{z_{0}}+\delta_{z_{1}}+\delta_{z_{2}})$ as an
initial measure $\alpha$ in (43), where $\delta_{x}$ is a unit mass at $x,$
then

$\mu_{k}:=\mathbf{M}_{\mathbf{p}_{s}}^{k-1}(\mu_{1})=\frac{1}{3^{k-1}}\sum_{i\in
M^{k-1}}\mu_{1}\circ f_{i}^{-1}=\frac{1}{3^{k}}\sum_{i\in
M^{k-1}}\left(\delta_{f_{i}(z_{0})}+\delta_{f_{i}(z_{1})}+\delta_{f_{i}(z_{2})}\right)$
(44)

is a probability measure supported on $A_{k}\subset S$ and
$\mu_{k}\overset{w}{\rightarrow}\mu.$
The discrete measure $\mu_{k}$ is the approximation of the invariant measure
$\mu$ that our algorithm takes at iteration $k.$

Lemmas 28 and 30 (Lemma 28 is proved in [23]), provide precise relationships
between the measures $\mu_{k}$ and $\mu.$

###### Lemma 28

(i) Let $\\{S_{i}:i\in I\subset M^{k}\\},$ $k\in\mathbb{N}^{+},$ be a
collection of $k$-cylinder sets of $S.$ Then,

$\mu\left(\bigcup\limits_{i\in I}S_{i}\right)\leq\mu_{k}\left(\bigcup_{i\in
I}S_{i}\right)$

(ii) Let $A\subset S,$ $k\in\mathbb{N}^{+},$ and let $I=\\{i\in
M^{k}:S_{i}\cap A\neq\varnothing\\}.$ Then,

$\mu_{k}(A)\leq\mu\left(\bigcup_{i\in I}S_{i}\right)$

###### Remark 29

The comparisons between the measures $\mu$ and $\mu_{k}$ on collections of
cylinders and sets given in the lemma above are passed to enlarged and reduced
balls in part (i) of the next lemma. Since our algorithms compute only
$\mu_{k}$-densities of balls with centres in $A_{k}$ (see (42)) and with some
point of $A_{k}$ in their boundaries, in part (ii) of this lemma we
approximate the $\mu$-measure of a ball centred at $x$ with the
$\mu_{k}$-measure of a ball with its same centre and with a point of $A_{k}$
at its boundary.
In order to obtain more accurate estimates of
$\underline{\theta}_{\mu}^{s}(z_{0})$ and $\overline{\theta}_{\mu}^{s}(z_{0})$
(as we also do in [22] and [23] for the estimation of $P^{s}(S)$ and
$C^{s}(S)),$ it is necessary to consider open balls when searching balls of
minimal $\mu_{k}$-density (see (46)), whereas in the search of balls with
maximal $\mu_{k}$-density, the approximating balls must be taken to be closed
balls (see (47)). In the definition of $\underline{\theta}_{\mu}^{s}(\cdot)$
and $\overline{\theta}_{\mu}^{s}(\cdot),$ the use of open or closed balls has
no relevance because the $\mu$-measure of the boundary of any ball is null.
However, in the case of densities of the discrete measures $\mu_{k},$ the
values obtained in one or the other case do actually matter, mainly if $k$ is
not large.

From now on, we shall use the notation
$\mathring{B}(x,d):=\\{y\in\mathbb{R}^{2}:|x-y|<d\\}$ and
$\mathring{\theta}_{\alpha}^{s}$ for the $s$-density of $\alpha$ defined using
open balls.

###### Lemma 30

Let $k>0,$ $x\in\mathbb{R}^{2},$ and $2^{-k}<d\leq\max_{i\in
M}\left\|z_{i}-x\right\|.$ Then,
(i)
$\mu_{k}(B(x,d-2^{-k}))\leq\mu(B(x,d))\leq\mu_{k}(\mathring{B}(x,d+2^{-k}))$
(ii) If $B(x,d)\cap A_{k}\neq\varnothing,$ then there are points $y_{k}$ and
$z_{k}$ in $A_{k}$ such that

$\mu_{k}\left(\mathring{B}(x,d_{y_{k}})\right)\leq\mu(B(x,d))\leq\mu_{k}(B(x,d_{z_{k}})),$

where $d_{y_{k}}:=\left|y_{k}-x\right|,$ $d_{z_{k}}:=\left|z_{k}-x\right|,$
and $\\{d_{y_{k}},d_{z_{k}}\\}\in[d-2^{-k},d+2^{-k}].$

