# The scenery flow of self-similar measures with weak separation condition

Aleksi Pyörälä  Research Unit of Mathematical Sciences, P.O.Box 8000,
FI-90014, University of Oulu, Finland<EMAIL_ADDRESS>
###### Abstract.

We show that self-similar measures on $\mathbb{R}^{d}$ satisfying the weak
separation condition are uniformly scaling. Our approach combines elementary
ergodic theory with geometric analysis of the structure given by the weak
separation condition.

###### 2010 Mathematics Subject Classification:

Primary 37A10; Secondary 28A80, 28D05

The research of this project has been supported by the Academy of Finland. I
am thankful to my supervisors Ville Suomala and Meng Wu for their helpful
comments and suggestions.

##  1\. Introduction

Taking tangents and obtaining information on an object by studying its small-
scale structure is a classical idea in analysis. Much like tangent spaces can
be used to study the local structure of differentiable manifolds, tangent
measures can be used to study the local structure of a Radon measure $\mu$.
Tangent measures are defined as the weak-∗ accumulation points of the
_scenery_ of $\mu$ at $x$, that is, the flow $(\mu_{x,t})_{t\geq 0}$, where

$\mu_{x,t}(A)=\frac{\mu(e^{-t}A+x)}{\mu(B(x,e^{-t}))}$

for measurable $A\subseteq B(0,1)$, and $B(x,e^{-t})$ denotes the closed ball
centered at $x$ and of radius $e^{-t}$.

The collection of tangent measures around a point can be very large, making it
difficult to infer from this collection any information on the original
measure. For example, O’Neil [27] has shown that there exist Radon measures on
$\mathbb{R}^{d}$ which possess every non-zero Radon measure as a tangent
measure, in almost every point. This raises the question whether some tangent
measures are more relevant than others in terms of studying the structure of
the original measure.

Recently, it has been observed that instead of individual tangent measures, it
is often more useful to study the statistical behaviour of the scenery
$(\mu_{x,t})_{t\geq 0}$. To establish these statistics, one views the scenery
as the orbit of $\mu$ under the continuous magnification operation at $x$.
This idea gives rise to the notion of _tangent distributions_ , defined as the
accumulation points of the _scenery flow_

$\left(\frac{1}{T}\int_{0}^{T}\delta[\mu_{x,t}]\,dt\right)_{T>0}.$

Here and throughout, $\delta[y]$ denotes the Dirac measure at the point $y$.
Tangent distributions at $x$ are supported on the collection of all tangent
measures at $x$, and are closely connected to Furstenberg’s _CP-processes_
[11] and their coordinate-independendent generalizations, Hochman’s _fractal
distributions_ [15]. The theory of tangent distributions was greatly developed
by Hochman and Shmerkin in [18] and further by Hochman in [15].

It is often possible to study geometric properties of a measure by studying
the structure of its tangent distributions. In particular, measures for which
the scenery flow converges, as $T\rightarrow\infty$, to a common distribution
in almost every point in their support, are geometrically much more regular
than arbitrary measures. For example, they are always exact dimensional [15].
Measures of this kind are called _uniformly scaling_ , a concept first
introduced by Gavish [13].

During recent years, certain dynamical properties of tangent distributions
have proved to be a powerful tool in establishing fine-structure properties of
fractal measures. For example, when the scenery flow converges to an _ergodic
fractal distribution_ (see Definition 5.1) in almost every point, as is often
the case for uniformly scaling measures due to a remarkable result of Hochman
[15], the measure satisfies a strong version of the classical Marstrand’s
projection theorem: If $\mu$ is a uniformly scaling measure on
$\mathbb{R}^{d}$ generating an ergodic fractal distribution, then for every
$k$ and $\varepsilon>0$ there exists an open dense set
$\mathcal{U}_{\varepsilon}$ in the space of linear projections from
$\mathbb{R}^{d}$ to $\mathbb{R}^{k}$ such that for all
$\pi\in\mathcal{U}_{\varepsilon}$,

(1.1) $\dim\pi\mu>\min\\{k,\dim\mu\\}-\varepsilon.$

This is due to Hochman and Shmerkin [18, Theorem 8.2]. Here and in the sequel
$\dim$ refers to Hausdorff dimension.

In [8], Ferguson, Fraser and Sahlsten applied this projection theorem to prove
Falconer’s distance set conjecture in some special cases: If $\mu$ is a
uniformly scaling measure which generates an ergodic fractal distribution and
$\mathcal{H}^{1}({\rm spt}\,\mu)>0$, writing

$D({\rm spt}\,\mu)=\\{|x-y|:\ x,y\in{\rm spt}\,\mu\\}$

for the distance set of the support of $\mu$, we have

(1.2) $\dim D({\rm spt}\,\mu)\geq\min\\{1,\dim\mu\\}.$

This is [8, Theorem 1.7].

Another application of the uniform scaling property is found in the prevalence
of normal numbers in the support of the measure. Recall that a number
$x\in[0,1]$ is $a$-_normal_ if the sequence $\\{a^{k}x\mod
1\\}_{k\in\mathbb{N}}$ equidistributes for Lebesgue measure, and a measure
$\mu$ is called _pointwise_ $a$-_normal_ if $\mu$-almost every $x$ is
$a$-normal. In a breakthrough paper [19, Theorem 1.1], Hochman and Shmerkin
established a condition for a uniformly scaling measure to be pointwise normal
in terms of dynamical properties of its tangent distribution: If $\mu$ is
uniformly scaling, generates an ergodic fractal distribution $P$ and the pure-
point spectrum of $P$ does not contain a non-zero integer multiple of
$\frac{1}{\log a}$, where $a>1$ is a Pisot number, then $\mu$ is pointwise
$a$-normal.

Motivated in part by this progression, much attention has lately been given to
the problem of understanding the scenery flow for various classes of fractal
measures, and whether they are uniformly scaling and generate ergodic fractal
distributions. Examples of uniformly scaling measures include ergodic measures
in the one-sided shift space [10], the occupation measure on Brownian motion
in dimension at least $3$ [13], self-affine measures on Bedford-McMullen-type
carpets [8] and self-similar measures with the _open set condition_ [13].
Recall that a measure $\mu$ on $\mathbb{R}^{d}$ is _self-similar_ if there
exists a finite family of strictly contracting, angle-preserving affine maps
$\\{\varphi_{i}\\}_{i=1}^{m}$ and a probability vector $(p_{i})_{i=1}^{m}$
with positive entries such that

(1.3) $\mu=\sum_{i=1}^{m}p_{i}\cdot\mu\circ\varphi_{i}^{-1}.$

For measures on $\mathbb{R}$ that satisfy (1.3) for some equicontractive
family $\\{\varphi_{i}\\}_{i=1}^{m}$, De-Jun Feng [6] has shown that the open
set condition can be relaxed to the strictly weaker _finite type condition_.
The separation conditions play crucial roles in [13] and [6], and this gives
rise to the natural question: Are all self-similar measures uniformly scaling?

The main result of this paper, Theorem 1.1, provides an affirmative answer to
this question under a separation condition called the _weak separation
condition_. We say that the family $\\{\varphi_{i}\\}_{i=1}^{m}$ (or the
associated self-similar measure) in (1.3) satisfies the weak separation
condition if the identity is not an accumulation point of the topological
group generated by $F^{-1}F$, where $F$ is the semigroup with identity
generated by $\\{\varphi_{i}\\}_{i=1}^{m}$. (In Section 2, Definition 2.1, we
adapt a different but equivalent characterization that is more explicitly
reflected in the geometry of the self-similar measure.) We remark that this
condition is strictly weaker than the conditions present in [13] and [6]; see
[30, 25, 14] for discussion on the relationship between the three conditions.

######  Theorem 1.1.

If $\mu$ is a self-similar measure satisfying the weak separation condition,
then $\mu$ is uniformly scaling and generates an ergodic fractal distribution.

As an immediate application to the geometry of the measure, this allows us to
deduce (1.1) and (1.2) for self-similar measures satisfying the weak
separation condition. Combining Theorem 1.1 with the methods of Hochman-
Shmerkin [19], we also obtain an application to their pointwise normality; see
Corollary 5.4 in Section 5.

The idea in the proof of Theorem 1.1 is to approximate the flow
$(\mu_{x,t})_{t\geq 0}$ with another flow that behaves statistically in a
similar manner as the orbit of a point in the underlying shift space. We
construct this flow by first producing a fixed reference measure $\nu$ on
$\mathbb{R}^{d}$ via the weak separation condition (in (3.4)). Then, drawing
from certain recurrence properties in the underlying symbolic space, we may
ensure that $\mu_{x,t}$ is “close to” $\nu$ for a large proportion of scales
$t$, in a quite strong sense which also ensures that the magnifications of
$\mu_{x,t}$ are close to those of $\nu$. This is made precise in Propositions
3.3 and 3.4. We may then use tools from ergodic theory to deduce that the
approximating flow equidistributes for a limiting measure on the space of
measures independent of the point $x$. Finally, a compactness argument allows
us to deduce the convergence of the scenery flow of $\mu$. It then follows
immediately from results of Hochman that the generated distribution is an
ergodic fractal distribution.

In the presence of the open set condition, approximating $(\mu_{x,t})_{t\geq
0}$ with another flow is not necessary to gain access to tools from ergodic
theory. Indeed, it follows from the open set condition and the relation (1.3)
that $\mu$ is homogeneous (in the sense of Gavish [13]), i.e. any measure in
the (weak-∗) closure of the family $\\{\mu_{x,t}:\ x\in\mathbb{R}^{d},t\geq
0\\}$ equals a restriction of a scaled and translated copy of $\mu$ itself.
Drawing from the CP-process machinery of Furstenberg, Gavish [13] showed that
this closure always contains many uniformly scaling measures, in fact when
$\mu$ is _any_ Radon measure on $\mathbb{R}^{d}$. Combining this with the
homogeneity of $\mu$, Gavish concluded that self-similar measures with the
open set condition are uniformly scaling.

Another argument was presented by Hochman in [15] assuming the _strong
separation condition_ under which the measures in the sum of (1.3) have
disjoint supports. In particular, if $K$ denotes the support of $\mu$, then
the map $T:K\rightarrow K$, $x\mapsto\varphi_{i}^{-1}(x)$ where $1\leq i\leq
m$ is the unique number such that $x\in\varphi_{i}(K)$, is well-defined and
preserves $\mu$. As described in [15, Section 4], the scenery flow of $\mu$
now arises as a factor of a suspension flow of the dynamical system
$(K,\mu,T)$ (or of a certain skew-product dynamical system, when the linear
parts of $\varphi_{i}$ are not positive scalars). Applying tools from ergodic
theory on this suspension flow allowed Hochman to establish the uniform
scaling property for self-similar measures satisfying the strong separation
condition.

Since a self-similar measure satisfying the weak separation condition is not
necessarily homogeneous, nor is it in general preserved under any obvious map
on $\mathbb{R}^{d}$, the existing arguments do not directly apply in our
setting. On the other hand, our argument relies heavily on the weak separation
condition in providing the reference measure $\nu$, which leaves open the
question whether self-similar measures without this separation condition are
uniformly scaling.

The paper is organized as follows. In Section 2, we set up our notation and
recall the basic theory of dynamical systems and iterated function systems. In
Section 3 we prove Theorem 1.1 in the simplified case when the defining IFS
consists of only homotheties, i.e. functions composed of scaling and
translation operations. Afterwards, we discuss the minor changes required to
prove the general case. Section 4 is devoted to the proofs of Propositions 3.3
and 3.4, the main tools required in the proof of Theorem 1.1. In Section 5 we
briefly discuss the second assertion of Theorem 1.1, a generalization of
Theorem 1.1 for ergodic Markov measures, and its application to pointwise
normality of self-similar measures.

##  2\. Preliminaries

In this paper, a measure always refers to a Radon measure on a metrizable
topological space. For a compact subset $A$ of a metric space $X$, write
$\mathcal{P}(A)$ for the space of probability measures on $X$ for which ${\rm
spt}\,\mu\subseteq A$. This space is endowed with the compact weak-∗ topology
which we metrize with the Prokhorov metric $d$. We use the notation
$\|\cdot\|$ to denote the total variation norm. It is well-known that
convergence in the metric induced by the total variation norm implies
convergence in the Prokhorov metric. Finally, the Euclidean norm is denoted by
$|\cdot|$ and the supremum norm by $\|\cdot\|_{\infty}$.

For a measure $\nu$ and all non-null measurable sets $A$, write
$\nu|_{A}:B\mapsto\nu(B\cap A)$ for the restriction of $\nu$ to $A$ and
$\nu_{A}=\nu(A)^{-1}\nu|_{A}$ for the normalized restriction. For a
$\nu$-measurable function $f$, write $f\nu=\nu\circ f^{-1}$ for the image (or
push-forward) measure, and if $f$ is integrable, write
$fd\nu:A\mapsto\int\chi[A](x)f(x)\,d\nu(x)$ for the weighted measure, where
$\chi[A]$ stands for the indicator function of $A$.

