# Families of infinite parabolic IFS with overlaps: the approximating method

Liangang Ma  Dept. of Mathematical Sciences, Binzhou University, Huanghe 5th
Road No. 391, Binzhou 256600, Shandong, P. R. China<EMAIL_ADDRESS>
###### Abstract.

This work is devoted to the study of families of infinite parabolic iterated
function systems (PIFS) on a closed interval parametrized by vectors in
$\mathbb{R}^{d}$ with overlaps. We show that the Hausdorff dimension and
absolute continuity of ergodic projections through a family of infinite PIFS
are decided _a.e._ by the growth rate of the entropy and Lyapunov exponents of
the families of truncated PIFS with respect to the concentrating measures,
under transversality of the family essentially. We also give an estimation on
the upper bound of the Hausdorff dimension of parameters where the
corresponding ergodic projections admit certain dimension drop. The setwise
topology on the space of finite measures enables us to approximate a family of
infinite systems by families of their finite truncated sub-systems, which
plays the key role throughout our work.

The work is supported by ZR2019QA003 from SPNSF and 12001056 from NSFC

##  1\. Introduction

This work should be understood in the background of families of overlapping
iterated function systems. A typical family of overlapping _iterated function
system_ (IFS) on a compact metric space consists of a flow of finitely many
strictly contractive endomorphisms on the space parametrized by some (time)
parameter. The dimension of attractors and measures supported on them
(especially the invariant or ergodic ones) is the focus in the theory of
families of IFS. Besides calculating the dimension of attractors and measures
supported on the attractors, another important problem is to decide whether
the projective measures on the attractors from measures on the symbolic spaces
are singular or absolutely continuous with respect to the Lebesgue measure,
refer to [BRS, Hoc1, Hoc3, PSS, Shm2, Shm3, SS1, SSS]. See also [Fur] by
Furstenberg, [Hoc4] by Hochman, [PS1] by Peres-Solomyak and [Shm4] by Shmerkin
for more interesting questions on families of overlapping IFS. These problems
are already very difficult in the case of Bernoulli convolutions, see for
example [BV1, BV2, Hoc1, LPS, Shm1, Sol3, SS2, Var1, Var2, Var3] for recent
progress on the topic.

Note that most of the above research are done on families of finite IFS, that
is, every individual IFS in the family is constituted by finitely many maps.
In this work we try to deal with families of infinite IFS. We choose the
families of parabolic iterated function systems investigated by K. Simon, B.
Solomyak and M. Urbański in [SSU1, SSU2] to demonstrate our ideas, which are
probably applicable to some other families of infinite IFS. Our technique here
is to utilize the sequence of concentrating measures on the attractors of
families of finite sub-systems to approximate the measures on the attractors
of families of infinite systems.

Let $X\subset\mathbb{R}$ be a closed interval with its Borel $\sigma$-algebra
$\mathcal{B}$. Let $\mathcal{\hat{M}}(X)$ be the collection of all the finite
Borel measures on $(X,\mathcal{B})$. For a set $A\subset\mathbb{R}^{d}$ with
$d\in\mathbb{N}$, let $HD(A)$ be its Hausdorff dimension [Fal2].

######  1.1 Definition.

For a measure $\nu\in\mathcal{\hat{M}}(X)$, the _lower_ and _upper Hausdorff
dimension_ of $\nu$ are defined respectively to be:

$dim_{*}\nu=\inf\\{HD(A):\nu(A)>0,A\in\mathcal{B}\\},$

and

$dim^{*}\nu=\inf\\{HD(A):\nu(A)=\nu(X),A\in\mathcal{B}\\}.$

The two values indicate the distribution of a measure on $X$ from two
complementary points of views, with the possibility of the lower one being
strictly less than the upper one. However, they coincide with each other for
ergodic measures with respect to transformations of proper regularity (for
example the $C^{1}$ diffeomorphisms) on $X$. See for example [HS, SimS2, You]
on calculation of the Hausdorff dimension of measures in various
circumstances. Following P. Mattila, M. Morán and J. M. Rey [MMR], the author
discovers some semi-continuity of the measure-dimension mappings $dim_{*}$ and
$dim^{*}$ on $\mathcal{\hat{M}}(X)$ equipped with the setwise topology [FKZ,
Las], which is crucial in our approximating process.

Now we gradually introduce the families of parabolic iterated function
systems. For $\vartheta\in(0,1]$, a point $v\in X$ is called an _indifferent
point_ of a $C^{1+\vartheta}$ map $s:X\rightarrow X$ if

$|s^{\prime}(v)|=1$.

######  1.2 Definition.

A $C^{1+\vartheta}$ map $s:X\rightarrow X$ is called _parabolic_ it satisfies
the following conditions:

  * •

$s$ has only one indifferent point $v$ and it is fixed, that is, $s(v)=v$.

  * •

$s$ is _contractive_ on $X\setminus\\{v\\}$, that is, $0<|s^{\prime}(x)|<1$
for any $x\in X\setminus\\{v\\}$.

  * •

$s$ is _well-behaved_ around $v$, that is, $s^{\prime}(x)$ is monotone on each
component of $X\setminus\\{v\\}$.

  * •

There exist $L_{1}\geq 1$ and $\beta<\cfrac{\vartheta}{1-\vartheta}$ such that

$\cfrac{1}{L_{1}}\leq\liminf_{x\rightarrow
v}\cfrac{|s^{\prime}(x)-s^{\prime}(v)|}{|x-v|^{\beta}}\leq\limsup_{x\rightarrow
v}\cfrac{|s^{\prime}(x)-s^{\prime}(v)|}{|x-v|^{\beta}}\leq L_{1}$.

It is called _hyperbolic_ if $0<|s^{\prime}(x)|<1$ for any $x\in X$. Parabolic
maps with finitely many indifferent points can be handled in our context, with
some technical modifications on the proofs.

######  1.3 Definition.

For $\vartheta\in(0,1]$ and a countable index set $I$ (with at least two
elements), a _parabolic iterated function system_ (PIFS) is a collection of
$C^{1+\vartheta}$ maps

$S=\\{s_{i}:X\rightarrow X\\}_{i\in I}$

satisfying the following conditions:

  * •

There is one and only one $i_{1}\in I$ such that the map $s_{i_{1}}$ is a
parabolic map with its indifferent point $v$.