Proof. (i) Let

$H_{k}:=\\{i\in M^{k}:B(x,d-2^{-k})\cap S_{i}\neq\varnothing\\}$

For any $i\in H_{k},$ $S_{i}\subset B(x,d)$ holds, so $\cup_{i\in
H_{k}}S_{i}\subset B(x,d).$ Using Lemma 28 (ii), we have

$\mu_{k}(B(x,d-2^{-k}))\leq\mu(\cup_{i\in H_{k}}S_{i})\leq\mu(B(x,d)).$

Let

$G_{k}:=\\{i\in M^{k}:S_{i}\subset\mathring{B}(x,d+2^{-k})\\}.$

Then, $\mathring{B}(x,d)\cap S\subset\cup_{i\in G_{k}}S_{i}$ and $\cup_{i\in
G_{k}}S_{i}\subset\mathring{B}(x,d+2^{-k}).$ Using Lemma 28 (i), we get

$\mu(B(x,d))=\mu(\mathring{B}(x,d)\cap S)\leq\mu(\cup_{i\in
G_{k}}S_{i})\leq\mu_{k}(\cup_{i\in
G_{k}}S_{i})\leq\mu_{k}(\mathring{B}(x,d+2^{-k}))$

(ii) Let $d^{\ast}=\max_{i\in M}\left\|z_{i}-x\right\|.$ If $S\subset B(x,d),$
then $d=d^{\ast}$ and
$\mu(B(x,d^{\ast}))=1=\mu_{k}(B(x,d^{\ast}))>\mu_{k}((\mathring{B}(x,d^{\ast})),$
so property (ii) holds for $d_{y_{k}}=d_{z_{k}}=d^{\ast}.$ Let us now assume
that $S\nsubseteq B(x,d).$ We prove first that

$F_{k}:=\\{i\in M^{k}:\partial B(x,d)\cap
S_{i}\neq\varnothing\\}\neq\varnothing.$ (45)

If $F_{k}=\varnothing,$ then

$\cup_{i\in M^{k}}S_{i}\subset\mathring{B}(x,d)\cup(B(x,d))^{c}.$

We know that $(\cup_{i\in M^{k}}S_{i})\cap\mathring{B}(x,d)\neq\varnothing$
because $B(x,d)\cap A_{k}\neq\varnothing$ and $F_{k}=\varnothing,$ and we also
know that $(\cup_{i\in M^{k}}S_{i})\cap(B(x,d))^{c}\neq\varnothing$ because
$S\nsubseteq B(x,d)$ and $F_{k}=\varnothing.$ This contradicts that
$\cup_{i\in M^{k}}S_{i}$ is a connected set, and (45) must hold.
Using (i), we have that

$\mu(B(x,d))\leq\mu_{k}(B(x,d+2^{-k}))=\mu_{k}(B(x,d_{z_{k}})),$

where $z_{k}$ satisfies $d_{z_{k}}=\left\|z_{k}-x\right\|$ with

$d_{z_{k}}=\max\\{\left\|y-x\right\|:\text{ }y\in A_{k}\cap B(x,d+2^{-k})\\}.$

The inequality $d_{z_{k}}\leq d+2^{-k}$ is obvious, and $d_{z_{k}}\geq
d-2^{-k}$ follows because $F_{k}\neq\varnothing$ and each $k$-cylinder
$S_{i},$ $i\in M^{k}$ contains some point in $A_{k}.$
Using the first inequality in (i), we have

$\mu(B(x,d))\geq\mu_{k}(B(x,d-2^{-k}))=\mu_{k}(\mathring{B}(x,d_{y_{k}})),$

where $y_{k}$ satisfies $d_{y_{k}}=\left\|y_{k}-x\right\|$ with

$d_{y_{k}}=\min\\{\left\|y-x\right\|:\text{ }y\in
A_{k}\cap\left(B(x,d-2^{-k})\right)^{c}\\}.$

The inequality $d_{y_{k}}\geq d-2^{-k}$ is obvious, and $d_{y_{k}}\leq
d+2^{-k}$ follows because $F_{k}\neq\varnothing.$