###  2.1. The scenery flow

Following the notation of [15], for $t\geq 0$ and $x\in\mathbb{R}^{d}$, we
write $S_{t}:\ y\mapsto e^{t}y$ for the exponential scaling map and $T_{x}:\
y\mapsto y-x$ for the translation which takes $x$ to the origin. The function
$S$ induces an additive action of $[0,+\infty)$ on $\mathcal{P}(B(0,1))$,
given for every $t$ by $S_{t}^{*}:\nu\mapsto C(S_{t}\nu)|_{B(0,1)}$, where $C$
is such that $S_{t}^{*}\nu$ is a probability measure. Here and throughout,
$B(0,1)$ denotes the closed unit ball. We sometimes refer to $S^{*}$ as the
“zoom-in” operation and point out that in [15] it was denoted by $S^{\square}$
in order to emphasize restriction to $B(0,1)$. For the composed scaling and
translation operation, we use the short-hand notation

$\nu_{x,t}:=S_{t}^{*}T_{x}\nu.$

Using the introduced notation, we recall the definition of the scenery flow of
a measure $\nu$ at a point $x\in{\rm spt}\,\nu$ as the flow

$\left(\frac{1}{T}\int_{0}^{T}\delta[\nu_{x,t}]\,dt\right)_{T>0}.$

Note that this is a flow on $\mathcal{P}(\mathcal{P}(B(0,1)))$. The elements
of $\mathcal{P}(\mathcal{P}(B(0,1)))$ are termed _distributions_ to emphasize
that they are measures on the space of measures. The space of distributions is
endowed with the weak-∗ topology. As a consequence of the compactness of this
topology, the set of accumulation points of the scenery flow at any point is
non-empty. Recall that a measure $\nu$ is termed uniformly scaling if there
exists a distribution $P$ such that the scenery flow of $\nu$ converges to $P$
$\nu$-almost everywhere, as $T\rightarrow\infty$.

###  2.2. The symbolic space

Let $\Gamma=\\{1,\ldots,m\\}$ be an alphabet with $m\geq 2$, and let
$\Phi=\\{\varphi_{i}\\}_{i\in\Gamma}$ be a finite family of contractions on
$\mathbb{R}^{d}$. The family $\Phi$ is called an iterated function system
(IFS). It is well-known (see e.g. [4]) that there is a unique compact set $K$,
called the _attractor_ of $\Phi$, such that

$K=\bigcup_{i\in\Gamma}\varphi_{i}(K).$

If the functions $\varphi_{i}$ are of the form
$\varphi_{i}(x)=\rho_{i}R_{i}x+a_{i}$, where $0<\rho_{i}<1$, $R_{i}$ is an
orthogonal matrix on $\mathbb{R}^{d}$ and $a_{i}\in\mathbb{R}^{d}$, the IFS
$\Phi$ is called _self-similar_ and its attractor a self-similar set. Since
our study does not depend on the coordinate basis of $\mathbb{R}^{d}$, in
order to simplify notation we may and do suppose that $0\in K$ and $K\subset
B(0,1)$, where the inclusion is strict.

Write $\Gamma^{*}=\bigcup_{n}\Gamma^{n}$ for the set of finite vectors, or
_words_ , composed of elements of $\Gamma$. We usually denote finite words by
characters $\mathtt{a}$, $\mathtt{b}$ and $\mathtt{c}$. For a finite word
$\mathtt{a}=(i_{0},i_{1},\ldots,i_{n})\in\Gamma^{*}$, write
$\varphi_{\mathtt{a}}=\varphi_{i_{0}}\circ\cdots\circ\varphi_{i_{n}}$ and
$K_{\mathtt{a}}=\varphi_{\mathtt{a}}(K)$. The notation $|\mathtt{a}|$ stands
for the number of elements in the word $\mathtt{a}$ and is called the length
of $\mathtt{a}$. For finite words $\mathtt{a}=(i_{0},\ldots,i_{n})$ and
$\mathtt{b}=(j_{0},\ldots,j_{m})$, we let
$\mathtt{a}\mathtt{b}:=(i_{0},\ldots,i_{n},j_{0},\ldots,j_{m})\in\Gamma^{|\mathtt{a}|+|\mathtt{b}|}$
denote their concatenation.

Elements of $\Gamma^{\mathbb{N}}$ are called infinite words and we often
denote them by characters $\mathtt{i}$ and $\mathtt{j}$. For a word
$\mathtt{i}$, either infinite or of length $|\mathtt{i}|\geq k$, we write
$\mathtt{i}|_{k}\in\Gamma^{k}$ for its projection to the first $k$
coordinates. Given $\mathtt{a}\in\Gamma^{*}$, we call elements of the set
$\\{\mathtt{b}\in\Gamma^{\mathbb{N}}\cup\bigcup_{k\geq|\mathtt{a}|}\Gamma^{k}:\
\mathtt{b}|_{|\mathtt{a}|}=\mathtt{a}\\}$ _descendants_ of $\mathtt{a}$; the
words $\mathtt{a}|_{k}$, $k=1,2,\ldots,|\mathtt{a}|$ are called the
_ancestors_ of $\mathtt{a}$. These notions extend in the obvious way to the
sets $K_{\mathtt{a}}$ and the functions $\varphi_{\mathtt{a}}$. For a finite
word $\mathtt{b}\in\Gamma^{*}$, we write $[\mathtt{b}]$ for the cylinder set
$\\{\mathtt{i}\in\Gamma^{\mathbb{N}}:\
\mathtt{i}|_{|\mathtt{b}|}=\mathtt{b}\\}$. We also use the term “cylinder” to
refer to the sets $K_{\mathtt{a}}$. The collection of all cylinder sets
generate the topology with which we equip $\Gamma^{\mathbb{N}}$.

Let $\sigma:(i_{0},i_{1},\ldots)\mapsto(i_{1},i_{2},\ldots)$ denote the
continuous left-shift on $\Gamma^{\mathbb{N}}$. We say that a measure $\nu$ on
$\Gamma^{\mathbb{N}}$ is _invariant_ with respect to $\sigma$ if
$\sigma\nu=\nu$, and _ergodic_ , if $\nu(E)\in\\{0,1\\}$ for all sets $E$
satisfying $\sigma^{-1}E=E$.

It is sometimes convenient to consider the _two-sided extension_
$(\Gamma^{\mathbb{Z}},\sigma)$ of the dynamical system
$(\Gamma^{\mathbb{N}},\sigma)$, where
$\sigma(\ldots,i_{-1};i_{0},i_{1},\ldots)=(\ldots,i_{0};i_{1},i_{2},\ldots)$
is also used to denote the left-shift on $\Gamma^{\mathbb{Z}}$ which, we note,
is now invertible. For
$\mathtt{i}=(\ldots,i_{-1};i_{0},i_{1},\ldots)\in\Gamma^{\mathbb{Z}}$ and
$n\leq m\in\mathbb{Z}$, write
$\mathtt{i}|_{n}^{m}=(i_{n},i_{n+1}\ldots,i_{m-1},i_{m})\in\Gamma^{*}$ and
$[\mathtt{i}]_{n}^{m}=\\{(\ldots,j_{-1};j_{0},j_{1},\ldots)\in\Gamma^{\mathbb{Z}}:\
(j_{n},j_{n+1},\ldots,j_{m-1},j_{m})=\mathtt{i}|_{n}^{m}\\}$. The topology of
$\Gamma^{\mathbb{Z}}$ is again generated by the cylinder sets
$[\mathtt{i}]_{n}^{m}$. There exists a natural bijection between the spaces of
$\sigma$-invariant probability measures on $\Gamma^{\mathbb{N}}$ and
$\Gamma^{\mathbb{Z}}$, given by $\nu\mapsto\nu^{*}$, where
$\nu^{*}([\mathtt{i}]_{n}^{m})=\nu([\mathtt{i}|_{n}^{m}])$ for every
$\mathtt{i}\in\Gamma^{\mathbb{Z}}$ and $n\leq m$. It is straightforward to
check that this bijection maps ergodic measures to ergodic ones.

Write $\pi$ for the surjective map $\Gamma^{\mathbb{N}}\rightarrow K$,

$\mathtt{i}\mapsto\lim_{k\rightarrow\infty}\varphi_{\mathtt{i}|_{k}}(0),$

which we call the natural projection. Slightly abusing notation, we often
write $T_{\mathtt{i}}:=T_{\pi(\mathtt{i})}$ for the translation taking
$\pi(\mathtt{i})$ to the origin.

Fix a probability vector $p=(p_{i})_{i\in\Gamma}$ and write
$\bar{\mu}=p^{\mathbb{N}}$ for the associated Bernoulli measure on
$\Gamma^{\mathbb{N}}$. It is well-known that the measure $\pi\bar{\mu}=:\mu$
is a Radon measure supported on the self-similar set $K$ and satisfies

$\mu=\sum_{i\in\Gamma}p_{i}\cdot\varphi_{i}\mu.$

This measure is called the _self-similar measure_ associated to the vector
$p$. Without loss of generality, we assume that $p$ has strictly positive
entries: Those $i$ with $p_{i}=0$ can be removed from $\Gamma$ without
affecting $\mu$, and if only one entry is positive, $\mu$ is a point mass and
trivially uniformly scaling.

###  2.3. The weak separation condition

To make the structure of self-similar sets and measures more tractable, there
are numerous separation conditions one can impose on the defining IFS. These
conditions are meant to limit the ways the cylinders $K_{\mathtt{a}}$ can
overlap with one another, making it easier to understand the geometric
properties of self-similar measures by studying the dynamical system
$(\Gamma^{\mathbb{N}},\mu,\sigma)$. Understanding the geometric properties of
self-similar sets and measures without assuming any separation condition is an
interesting and challenging problem in fractal geometry; see [16, 28, 29, 17]
and the references therein for some recent breakthroughs on the topic.

The _weak separation condition_ was first introduced by Lau and Ngai in [24,
Definition 6.2], where the authors studied the multifractal formalism for
self-similar measures satisfying the condition. It allows overlaps between the
cylinders $K_{\mathtt{a}}$ but in a certain limited manner: Roughly speaking,
for any $r>0$, any closed ball of radius $r$ may intersect at most $M$
distinct cylinders of diameter $\approx r$, for some $M$ independent of $r$.
We point out that no bound is imposed on the number of _exact overlaps_ :
Given $\mathtt{a}\in\Gamma^{*}$, there might be numerous words
$\mathtt{b}\in\Gamma^{*}$ for which $K_{\mathtt{a}}=K_{\mathtt{b}}$. Before we
formulate the separation condition precisely, some more notation is in order.

Let $\Phi=\\{\varphi_{i}(x)=\rho_{i}R_{i}x+a_{i}\\}_{i\in\Gamma}$ be a self-
similar IFS. For a finite word $\mathtt{a}=(i_{0},\ldots,i_{n})\in\Gamma^{*}$,
write $\mathtt{a}^{-}=(i_{0},\ldots,i_{n-1})$,
$\rho_{\mathtt{a}}=\rho_{i_{0}}\cdots\rho_{i_{n}}$,
$B_{\mathtt{a}}=\varphi_{\mathtt{a}}(B(0,1))$, and denote

$\mathcal{N}(\mathtt{a})=\\{\varphi_{\mathtt{a}}^{-1}\circ\varphi_{\mathtt{b}}:\
\mathtt{b}\in\Gamma^{*},\
\rho_{\mathtt{b}}\leq\rho_{\mathtt{a}}<\rho_{\mathtt{b}^{-}},\
K_{\mathtt{b}}\cap B_{\mathtt{a}}\neq\emptyset\\}.$

We call the collection $\mathcal{N}(\mathtt{a})$ the _neighbourhood system_ of
$\mathtt{a}$. The sets $\varphi_{\mathtt{a}}\circ f(K)$,
$f\in\mathcal{N}(\mathtt{a})$, are called the _neighbours_ of
$K_{\mathtt{a}}$.

######  Definition 2.1.

The IFS $\Phi$ satisfies the _weak separation condition_ , if

(2.1) $\sup_{\mathtt{a}\in\Gamma^{*}}\\#\mathcal{N}(\mathtt{a})<\infty.$

######  Remark 2.2.

If $K$ is not contained in an affine hyperplane and we write
$\mathcal{N}^{\prime}(\mathtt{a})=\\{f\in\mathcal{N}(\mathtt{a}):\
f(K)\cap{\rm int}\,{\rm conv}\,K\neq\emptyset\\}$, where ${\rm int}$ denotes
the interior and ${\rm conv}\,$ denotes the convex hull, then it is not
difficult to see that (2.1) is equivalent to the condition
$\sup_{\mathtt{a}\in\Gamma^{*}}\\#\mathcal{N}^{\prime}(\mathtt{a})<\infty$. In
fact111We thank Alex Rutar for suggesting this remark., when ${\rm int}\,{\rm
conv}\,K\neq\emptyset$, it is an interesting question whether (2.1) is
equivalent to the a-priori stronger condition

(2.2)
$\\#\left(\bigcup_{\mathtt{a}\in\Gamma^{*}}\mathcal{N}^{\prime}(\mathtt{a})\right)<\infty.$

This question was first raised by Lau and Ngai in [25]. Some evidence for the
equivalence was provided in [14] where the authors showed that (2.1) implies
(2.2) when $K=[0,1]$.