  * •

The other maps $\\{s_{i}\\}_{i\in I\setminus\\{i_{1}\\}}$ are all hyperbolic.

The collection of PIFS $S=\\{s_{i}:X\rightarrow X\\}_{i\in I}$ satisfying

$\cup_{i\in I\setminus\\{i_{1}\\}}s_{i}(X)\subset X^{o}\setminus\\{v\\}$

is termed $\Gamma_{X}(\vartheta)$. For an integer $d\in\mathbb{N}$ and an open
set $U\subset\mathbb{R}^{d}$, we focus on families of PIFS instead of a single
system in $\Gamma_{X}(\vartheta)$ parametrized by the vector-time parameter
$\mathbb{t}\in U$. The parabolic map is always fixed at any time in the
family. Since families of finite PIFS (that is, $\\#I<\infty$) has been
considered by Simon-Solomyak-Urbański, we focus on the families of infinite
PIFS ($\\#I=\infty$) in the following. That is, we focus on families of PIFS

$S^{\mathbb{t}}=\\{s_{i}^{\mathbb{t}}:X\rightarrow X\\}_{i\in\mathbb{N}}$,

such that $S^{\mathbb{t}}\in\Gamma_{X}(\vartheta)$ with the parabolic map
$s_{1}^{\mathbb{t}}=s_{1}$ remaining the same at any time $\mathbb{t}\in U$.
Let

$\pi_{\mathbb{t}}:I^{\infty}\rightarrow X$

be the projection map and $J_{\mathbb{t}}=\pi_{\mathbb{t}}(I^{\infty})$ be the
attractor ([Hut]) at time $\mathbb{t}\in U$. To achieve some reasonable
conclusions on the parametrized family, obviously some control on the
dependence of the family with respect to $\mathbb{t}$ is necessary. A family
of PIFS $S^{\mathbb{t}}=\\{s_{i}^{\mathbb{t}}:X\rightarrow
X\\}_{i\in\mathbb{N}}$ is said to satisfy the _continuity condition_ if for
any fixed $i\in\mathbb{N}$, the map $s_{i}^{\mathbb{t}}$ depends continuously
on the parameter $\mathbb{t}\in U$ in the Banach space $C^{1+\vartheta}$
equipped with the supreme norm. Let $\mathfrak{L}^{d}$ be the Lebesgue measure
on $\mathbb{R}^{d}$ for $d\in\mathbb{N}$. The family is said to satisfy the
_transversality condition_ if there exists a constant $C_{1}$ such that

$\mathfrak{L}^{d}\\{\mathbb{t}\in
U:|\pi_{\mathbb{t}}(\omega)-\pi_{\mathbb{t}}(\tau)|\leq r\\}\leq C_{1}r$

for any $\omega,\tau\in\mathbb{N}^{\infty},\omega_{1}\neq\tau_{1}$ and any
$r>0$.

While the continuity condition comes more naturally, the transversality
condition has been recognized to be an effective condition to achieve some
uniform conclusions in various contexts of flows of dynamical systems. For
example, see [Fal1, PolS, PS2, PS3, SimS1, Sol1, Sol2].

The following are some classical notions on the symbolic dynamics on
$\mathbb{N}^{\infty}$. For an infinite word
$\omega=\omega_{1}\omega_{2}\cdots\in\mathbb{N}^{\infty}$, let

$\sigma(\omega)=\omega_{2}\omega_{3}\cdots$

be the shift map. For a finite $k$-word $\tau\in\mathbb{N}^{k}$, let

$[\tau]=\\{\omega\in\mathbb{N}^{\infty}:\omega|_{k}=\tau\\}$

be a _cylinder set_ , in which
$\omega|_{k}=\omega_{1}\omega_{2}\cdots\omega_{k}$ is the $k$-th restriction
of the infinite word $\omega$. Let

$\mathbb{N}^{*}=\cup_{k=1}^{\infty}\mathbb{N}^{k}$

be the collection of all the finite words. Now consider the $\sigma$-algebra
$\mathcal{B}_{\mathbb{N}^{\infty}}$ generated by all the cylinder sets on
$\mathbb{N}^{\infty}$. Let $h_{\mu}(\sigma)$ be the entropy of the shift map
$\sigma$ with respect to the partition of $\mathbb{N}^{\infty}$ by cylinder
sets $\\{[i]\\}_{i=1}^{\infty}$ for a measure
$\mu\in\mathcal{\hat{M}}(\mathbb{N}^{\infty})$ ([Hoc2, Wal]).

One can treat $\mu$ as the law of an infinite discrete-time _stochastic
process_

$\mathbb{Y}=\\{Y_{i}\\}_{i\in\mathbb{N}}$,

in which $Y_{i}$ is a discrete random variable on $\mathbb{N}$ for
$i\in\mathbb{N}$ under the natural projection with law
$\mu(\mathbb{N}\times\mathbb{N}\times\cdots\times
Y_{i}\times\mathbb{N}\times\cdots)$. The sequence of infinite random variables
is called _independent_ if the finite variables

$\\{Y_{1},Y_{2},\cdots,Y_{n}\\}$

are independent for any $n\in\mathbb{N}$, see [Tsi, Definition 3a(1),(b)]. The
stochastic process $\mathbb{Y}$ is a _Markov process_ under the hypothesis of
independence, refer to [BHP, FKZ, Hai, HL].

In the following we focus on the measures on $J_{\mathbb{t}}$ projected from
probability measures on $\mathbb{N}^{\infty}$ through $\pi_{\mathbb{t}}$ for
$\mathbb{t}\in U$. For a measure $\mu$ on the symbolic space
$\mathbb{N}^{\infty}$, consider its projection under $\pi_{\mathbb{t}}$:

$\nu_{\mathbb{t}}=\mu\circ\pi_{\mathbb{t}}^{-1}$

for any $\mathbb{t}\in U$. We are particularly interested in the cases when
$\mu$ is invariant or ergodic with respect to the shift map $\sigma$ on
$\mathbb{N}^{\infty}$. If $\mu$ is invariant then $\nu_{\mathbb{t}}$ is of
pure type, see for example [JW]. Let

$\lambda^{\mathbb{t}}_{\mu}(\sigma)=-\int_{\mathbb{N}^{\infty}}\log|s_{\omega_{1}}^{\prime}(\pi_{\mathbb{t}}\circ\sigma(\omega))|d\mu(\omega)$

be the average _Lyapunov exponent_ of the family of PIFS
$S^{\mathbb{t}}=\\{s_{i}^{\mathbb{t}}:X\rightarrow X\\}_{i\in\mathbb{N}}$ at
time $\mathbb{t}\in U$.