Theorem 26 allows us to characterise $Spec(\alpha,S)$ for
$\alpha\in\\{\mu,\mathcal{\ }P^{s}\lfloor_{S},$ $C^{s}\lfloor_{S}\\}$ through
only four numbers, namely,
$\underline{\theta}_{\mu}^{s}(z_{0}),\overline{\theta}_{\mu}^{s}(z_{0}),$
$P^{s}(S)$ and $C^{s}(S).$ Thanks to previous numerical work that uses the
measures $\mu_{k}$ and the sets $A_{k}$ (see (44) and (42)) as approximations
of $\mu$ and $S,$ respectively, we have estimates given by our algorithms
$P_{k}$ of $P^{s}(S)$ (see [22]) and $C_{k}$ of $C^{s}(S)$ (see [23]) and
precise error bounds for such estimates. We show in Theorem 31 below how to
obtain estimates $\underline{\xi}_{k}$ of
$\underline{\theta}_{\mu}^{s}(z_{0})$ and $\overline{\xi}_{k}$ of
$\overline{\theta}_{\mu}^{s}(z_{0}),$ that such estimates converge to the real
values, and we give accurate bounds for them, that is
$\underline{\theta}_{\mu}^{s}(z_{0})\in[\underline{\xi}_{k}^{\inf},\underline{\xi}_{k}^{\sup}]$
and
$\overline{\theta}_{\mu}^{s}(z_{0})\in[\overline{\xi}_{k}^{\inf},\overline{\xi}_{k}^{\sup}]$
(see the definition of $\underline{\xi}_{k},$ $\overline{\xi}_{k}$ and of the
intervals $[\underline{\xi}_{k}^{\inf},\underline{\xi}_{k}^{\sup}]$ and
$[\overline{\xi}_{k}^{\inf},\overline{\xi}_{k}^{\sup}]$ in Theorem 31). This
allows us to implement an algorithm along the lines of those developed for the
estimation of $C^{s}(S)$ and $P^{s}(S)$ (see [22, 23]).

###### Theorem 31

For $k>1,$ let

$\underline{\xi}_{k}:=\min\left\\{\mathring{\theta}_{\mu_{k}}^{s}(z_{0},d):\text{
}d=\left|x-z_{0}\right|,\text{ }x\in A_{k},\text{
}d\in[\frac{1}{2}-2^{-k},1]\right\\}$ (46)

and

$\overline{\xi}_{k}:=\max\left\\{\theta_{\mu_{k}}^{s}(z_{0},d):\text{
}d=\left|x-z_{0}\right|,\text{ }x\in A_{k},\text{
}d\in[\frac{1}{2}-2^{-k},1]\right\\}$ (47)

be the estimates of $\underline{\theta}_{\mu}^{s}(z_{0})$ and
$\overline{\theta}_{\mu}^{s}(z_{0}),$ respectively. Let $d_{k}$ be such that
$\mathring{\theta}_{\mu_{k}}^{s}(z_{0},d_{k})=\underline{\xi}_{k},$ and let
$D_{k}$ be such that $\theta_{\mu_{k}}^{s}(z_{0},D_{k})=\overline{\xi}_{k}.$
Then,

$\\{\underline{\theta}_{\mu}^{s}(z_{0}),\underline{\xi}_{k}\\}\in[\underline{\xi}_{k}^{\inf},\underline{\xi}_{k}^{\sup}],$
(48)

and

$\\{\overline{\theta}_{\mu}^{s}(z_{0}),\overline{\xi}_{k}\\}\in[\overline{\xi}_{k}^{\inf},\overline{\xi}_{k}^{\sup}],$
(49)

where

$\underline{\xi}_{k}^{\inf}=\underline{K}_{k}\underline{\xi}_{k},\quad\underline{K}_{k}=(1-2^{1-k})^{s},\quad\underline{\xi}_{k}^{\sup}=\frac{\mu_{k}(\mathring{B}(z_{0},d_{k}+2^{-k}))}{(2d_{k})^{s}},$
(50)
$\overline{\xi}_{k}^{\inf}=\frac{\mu_{k}(B(z_{0},D_{k}-2^{-k}))}{(2D_{k})^{s}},\quad\overline{K}_{k}=(1+2^{1-k})^{s},\quad\overline{\xi}_{k}^{\sup}=\overline{K}_{k}\overline{\xi}_{k}.$
(51)