While appearing different in form, Definition 2.1 coincides with that of Lau
and Ngai; this is proved in Zerner’s paper [30, Theorem 1], where multiple
equivalent formulations (including the one in the introduction) for the weak
separation condition are listed. We refer the reader to [26, 7, 21, 9] for
further results on the structure of weakly separated self-similar sets and
measures.

In our study of the scenery flow, we aim to establish regularity in
$(\mu_{\mathtt{i},t})_{t\geq 0}$ through the existence of a recurring
neighbourhood system in the sequence
$(\mathcal{N}(\mathtt{i}|_{k}))_{k\in\mathbb{N}}$. To study this recurrence
with the help of the theory of dynamical systems, we let
$\mathtt{a}_{0}\in\Gamma^{*}$ be a word for which

(2.3)
$\\#\mathcal{N}(\mathtt{a}_{0})=\sup_{\mathtt{a}\in\Gamma^{*}}\\#\mathcal{N}(\mathtt{a})$

and call the _maximal neighbourhood (system)_ the collection

(2.4) $\mathcal{N}_{0}:=\mathcal{N}(\mathtt{a}_{0}).$

The following observation regarding the recurrence of $\mathcal{N}_{0}$ in the
zoom-in process was made by Feng and Lau in [7]. It is one of the key tools in
our proof.

######  Lemma 2.3.

For any $\mathtt{c}\in\Gamma^{*}$,

$\\{\varphi_{\mathtt{b}}:\ \mathtt{b}\in\Gamma^{*},\
\rho_{\mathtt{b}}\leq\rho_{\mathtt{c}\mathtt{a}_{0}}\leq\rho_{\mathtt{b}^{-}},\
K_{\mathtt{b}}\cap
B_{\mathtt{c}\mathtt{a}_{0}}\neq\emptyset\\}=\varphi_{\mathtt{c}\mathtt{a}_{0}}\mathcal{N}_{0},$

where we write
$\varphi_{\mathtt{c}\mathtt{a}_{0}}\mathcal{N}_{0}=\\{\varphi_{\mathtt{c}\mathtt{a}_{0}}\circ
f:\ f\in\mathcal{N}_{0}\\}$.

$K$$K_{\mathtt{c}_{1}\mathtt{a}_{0}}$$K_{\mathtt{c}_{2}\mathtt{a}_{0}}$$\\{f(K):\
f\in\mathcal{N}_{0}\\}$ Figure 1. Here
$\mathtt{c}_{1},\mathtt{c}_{2}\in\Gamma^{*}$. The neighbourhood system of any
cylinder whose symbolic coding ends with $\mathtt{a}_{0}$ is
$\mathcal{N}_{0}$.

###### Proof.

We have

$\displaystyle\\{\varphi_{\mathtt{b}}:\ \mathtt{b}\in\Gamma^{*},\
\rho_{\mathtt{b}}\leq\rho_{\mathtt{c}\mathtt{a}_{0}}\leq\rho_{\mathtt{b}^{-}},\
K_{\mathtt{b}}\cap B_{\mathtt{c}\mathtt{a}_{0}}\neq\emptyset\\}$
$\displaystyle\supset\ $ $\displaystyle\\{\varphi_{\mathtt{c}\mathtt{a}}:\
\mathtt{a}\in\Gamma^{*},\
\rho_{\mathtt{c}\mathtt{a}}\leq\rho_{\mathtt{c}\mathtt{a}_{0}}\leq\rho_{\mathtt{c}\mathtt{a}^{-}},\
K_{\mathtt{c}\mathtt{a}}\cap B_{\mathtt{c}\mathtt{a}_{0}}\neq\emptyset\\}$
$\displaystyle=\ $
$\displaystyle\\{\varphi_{\mathtt{c}}\circ\varphi_{\mathtt{a}_{0}}\circ\varphi_{\mathtt{a}_{0}}^{-1}\circ\varphi_{\mathtt{a}}:\
\mathtt{a}\in\Gamma^{*},\
\rho_{\mathtt{a}}\leq\rho_{\mathtt{a}_{0}}\leq\rho_{\mathtt{a}^{-}},\
K_{\mathtt{a}}\cap B_{\mathtt{a}_{0}}\neq\emptyset\\}$ $\displaystyle=\ $
$\displaystyle\varphi_{\mathtt{c}\mathtt{a}_{0}}\mathcal{N}_{0}.$

This is equivalent to

(2.5)
$\mathcal{N}_{0}\subseteq\\{\varphi_{\mathtt{c}\mathtt{a}_{0}}^{-1}\circ\varphi_{\mathtt{b}}:\
\mathtt{b}\in\Gamma^{*},\
\rho_{\mathtt{b}}\leq\rho_{\mathtt{c}\mathtt{a}_{0}}\leq\rho_{\mathtt{b}^{-}},\
K_{\mathtt{b}}\cap B_{\mathtt{c}\mathtt{a}_{0}}\neq\emptyset\\}.$

Recall from (2.3) and (2.4) that $\mathcal{N}_{0}$ has cardinality at least
that of the right-hand side of (2.5). Hence (2.5) holds with equality. ∎

##  3\. The scenery flow

For simplicity, we first prove the following special case of Theorem 1.1, and
afterwards discuss the minor modifications required in the proof in order to
lift these restrictions on the IFS.

######  Theorem 3.1.

If $\Phi=\\{\phi_{i}(x)=\rho x+a_{i}\\}_{i\in\Gamma}$ is an IFS of
equicontractive homotheties on $\mathbb{R}^{d}$ satisfying the weak separation
condition, then any self-similar measure $\mu$ associated to $\Phi$ is
uniformly scaling.

In the setting of Theorem 3.1, the neighbourhood system of a finite word
$\mathtt{a}\in\Gamma^{*}$ assumes the slightly simplified form

$\mathcal{N}(\mathtt{a})=\\{\varphi_{\mathtt{a}}^{-1}\circ\varphi_{\mathtt{b}}:\
\mathtt{b}\in\Gamma^{|\mathtt{a}|},\ K_{\mathtt{b}}\cap
B_{\mathtt{a}}\neq\emptyset\\}.$

If $(r_{n})_{n\in\mathbb{N}}\subseteq[0,+\infty)$ is a sequence such that

(3.1) $\lim_{n\rightarrow\infty}\frac{r_{n}}{n}\in(0,+\infty),$

then the sequence

(3.2)
$\left(\frac{1}{r_{n}}\int_{0}^{r_{n}}\delta[\mu_{\mathtt{i},t}]\,dt\right)_{n\in\mathbb{N}}$

is asymptotic to the scenery flow of $\mu$ at $\pi(\mathtt{i})$. Our strategy
is for every $\varepsilon>0$ and almost every
$\mathtt{i}\in\Gamma^{\mathbb{N}}$ to choose the numbers
$r_{n}=r_{n}(\mathtt{i})$ in such a way that the tail of (3.2) is within
Prokhorov distance $\varepsilon$ from a distribution $P_{\varepsilon}$
independent of $\mathtt{i}$. Provided that $(r_{n})_{n\in\mathbb{N}}$ also
satisfies (3.1), it follows that tail of the scenery flow is within distance
$2\varepsilon$ from $P_{\varepsilon}$. Taking $\varepsilon\rightarrow 0$ along
a countable sequence and using the compactness of the space of distributions,
we then obtain a set of full $\mu$-measure in which the scenery flow
converges.

To find a distribution $P_{\varepsilon}$ which nearly captures the statistics
of the scenery of $\mu$, we want to approximate $(\mu_{\mathtt{i},t})_{t\geq
0}$ with a flow whose dynamics can be more easily traced back to
$(\Gamma^{\mathbb{N}},\bar{\mu},\sigma)$. Since our prior hope of
accomplishing this is to use the recurrence of $\mathcal{N}_{0}$ in
$(\mathcal{N}(\mathtt{i}|_{k}))_{k\in\mathbb{N}}$ which follows from the
recurrent visits of $(\sigma^{k}\mathtt{i})_{k\in\mathbb{N}}$ in
$[\mathtt{a}_{0}]$, our first task is to exhibit the scenery measures using
the neighbourhood systems appearing around $\pi(\mathtt{i})$. The following
representation is immediate from the self-similarity of $\mu$; recall that
$T_{\mathtt{i}}$ denotes the translation $x\mapsto x-\pi(\mathtt{i})$.

######  Lemma 3.2.

Let $\mathtt{i}\in\Gamma^{\mathbb{N}}$. For every $k\in\mathbb{N}$, there
exists a probability vector
$(q_{f}(\mathtt{i}|_{k}))_{f\in\mathcal{N}(\mathtt{i}|_{k})}$ such that

$\mu_{\mathtt{i},t}=S_{t+k\log\rho}^{*}T_{\sigma^{k}\mathtt{i}}\sum_{f\in\mathcal{N}(\mathtt{i}|_{k})}q_{f}(\mathtt{i}|_{k})\cdot
f\mu$

whenever $t>0$ is such that $B(\pi(\mathtt{i}),e^{-t})\subseteq
B_{\mathtt{i}|_{k}}$.

###### Proof.

An elementary calculation shows that

(3.3)
$S_{k\log\rho^{-1}}T_{\mathtt{i}}=T_{\sigma^{k}\mathtt{i}}\varphi_{\mathtt{i}|_{k}}^{-1}.$

Since we suppose that $B(\pi(\mathtt{i}),e^{-t})\subseteq
B_{\mathtt{i}|_{k}}$, using self-similarity, (3.3) and the additivity of the
action induced by $S^{*}$ we can write

$\displaystyle\mu_{\mathtt{i},t}$
$\displaystyle=S_{t+k\log\rho}^{*}S_{k\log\rho^{-1}}T_{\mathtt{i}}\sum_{\mathtt{a}\in\Gamma^{k},\
K_{\mathtt{a}}\cap
B_{\mathtt{i}|_{k}}\neq\emptyset}p_{\mathtt{a}}\cdot\varphi_{\mathtt{a}}\mu$
$\displaystyle=S^{*}_{t+k\log\rho}T_{\sigma^{k}\mathtt{i}}\sum_{\mathtt{a}\in\Gamma^{k},\
K_{\mathtt{a}}\cap
B_{\mathtt{i}|_{k}}\neq\emptyset}p_{\mathtt{a}}\cdot(\varphi_{\mathtt{i}|_{k}}^{-1}\circ\varphi_{\mathtt{a}})\mu$

and using the definition of $\mathcal{N}(\mathtt{i}|_{k})$, the sum in this
representation can be reordered as

$\sum_{f\in\mathcal{N}(\mathtt{i}|_{k})}\left(\sum_{\mathtt{a}\in\Gamma^{k},\
\varphi_{\mathtt{i}|_{k}}^{-1}\circ\varphi_{\mathtt{a}}=f}p_{\mathtt{a}}\right)\cdot
f\mu.$

Normalizing gives the statement. ∎

Motivated by the recurrence of $\mathcal{N}_{0}$ in
$(\mathcal{N}(\mathtt{i}|_{k}))_{k\in\mathbb{N}}$ and the prescribed
representation for the scenery measures, we define the measure

(3.4) $\nu=\sum_{f\in\mathcal{N}_{0}}f\mu.$

This is the recurring “frame” to which we compare the scenery measures.
Indeed, for those $k$ for which
$\mathcal{N}(\mathtt{i}|_{k})=\mathcal{N}_{0}$, Lemma 3.2 tells us that
$\mu_{\mathtt{i},t}\ll\nu_{\sigma^{k}\mathtt{i},t+k\log\rho}$ with density
$\zeta=\zeta(\mathtt{i}|_{k})$ determined by the weights
$q_{f}(\mathtt{i}|_{k})$. While the exact form of the density depends on the
entirety of $\mathtt{i}|_{k}$, the content of the following proposition is
that much less information on $\mathtt{i}|_{k}$ is required to control
$\zeta(\mathtt{i}|_{k})$ on a sufficient level.

######  Proposition 3.3.

There exists a finite word $\mathtt{b}_{0}\in\Gamma^{*}$ and a family of
$\nu$-integrable functions $\\{\zeta_{h}:\
h\in\mathcal{F}\subseteq\mathcal{N}_{0}\\}$, each bounded from below by a
constant $C_{h}>0$, such that for almost every
$\mathtt{i}\in\Gamma^{\mathbb{N}}$ and all $k\in\mathbb{N}$ for which
$\sigma^{k-|\mathtt{a}_{0}\mathtt{b}_{0}|}\mathtt{i}\in[\mathtt{a}_{0}\mathtt{b}_{0}]$,
there exists a function $\zeta=\zeta(\mathtt{i}|_{k})$ which is a convex
combination of the functions $\zeta_{h}$ such that

$\mu_{\mathtt{i},t}=\left(\zeta
d\nu\right)_{\sigma^{k}\mathtt{i},t+k\log\rho}$

for any $t>0$ satisfying $B(\pi(\mathtt{i}),e^{-t})\subseteq
B_{\mathtt{i}|_{k}}$.