Our first main result deals with the two basic questions regarding the
projective measures $\\{\nu_{\mathbb{t}}\\}_{\mathbb{t}\in U}$, these are,
deciding their dimensions $dim_{*}\nu_{\mathbb{t}},dim^{*}\nu_{\mathbb{t}}$
and whether they are absolutely continuous or singular with respect to the
Lebesgue measure for $\mathbb{t}\in U$. Let
$\mathbb{N}_{n}=\\{1,2,\cdots,n\\}$ be the $n$-th truncation of $\mathbb{N}$
for $n\in\mathbb{N}$. We leave the technical concept-the $n$-th concentrating
measure $\mu_{n}$ (supported on $\mathbb{N}_{n}^{\infty}$) of a finite measure
$\mu$ (supported on $\mathbb{N}^{\infty}$) to Section 4. The following result
generalizes [SSU1, Theorem 2.3] from families of finite PIFS to families of
infinite PIFS.

######  1.4 Theorem.

Let

$\big{\\{}S^{\mathbb{t}}=\\{s_{i}^{\mathbb{t}}:X\rightarrow
X\\}_{i\in\mathbb{N}}\in\Gamma_{X}(\vartheta)\big{\\}}_{\mathbb{t}\in U}$

be a family of infinite parabolic iterated function systems satisfying the
continuity and transversality condition with respect to the vector-time
parameter $\mathbb{t}\in U$. For an ergodic probability measure $\mu$ on the
symbolic space $\mathbb{N}^{\infty}$ with positive entropy $h_{\mu}(\sigma)$,
let $\mu_{n}$ be its $n$-th concentrating measure for any $n\in\mathbb{N}$. If
the sequence of infinite random variables in $\mathbb{Y}$ under the law $\mu$
is independent and $\mu(\cup_{n=1}^{\infty}\mathbb{N}_{n}^{\infty})=1$, then

  1. (i).

For Lebesgue _a.e._ $\mathbb{t}\in U$,
$dim_{*}\nu_{\mathbb{t}}=dim^{*}\nu_{\mathbb{t}}=\lim_{n\rightarrow\infty}\min\Big{\\{}\cfrac{h_{\mu_{n}}(\sigma)}{\lambda^{\mathbb{t}}_{\mu_{n}}(\sigma)},1\Big{\\}}$,

  2. (ii).

$\nu_{\mathbb{t}}$ is absolutely continuous for Lebesgue _a.e._
$\mathbb{t}\in\Big{\\{}\mathbb{t}:\limsup_{n\rightarrow\infty}\big{\\{}\cfrac{h_{\mu_{n}}(\sigma)}{\lambda^{\mathbb{t}}_{\mu_{n}}(\sigma)}\big{\\}}>1\Big{\\}}$,

in which $\nu_{\mathbb{t}}=\mu\circ\pi_{\mathbb{t}}^{-1}$ is the projective
measure at time $\mathbb{t}\in U$.

######  1.5 Remark.

We are not sure whether the the hypothesis of independence of the sequence of
infinite random variables under the law $\mu$ can be removed or not, although
it seems so alluring to us.

######  1.6 Remark.

As to the approximation of $\mu$, better results are possible if one does the
approximation by the sequence of restrictions
$\\{\mu|_{\mathbb{N}_{n}^{\infty}}\\}_{n=1}^{\infty}$ instead of the sequence
of concentrating measures $\\{\mu_{n}\\}_{n=1}^{\infty}$. However, the
restrictions are not probabilities, while the concentrating measures take the
advantage that they are all probability measures.

######  1.7 Remark.

Similar to [SSU1, Theorem 2.3], we can only guarantee the result holds _a.e._
instead of everywhere in Theorem 1.4, as it is still impossible to rule out
the exact overlapping between images of
$\\{s^{\mathbb{t}}_{\omega}\\}_{\omega\in\mathbb{N}^{*}}$ for us for fixed
$\mathbb{t}\in U$. There is an important conjecture that a dimension drop
implies exact overlapping, which has been confirmed in the case of finite
Bernoulli convolutions, see for example [Hoc1, Var3].

Note that the proof of [SSU1, Theorem 2.3] does not apply to Theorem 1.4.
There are two main obstacles on application of Simon-Solomyak-Urbański’s
techniques to the families of infinite PIFS. The first one is that
$h_{\mu}(\sigma)$ can explode in case of $\\#I=\infty$, see [Hoc2]. If
$h_{\mu}(\sigma)=\infty$ for some ergodic
$\mu\in\mathcal{\hat{M}}(\mathbb{N}^{\infty})$ with respect to the shift map,
the Shannon-McMillan-Breiman Theorem is not known to hold in this case [Hay].
The second one is the regularity of the families of infinite PIFS. The
collection of PIFS $S=\\{s_{i}:X\rightarrow X\\}_{i\in I}$ satisfying the
following conditions is called $\Gamma_{X}(\vartheta,V,\gamma,u,M)$ in [SSU1].

  1. (a)

There exists an open connected neighbourhood $V$ of $v$, such that

$\cup_{i\in I\setminus\\{i_{1}\\}}s_{i}(X)\cap V=\emptyset$.

  2. (b)

There exists some $\gamma\in(0,1)$, such that

$\sup_{i\in I\setminus\\{i_{1}\\}}\\{\|s_{i}^{\prime}\|\\}\leq\gamma$.

  3. (c)

There exists some $u\in(0,1)$, such that

$\inf_{i\in I,x\in X}\\{|s_{i}^{\prime}(x)|\\}\geq u$.

  4. (d)

There exists $M>0$ such that

$\|S^{\prime}\|_{\vartheta}=\sup_{i\in I}\sup_{x,y\in
X}\Big{\\{}\cfrac{|s_{i}^{\prime}(x)-s_{i}^{\prime}(y)|}{|x-y|^{\vartheta}}\Big{\\}}\leq
M$.

For a finite PIFS, $S\in\Gamma_{X}(\vartheta)$ is enough to guarantee
$S\in\Gamma_{X}(\vartheta,V,\gamma,u,M)$ for some parameters $V,\gamma,u,M$.
However, this is not true for an infinite PIFS. Our technique of approximating
measures on the infinite symbolic space by the sequence of concentrating
measures on its finite subspaces successfully overcomes these obstacles.