Proof. That
$\underline{\xi}_{k}\in[\underline{\xi}_{k}^{\inf},\underline{\xi}_{k}^{\sup}]$
and
$\overline{\xi}_{k}\in[\overline{\xi}_{k}^{\inf},\overline{\xi}_{k}^{\sup}]$
is obvious from the definitions.
We prove first that
$\underline{\theta}_{\mu}^{s}(z_{0})\in[\underline{\xi}_{k}^{\inf},\underline{\xi}_{k}^{\sup}].$
Using Lemma 30 (i) and (39), we obtain

$\underline{\theta}_{\mu}^{s}(z_{0})\leq\frac{\mu(B(z_{0},d_{k}))}{(2d_{k})^{s}}\leq\frac{\mu_{k}(\mathring{B}(z_{0},d_{k}+2^{-k}))}{(2d_{k})^{s}}=\underline{\xi}_{k}^{\sup}.$

Let $d\in[\frac{1}{2},1]$ be such that
$\underline{\theta}_{\mu}^{s}(z_{0})=\frac{\mu(B(z_{0},d))}{(2d)^{s}}.$ Lemma
30 (ii) guarantees the existence of $y_{k}\in A_{k}$ such that
$\mu(B(z_{0},d))\geq\mu_{k}(\mathring{B}(z_{0},d_{y_{k}})),$ where
$d_{y_{k}}:=\left|y_{k}-z_{0}\right|\in[d-2^{-k},d+2^{-k}]\subset[\frac{1}{2}-2^{-k},1].$
This, together with (46) and the inequality $d\geq\frac{1}{2}$ gives

$\displaystyle\underline{\theta}_{\mu}^{s}(z_{0})$
$\displaystyle=\frac{\mu(B(z_{0},d))}{(2d)^{s}}\geq\frac{\mu_{k}(\mathring{B}(z_{0},d_{y_{k}}))}{(2d)^{s}}=\left(\frac{d_{y_{k}}}{d}\right)^{s}\frac{\mu_{k}(\mathring{B}(z_{0},d_{y_{k}}))}{(2d_{y_{k}})^{s}}$
$\displaystyle\geq\left(\frac{d_{y_{k}}}{d}\right)^{s}\underline{\xi}_{k}\geq\left(\frac{d-2^{-k}}{d}\right)^{s}\underline{\xi}_{k}\geq\underline{\xi}_{k}^{\inf}.$

The proof that
$\overline{\theta}_{\mu}^{s}(z_{0})\in[\overline{\xi}_{k}^{\inf},\overline{\xi}_{k}^{\sup}]$
is analogous. Using Lemma 30 (i) and (40), we obtain

$\overline{\theta}_{\mu}^{s}(z_{0})\geq\frac{\mu(B(z_{0},D_{k}))}{(2D_{k})^{s}}\geq\frac{\mu_{k}(B(z_{0},D_{k}-2^{-k}))}{(2D_{k})^{s}}=\overline{\xi}_{k}^{\inf}.$

Let $D\in[\frac{1}{2},1]$ be such that
$\overline{\theta}_{\mu}^{s}(z_{0})=\frac{\mu(B(z_{0},D))}{(2D)^{s}}.$ Lemma
30 (ii) guarantees the existence of $z_{k}\in A_{k}$ such that
$\mu(B(z_{0},D))\leq\mu_{k}(B(z_{0},d_{z_{k}})),$ where
$d_{z_{k}}:=\left|z_{k}-z_{0}\right|\in[D-2^{-k},D+2^{-k}]\subset[\frac{1}{2}-2^{-k},1].$
This, together with (47) and the inequality $D\geq\frac{1}{2}$ gives

$\displaystyle\overline{\theta}_{\mu}^{s}(z_{0})$
$\displaystyle=\frac{\mu(B(z_{0},D))}{(2D)^{s}}\leq\frac{\mu_{k}(B(z_{0},d_{z_{k}}))}{(2D)^{s}}$
$\displaystyle=\left(\frac{d_{z_{k}}}{D}\right)^{s}\frac{\mu_{k}(B(z_{0},d_{z_{k}}))}{(2d_{z_{k}})^{s}}\leq\left(\frac{d_{z_{k}}}{D}\right)^{s}\overline{\xi}_{k}$
$\displaystyle\leq\left(\frac{D+2^{-k}}{D}\right)^{s}\overline{\xi}_{k}\leq\overline{\xi}_{k}^{\sup}.$

We present in Table 1 the estimates $\underline{\xi}_{k},$ and
$\overline{\xi}_{k}$ of $\underline{\theta}_{\mu}^{s}(z_{0})$ and
$\overline{\theta}_{\mu}^{s}(z_{0})$ (see (48) and (49) for definitions),
respectively, and the corresponding lower and upper bounds in the 100%
confidence intervals
$[\underline{\xi}_{k}^{\inf},\underline{\xi}_{k}^{\sup}],$
$[\overline{\xi}_{k}^{\inf},\overline{\xi}_{k}^{\sup}]$ (see (51),(49))
obtained by our algorithm for $k=14$ (see the definition these values in (46),
(47), (50) and (51)). We also provide the radii, $d_{k}$ and $D_{k},$ of the
$\mu_{k}$-optimal balls.