The proof requires some geometric work and can be found in Section 4.

We write

$\mathcal{W}=\\{\zeta(\mathtt{i}|_{k}):\ \mathtt{i}\ {\rm and}\ k\
\text{satisfy the assumptions of Proposition \ref{nu_p}}\\}$

for the countable collection of all functions given by Proposition 3.3. Note
that as a bounded subset of the linear span of a finite family of functions,
the closure of $\mathcal{W}$ is compact in the topology induced by the
supremum norm. Combining this compactness with an application of the Lebesgue-
Besicovitch differentiation theorem and some elementary continuity properties
of $\zeta\mapsto(\zeta d\nu)_{\mathtt{i},t}$ we can deduce the following:

######  Proposition 3.4.

For all $\varepsilon>0$, there exists a set
$\mathcal{J}=\mathcal{J}(\varepsilon)\subseteq\Gamma^{\mathbb{N}}$ and an
integer $N=N(\varepsilon)$ such that $\bar{\mu}(\mathcal{J})>0$ and for any
$\zeta\in\mathcal{W}$ and every $\mathtt{i}\in\mathcal{J}$,

(3.5) $d\left(\frac{1}{T}\int_{0}^{T}\delta[\mu_{\mathtt{i},t}]\,dt,\
\frac{1}{T}\int_{0}^{T}\delta[(\zeta
d\nu)_{\mathtt{i},t}]\,dt\right)<\varepsilon$

for all $T\geq N$.

The proof can be found in Section 4.

These propositions together give us a way to approximate the statistics of
$(\mu_{\mathtt{i},t})_{t\geq 0}$ using only the statistics of the recurrence
of $(\sigma^{k}\mathtt{i})_{k\in\mathbb{N}}$ in
$[\mathtt{a}_{0}\mathtt{b}_{0}]$.

###  3.1. The sequence $(r_{n})_{n}$

Let $\varepsilon>0$. We now move on to defining the distribution of the random
sequence $(r_{n})_{n\in\mathbb{N}}$; recall (3.2) and the discussion
thereafter. Let $\mathtt{b}_{0}$ be the finite word given by Proposition 3.3,
and let $\mathcal{J}=\mathcal{J}(\varepsilon)$ be the set given by Proposition
3.4. Since everything we define in the following depends on the set
$\mathcal{J}$ and hence on $\varepsilon$, we suppress this dependence from our
notation.

We will first define, for almost every $\mathtt{i}\in\Gamma^{\mathbb{N}}$, a
sequence $(t_{n})_{n\in\mathbb{N}}\subseteq\mathbb{N}$ such that for every
$n$,

  * i)

$\sigma^{t_{n}}\mathtt{i}\in\mathcal{J}$ and

  * ii)

$\sigma^{t_{n}-|\mathtt{a}_{0}\mathtt{b}_{0}|}\mathtt{i}\in[\mathtt{a}_{0}\mathtt{b}_{0}]$

In particular, the sequence $(\sigma^{t_{n}}\mathtt{i})_{n}$ must also include
some information on the “past” of each element. One way to deal with this is
to consider the two-sided extension of the space $\Gamma^{\mathbb{N}}$. Since
there is a natural bijection between the spaces of invariant measures on
$\Gamma^{\mathbb{N}}$ and $\Gamma^{\mathbb{Z}}$, it is convenient to denote
the extended Bernoulli measure on $\Gamma^{\mathbb{Z}}$ also by $\bar{\mu}$.
Since the natural projection $\pi$ extends to $\Gamma^{\mathbb{Z}}$ by
$\mathtt{i}\mapsto\pi(\mathtt{i}^{+})$, where $(\cdot)^{+}$ denotes the
projection to the positive coordinates, and both the Bernoulli measure and the
extended one project onto the self-similar measure $\mu$, we do not usually
make a distinction between one-sided and two-sided infinite words.

We let

$\mathcal{J}^{\prime}=\\{\mathtt{i}\in\Gamma^{\mathbb{Z}}:\
\mathtt{i}|_{-|\mathtt{a}_{0}\mathtt{b}_{0}|}^{-1}=\mathtt{a}_{0}\mathtt{b}_{0},\
\mathtt{i}^{+}\in\mathcal{J}\\}$

and define the sequence $(t_{n})_{n\in\mathbb{N}}$ as the return times of
$\mathtt{i}$ to the set $\mathcal{J}^{\prime}$: write

$\tau:\ \mathtt{j}\mapsto\min\\{n\geq 1:\
\sigma^{n}\mathtt{j}\in\mathcal{J}^{\prime}\\}$

for the time of first visit in $\mathcal{J}^{\prime}$, defined
$\bar{\mu}$-almost everywhere, and set

$\displaystyle t_{0}$ $\displaystyle=0,$ $\displaystyle t_{n}(\mathtt{i})$
$\displaystyle=t_{n-1}(\mathtt{i})+\tau(\sigma^{t_{n-1}}\mathtt{i})\
\text{for}\ n\geq 1.$

Note that $\bar{\mu}(\mathcal{J}^{\prime})>0$. We also point out the
following:

######  Lemma 3.5.

The function $\tau$ is $\bar{\mu}$-integrable over $\mathcal{J}^{\prime}$.

###### Proof.

Since $\bar{\mu}$ is ergodic, this is implied by Kac’s theorem [20, Theorem
2]. ∎

Finally, define

$\displaystyle r_{n}(\mathtt{i})$ $\displaystyle=\inf\\{r\geq 0:\
B(\pi(\mathtt{i}),e^{-r})\subseteq B_{\mathtt{i}|_{t_{n}(\mathtt{i})}}\\}$
$\displaystyle=t_{n}(\mathtt{i})\log\rho^{-1}+\alpha_{0}(\sigma^{t_{n}(\mathtt{i})}\mathtt{i}).$

Here

$\displaystyle\alpha_{0}:\ \mathtt{j}\mapsto\inf\\{r\geq 0:\
B(\pi(\mathtt{j}),e^{-r})\subseteq B(0,1)\\}$

is the correction that arises because the natural projections of
$\mathtt{j}\in\Gamma^{\mathbb{N}}$ are not necessarily in the “center” of
$B(0,1)$. Since we suppose that $K$ is strictly included in $B(0,1)$, for a
large enough integer $M$ we have $\alpha_{0}(\mathtt{j})\in[0,M]$ for all
$\mathtt{j}\in\Gamma^{\mathbb{N}}$.

Let $N$ be the integer given by Proposition 3.4. Note that since we have

$\displaystyle r_{n+1}(\mathtt{i})-r_{n}(\mathtt{i})$
$\displaystyle=(t_{n+1}(\mathtt{i})-t_{n}(\mathtt{i}))\log\rho^{-1}+\alpha_{0}(\sigma^{t_{n+1}}\mathtt{i})-\alpha_{0}(\sigma^{t_{n}}\mathtt{i})$
$\displaystyle\geq\log\rho^{-1}-M,$

by replacing the alphabet $\Gamma$ with a high-level iteration
$\Gamma^{\lceil(M+N)/\log\rho^{-1}\rceil}$ we can suppose that the difference
$r_{n+1}(\mathtt{i})-r_{n}(\mathtt{i})\geq N$ for almost every
$\mathtt{i}\in\mathcal{J}^{\prime}$. We will later see that this sequence
satisfies the condition (3.1).

We are now ready to prove Theorem 3.1.

###  3.2. Proof of Theorem 3.1

In the proof, we require the following application of Birkhoff’s ergodic
theorem. We postpone its short proof to the end of the section.

######  Claim 3.6.

For any $\bar{\mu}$-integrable (distribution- or real-valued) $g$ and almost
every $\mathtt{i}$,

$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=0}^{n-1}g(\sigma^{t_{k}}\mathtt{i})=\int
g\,d\bar{\mu}_{\mathcal{J}^{\prime}}.$

Define the functions $\psi,\eta$ for $\bar{\mu}$-almost every $\mathtt{j}$ by

(3.6) $\displaystyle\begin{split}&\psi:\
\mathtt{j}\mapsto\int_{\alpha_{0}(\mathtt{j})}^{\tau(\mathtt{j})\log\rho^{-1}+\alpha_{0}(\sigma^{\tau(\mathtt{j})}\mathtt{j})}\delta[\mu_{\mathtt{j},t}]\,dt,\\\
&\eta:\
\mathtt{j}\mapsto\tau(\mathtt{j})\log\rho^{-1}+\alpha_{0}(\sigma^{\tau(\mathtt{j})}\mathtt{j})-\alpha_{0}(\mathtt{j}).\end{split}$

The measurability of these functions is standard, and integrability over
$\mathcal{J}^{\prime}$ follows from that of $\tau$ (Lemma 3.5) and the
assumption $\alpha_{0}(\mathcal{J})\subseteq[0,M]$.

###### Proof of Theorem 3.1.

Let now $\mathtt{i}\in\bigcup_{n=0}^{\infty}\sigma^{-n}\mathcal{J}^{\prime}$
be such that Claim 3.6 holds for the functions $\psi$ and $\eta$. The set of
these words has full $\bar{\mu}$-measure. Since the scale we start the zoom-in
process at has no effect on the generated distribution, we may suppose that
$\mathtt{i}\in\mathcal{J}^{\prime}$. To make the notation more clear, from now
on we suppress the dependance of $r_{k}$ and $t_{k}$ on $\mathtt{i}$. Recall
that the set $\mathcal{J}^{\prime}$ was defined so that whenever
$\mathtt{i}\in\mathcal{J}^{\prime}$, we have $\mathtt{i}^{+}\in\mathcal{J}$
and
$(\sigma^{-|\mathtt{a}_{0}\mathtt{b}_{0}|}\mathtt{i})^{+}\in[\mathtt{a}_{0}\mathtt{b}_{0}]$.

Now, we can write the sequence (3.2) as

$\displaystyle\frac{1}{r_{n}}\int_{0}^{r_{n}}\delta[\mu_{\mathtt{i},t}]\,dt$
$\displaystyle=\frac{1}{r_{n}}\sum_{k=0}^{n-1}\int_{r_{k}}^{r_{k+1}}\delta[\mu_{\mathtt{i},t}]\,dt$
$\displaystyle=\frac{1}{r_{n}}\sum_{k=0}^{n-1}\int_{t_{k}\log\rho^{-1}+\alpha_{0}(\sigma^{t_{k}}\mathtt{i})}^{t_{k+1}\log\rho^{-1}+\alpha_{0}(\sigma^{t_{k+1}}\mathtt{i})}\delta[(\zeta(\mathtt{i}|_{k})d\nu)_{\sigma^{t_{k}}\mathtt{i},t+t_{k}\log\rho}]\,dt$
(3.7)
$\displaystyle=\frac{1}{r_{n}}\sum_{k=0}^{n-1}\int_{\alpha_{0}(\sigma^{t_{k}}\mathtt{i})}^{\tau(\sigma^{t_{k}}\mathtt{i})\log\rho^{-1}+\alpha_{0}(\sigma^{t_{k+1}}\mathtt{i})}\delta[(\zeta(\mathtt{i}|_{k})d\nu)_{\sigma^{t_{k}}\mathtt{i},t}]\,dt,$

using Proposition 3.3 and writing out the definitions of $r_{k}$ in the second
equality, and performing a change of variable $t\mapsto t-t_{k}\log\rho$ in
the last one. Combining this with Proposition 3.4 we get that

(3.8) $d\left(\frac{1}{r_{n}}\int_{0}^{r_{n}}\delta[\mu_{\mathtt{i},t}]\,dt,\
\frac{1}{r_{n}}\sum_{k=0}^{n-1}\int_{\alpha_{0}(\sigma^{t_{k}}\mathtt{i})}^{\tau(\sigma^{t_{k}}\mathtt{i})\log\rho^{-1}+\alpha_{0}(\sigma^{t_{k+1}}\mathtt{i})}\delta[\mu_{\sigma^{t_{k}}\mathtt{i},t}]\,dt\right)<\varepsilon$

for all large enough $n$. Write

$Q_{n}:=\frac{1}{r_{n}}\sum_{k=0}^{n-1}\int_{\alpha_{0}(\sigma^{t_{k}}\mathtt{i})}^{\tau(\sigma^{t_{k}}\mathtt{i})\log\rho^{-1}+\alpha_{0}(\sigma^{t_{k+1}}\mathtt{i})}\delta[\mu_{\sigma^{t_{k}}\mathtt{i},t}]\,dt.$

Using the functions $\psi$ and $\eta$ defined in (3.6), we have for $Q_{n}$
the representation

$\displaystyle Q_{n}$
$\displaystyle=\frac{1}{\sum_{k=0}^{n-1}(r_{k+1}-r_{k})}\sum_{k=0}^{n-1}\psi(\sigma^{t_{k}}\mathtt{i})=\left(\frac{1}{\frac{1}{n}\sum_{k=0}^{n-1}\eta(\sigma^{t_{k}}\mathtt{i})}\right)\left(\frac{1}{n}\sum_{k=0}^{n-1}\psi(\sigma^{t_{k}}\mathtt{i})\right).$

It follows from Claim 3.6 that the sequence $(Q_{n})_{n\in\mathbb{N}}$
converges to a limiting distribution $P_{\varepsilon}$ as
$n\rightarrow\infty$. Combining this with (3.8), it follows that there exists
an integer $n_{\varepsilon}$ such that

$d\left(\frac{1}{r_{n}}\int_{0}^{r_{n}}\delta[\mu_{\mathtt{i},t}]\,dt,\
P_{\varepsilon}\right)<2\varepsilon\ \text{for all}\ n\geq n_{\varepsilon}.$

Since Claim 3.6 applied for $\eta$ shows that

$\lim_{n\rightarrow\infty}\frac{r_{n}}{n}=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=0}^{n-1}\eta(\sigma^{t_{k}}\mathtt{i})=\int\eta\,d\bar{\mu}_{\mathcal{J}^{\prime}}\in(0,+\infty),$

it follows that the sequence

$\left(\frac{1}{r_{n}}\int_{0}^{r_{n}}\delta[\mu_{\mathtt{i},t}]\,dt\right)_{n\in\mathbb{N}}$

is asymptotic to the scenery flow of $\mu$ at $\pi(\mathtt{i})$.