Our next main result deals with local dimension of the exceptional parameters
of a family of infinite PIFS $\\{S^{\mathbb{t}}\\}_{\mathbb{t}\in U}$. Let
$\mu$ be a measure on the symbolic space $\mathbb{N}^{\infty}$ with its
concentrating measures $\\{\mu_{n}\\}_{n\in\mathbb{N}}$. For a subset
$G\subset U$ and $0<\alpha<1$, in case the sequence
$\Big{\\{}\cfrac{h_{\mu_{n}}(\sigma)}{\lambda^{\mathbb{t}}_{\mu_{n}}(\sigma)}\Big{\\}}_{n=1}^{\infty}$
admits a limit for any $\mathbb{t}\in G$, let

$E_{\alpha,G}:=\Big{\\{}\mathbb{t}\in
G:dim^{*}\nu_{\mathbb{t}}<\lim_{n\rightarrow\infty}\min\big{\\{}\cfrac{h_{\mu_{n}}(\sigma)}{\lambda^{\mathbb{t}}_{\mu_{n}}(\sigma)},\alpha\big{\\}}\Big{\\}}$

be the level-$\alpha$ exceptional parameters in $G$. Let

$K_{\alpha,G}=\min\Big{\\{}\sup_{\mathbb{t}\in
G}\lim_{n\rightarrow\infty}\cfrac{h_{\mu_{n}}(\sigma)}{\lambda^{\mathbb{t}}_{\mu_{n}}(\sigma)},\alpha\Big{\\}}+d-1$.

For a parameter set $A\subset U\subset\mathbb{R}^{d}$, let $N_{r}(A)$ be the
least number of open balls of radius $r>0$ needed to cover the set $A$. A
family of PIFS $S^{\mathbb{t}}=\\{s_{i}^{\mathbb{t}}:X\rightarrow
X\\}_{i\in\mathbb{N}}$ is said to satisfy the _strong transversality
condition_ if there exists a constant $C_{2}>0$ such that

$N_{r}\\{\mathbb{t}\in U:|\pi_{\mathbb{t}}(\omega)-\pi_{\mathbb{t}}(\tau)|\leq
r\\}\leq C_{2}r^{1-d}$

for any $\omega,\tau\in\mathbb{N}^{\infty},\omega_{1}\neq\tau_{1}$ and any
$r>0$. Obviously this condition is stronger than the transversality condition.
We have the following estimation on the upper bound of the Hausdorff dimension
of the local level-$\alpha$ exceptional set.

######  1.8 Theorem.

Let

$\big{\\{}S^{\mathbb{t}}=\\{s_{i}^{\mathbb{t}}:X\rightarrow
X\\}_{i\in\mathbb{N}}\in\Gamma_{X}(\vartheta)\big{\\}}_{\mathbb{t}\in U}$

be a family of infinite parabolic iterated function systems satisfying the
continuity and strong transversality condition with respect to the vector-time
parameter $\mathbb{t}\in U$. For an ergodic probability measure $\mu$ on the
symbolic space $\mathbb{N}^{\infty}$ with
$\mu(\cup_{n=1}^{\infty}\mathbb{N}_{n}^{\infty})=1$, if the sequence of
infinite random variables in $\mathbb{Y}$ under the law $\mu$ is independent,
then

(1.1) $HD(E_{\alpha,G})\leq K_{\alpha,G}$

for any $0<\alpha<1$ and $G\subset U$ such that
$\lim_{n\rightarrow\infty}\cfrac{h_{\mu_{n}}(\sigma)}{\lambda^{\mathbb{t}}_{\mu_{n}}(\sigma)}$
exists for any $\mathbb{t}\in G$, in which $\\{\mu_{n}\\}_{n\in\mathbb{N}}$
are the concentrating measures.

This result is a generalization of [SSU1, Theorem 5.3] from families of finite
PIFS to families of infinite PIFS. See also [Kau, Theorem].

##  2\. The setwise topology and semi-continuity of the measure-dimension
mappings under it

Since we are considering families of IFS in this work, it would be very
helpful if the measure-dimension mappings

$dim_{*}:\mathcal{\hat{M}}(X)\rightarrow[0,\infty)$

or

$dim^{*}:\mathcal{\hat{M}}(X)\rightarrow[0,\infty)$

admits some continuity property. To discuss the continuity problem, some
appropriate topology on $\mathcal{\hat{M}}(X)$ is needed. Since the measures
$\\{\nu_{\mathbb{t}}\\}_{\mathbb{t}\in U}\subset\mathcal{\hat{M}}(X)$ in
Theorem 1.4 inherit the parametrization naturally from parametrization of the
PIFS, one may try to endow the Euclidean metric on
$\\{\nu_{\mathbb{t}}\\}_{\mathbb{t}\in U}$ through the parameter
$\mathbb{t}\in U\subset\mathbb{R}^{d}$ to discuss the continuity of the
measure-dimension mappings. The measure-dimension mappings are not always
continuous under the Euclidean metric on
$\\{\nu_{\mathbb{t}}\\}_{\mathbb{t}\in U}$ in general consideration, although
lower semi-continuity of them can be expected. One can see from the family of
Bernoulli convolutions
$\\{\nu_{\lambda}\\}_{\lambda}\subset\mathcal{\hat{M}}([0,1])$ with the
contraction ratio $\lambda\in(0,1)$. The measure-dimension mappings

$dim_{*}=dim^{*}:(0,1)\rightarrow[0,1]$

are not always continuous since there are dimension drops from $1$ at the
inverses of the Pisot parameters. However, they are lower semi-continuous [HS,
Theorem 1.8].

Since we are dealing with measures originated from families of the infinite
systems and their finite sub-systems simultaneously in this work, we will
abandon the Euclidean metric on $\\{\nu_{\mathbb{t}}\\}_{\mathbb{t}\in
U}\subset\mathcal{\hat{M}}(X)$ to discuss the continuity problem. A well-known
topology on $\mathcal{\hat{M}}(X)$ is the weak topology.

######  2.1 Definition.

A sequence of bounded measures
$\\{\nu_{n}\in\mathcal{\hat{M}}(X)\\}_{n=1}^{\infty}$ is said to converge
_weakly_ to $\nu\in\mathcal{\hat{M}}(X)$, if

$\lim_{n\rightarrow\infty}\int_{X}f(x)d\nu_{n}=\int_{X}f(x)d\nu$

for any bounded continuous $f:X\rightarrow\mathbb{R}$.