See in Fig. 2(a) the graph of the function $\theta_{\mu_{14}}^{s}(z_{0},d)$ as
a function of $d\in[\varepsilon,1],$ and in Fig. 2(b) the points
$(g(d),\theta_{\mu_{14}}^{s}(z_{0},d)),$ where
$g(d):=\varepsilon+\frac{\varepsilon-1}{\log(\varepsilon)}(\log(d)-\log(\varepsilon))$
and $\varepsilon=0.05.$ This is a suitable logarithmic scale, [10], which
allows us to see the periodicity of this function at such a scale.

(a) Values of $\theta_{\mu_{14}}^{s}(z_{0},d)$ for $d\in[\varepsilon,1]$ and
$\varepsilon=0.05.$

(b) Values of $(g(d),\theta_{\mu_{14}}^{s}(z_{0},d)),$ where
$g(d):=\varepsilon+\frac{\varepsilon-1}{\log(\varepsilon)}(\log(d)-\log(\varepsilon))$
and $\varepsilon=0.05.$

Figure 2:  Densities at $z_{0}$

We present in Table 2 the estimates $P_{k}$ of $P^{s}(S)$ and $C_{k}$ of
$C^{s}(S)$ obtained by our algorithms for $k=14.$ The lower and upper bounds
of $P^{s}(S)$ are denoted by $P_{k}^{\inf}$ and $P_{k}^{\sup},$ respectively,
and the bounds of $C^{s}(S)$ are denoted by $C_{k}^{\inf}$ and $C_{k}^{\sup},$
respectively. These results were computed in [22] and [23]), respectively.
Recall that $\left(P^{s}(S)\right)^{-1}$ and $\left(C^{s}(S)\right)^{-1}$ are
the $\mu$-densities of the balls of minimum and maximum $\mu$-density in the
set of typical balls. The estimates $P_{k}$ and $C_{k}$ are obtained by
replacing $S$ with $A_{k}$ and $\mu$ with $\mu_{k}.$ Again, we have used open
balls in the estimation of the density of the ball of minimum
$\mu_{k}$-density, and closed balls for the density of the ball of maximum
$\mu_{k}$-density. The centre and radius of the open ball of minimum
$\mu_{k}$-density are denoted by $x_{k}^{\ast}$ and $d_{k},$ respectively, and
the centre and radius corresponding to the closed ball of maximum
$\mu_{k}$-density are denoted by $y_{k}^{\ast}$ and $D_{k}.$ The table also
contains the corresponding optimal $\mu_{k}$-densities and their bounds. The
upper bound $P_{k}^{\sup}:=K_{k}^{P}P_{k}$ of $P^{s}(S)$ is slightly improved
here with respect to the one given in [22]. Here
$K_{k}^{P}:=(1-\frac{2^{5-k}}{\sqrt{3}})^{-s}$ instead of the value
$K_{k}^{P}=(1-\frac{2^{6-k}}{\sqrt{3}})^{-s}$ used in [22]. This gives the
value $P_{14}^{\sup}=1.671292$ given in Table 2 instead of the value
$P_{14}^{\sup}=1.668305$ given in Table 1 in [22].
The results of the following corollary are based on the estimates of Tables 1
and 2.