We have now found, for every $\varepsilon>0$, a distribution $P_{\varepsilon}$
and an integer $T_{\varepsilon}$ such that for almost every $\mathtt{i}$,

(3.9)
$d\left(\frac{1}{T}\int_{0}^{T}\delta[\mu_{\mathtt{i},t}]\,dt,P_{\varepsilon}\right)<2\varepsilon\
\text{for all}\ T\geq T_{\varepsilon}.$

Letting $\varepsilon\rightarrow 0$ along a countable sequence and taking the
intersection of the sets of full $\bar{\mu}$-measure in which (3.9) holds, we
obtain a set of full measure in which (3.9) holds for every $\varepsilon$.
Using the compactness of $\mathcal{P}(\mathcal{P}(B(0,1)))$, it follows that
the scenery flow of $\mu$ converges almost everywhere. Moreover, since the
distributions $P_{\varepsilon}$ are independent of $\mathtt{i}$, so is the
limit of the scenery flow. This concludes the proof of Theorem 3.1. ∎

###### Proof of Claim 3.6.

Note first that the claim is equivalent to

(3.10)
$\displaystyle\lim_{n\rightarrow\infty}\frac{1}{\bar{\mu}(\mathcal{J}^{\prime})n}\sum_{k=0}^{\lfloor\bar{\mu}(\mathcal{J}^{\prime})n\rfloor}g(\sigma^{t_{k}}\mathtt{i})=\int
g\,d\bar{\mu}_{\mathcal{J}^{\prime}}.$

Using Birkhoff’s ergodic theorem for $\sigma$ and $\chi[\mathcal{J}^{\prime}]$
in the form

$\lim_{n\rightarrow\infty}\frac{\\#\\{0\leq\ell\leq n-1:\
\sigma^{\ell}\mathtt{i}\in\mathcal{J}^{\prime}\\}}{n}=\bar{\mu}(\mathcal{J}^{\prime}),$

we see that the sequence on the left-hand side of (3.10) is asymptotic to the
sequence

$\frac{1}{\bar{\mu}(\mathcal{J}^{\prime})n}\sum_{k=0}^{\\#\\{0\leq\ell\leq
n-1:\
\sigma^{\ell}\mathtt{i}\in\mathcal{J}^{\prime}\\}}g(\sigma^{t_{k}}\mathtt{i}).$

Now, using the fact that $\sigma^{\ell}\mathtt{i}\in\mathcal{J}^{\prime}$ if
and only if $\ell=t_{k}$ for some $k$, which follows from the definition of
$(t_{k})_{k}$ as the return times to the set $\mathcal{J}^{\prime}$, we can
write

$\frac{1}{\bar{\mu}(\mathcal{J}^{\prime})n}\sum_{k=0}^{\\#\\{0\leq\ell\leq
n-1:\
\sigma^{\ell}\mathtt{i}\in\mathcal{J}^{\prime}\\}}g(\sigma^{t_{k}}\mathtt{i})=\frac{1}{\bar{\mu}(\mathcal{J}^{\prime})}\frac{1}{n}\sum_{\ell=0}^{n-1}\chi[\mathcal{J}^{\prime}](\sigma^{\ell}\mathtt{i})g(\sigma^{\ell}\mathtt{i}).$

Finally, using Birkhoff’s ergodic theorem for $\chi[\mathcal{J}^{\prime}]g$
and $\sigma$, we obtain (3.10). ∎

###  3.3. On the proof of Theorem 1.1

In the proof of Theorem 3.1 it was assumed that the IFS $\Phi$ consists of
equicontractive homotheties. Only minor modifications in the proof are
required in order to dispose of this requirement. We provide an overview of
the needed modifications; details are left to the interested reader. For now,
suppose that $\Psi=\\{\varphi_{i}(x)=\rho_{i}R_{i}x+a_{i}\\}_{i\in\Gamma}$ is
a self-similar IFS satisfying the weak separation condition and $\mu$ is an
associated self-similar measure.

In Proposition 3.3, to obtain the stated identity we need to replace the
measure $(\zeta d\nu)_{\sigma^{k},t+k\log\rho}$ by $(R_{\mathtt{i}|_{k}}(\zeta
d\nu))_{\sigma^{k}\mathtt{i},t+\log\rho_{\mathtt{i}|_{k}}}$, where
$R_{\mathtt{i}|_{k}}$ is the matrix component of $\varphi_{\mathtt{i}|_{k}}$.
The statement of Proposition 3.4 extends as it is for distributions generated
by $R\nu$ and $R(\zeta d\nu)$, for any orthogonal matrix $R$.

In the ergodic-theoretic part of Subsection 3.2, one must take into account
the additional dynamics coming from the matrix components of the similarities.
Instead of the system $(\Gamma^{\mathbb{N}},\bar{\mu},\sigma)$, we need to
consider the skew-product $(\Gamma^{\mathbb{N}}\times
G,\bar{\mu}\times\gamma,F)$, where $G$ is the closure of the topological group
generated by the matrix components of $\varphi_{i}$, $\gamma$ is the right-
invariant Haar measure on $G$ and $F$ is the transformation
$(\mathtt{i},R)\mapsto(\sigma\mathtt{i},R\cdot R_{\mathtt{i}|_{1}})$. The
ergodicity of this kind of skew-product system is due to Kakutani [22]. Going
through Section 3 with these modifications, one sees that for $\gamma$-almost
every $R$, the measure $R\mu$ is uniformly scaling. This property is clearly
preserved under action by $R^{-1}$.

The fact that $\mu$ generates an ergodic fractal distribution is immediate
from the results of Hochman [15]. Since some additional terminology needs to
be introduced before the short deduction, we postpone it to Section 5 in order
not to disrupt the main line of argument behind the uniform scaling property.

##  4\. The scenery of the self-similar measure

In this section, we prove Propositions 3.3 and 3.4. Although the proofs are
written in the setting of Theorem 3.1 for the sake of notational simplicity,
after modifying the statements of the propositions according to the highlights
in Subsection 3.3, the arguments extend with little work to the setting of
Theorem 1.1.

We adopt the notation

(4.1) $\mu_{\mathtt{a}}:=(\varphi_{\mathtt{a}}^{-1}\mu)_{B(0,1)}$

for every $\mathtt{a}\in\Gamma^{*}$, for the “symbolic magnifications” of the
self-similar measure.

###  4.1. Proof of Proposition 3.3

Recall Lemma 3.2 and that our motivation is to find a way to describe the
probability vector
$(q_{f}(\mathtt{i}|_{k}))_{f\in\mathcal{N}(\mathtt{i}|_{k})}$ without
requiring information on the entirety of the word $\mathtt{i}|_{k}$. Recall
also the choice of the word $\mathtt{a}_{0}$ from (2.3) and the definition of
the maximal neighbourhood $\mathcal{N}_{0}$ from (2.4). Through the work in
this subsection, we will find a finite word $\mathtt{b}_{0}$ such that for any
$\mathtt{c}\in\Gamma^{*}$,
$\mathcal{N}(\mathtt{c}\mathtt{a}_{0}\mathtt{b}_{0})=\mathcal{N}_{0}$ and the
weights $q_{f}(\mathtt{c}\mathtt{a}_{0}\mathtt{b}_{0})$,
$f\in\mathcal{N}_{0}$, are bounded from below by a constant independent of
$\mathtt{c}$.

The key geometric observation in the proof is the following Claim which
provides us with the word $\mathtt{b}_{0}$.

######  Claim 4.1.

There exists a finite word $\mathtt{b}_{0}$ and a collection
$\mathcal{F}\subseteq\mathcal{N}_{0}$ such that
$\mathcal{N}(\mathtt{b}_{0})=\mathcal{N}_{0}$ and

  * (i)

for every $h\in\mathcal{F}$, every neighbour of $K_{\mathtt{b}_{0}}$ is a
descendant of $h(K)$, that is, there exists a word
$\mathtt{b}_{h}\in\Gamma^{*}$ such that
$h\circ\varphi_{\mathtt{b}_{h}}=\varphi_{\mathtt{b}_{0}}$ and
$\mathcal{N}(\mathtt{b}_{h})=\mathcal{N}_{0}$,

  * (ii)

for every $h\in\mathcal{N}_{0}\setminus\mathcal{F}$, $B_{\mathtt{b}_{0}}\cap
h(K)=\emptyset$.

###### Proof.

In the case where $K_{\mathtt{a}_{0}}\cap h(K)=\emptyset$ for every
$h\in\mathcal{N}_{0}\setminus\\{{\rm Id}\\}$ the claim is close to trivial:
set $\mathcal{F}=\\{{\rm Id}\\}$ and choose $\mathtt{b}_{0}\in\Gamma^{*}$ in
such a way that
$B_{\mathtt{b}_{0}}\cap\bigcup_{h\in\mathcal{N}_{0}\setminus\mathcal{F}}h(K)=\emptyset$
as we may since the union is closed.

Suppose then that $K_{\mathtt{a}_{0}}\cap h_{1}(K)\neq\emptyset$ for some
$h_{1}\in\mathcal{N}_{0}\setminus\\{{\rm Id}\\}$. The set $\mathcal{F}$ and
the word $\mathtt{b}_{0}$ are now found through the following recursive
construction. First, we claim that a descendant of $h_{1}(K)$ overlaps exactly
with a descendant of $K$.

######  Claim 4.2.

There exist finite words $\mathtt{b},\mathtt{c}\in\Gamma^{*}$ such that

$\varphi_{\mathtt{b}}=h_{1}\circ\varphi_{\mathtt{c}}.$

###### Proof of Claim 4.2.

Since we assume $K_{\mathtt{a}_{0}}\cap h_{1}(K)\neq\emptyset$, there exists
$\mathtt{c}\in\Gamma^{|\mathtt{a}_{0}|}$ such that $h_{1}(K_{\mathtt{c}})\cap
K_{\mathtt{a}_{0}}\neq\emptyset$. Because
$h_{1}\in\mathcal{N}_{0}=\mathcal{N}(\mathtt{a}_{0})$, by definition of
$\mathcal{N}$ there exists $\mathtt{a}\in\Gamma^{|\mathtt{a}_{0}|}$ such that
$h_{1}=\varphi_{\mathtt{a}_{0}}^{-1}\circ\varphi_{\mathtt{a}}$, so we have

$\varphi_{\mathtt{a}_{0}}^{-1}\circ\varphi_{\mathtt{a}\mathtt{c}}(K)\cap\varphi_{\mathtt{a}_{0}}(K)\neq\emptyset.$

Mapping both sets by $\varphi_{\mathtt{a}_{0}}$, we can use Lemma 2.3 to
deduce that
$\varphi_{\mathtt{a}\mathtt{c}}\in\varphi_{\mathtt{a}_{0}\mathtt{a}_{0}}\mathcal{N}_{0}$,
or equivalently,
$h_{1}\circ\varphi_{\mathtt{c}}\in\varphi_{\mathtt{a}_{0}}\mathcal{N}_{0}$.

Since $\mathcal{N}_{0}$ consists of functions of the form
$\varphi_{\mathtt{a}_{0}}^{-1}\circ\varphi_{\mathtt{b}}$ for
$\mathtt{b}\in\Gamma^{|\mathtt{a}_{0}|}$, there exists $\mathtt{b}$ such that
$h_{1}\circ\varphi_{\mathtt{c}}=\varphi_{\mathtt{a}_{0}}\circ\varphi_{\mathtt{a}_{0}}^{-1}\circ\varphi_{\mathtt{b}}=\varphi_{\mathtt{b}}$.
The claim is satisfied with the words $\mathtt{b}$ and $\mathtt{c}$. ∎

Write $\mathtt{b}_{1}:=\mathtt{b}$ and $\mathtt{c}_{1,1}:=\mathtt{c}$ for the
words $\mathtt{b}$ and $\mathtt{c}$ given by Claim 4.2.