Denote the convergence in this sense by $\nu_{n}\stackrel{{\scriptstyle
w}}{{\rightarrow}}\nu$ as $n\rightarrow\infty$. However, The measure-dimension
mappings $dim^{*}$ and $dim_{*}$ do not admit any semi-continuity under the
weak topology on $\mathcal{\hat{M}}(X)$ [Ma, Theorem 3.1]. It turns out that
the setwise topology is the most ideal topology for us to discuss the
continuity of the measure-dimension mappings.

######  2.2 Definition.

A sequence of measures $\\{\nu_{n}\in\mathcal{\hat{M}}(X)\\}_{n=1}^{\infty}$
is said to converge _setwisely_ to $\nu\in\mathcal{\hat{M}}(X)$, if

$\lim_{n\rightarrow\infty}\nu_{n}(B)=\nu(B)$

for any $B\in\mathcal{B}$.

One is recommended to refer to [Doo, FKZ, GR, HL, Las, LY] for more equivalent
descriptions of the setwise topology on $\mathcal{\hat{M}}(X)$ and its various
applications. Denote the convergence in this sense by
$\nu_{n}\stackrel{{\scriptstyle s}}{{\rightarrow}}\nu$ as
$n\rightarrow\infty$. The measure-dimension mappings $dim^{*}$ and $dim_{*}$
are lower semi-continuous and upper semi-continuous under the setwise topology
respectively on $\mathcal{\hat{M}}(X)$.

######  2.3 Theorem.

The measure-dimension mapping $dim^{*}$ is lower semi-continuous, while
$dim_{*}$ is upper semi-continuous under the setwise topology on
$\mathcal{\hat{M}}(X)$, that is, if $\nu_{n}\stackrel{{\scriptstyle
s}}{{\rightarrow}}\nu$ in $\mathcal{\hat{M}}(X)$ as $n\rightarrow\infty$, then

(2.1) $\liminf_{n\rightarrow\infty}dim^{*}\nu_{n}\geq dim^{*}\nu$

while

(2.2) $\limsup_{n\rightarrow\infty}dim_{*}\nu_{n}\leq dim_{*}\nu.$

######  2.4 Remark.

Similar results appear in [Ma, Theorem 2.8, 2.9] when one considers the
measure-dimension mappings $dim^{*}$ and $dim_{*}$ on the probability space

$\mathcal{M}(X)=\\{\nu:\nu(X)=1,\nu\in\mathcal{\hat{M}}(X)\\}$

with $X$ being an arbitrary metric space. The proofs apply to the measure-
dimension mappings on the space of finite measures $\mathcal{\hat{M}}(X)$
here, or even on the space of infinite measures.

Considering generalizations of measures, Theorem 2.3 is still true on the
space of _finitely additive measures_ (a finitely additive measure $\nu$ on
$(X,\mathcal{B})$ is a function from $\mathcal{B}$ to $[0,\infty)$ satisfying
finite additivity instead of $\sigma$-additivity), but not true for a _signed
measure_ on $(X,\mathcal{B})$ under the Definition 1.1. Obviously a new
meterage for the dimension of signed measures is needed. Let
$\mathcal{\hat{M}}_{s}(X)$ be the collection of all the finite signed Borel
measures on $X$.

######  2.5 Definition.

For a finite signed measure $\nu\in\mathcal{\hat{M}}_{s}(X)$, the _lower_ and
_upper Hausdorff dimension_ of $\nu$ are defined respectively to be:

$dim_{s}\nu=\inf\\{HD(A):\nu(A)>0\mbox{ or }\nu(A)<0,A\in\mathcal{B}\\},$

and

$dim^{s}\nu=\max\Big{\\{}\inf\big{\\{}HD(A):\nu(A)=\max\\{\nu(B)\\}_{B\in\mathcal{B}}\big{\\}},\inf\big{\\{}HD(A):\nu(A)=\min\\{\nu(B)\\}_{B\in\mathcal{B}}\big{\\}}\Big{\\}}.$

Under this modified definition the measure-dimension mappings $dim_{s}$ and
$dim^{s}$ from $\mathcal{\hat{M}}_{s}(X)$ to $[0,\infty)$ still bear some
continuity.

######  2.6 Proposition.

The measure-dimension mapping $dim^{s}$ is lower semi-continuous, while
$dim_{s}$ is upper semi-continuous under the setwise topology on
$\mathcal{\hat{M}}_{s}(X)$, that is, if $\nu_{n}\stackrel{{\scriptstyle
s}}{{\rightarrow}}\nu$ in $\mathcal{\hat{M}}_{s}(X)$ as $n\rightarrow\infty$,
then

(2.3) $\liminf_{n\rightarrow\infty}dim^{s}\nu_{n}\geq dim^{s}\nu$

while

(2.4) $\limsup_{n\rightarrow\infty}dim_{s}\nu_{n}\leq dim_{s}\nu.$

###### Proof.

The conclusion $dim_{s}$ is upper semi-continuous follows a similar argument
as the proof of [Ma, Theorem 2.9]. Now we show $dim^{s}$ is lower semi-
continuous under the setwise topology on $\mathcal{\hat{M}}_{s}(X)$. Without
loss of generality we assume

$\inf\big{\\{}HD(A):\nu(A)=\max\\{\nu(B)\\}_{B\in\mathcal{B}}\big{\\}}\geq\inf\big{\\{}HD(A):\nu(A)=\min\\{\nu(B)\\}_{B\in\mathcal{B}}\big{\\}}$

and

$\inf\big{\\{}HD(A):\nu(A)=\max\\{\nu(B)\\}_{B\in\mathcal{B}}\big{\\}}>0$,
$\max\\{\nu(B)\\}_{B\in\mathcal{B}}>0$.