###### Corollary 32

Let $S$ be the Sierpinski gasket.
(i) For any $\alpha\in\mathcal{M}^{s}\lfloor_{S},$ $Spec(\alpha,S)$ is the
union of two closed disjointed intervals.
(ii)

$\displaystyle Spec(\mu,S)$
$\displaystyle\sim\left[0.2997,0.3567\right]\cup\left[0.5994,0.9951\right]$
$\displaystyle\left[0.2998,0.3566\right]\cup\left[0.5999,0.9944\right]$
$\displaystyle\subset
Spec(\mu,S)\subset\left[0.2996,0.3568\right]\cup\left[0.5983,0.9970\right]$

(iii)

$\displaystyle Spec(P^{s}\lfloor_{S},S)$
$\displaystyle\sim\left[0.5,0.5951\right]\cup\left[1,1.6602\right]$
$\displaystyle[0.5010,0.5945]\cup[1,1.6578]$ $\displaystyle\subset
Spec(P^{s}\lfloor_{S},S)\subset[0.4995,0.5963]\cup[1,1.6662]$

(iv)

$\displaystyle Spec(C^{s}\lfloor_{S},S)$
$\displaystyle\sim\left[0.3012,0.3584\right]\cup\left[0.6023,1\right]$
$\displaystyle[0.3015,0.3577]\cup[0.6032,1]$ $\displaystyle\subset
Spec(C^{s}\lfloor_{S},S)\subset[0.3005,0.3588]\cup[0.6002,1]$

Proof. We know (see (38) in Theorem 14) that

$Spec(\mu,S)=\left[\underline{\theta}_{\mu}^{s}(z_{0}),\overline{\theta}_{\mu}^{s}(z_{0})\right]\cup\left[\frac{1}{P^{s}(S)},\frac{1}{C^{s}(S)}\right],$
(52)

and that (see (19))

$Spec(\alpha,S)=\alpha(S)Spec(\mu,S),\text{ \
}\alpha\in\mathcal{M}^{s}\lfloor_{S}.$ (53)

The two intervals in $Spec(\alpha,S),$ $\alpha\in\mathcal{M}^{s}\lfloor_{S}$
are disjointed if $\overline{\theta}_{\mu}^{s}(z_{0})<\frac{1}{P^{s}(S)}.$
Such a condition holds (see Theorem 31, and Tables 1 and 2) because

$\overline{\theta}_{\mu}^{s}(z_{0})\leq\overline{\xi}_{14}^{\sup}<0.3568$

and

$\frac{1}{P^{s}(S)}\geq\frac{1}{P_{14}^{\sup}}>0.5983.$

Using (52), Theorem 31 and (53), we have that

$\displaystyle Spec(\mu,S)$
$\displaystyle\sim[\underline{\xi}_{14},\overline{\xi}_{14}]\cup\left[\frac{1}{P_{14}},\frac{1}{C_{14}}\right],$
$\displaystyle\left[\underline{\xi}_{14}^{\sup},\overline{\xi}_{14}^{\inf}\right]\cup\left[\frac{1}{P_{14}^{\inf}},\frac{1}{C_{14}^{\sup}}\right]$
$\displaystyle\subset
Spec(\mu,S)\subset\left[\underline{\xi}_{14}^{\inf},\overline{\xi}_{14}^{\sup}\right]\cup\left[\frac{1}{P_{14}^{\sup}},\frac{1}{C_{14}^{\inf}}\right],$
$\displaystyle Spec(P^{s}\lfloor_{S},S)$
$\displaystyle\sim[P_{14}\underline{\xi}_{\mu_{14}},P_{14}\overline{\xi}_{\mu_{14}}]\cup\left[1,\frac{P_{14}}{C_{14}}\right],$
$\displaystyle\left[P_{14}^{\sup}\underline{\xi}_{14}^{\sup},P_{14}^{\inf}\overline{\xi}_{14}^{\inf}\right]\cup\left[1,\frac{P_{14}^{\inf}}{C_{14}^{\sup}}\right]$
$\displaystyle\subset
Spec(P^{s}\lfloor_{S},S)\subset\left[P_{14}^{\inf}\underline{\xi}_{14}^{\inf},P_{14}^{\sup}\overline{\xi}_{14}^{\sup}\right]\cup\left[1,\frac{P_{14}^{\sup}}{C_{14}^{\inf}}\right],$

(and, analogously, for $Spec(C^{s}\lfloor_{S},S)),$ and the proof is completed
using the corresponding estimates of Tables 1 and 2.