Suppose now that for an integer $n\geq 1$ and every $1\leq k\leq n$ we have
found functions $h_{k}$ and words $\mathtt{b}_{n},\mathtt{c}_{n,k}$ such that
$\varphi_{\mathtt{b}_{n}}=h_{k}\circ\varphi_{\mathtt{c}_{n,k}}$ for every $k$.
Now, if there exists a function $h_{n+1}\in\mathcal{N}_{0}\setminus\\{{\rm
Id},h_{1},\ldots,h_{n}\\}$ for which $K_{\mathtt{b}_{n}\mathtt{a}_{0}}\cap
h_{n+1}(K)\neq\emptyset$, proceed as in the proof of Claim 4.2 with
$\mathtt{a}_{0}$ replaced by $\mathtt{b}_{n}\mathtt{a}_{0}$ to find words
$\mathtt{b},\mathtt{c}$ such that
$\varphi_{\mathtt{b}_{n}\mathtt{b}}=h_{n+1}\circ\varphi_{\mathtt{c}}=h_{k}\circ\varphi_{\mathtt{c}_{n,k}\mathtt{b}}$
for every $1\leq k\leq n$; the latter equality follows from the hypothesis on
$\mathtt{b}_{n}$ and $\mathtt{c}_{n,k}$. Set
$\mathtt{b}_{n+1}:=\mathtt{b}_{n}\mathtt{b}$,
$\mathtt{c}_{n+1,n+1}:=\mathtt{c}$ and
$\mathtt{c}_{n+1,k}:=\mathtt{c}_{n,k}\mathtt{b}$ for each $1\leq k\leq n$.

Continue this process until we have reached an integer $n<\\#\mathcal{N}_{0}$
for which either $\mathcal{N}_{0}=\\{{\rm Id},h_{1},\ldots,h_{n}\\}$ or
$K_{\mathtt{b}_{n}\mathtt{a}_{0}}\cap h(K)=\emptyset$ for each
$h\in\mathcal{N}_{0}\setminus\\{{\rm Id},h_{1},\ldots,h_{n}\\}$. Write
$\mathcal{F}=\\{{\rm Id},h_{1},\ldots,h_{n}\\}$ and note that by construction
of $\mathcal{F}$, for every $h\in\mathcal{F}$ there exists
$\mathtt{b}_{h}^{\prime}\in\Gamma^{*}$ such that
$\varphi_{\mathtt{b}_{n}}=h\circ\varphi_{\mathtt{b}_{h}^{\prime}}$. By passing
onto a descendant of $\mathtt{b}_{n}$ we may assume that
$B_{\mathtt{b}_{n}}\cap\bigcup_{h\in\mathcal{N}_{0}\setminus\mathcal{F}}h(K)=\emptyset$
since the union is closed. Finally, set
$\mathtt{b}_{0}=\mathtt{b}_{n}\mathtt{a}_{0}$ and
$\mathtt{b}_{h}=\mathtt{b}_{h}^{\prime}\mathtt{a}_{0}$ for each
$h\in\mathcal{F}$ to obtain the lemma.

∎

###### Proof of Proposition 3.3.

We first consider the claimed representation for symbolic magnifications of
$\mu$. Using self-similarity, we first write

$(\varphi_{\mathtt{i}|_{k}}^{-1}\mu)|_{B(0,1)}=\varphi_{\mathtt{i}|_{k}}^{-1}\sum_{\mathtt{a}\in\Gamma^{k},\
K_{\mathtt{a}}\cap
B_{\mathtt{i}|_{k}}\neq\emptyset}p_{\mathtt{a}}\cdot(\varphi_{\mathtt{a}}\mu)|_{B(0,1)}.$

Using the equality $\\{\varphi_{\mathtt{a}}:\ \mathtt{a}\in\Gamma^{k},\
K_{\mathtt{a}}\cap
B_{\mathtt{i}|_{k}}\\}=\varphi_{\mathtt{i}|_{k}}\mathcal{N}_{0}$ which is
Lemma 2.3, we can write the above measure as

(4.2) $\sum_{f\in\mathcal{N}_{0}}\left(\sum_{\mathtt{a}\in\Gamma^{k},\
\varphi_{\mathtt{a}}=\varphi_{\mathtt{i}|_{k}}\circ
f}p_{\mathtt{a}}\right)\cdot(f\mu)|_{B(0,1)}.$

Recall that $k$ is such that
$\sigma^{k-|\mathtt{a}_{0}\mathtt{b}_{0}|}\mathtt{i}\in[\mathtt{a}_{0}\mathtt{b}_{0}]$.
Writing $k^{\prime}=k-|\mathtt{b}_{0}|$, we further reorder the above sum
using the following claim. It is a simple consequence of Claim 4.1 and says
that each neighbour of $K_{\mathtt{i}|_{k}}$ is a descendant of
$\varphi_{\mathtt{i}|_{k^{\prime}}}\circ h(K)$ if and only if
$h\in\mathcal{F}$.

######  Claim 4.3.

For each $f\in\mathcal{N}_{0}$, we may write

$\\{\mathtt{a}\in\Gamma^{*}:\
\varphi_{\mathtt{a}}=\varphi_{\mathtt{i}|_{k}}\circ
f\\}=\bigcup_{h\in\mathcal{F}}\\{\mathtt{b}\mathtt{c}\in\Gamma^{*}:\
\varphi_{\mathtt{b}}=\varphi_{\mathtt{i}|_{k^{\prime}}}\circ h,\
h\circ\varphi_{\mathtt{c}}=\varphi_{\mathtt{b}_{0}}\circ f\\},$

where the union is disjoint. In particular, note that $\mathtt{b}$ and
$\mathtt{c}$ can be chosen independently of each other.

###### Proof of Claim 4.3.

First we observe that $\varphi_{\mathtt{i}|_{k}}\circ f(K)$ coincides with a
descendant of $K_{\mathtt{i}|_{k^{\prime}}}$. Indeed, because
$\mathtt{i}|_{k}=\mathtt{i}|_{k^{\prime}}\mathtt{b}_{0}$, this is equivalent
to $\varphi_{\mathtt{b}_{0}}\circ f(K)$ coinciding with a descendant of $K$,
which in turn follows by Lemma 2.3 because
$f\in\mathcal{N}_{0}=\mathcal{N}(\mathtt{b}_{0})$. From this we can deduce
that in particular, any ancestor of $\varphi_{\mathtt{i}|_{k}}\circ f(K)$ must
intersect $B_{\mathtt{i}|_{k^{\prime}}}$. Using this and the fact that
$\varphi_{i}$ are similarities, we may write

$\displaystyle\\{\mathtt{a}:\
\varphi_{\mathtt{a}}=\varphi_{\mathtt{i}|_{k}}\circ f\\}$
$\displaystyle=\\{(\mathtt{b},\mathtt{c})\in\Gamma^{k^{\prime}}\times\Gamma^{|\mathtt{b}_{0}|}:\
K_{\mathtt{b}\mathtt{c}}=\varphi_{\mathtt{i}|_{k}}\circ f(K)\\}$
$\displaystyle=\\{(\mathtt{b},\mathtt{c})\in\Gamma^{k^{\prime}}\times\Gamma^{|\mathtt{b}_{0}|}:\
K_{\mathtt{b}}\cap B_{\mathtt{i}|_{k^{\prime}}}\neq\emptyset,\
K_{\mathtt{b}\mathtt{c}}=\varphi_{\mathtt{i}|_{k}}\circ f(K)\\}.$

On the other hand, because
$\mathcal{N}(\mathtt{i}|_{k^{\prime}})=\mathcal{N}_{0}$, each neighbour of
$K_{\mathtt{i}|_{k^{\prime}}}$ coincides with
$\varphi_{\mathtt{i}|_{k^{\prime}}}\circ h(K)$ for some $h\in\mathcal{N}_{0}$,
by Lemma 2.3. Finally, by the properties of $\mathtt{b}_{0}$ given by Claim
4.1, as a neighbour of $K_{\mathtt{i}|_{k}}$ the set
$\varphi_{\mathtt{i}|_{k}}\circ f(K)$ coincides with a descendant of
$\varphi_{\mathtt{i}|_{k^{\prime}}}\circ h(K)$ for each $h\in\mathcal{F}$, and
for $h\not\in\mathcal{F}$,

$B_{\mathtt{i}|_{k}}\cap\varphi_{\mathtt{i}|_{k^{\prime}}}\circ
h(K)=\varphi_{\mathtt{i}|_{k^{\prime}}}(B_{\mathtt{b}_{0}}\cap
h(K))=\emptyset.$

Thus, we may write

$\displaystyle\\{(\mathtt{b},\mathtt{c})\in\Gamma^{k^{\prime}}\times\Gamma^{|\mathtt{b}_{0}|}:\
K_{\mathtt{b}}\cap B_{\mathtt{i}|_{k^{\prime}}}\neq\emptyset,\
K_{\mathtt{b}\mathtt{c}}=\varphi_{\mathtt{i}|_{k}}\circ f(K)\\}$
$\displaystyle=\ $
$\displaystyle\bigcup_{h\in\mathcal{F}}\\{(\mathtt{b},\mathtt{c})\in\Gamma^{k^{\prime}}\times\Gamma^{|\mathtt{b}_{0}|}:\
K_{\mathtt{b}}=\varphi_{\mathtt{i}|_{k^{\prime}}}\circ h(K),\
\varphi_{\mathtt{i}|_{k^{\prime}}}\circ
h\circ\varphi_{\mathtt{c}}(K)=\varphi_{\mathtt{i}|_{k}}\circ f(K)\\}.$

Using the fact that $\varphi_{i}$ are similarities, the above can again be
written as

$\displaystyle\bigcup_{h\in\mathcal{F}}\\{\mathtt{b}\mathtt{c}\in\Gamma^{*}:\
\varphi_{\mathtt{b}}=\varphi_{\mathtt{i}|_{k^{\prime}}}\circ h,\
\varphi_{\mathtt{i}|_{k^{\prime}}}\circ
h\circ\varphi_{\mathtt{c}}=\varphi_{\mathtt{i}|_{k^{\prime}}\mathtt{b}_{0}}\circ
f\\}$ $\displaystyle=\ $
$\displaystyle\bigcup_{h\in\mathcal{F}}\\{\mathtt{b}\mathtt{c}\in\Gamma^{*}:\
\varphi_{\mathtt{b}}=\varphi_{\mathtt{i}|_{k^{\prime}}}\circ h,\
h\circ\varphi_{\mathtt{c}}=\varphi_{\mathtt{b}_{0}}\circ f\\},$

which is what was claimed. ∎

Using Claim 4.3, we write the sum (4.2) as

$\displaystyle\sum_{f\in\mathcal{N}_{0}}\left(\sum_{\mathtt{a}\in\Gamma^{k},\
\varphi_{\mathtt{a}}=\varphi_{\mathtt{i}|_{k}}\circ
f}p_{\mathtt{a}}\right)\cdot(f\mu)|_{B(0,1)}$ $\displaystyle=\ $
$\displaystyle\sum_{h\in\mathcal{F}}\sum_{f\in\mathcal{N}_{0}}\left(\sum_{\mathtt{b}\mathtt{c}\in\Gamma^{*},\
\varphi_{\mathtt{b}}=h\circ\varphi_{\mathtt{i}|_{k^{\prime}}},\
h\circ\varphi_{\mathtt{c}}=\varphi_{\mathtt{b}_{0}}\circ
f}p_{\mathtt{b}\mathtt{c}}\right)\cdot(f\mu)|_{B(0,1)}$ $\displaystyle=\ $
$\displaystyle\sum_{h\in\mathcal{F}}\tilde{q}_{h}(\mathtt{i}|_{k})\sum_{f\in\mathcal{N}_{0}}\left(\sum_{\mathtt{c}\in\Gamma^{*},\
h\circ\varphi_{\mathtt{c}}=h\circ\varphi_{\mathtt{b}_{h}}\circ
f}p_{\mathtt{c}}\right)\cdot(f\mu)|_{B(0,1)}$ (4.3) $\displaystyle=\ $
$\displaystyle\sum_{h\in\mathcal{F}}\tilde{q}_{h}(\mathtt{i}|_{k})\sum_{f\in\mathcal{N}_{0}}\left(\sum_{\mathtt{c}\in\Gamma^{*},\
\varphi_{\mathtt{c}}=\varphi_{\mathtt{b}_{h}}\circ
f}p_{\mathtt{c}}\right)\cdot(f\mu)|_{B(0,1)}$

where

$\tilde{q}_{h}(\mathtt{i}|_{k})=\sum_{\mathtt{b}\in\Gamma^{k^{\prime}},\
\varphi_{\mathtt{b}}=h\circ\varphi_{\mathtt{i}|_{k^{\prime}}}}p_{\mathtt{b}}.$

On the third row, we have summed over $\mathtt{b}\in\Gamma^{k^{\prime}}$ for
every fixed $\mathtt{c}$ and used the equality
$\varphi_{\mathtt{b}_{0}}=h\circ\varphi_{\mathtt{b}_{h}}$ for every
$h\in\mathcal{F}$, given by Claim 4.1.