Suppose conversely that (2.3) does not hold. Then we can find a subsequence
$\\{n_{i}\\}_{i=1}^{\infty}$ and a real
$0<a<\inf\big{\\{}HD(A):\nu(A)=\max\\{\nu(B)\\}_{B\in\mathcal{B}}\big{\\}}$
such that

(2.5)
$\lim_{i\rightarrow\infty}dim^{s}\nu_{n_{i}}<a<dim^{s}\nu=\inf\big{\\{}HD(A):\nu(A)=\max\\{\nu(B)\\}_{B\in\mathcal{B}}\big{\\}}.$

Now apply the Hahn decomposition theorem to the measure $\nu$, let
$X^{+}\subset X$ be a positive set of $\nu$ with
$\nu(X^{+})=\max\\{\nu(B)\\}_{B\in\mathcal{B}}$. Under the above assumption we
have

$HD(X^{+})\geq dim^{s}\nu>a>dim^{s}\nu_{n_{i}}$

for any $i$ large enough. Since
$\lim_{i\rightarrow\infty}\nu_{n_{i}}(X^{+})=\nu(X^{+})>0$, considering (2.5),
for any $i$ large enough, there must exist $X^{+}_{i}\subset X^{+}$, such that

$\nu_{n_{i}}(X^{+}_{i})=0$ and $HD(X^{+}_{i})=HD(X^{+})>a>HD(X^{+}\setminus
X^{+}_{i})$.

Now let $X^{+}_{\infty}=\cup_{i=N}^{\infty}(X^{+}\setminus X^{+}_{i})$ for
some integer $N$ large enough. One can see that

$X^{+}\setminus X^{+}_{\infty}\neq\emptyset$

as

(2.6) $HD(X^{+}_{\infty})\leq a<HD(X^{+}).$

Moreover, we have

$\nu(X^{+}\setminus
X^{+}_{\infty})=\lim_{i\rightarrow\infty}\nu_{n_{i}}(X^{+}\setminus
X^{+}_{\infty})=0$,

which forces

(2.7) $\nu(X^{+}_{\infty})=\nu(X^{+}).$

Now (2.6) together with (2.7) contradict the fact that $dim^{s}\nu>a$, which
finishes the proof.

∎

##  3\. Absolute continuity of convergent sequences of measures under the
setwise topology

In this section we discuss the relationship between absolute continuity of a
sequence of measures and absolute continuity of its limit measure (under the
weak or setwise topology, in case the sequence converges). We start from the
following basic result.

######  3.1 Proposition.

For a sequence of measures $\nu_{n}\stackrel{{\scriptstyle
s}}{{\rightarrow}}\nu$ in $\mathcal{\hat{M}}(X)$ as $n\rightarrow\infty$, if
there exists a subsequence $\\{\nu_{n_{i}}\\}_{i=1}^{\infty}$ such that
$\nu_{n_{i}}$ is absolutely continuous with respect to some
$\varrho\in\mathcal{\hat{M}}(X)$ for any $1\leq i<\infty$, then $\nu$ is
absolutely continuous with respect to $\varrho$.

###### Proof.

We do this by reduction to absurdity. Suppose that $\nu$ is not absolutely
continuous with respect to $\varrho$, then there exists $A\in\mathcal{B}$ such
that $\nu(A)>0$ while $\varrho(A)=0$. Since $\nu_{n}\stackrel{{\scriptstyle
s}}{{\rightarrow}}\nu$ as $n\rightarrow\infty$, there exists $N$ large enough
such that $\nu_{n_{N}}(A)>0$. Considering $\varrho(A)=0$, this contradicts the
fact that $\nu_{n_{N}}$ is absolutely continuous with respect to $\varrho$.

∎

Due to Proposition 3.1, we make the following definition formally.

######  3.2 Definition.

A sequence of measures $\\{\nu_{n}\\}_{n=1}^{\infty}$ is said to be
_absolutely continuous_ with respect to some $\varrho$ on
$\mathcal{\hat{M}}(X)$ if it contains a subsequence such that every measure in
the subsequence is absolutely continuous with respect to $\varrho$.

In fact Proposition 3.1 holds for any topological ambient space $X$. However,
it is possible that a sequence of absolutely continuous measures converges
weakly to a measure which is not absolutely continuous, as one can see from
the Example 3.3. Let $\mathfrak{L}^{d}|_{A}$ be the restriction of
$\mathfrak{L}^{d}$ on a set $A\subset\mathbb{R}^{d}$.

######  3.3 Example.

Consider the Cantor middle-third set $J$ appears as the attractor of the
homogeneous IFS

$S=\big{\\{}s_{1}(x)=\frac{1}{3}x,s_{2}(x)=\frac{1}{3}x+\frac{2}{3}\big{\\}}$

on $X=[0,1]$. Let

$\nu_{n}=\frac{1}{2^{n}}\mathfrak{L}^{1}|_{\cup_{\omega\in\mathbb{N}_{2}^{n}}s_{\omega}(X)}$.

Let $\nu$ be the unique probability such that

$\nu=\frac{1}{2}\nu\circ s_{1}^{-1}+\frac{1}{2}\nu\circ s_{2}^{-1}$.

Unfortunately, the inverse of Proposition 3.1 is not always true, as one can
see from the following example. Let $\delta_{\\{x\\}}$ be the Dirac
probability measure at the point $x\in X$.

######  3.4 Example.

For $X=[0,1]$, let

$\nu_{n}=\mathfrak{L}^{1}|_{X}+\frac{1}{n}\delta_{\\{1\\}}$

be a sequence of measures in $\mathcal{\hat{M}}(X)$.

It is easy to justify that $\nu_{n}\stackrel{{\scriptstyle
s}}{{\rightarrow}}\mathfrak{L}^{1}|_{X}$ as $n\rightarrow\infty$ in Example
3.4. However, there are no measure absolutely continuous with respect to
$\mathfrak{L}^{1}|_{X}$ at all in the sequence $\\{\nu_{n}\\}_{n=1}^{\infty}$.
One can see that in Example 3.4 the measures in the sequence are all of mixed
type. In fact this is the only reason which prevents the inverse of
Proposition 3.1 to be true.

######  3.5 Proposition.

For a sequence of measures $\nu_{n}\stackrel{{\scriptstyle
s}}{{\rightarrow}}\nu$ in $\mathcal{\hat{M}}(X)$ as $n\rightarrow\infty$ with
$\nu(X)>0$, if $\nu$ is absolutely continuous with respect to some
$\varrho\in\mathcal{\hat{M}}(X)$ and the measures in the sequence are all of
pure type with respect to $\varrho$, then there exists $N\in\mathbb{N}$ large
enough such such that $\nu_{n}$ is absolutely continuous with respect to
$\varrho\in\mathcal{\hat{M}}(X)$ for any $n\geq N$.