$\underline{\xi}_{14}$
$[\underline{\xi}_{14}^{\inf},\underline{\xi}_{14}^{\sup}]$ $d_{14}$ 0.299714
[0.299656,0.299763] 0.642272 $\overline{\xi}_{14}$
$[\overline{\xi}_{14}^{\inf},\overline{\xi}_{14}^{\sup}]$ $D_{14}$ 0.356687
[0.356645,0.356756] 0.913663

Table 1:  Extreme densities at $z_{0}$
Estimates of $\underline{\theta}_{\mu}^{s}(z_{0})$ and
$\overline{\theta}_{\mu}^{s}(z_{0}),$ bounds and radii, $d_{k}$ and $D_{k},$
of the $\mu_{k}$-optimal balls for $k=14.$

$x_{14}^{\ast}$ $d_{14}$ $P_{14}$ $[P_{14}^{\inf},P_{14}^{\sup}]$
$\left(P_{14}\right)^{-1}=\mathring{\theta}_{\mu_{14}}^{s}(x_{14}^{\ast},d_{14})$
$\
\left[\left(P_{14}^{\sup}\right)^{-1},\left(P_{14}^{\inf}\right)^{-1}\right]$
(0.5,0) 0.160543 1.668305 [1.667178, 1.671292] 0.599411 [0.598339, 0.599816]
$y_{14}^{\ast}$ $D_{14}$ $C_{14}$ $[C_{14}^{\inf},C_{14}^{\sup}]$
$\left(C_{14}\right)^{-1}=\theta_{\mu_{14}}^{s}(y_{14}^{\ast},D_{14})$
$\left[\left(C_{14}^{\sup}\right)^{-1},\left(C_{14}^{\inf}\right)^{-1}\right]$
$\left(\frac{5}{16},\frac{\sqrt{3}}{16}\right)$ 0.145957 1.004903
[1.003109,1.005611] 0.995121 [0.994420, 0.996901]

Table 2: Packing and Centred measure estimates of $S$
Centres and radii of the balls $\mathring{B}(x_{14}^{\ast},d_{14})$ and
$B(y_{14}^{\ast},D_{14})$ of minimum and maximum $\mu_{14}$-densities,
estimates $P_{14}$ and $C_{14}$ of $P^{s}(S)$ and $C^{s}(S),$ and bounds. The
last two columns in the table are the $\mu_{14}$-densities of the optimal
balls (inverses of $P^{s}(S)$ and $C^{s}(S))$ and their bounds.

###### Acknowledgement 33

This work was supported by the Universidad Conmplutense de Madrid and the
Banco de Santander (PR108/20-14).