Define now the functions

$\zeta_{h}=\sum_{f\in\mathcal{N}_{0}}\left(\sum_{\mathtt{c}\in\Gamma^{*},\
\varphi_{\mathtt{c}}=\varphi_{\mathtt{b}_{h}}\circ
f}p_{\mathtt{c}}\right)\cdot\frac{d(f\mu)|_{B(0,1)}}{d\nu}$

for $h\in\mathcal{F}$, where the last expression denotes the Radon-Nikodym
derivative of $(f\mu)|_{B(0,1)}$ with respect to $\nu$. Because
$\mathcal{N}(\mathtt{b}_{h})=\mathcal{N}_{0}$, for every $h\in\mathcal{F}$ we
have

$\zeta_{h}(x)\geq
C_{h}:=\min_{f\in\mathcal{N}_{0}}\sum_{\mathtt{c}\in\Gamma^{*},\
\varphi_{\mathtt{c}}=\varphi_{\mathtt{b}_{h}}\circ f}p_{\mathtt{c}}>0$

for $\nu$-a.e. $x\in\bigcup_{f\in\mathcal{N}_{0}}f(K)\cap B(0,1)$. Normalizing
the coefficients $\tilde{q}_{h}(\mathtt{i}|_{k})$ in (4.1), writing
$(q_{h}(\mathtt{i}|_{k}))_{h\in\mathcal{F}}$ for the obtained probability
vector and defining
$\zeta:=\zeta(\mathtt{i}|_{k})=\sum_{h\in\mathcal{F}}q_{h}(\mathtt{i}|_{k})\zeta_{h}$,
we obtain from (4.1) the representation

(4.4) $\mu_{\mathtt{i}|_{k}}=(\zeta d\nu)_{B(0,1)}.$

An elementary calculation shows that
$S_{k\log\rho^{-1}}T_{\mathtt{i}}=T_{\sigma^{k}\mathtt{i}}\varphi_{\mathtt{i}|_{k}}^{-1}$.
For any $t>0$ for which $B(\pi(\mathtt{i}),e^{-t})\subseteq
B_{\mathtt{i}|_{k}}$, using (4.4) we have

$\displaystyle\mu_{\mathtt{i},t}$
$\displaystyle=S^{*}_{t+k\log\rho}S_{-k\log\rho}T_{\mathtt{i}}\mu$
$\displaystyle=S_{t+k\log\rho}^{*}T_{\sigma^{k}\mathtt{i}}\mu_{\mathtt{i}|_{k}}$
$\displaystyle=(\zeta d\nu)_{\sigma^{k}\mathtt{i},t+k\log\rho}$

which is what was claimed. ∎

###  4.2. Proof of Proposition 3.4

Proposition 3.4 is a consequence of the uniform continuity of the function
$\zeta\mapsto(\zeta d\nu)_{\mathtt{i},t}$ which is established in the
following lemmas.

The first one is a simple application of the Lebesgue-Besicovitch
differentiation theorem.

######  Lemma 4.4.

If $\eta$ and $\gamma$ are Radon measures and $0\neq\eta\ll\gamma$, then for
$\eta$-almost every $x$,

$\lim_{t\rightarrow\infty}\|\eta_{x,t}-\gamma_{x,t}\|=0.$

###### Proof.

See [15, Proposition 3.8]. ∎

Write $\phi=\frac{d\,\mu}{d\,\nu}$ and let $\mathcal{V}_{n}$ be a partition of
the image of $\phi$ into sets with diameter at most $1/n$. For each
$V\in\mathcal{V}_{n}$ with $\mu(\phi^{-1}V)>0$, take the set of full
$\mu$-measure $V^{\prime}\subseteq\phi^{-1}V$ in which

$\lim_{r\rightarrow
0}\frac{\mu(B(y,r)\cap\phi^{-1}V)}{\mu(B(y,r))}=\lim_{r\rightarrow
0}\frac{\nu(B(y,r)\cap\phi^{-1}V)}{\nu(B(y,r))}=1$

for all $y\in V^{\prime}$. Choose a point $x$ from the set
$\bigcap_{n\in\mathbb{N}}\bigcup_{V\in\mathcal{V}_{n}}V^{\prime}$ of full
$\mu$-measure.

Now, let $\varepsilon>0$ be arbitrary and choose a partition $\mathcal{V}_{n}$
with $1/n<\delta$. Let $V\in\mathcal{V}_{n}$ be such that $x\in V^{\prime}$.
For $\mu$-a.e. $y\in B(x,r)\cap\phi^{-1}V$, using the density theorem and
definition of $\phi$, we have

$\lim_{t\rightarrow
0}\frac{\mu(B(y,t)\cap\phi^{-1}V)}{\nu(B(y,t))}=\phi(y)\in[\phi(x)-1/n,\phi(x)+1/n].$

Applying the Vitali covering theorem, for every $A\subseteq B(x,r)$, when $r$
is small we have

$\displaystyle\mu(A)$ $\displaystyle=\mu(A\cap\phi^{-1}V)(1+o(r))$
$\displaystyle=\sum_{i}\mu(B(x_{i},r_{i})\cap\phi^{-1}V)$
$\displaystyle\approx\sum_{i}\nu(B(x_{i},r_{i}))$
$\displaystyle\approx\nu(A\cap\phi^{-1}V)$ $\displaystyle\approx\nu(A).$

Recall that the set $\mathcal{W}$ consists of all the functions
$\zeta=\zeta(\mathtt{i}|_{k})$ given by Proposition 3.3 for
$\mathtt{i}\in\Gamma^{\mathbb{N}}$, $k\in\mathbb{N}$, and that $\zeta\geq
C:=\min_{h\in\mathcal{F}}C_{h}>0$ for each $\zeta\in\mathcal{W}$. Write

$\mathcal{A}=\bigcap_{\zeta\in\mathcal{W}}\\{\mathtt{i}\in\Gamma^{\mathbb{N}}:\
\lim_{t\rightarrow\infty}\|\mu_{\mathtt{i},t}-(\zeta
d\nu)_{\mathtt{i},t}\|=0\\}.$

Since $\mu\ll\zeta d\nu$ and $\mathcal{W}$ is countable, we deduce from Lemma
4.4 that $\bar{\mu}(\mathcal{A})=1$. In the following, the lower bound for the
functions in $\mathcal{W}$ comes to play: while the previous lemma asserts
that $\mu_{\mathtt{i},t}$ and $(\zeta d\nu)_{\mathtt{i},t}$ have similar
asymptotic behaviour in $t$, the following two lemmas show that this
similarity is actually uniform over $\mathcal{W}$, when $\mathtt{i}$ is given.

######  Lemma 4.5.

For $\mathtt{i}\in\mathcal{A}$ and $t\geq 0$, the function
$f^{\mathtt{i},t}:\mathcal{W}\rightarrow\mathbb{R}$,

$\zeta\mapsto\|\mu_{\mathtt{i},t}-(\zeta d\nu)_{\mathtt{i},t}\|$

is continuous (when $\mathcal{W}$ is equipped with the metric induced by the
supremum norm) and the modulus of continuity is independent of $\mathtt{i}$
and $t$.

###### Proof.

Fix $\mathtt{i}$ and $t$. Let $\varepsilon>0$ be given and let $\delta>0$ be
such that whenever $\zeta_{1},\zeta_{2}\in\mathcal{W}$ and
$\|\zeta_{1}-\zeta_{2}\|_{\infty}<\delta$, we have
$\zeta_{i}\in[(1-\varepsilon)\zeta_{j},(1+\varepsilon)\zeta_{j}]$ for
$i,j\in\\{1,2\\}$. Choosing such a $\delta$ is possible because the functions
$\zeta_{1},\zeta_{2}$ are bounded from below by $C>0$.

Now, if $\|\zeta_{1}-\zeta_{2}\|_{\infty}<\delta$, we have for
$i,j\in\\{1,2\\}$ and all measurable $A\subseteq B(0,1)$,

$\displaystyle(\zeta_{i}d\nu)_{\mathtt{i},t}(A)-(\zeta_{j}d\nu)_{\mathtt{i},t}(A)$
$\displaystyle=\frac{\int_{e^{-t}A+\pi(\mathtt{i})}\zeta_{i}\,d\nu}{\int_{B(\pi(\mathtt{i}),e^{-t})}\zeta_{i}\,d\nu}-\frac{\int_{e^{-t}A+\pi(\mathtt{i})}\zeta_{j}\,d\nu}{\int_{B(\pi(\mathtt{i}),e^{-t})}\zeta_{j}\,d\nu}$
$\displaystyle\leq\frac{\int_{e^{-t}A+\pi(\mathtt{i})}\left((1+\varepsilon)\zeta_{i}-(1-\varepsilon)\zeta_{i}\right)\,d\nu}{\int_{B(\pi(\mathtt{i}),e^{-t})}(1+\varepsilon)\zeta_{i}\,d\nu}$
$\displaystyle\leq\frac{2\varepsilon}{1+\varepsilon}.$

Thus

$|f(\zeta_{1},t,\mathtt{i})-f(\zeta_{2},t,\mathtt{i})|\leq\|(\zeta_{1}d\nu)_{\mathtt{i},t}-(\zeta_{2}d\nu)_{\mathtt{i},t}\|<2\varepsilon.$

Since $\varepsilon$ was arbitrary and $\delta$ depends only on $\varepsilon$
and $C$, this completes the proof.

∎

######  Lemma 4.6.

Let $X$ be a subset of a compact metric space and let $f$ be a function on
$X\times\mathbb{R}$ such that $x\mapsto f(x,t)$ is continuous, with modulus of
continuity independent of $t$, and for every $x$,
$\lim_{t\rightarrow\infty}f(x,t)=0$. Then for every $\varepsilon>0$,

$\sup_{x\in X}\inf\\{t_{0}:\ f(x,t)\leq\varepsilon\ \text{for every}\ t\geq
t_{0}\\}<\infty.$

###### Proof.

Suppose otherwise, that there exists $c>0$ with the property that for every
$n\in\mathbb{N}$, there exists $x_{n}\in X$ and $t_{n}\geq n$ such that
$f(x_{n},t_{n})>c$. Let $(y_{n})_{n\in\mathbb{N}}\subseteq X$ be a Cauchy
subsequence of $(x_{n})_{n\in\mathbb{N}}$. Now, for every $\delta>0$ there
exist $N,M\in\mathbb{N}$ such that $|y_{n}-y_{N}|<\delta$,
$f(y_{N},t_{n})<c/2$ and $f(y_{n},t_{n})>c$ for all $n\geq M$. Since the
continuity of $x\mapsto f(x,t)$ was assumed to be uniform in $t$, this is a
contradiction. ∎

We are now able to deduce Proposition 3.4.

###### Proof of Proposition 3.4.

As a closed and bounded subset of the linear span of a finite family of
functions, the closure of $\mathcal{W}$ is compact. For
$\mathtt{i}\in\mathcal{A}$ and $t>0$, let $f^{\mathtt{i},t}$ denote the
function of Lemma 4.5. Now, for each $\mathtt{i}\in\mathcal{A}$ and
$\varepsilon>0$, apply Lemma 4.6 for the function $(\zeta,t)\mapsto
f^{\mathtt{i},t}(\zeta)$ to obtain the existence of an integer
$N(\mathtt{i},\varepsilon)<\infty$ such that $\|\mu_{\mathtt{i},t}-(\zeta
d\nu)_{\mathtt{i},t}\|<\varepsilon$ for every $\zeta\in\mathcal{W}$ and $t\geq
N(\mathtt{i},\varepsilon)$. Since
$\bar{\mu}(\bigcup_{N\in\mathbb{N}}\\{\mathtt{i}\in\mathcal{A}:\
N(\mathtt{i},\varepsilon)\leq N\\})=\bar{\mu}(\mathcal{A})=1$, we may choose
an integer $N^{\prime}$ so that if
$\mathcal{J}=\\{\mathtt{i}\in\Gamma^{\mathbb{N}}:\ \|\mu_{\mathtt{i},t}-(\zeta
d\nu)_{\mathtt{i},t}\|<\varepsilon\ \text{for all}\ t\geq N^{\prime}\\}$, we
have $\bar{\mu}(\mathcal{J})>0$.

To obtain the statement of the proposition, choose $N$ large enough with
respect to $N^{\prime}$ that

$d\left(\frac{1}{T}\int_{0}^{T}\delta[\mu_{\mathtt{i},t}]\,dt,\
\frac{1}{T}\int_{0}^{T}\delta[(\zeta
d\nu)_{\mathtt{i},t}]\,dt\right)<\varepsilon$

for all $T\geq N$ and $\mathtt{i}\in\mathcal{J}$. That such an $N$ exists can
be easily seen from the definition of the Prokhorov metric.

∎

##  5\. Discussion

We now recall the definition of a fractal distribution, and afterwards provide
the short deduction of the second assertion of Theorem 1.1.

Following the terminology of Hochman from [15], _fractal distributions_ are
distributions which are $S^{*}$-invariant and possess a spatial invariance
property called _quasi-Palm_. They generalize the notion of CP-distributions
introduced by Furstenberg to a coordinate-free setting, and, in the context of
scenery flows, the well-known principle of Preiss that “tangent measures to
tangent measures are tangent measures”.