###### Proof.

Suppose in the contrary that the conclusion is not true, then we can find a
subsequence $\\{n_{i}\\}_{i=1}^{\infty}\subset\mathbb{N}$ such that any
measure in the sequence $\\{\nu_{n_{i}}\\}_{i=1}^{\infty}$ is singular with
respect to $\varrho$. Denote an essential support of $\nu_{n_{i}}$ by
$A_{i}\subset X$ with $\varrho(A_{i})=0$ for any $1\leq i<\infty$. Let
$A=\cup_{i=1}^{\infty}A_{i}$, according to the $\sigma$-additivity of
$\varrho$, we have

(3.1) $\varrho(A)=0.$

Note that since $\nu_{n}\stackrel{{\scriptstyle s}}{{\rightarrow}}\nu$ as
$n\rightarrow\infty$, we have

(3.2)
$\nu(A)=\lim_{i\rightarrow\infty}\nu_{n_{i}}(A)=\lim_{i\rightarrow\infty}\nu_{n_{i}}(X)=\nu(X)>0.$

Now (3.1) together with (3.2) contradict the fact that $\nu$ is absolutely
continuous with respect to $\varrho$. ∎

Proposition 3.5 also holds for any topological ambient space $X$, but it is
not true for convergent sequences of measures under the weak topology on
$\mathcal{\hat{M}}(X)$, as one can also see from our Example 3.3. The above
results show that the absolute continuity of convergent sequences of measures
and their limit measures are equivalent to each other under the setwise
topology on $\mathcal{\hat{M}}(X)$, while this relationship is not true under
the weak topology. This is another advantage that the setwise topology takes
over the weak topology on $\mathcal{\hat{M}}(X)$.

##  4\. The $n$-th concentrating measures of probability on the symbolic
spaces and some inherited properties

In this section we do the approximation of a probability measure $\mu$ on the
infinite symbolic space $\mathbb{N}^{\infty}$ by the sequence of its
concentrating measures $\\{\mu_{n}\\}_{n=1}^{\infty}$ supported on
$\\{\mathbb{N}_{n}^{\infty}\\}_{n=1}^{\infty}$ respectively. We will prove
that some properties are inherited by the concentrating measures from $\mu$,
under the hypothesis that the sequence of infinite random variables in the
infinite stochastic process

$\mathbb{Y}=\\{Y_{i}\\}_{i\in\mathbb{N}}$

is independent under the law $\mu$. As indicated in Remark 1.6, one may be
able to remove the independent hypothesis in due course if one tries the
approximation of $\mu$ by its restrictions
$\\{\mu|_{\mathbb{N}_{n}^{\infty}}\\}_{n=1}^{\infty}\subset\mathcal{\hat{M}}(\mathbb{N}_{n}^{\infty})$
instead of the concentrating measures
$\\{\mu_{n}\\}_{n=1}^{\infty}\subset\mathcal{M}(\mathbb{N}_{n}^{\infty})$.

######  4.1 Definition.

For a probability measure $\mu$ on the symbolic space $\mathbb{N}^{\infty}$,
define its $n$-th _concentrating measure_ $\mu_{n}$ for any $n\in\mathbb{N}$
to be the unique probability measure supported on $\mathbb{N}_{n}^{\infty}$
satisfying the following conditions.

  1. (1)

On the first level cylinders,

$\mu_{n}([i])=\mu([i])$ for $1\leq i\leq n-1$,
$\mu_{n}([n])=\sum_{i=n}^{\infty}\mu([i])$.

  2. (2)

On the second level cylinders,

$\begin{array}[]{l}\mu_{n}([ij])=\mu([ij])\mbox{ for }1\leq i,j\leq n-1,\\\
\mu_{n}([in])=\sum_{k=n}^{\infty}\mu([ik])\mbox{ for }1\leq i\leq n-1,\\\
\mu_{n}([nj])=\sum_{k=n}^{\infty}\mu([kj])\mbox{ for }1\leq j\leq n-1,\\\
\mu_{n}([nn])=\sum_{l=n}^{\infty}\sum_{k=n}^{\infty}\mu([kl]).\end{array}$

  3. (3)

For any $q$-th cylinder $[i_{1}i_{2}\cdots
i_{q}]\subset\mathbb{N}_{n}^{\infty}$ with
$i_{1},i_{2},\cdots,i_{q}\in\mathbb{N}_{n}$ and $q\in\mathbb{N}$, let

$\\#\\{1\leq k\leq q:i_{k}=n\\}=l$.

Number the indexes in $\\{1\leq k\leq q:i_{k}=n\\}$ in increasing order as

$j_{1}<j_{2}<\cdots<j_{l}$,

that is, $\\{j_{1},j_{2},\cdots,j_{l}\\}=\\{1\leq k\leq q:i_{k}=n\\}$. Then

$\begin{array}[]{ll}&\mu_{n}([i_{1}i_{2}\cdots i_{q}])\\\ =&\sum_{n\leq
r_{1},r_{2},\cdots,r_{l}<\infty}\mu([i_{1}\cdots
i_{j_{1}-1}r_{1}i_{j_{1}+1}\cdots i_{j_{2}-1}r_{2}i_{j_{2}+1}\cdots
i_{j_{l}-1}r_{l}i_{j_{l}+1}\cdots i_{q}]).\end{array}$

Since for any $q$-level cylinder $[i_{1}i_{2}\cdots
i_{q}]\subset\mathbb{N}_{n}^{\infty}$ with $i_{1},i_{2},\cdots
i_{q}\in\mathbb{N}_{n}$ and $q\in\mathbb{N}$, the conditions in Definition 4.1
(1)-(3) imply

$\mu_{n}([i_{1}i_{2}\cdots i_{q}])=\sum_{j=1}^{n}\mu_{n}([i_{1}i_{2}\cdots
i_{q}j])$

between cylinders of successive levels, the existence and uniqueness of the
measure $\mu_{n}$ are guaranteed by the _Kolmogorov extension theorem_ , see
for example [Tao, Theorem 2.4.3]. Moreover, $\mu_{n}$ is invariant for any
$n\in\mathbb{N}$ if $\mu$ is invariant with respect to the shift map $\sigma$
according to [Ber, Theorem 2]. In fact, more properties are inherited from
$\mu$ as the structure of the original measure is suitably preserved
considering the structure of the concentrating measures
$\\{\mu_{n}\\}_{n\in\mathbb{N}}$ on the cylinder sets, under the independent
hypothesis of $\mathbb{Y}$ with respect to $\mu$.