## References

  * [1] J. M. Marstrand, The $(\phi,s)$ regular subsets of $n$ space, Trans. Am. Math. Soc. 113 (1964) 369-392.
  * [2] S. P. Lalley, The packing and covering functions of some self-similar fractals, Indiana Univ. Math. J. 37(3) (1988) 699-710.
  * [3] A. Schief, Separation Properties for Self-Similar Sets, Proc. Amer. Math. Soc. 122(1) (1994) 111-115.
  * [4] M. Morán, Dynamical boundary of a self-similar set, Fundam. Math. 160 (1999) 1-14.
  * [5] P. Mattila, Geometry of sets and measures in Euclidean Spaces (Cambridge University Press, 1995).
  * [6] J. E. Hutchinson, Fractals and self-similarity, Ind. J. Math. 30 (1984) 713-747.
  * [7] T. Bedford, A. M. Fisher and M. Urbanski, The scenery flow for hyperbolic Julia sets, Proc. London Math. Soc. 3(85) (2002) 467-492.
  * [8] C. Bandt, The tangent distribution for self-similar measures, Lecture at the 5th Conference on Real Analysis and Measure Theory, Capri (1992).
  * [9] S. Graf, On Bandt’s tangential distribution for self-similar measures, Monatsh. Math. 120 (1995) 223-246.
  * [10] C. Bandt, Local Geometry of Fractals Given by Tangent Measure Distributions, Monatsh. Math. (2001) 260-280.
  * [11] M. Arbeiter and N. Patzsche, Random self-similar multifractals, Math. Nachr. 181 (1996) 5-42.
  * [12] A. Pyörälä, The scenery flow of self-similar measures with weak separation condition. Arxiv: 2103.14018v2 [math.DS] (2021).
  * [13] M. Gavish, Measures with uniform scaling scenery, Ergod. Theory Dyn. Syst. 31(1) (2011) 33-48.
  * [14] D. Preiss, Geometry of measures in $R^{n}.$ Distribution, rectifiability and densities, Ann. Math. 125 (1987) 537-643.
  * [15] P. Mörtens and D. Preiss, Tangent measure distributions of fractal measures, Math. Ann. 312 (1998) 53-93.
  * [16] R. Mauldin and S. Williams, Hausdorff Dimension in Graph Directed Constructions, Trans. Am. Math. Soc. 309(2) (1988) 811-829.
  * [17] P. Moran, Additive functions of intervals and Hausdorff measure, Math. Proc. Camb. Philos. Soc. 42(1) (1946) 15-23.
  * [18] D. Harte, Multifractals. Theory and applications. (Chapman & Hall/CRC, Boca Raton, FL., 2001).
  * [19] H. Cajar, Billingsley Dimension in Probability Spaces (Springer, 1980).
  * [20] C. M. Colebrook, The Hausdorff dimension of certain sets of non-normal numbers, Michigan Math. J. 17 (1970) 103-116.
  * [21] J. Li and M. Wu, The sets of divergence points of self-similar measures are residual, J. Math. Anal. Appl. 404(2) (2013) 429–437.
  * [22] M. Llorente, M. E. Mera and M. Morán, On the packing measure of the Sierpinski gasket. Nonlinearity 31 (2018) 2571-2589.
  * [23] M. Llorente, M. E. Mera and M. Morán, On the centered Hausdorff measure of the Sierpinski gasket, preprint 2021. http://ssrn.com/abstract=3970808.
  * [24] M. F. Barnsley, Fractals Everywhere (Courier Corporation, 2012).
  * [25] M. Llorente and M. Morán, An algorithm for computing the centered Hausdorff measures of self-similar sets, Chaos Solitons & Fractals, 45(3) (2012) 246-255.
  * [26] M. Morán and J. M. Rey, Geometry of self-similar measures, Ann. Acad. Sci. Fenn. Math. 22(2) (1997) 365–386.
  * [27] M. Morán and J.M. Rey Singularity of Self-Similar Measures with respect to Hausdorff Measures, Trans. Amer. Math. Soc. 350(6) (1998) 2297–2310.
  * [28] X. Saint Raymond and C. Tricot, Packing regularity of sets in $n$-space. Math. Proc. Cambridge Philos. Soc. 103(1) (1988) 133-145.
  * [29] C. Tricot, Rectifiable and Fractal Sets, in Fractal Geometry and Analysis, eds. J. Bélair and S. Dubuc (Proc. Montreal. Nato Adv. Sci. Inst.v 346, Kluber, Dordrecht, 1991) 364-403.
  * [30] M. Llorente and M. Morán, Advantages of the centered Hausdorff measure from the computability point of view, Math. Scand. 107(1) (2010) 103-122.
  * [31] D. Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups, Acta Math. 153(1) (1984) 259-277.
  * [32] C. Tricot, Sur la classification des ensembles boréliens de mesure de Lebesgue nulle,  These de doctorat, Geneve (1979).
  * [33] D. J. Feng, S. Hua and Z.Y. Wen, Some relations between packing premeasure and packing measure, Bull. London Math. Soc. 31(6) (1999) 665-670.
  * [34] F. Hausdorff, Dimension und äusseres, Math. Ann. 79 (1919) 157-179.
  * [35] P. Mattila, On the structure of self-similar fractals, Ann. Acad. Sci. Fenn. Math. Ser. A 7(2) (1982) 189-195.
  * [36] M. Morán, Computability of the Hausdorff and packing measures on self-similar sets and the self-similar tiling principle, Nonlinearity 18(2) (2005) 559-570.
  * [37] P. Walters, An Introduction to Ergodic Theory (Springer-Verlag, New York, 1982).
  * [38] M. Llorente and M. Morán, Self-similar sets with optimal coverings and packings, J. Math. Anal. Appl. 334(2) (2007) 1088-1095.
  * [39] E. Ayer and R. S. Strichartz, Exact Hausdorff measure and intervals of maximum density for Cantor sets in the line, Trans. Am. Math. Soc. 351(9) (1999) 3725-3741.
  * [40] D. J. Feng, Exact packing measure of linear Cantor set, Math. Natch. 248(1) (2003) 102-109.
  * [41] M. Llorente and M. Morán, Computability of the packing measure of totally disconnected self-similar sets, Ergod. Theory Dyn. Syst. 36(5) (2016) 1534-1556.
  * [42] M. Llorente, M. E. Mera and M. Morán, Rate of convergence: the packing and centered Hausdorff measures of totally disconnected self-similar sets, Chaos, Solitons & Fractals, 98 (2017) 220-232.