######  Definition 5.1.

Let $P$ be a distribution on the space of probability measures on $B(0,1)$. We
say that $P$ is _quasi-Palm_ if for any measurable $\mathcal{A}$,
$P(\mathcal{A})=1$ if and only if for every $r>0$, $P$-almost every $\nu$
satisfies

$\nu_{x,r}\in\mathcal{A}$

for $\nu$-almost every $x$ with $B(x,e^{-r})\subseteq B(0,1)$.

A _fractal distribution_ is a distribution which is $S^{*}$-invariant and
quasi-Palm. If the distribution is also ergodic with respect to $S^{*}$, we
say that it is an _ergodic fractal distribution_.

######  Remark 5.2.

This definition coincides with Hochman’s definition of a _restricted_ fractal
distribution in [15], and this particular formulation was used by Hochman and
Shmerkin in [19]. The definition of a fractal distribution in [15] considers
distributions supported on all Radon measures of $\mathbb{R}^{d}$ whose
support contains the origin, and does not require the condition
$B(x,e^{-r})\subseteq B(0,1)$. Since we only consider distributions arising
from the scenery flow, the restricted versions are more relevant in our
context. However, since it was shown in [15] that each restricted fractal
distribution extends uniquely to a fractal distribution on the space of all
Radon measures, the results of [15] that we use also apply with the above
definition.

We will now explain how to deduce the second assertion of Theorem 1.1 from the
results of [15]. Suppose that $\mu$ is a self-similar measure satisfying the
hypothesis of Theorem 3.1: With the modifications highlighted in Subsection
3.3, the following goes through also in the setting of Theorem 1.1.

Let $P$ denote the limit of the scenery flow of $\mu$. It is due to Hochman
[15, Theorem 1.7] that $P$ is a fractal distribution. It is not difficult to
see from the representation (3.2) that $P$ gives positive mass to the closed
set

(5.1) $\\{(\zeta d\nu)_{\mathtt{i},t}:\ \zeta\in\overline{\mathcal{W}},\
\mathtt{i}\in\Gamma^{\mathbb{N}},\ 0\leq t\leq R\\}$

for large enough $R$. In particular, there exists an ergodic component
$P^{\prime}$ of $P$ which gives positive mass to the set (5.1). Because
ergodic components of fractal distributions are also fractal distributions by
[15, Theorem 1.3], $P^{\prime}$ is an ergodic fractal distribution. From the
ergodic theorem and the quasi-Palm property it is not difficult to see that
typical measures for $P^{\prime}$ are uniformly scaling and generate
$P^{\prime}$. In particular, a measure in the set (5.1) generates
$P^{\prime}$. Finally, since each of these measures contains as an absolutely
continuous component a translated and scaled copy of $\mu$, Lemma 4.4 allows
us to deduce that the distribution generated by $\mu$ also equals
$P^{\prime}$. In particular, $\mu$ generates an ergodic fractal distribution.

######  Remark 5.3.

It suffices to consider any accumulation point of the scenery flow in place of
$P$ in the preceding paragraph. This way, after observing from (3.2) that an
accumulation point is supported on measures of the form (5.1), one can infer
another proof of Theorem 1.1 that relies on the deep results of [15] instead
of the more hands-on geometric analysis of Proposition 3.3.

###  5.1. Projections of Markov measures

Theorem 1.1 extends for natural projections of ergodic Markov measures on
$\Gamma^{\mathbb{N}}$. Write $\bar{\mu}$ for such a measure on
$\Gamma^{\mathbb{N}}$ and $\mu:=\pi\bar{\mu}$ for its natural projection on
the self-similar set. We go through the main observations one must make in
addition to the ones in the Bernoulli case; details are left to the interested
reader.

Let $\Sigma\subseteq\Gamma^{\mathbb{N}}$ denote the subshift of finite type
associated to $\bar{\mu}$, and $\Sigma^{*}$ the collection of its finite
words. First we note that
$\\#\\{\mu^{\mathtt{a}}:=\pi(\bar{\mu}_{[\mathtt{a}]}):\
\mathtt{a}\in\Sigma^{*}\\}\leq(\\#\Gamma)^{2}<\infty$ and that for any
$\mathtt{b}\in\Sigma^{*}$,
$\mu^{\mathtt{a}\mathtt{b}}=\mu^{\mathtt{c}\mathtt{b}}$ for all
$\mathtt{a},\mathtt{c}\in\Sigma^{*}$ for which
$\mathtt{a}\mathtt{b},\mathtt{c}\mathtt{b}\in\Sigma^{*}$. These are immediate
from the Markov property of $\bar{\mu}$. The definition of the neighbourhood
system of $\mathtt{a}\in\Sigma^{*}$ is replaced by

$\mathcal{N}(\mathtt{a}):=\\{\varphi_{\mathtt{a}}^{-1}\circ\varphi_{\mathtt{b}}:\
\mathtt{b}\in\Sigma^{*},\
\rho_{\mathtt{b}}\leq\rho_{\mathtt{a}}\leq\rho_{\mathtt{b}^{-}},\
K_{\mathtt{b}}\cap B_{\mathtt{a}}\neq\emptyset\\}.$

Arguing similarly as in the proof of Lemma 2.3, using the weak separation
condition and finiteness of the collection $\mathcal{M}:=\\{\mu^{\mathtt{a}}:\
\mathtt{a}\in\Sigma^{*}\\}$, we see that there exists a word
$\mathtt{a}_{0}^{\prime}\in\Sigma^{*}$ such that for any $\mathtt{b}$ for
which $\mathtt{b}\mathtt{a}_{0}^{\prime}\in\Sigma^{*}$,
$\mathcal{N}(\mathtt{b}\mathtt{a}_{0}^{\prime})=\mathcal{N}(\mathtt{a}_{0}^{\prime})=:\mathcal{N}_{0}$
and

(5.2)
$\mu_{\mathtt{b}\mathtt{a}_{0}^{\prime}}\sim\mu_{\mathtt{a}_{0}^{\prime}}.$

Recall (4.1) for the definition of the symbolic magnifications
$\mu_{\mathtt{a}}$. The equivalence (5.2) follows from the condition
$\mathcal{N}(\mathtt{b}\mathtt{a}_{0}^{\prime})=\mathcal{N}(\mathtt{a}_{0}^{\prime})$
when $\bar{\mu}$ is Bernoulli, and allows us to compare the scenery measures
to the uniform “frame” $\nu:=\mu_{\mathtt{a}_{0}^{\prime}}$ as in the case of
the Bernoulli measure. Now, if the collection
$\mathcal{F}\subseteq\mathcal{N}_{0}$ and the words
$\mathtt{b}_{h},\mathtt{b}_{0}\in\Sigma^{*}$ are as in Claim 4.1, we can go
through the proof of Proposition 3.3 to obtain the same result with
$\zeta(\mathtt{i}|_{k})$ being a convex combination of the functions
$0<\zeta_{h}^{\eta}=\frac{d\eta_{\mathtt{b}_{h}}}{d\nu}<\infty$, where
$h\in\mathcal{F}$ and $\eta\in\mathcal{M}$ is such that
$\eta_{\mathtt{b}_{h}}\neq 0$. The ergodic-theoretic work in Subsection 3.2
also goes through, as all properties of Bernoulli measures we use are shared
by ergodic Markov measures. The ergodicity of the skew-product system we
require in the rotating case follows from [23, Theorem 2.2].

###  5.2. Pointwise normality

In [19, Theorem 1.4], Hochman and Shmerkin proved that if
$\Phi=\\{\varphi_{i}(x)=\rho_{i}x+a_{i}\\}_{i\in\Gamma}$ is a self-similar IFS
on the real line with the open set condition and if $s>1$ is a Pisot number
such that $\frac{\log s}{\log\rho_{i}}\not\in\mathbb{Q}$ for some
$i\in\Gamma$, then any self-similar measure associated to $\Phi$ is pointwise
$s$-normal. The open set condition was required partly because of the uniform
scaling assumption.

Combining Theorem 1.1 with the methods of Hochman-Shmerkin, we obtain the
following:

######  Corollary 5.4.

Let $\Phi=\\{\varphi_{i}(x)=\rho_{i}x+a_{i}\\}_{i\in\Gamma}$ be a self-similar
IFS on $\mathbb{R}$ satisfying the weak separation condition, and let $\mu$ be
a self-similar measure associated to $\Phi$. If $s>1$ a Pisot number and
$\frac{\log s}{\log\rho_{i}}\not\in\mathbb{Q}$ for some $i\in\Gamma$, then
$\mu$ is pointwise $s$-normal.

###### Proof.

In the proof of [19, Theorem 1.4], the open set condition was only used to
deduce that (i) $\mu$ is uniformly scaling and generates an ergodic fractal
distribution $P$, and that (ii) given an eigenfunction of $P$, the phase
measure of $\mu$ is an atom. Recall [19, Section 4.3] for the definitions of
an eigenfunction and a phase measure. Since (i) is true by Theorem 1.1, we
will only explain how to deduce that the phase measure of $\mu$ is an atom in
the setting of Corollary 5.4; after this, the proof proceeds exactly as the
proof of [19, Theorem 1.4].

It follows from [19, Proposition 4.15] that given any eigenfunction, the phase
measure of $P$-almost every $\eta$ is an atom. Since $P$ gives positive mass
to the set (5.1), it follows that for some
$\zeta\in\overline{\mathcal{W}},\mathtt{i}\in\Gamma^{\mathbb{N}}$ and $t\geq
0$, the phase measure of $\eta=\zeta\,d\nu_{\mathtt{i},t}$ is an atom. Since
$\mu_{\mathtt{i},t}\ll\eta$, using the self-similarity of $\mu$ we find a
linear map $f$ such that $f\mu\ll\eta$. By [19, Corollary 4.17, part 1], the
phase measure of $f\mu$ is also an atom, and by applying [19, Corollary 4.17,
part 2] to the image of $f\mu$ through $f^{-1}$, we deduce that also the phase
measure of $\mu$ is an atom. ∎

We remark that refinements of [19, Theorem 1.4] have been obtained by Algom,
Rodriguez Hertz and Wang [1] for self-similar measures with the Rajchman
property, and by Dayan, Ganguly and Weiss [2] for certain self-similar
measures defined by contractions with integer contraction ratios. While these
works cover a large class of self-similar measures outside the assumptions of
Hochman-Shmerkin, it is nevertheless not difficult to construct self-similar
measures which satisfy the assumptions of Corollary 5.4 but do not satisfy the
assumptions of [19, Theorem 1.4], [1] or [2]. Consider, for example, the
Bernoulli convolution $\nu_{\rho}$, i.e. the self-similar measure associated
to the IFS $\\{x\mapsto\rho x-1,x\mapsto\rho x+1\\}$ with the uniform
probability vector. It is well-known that when $0<\rho<1$ is such that
$\rho^{-1}$ is a Pisot number, $\nu_{\rho}$ is not a Rajchman measure [3] but
it satisfies the weak separation condition [12, 24]. By Corollary 5.4,
$\nu_{\rho}$ is pointwise $s$-normal for any Pisot $s>1$ such that $\frac{\log
s}{\log\rho}\not\in\mathbb{Q}$.

###  5.3. Prospects

####  5.3.1. Weaker separation conditions

In our proof of Theorem 1.1, the weak separation condition played a crucial
role in providing the reference measure $\nu$ in (3.4). A natural question is
whether the statement of the theorem holds in the presence of weaker
separation conditions, for example, the _asymptotic weak separation condition_
introduced in [5].

######  Question 5.5.

Let $\Phi=\\{\varphi_{i}\\}_{i\in\Gamma}$ be a self-similar IFS satisfying the
asymptotic weak separation condition, that is,

$\lim_{n\rightarrow\infty}\frac{\log(\max_{\mathtt{a}\in\Gamma^{n}}\\#\mathcal{N}(\mathtt{a}))}{n}=0.$

Are self-similar measures associated to $\Phi$ uniformly scaling?

This condition lacks the existence of a maximal neighbourhood system, instead
allowing the cardinalities of the neighbourhood systems to grow
subexponentially in $n$. As a consequence, it is much more difficult to
understand the dynamic behaviour of the sequence
$(\mathcal{N}(\mathtt{i}|_{k}))_{k\in\mathbb{N}}$ for any
$\mathtt{i}\in\Gamma^{\mathbb{N}}$, and it seems that substantial refinements
are required in our argument for one to be able to say anything about the
scenery flow under this condition.

####  5.3.2. A description of the tangent distribution

In our proof of Theorem 1.1, the tangent distribution of $\mu$ arises from a
limiting argument. Therefore, it would be interesting to find a description of
the tangent distribution for at least one explicit self-similar measure which
satisfies the weak separation condition but not the open set condition. This
would require describing the distribution of the sequence
$(\zeta(\mathtt{i}|_{k}))_{k\in\mathbb{N}}$ in (3.2) in more detail than what
is required for the approximation (3.8).

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