######  4.2 Lemma.

For a measure $\mu$ on the symbolic space $\mathbb{N}^{\infty}$, if the
sequence of infinite random variables in $\mathbb{Y}$ under the law $\mu$ is
independent, then it is also independent under the concentrating law $\mu_{n}$
for any $n\in\mathbb{N}$.

###### Proof.

For any $q$-word $\omega=i_{1}i_{2}\cdots i_{q}\in\mathbb{N}_{n}^{q}$, let
$\\#\\{1\leq k\leq q:i_{k}=n\\}=l\geq 0$ and

$\\{j_{1},j_{2},\cdots,j_{l}\\}=\\{1\leq k\leq q:i_{k}=n\\}$.

Since $\mathbb{Y}$ is independent under the law $\mu$, we have

$\begin{array}[]{ll}&\mu_{n}([\omega])\\\ =&\sum_{n\leq r_{1}<\infty,n\leq
r_{2}<\infty,\cdots,n\leq r_{l}<\infty}\mu([i_{1}\cdots
i_{j_{1}-1}r_{1}i_{j_{1}+1}\cdots i_{j_{2}-1}r_{2}i_{j_{2}+1}\cdots
i_{j_{l}-1}r_{l}i_{j_{l}+1}\cdots i_{q}])\\\
=&\mu([i_{1}])\cdots\mu([i_{j_{1}-1}])\big{(}\sum_{n\leq
r_{1}<\infty}\mu([r_{1}])\big{)}\mu([i_{j_{1}+1}])\cdots\mu([i_{j_{2}-1}])\big{(}\sum_{n\leq
r_{2}<\infty}\mu([r_{2}])\big{)}\mu([i_{j_{2}+1}])\\\
&\cdots\mu([i_{j_{l}-1}])\big{(}\sum_{n\leq
r_{l}<\infty}\mu([r_{l}])\big{)}\mu([i_{j_{l}+1}])\cdots\mu([i_{q}])\\\
=&\mu_{n}([i_{1}])\cdots\mu_{n}([i_{j_{1}-1}])\mu_{n}([i_{j_{1}}])\mu_{n}([i_{j_{1}+1}])\cdots\mu_{n}([i_{j_{2}-1}])\mu_{n}([i_{j_{2}}])\mu_{n}([i_{j_{2}+1}])\cdots\\\
&\mu_{n}([i_{j_{l}-1}])\mu_{n}([i_{j_{l}}])\mu_{n}([i_{j_{l}+1}])\cdots\mu_{n}([i_{q}]),\end{array}$

which justifies the independence of the random variables in $\mathbb{Y}$ under
the law $\mu_{n}$ for any $n\in\mathbb{N}$. ∎

The ergodicity is also inherited by the concentrating measures
$\\{\mu_{n}\\}_{n\in\mathbb{N}}$ under the independent hypothesis of
$\mathbb{Y}$.

######  4.3 Lemma.

For an ergodic measure $\mu$ on the symbolic space $\mathbb{N}^{\infty}$, if
the sequence of infinite random variables in $\mathbb{Y}$ under the law $\mu$
is independent, then the concentrating measure $\mu_{n}$ is also ergodic on
$\mathbb{N}_{n}^{\infty}$ for any $n\in\mathbb{N}$ with respect to the shift
map $\sigma$.

###### Proof.

This is a standard proof in classical probability techniques, with all the
above preparations. First, due to Lemma 4.2 and the independence assumption,
the sequence of random variables $\\{Y_{i}\\}_{i=1}^{\infty}$ in $\mathbb{Y}$
under the law $\mu_{n}$ is independent for any $n\in\mathbb{N}$. Now let
$\mathcal{B}_{\mathbb{N}_{n}^{\infty}}$ be the $\sigma$-algebra generated by
the cylinder sets in $\mathbb{N}_{n}^{\infty}$ for some $n\in\mathbb{N}$. Let

$\mathcal{B}_{k}=\\{\mathbb{N}_{n}^{k}\times
B:B\in\mathcal{B}_{\mathbb{N}_{n}^{\infty}}\\}$.

The tail $\sigma$-algebra is define to be

$\mathcal{B}_{T}=\cap_{k\in\mathbb{N}}\mathcal{B}_{k}$.

Obviously the events

$\\{A\in\mathcal{B}_{\mathbb{N}_{n}^{\infty}}:\sigma^{-1}(A)=A\\}\subset\mathcal{B}_{T}$

are all tail events. So according to the Kolmogorov’s $0-1$ law (see for
example [Tsi, Proposition 3b9] or [Shi]),

$\mu_{n}(A)=0$ or $1$

for any event in
$\\{A\in\mathcal{B}_{\mathbb{N}_{n}^{\infty}}:\sigma^{-1}(A)=A\\}$. ∎

######  4.4 Remark.

Similar to Remark 1.5, we are wondering whether the result holds without the
hypothesis of independence on the sequence of infinite random variables with
law $\mu$, which essentially confines our main results to be under this
condition.

Considering the limit behaviour of the sequence of concentrating measures
$\\{\mu_{n}\\}_{n\in\mathbb{N}}$, we have the following simple but important
result.

######  4.5 Proposition.

For a probability measure $\mu$ on the symbolic space $\mathbb{N}^{\infty}$
such that $\mu(\cup_{n=1}^{\infty}\mathbb{N}_{n}^{\infty})=1$, let $\mu_{n}$
be its $n$-th concentrating measure on $\mathbb{N}_{n}^{\infty}$ for any
$n\in\mathbb{N}$. Then

$\mu_{n}\stackrel{{\scriptstyle s}}{{\rightarrow}}\mu$

as $n\rightarrow\infty$.

###### Proof.

It is obvious that for any finite word $\omega\in\mathbb{N}^{*}$, we have

$\lim_{n\rightarrow\infty}\mu_{n}([\omega])=\mu([\omega])$,

as $\mu_{n}([\omega])=\mu([\omega])$ for any $n$ large enough. Since $\mu$ is
supported on $\cup_{n=1}^{\infty}\mathbb{N}_{n}^{\infty}$, the convergence
extends to all measurable sets in $\mathbb{N}^{\infty}$ due to regularity of
the measures, or [FKZ, Theorem 2.3]. ∎

######  4.6 Remark.

The convergence of the sequence of concentrating measures
$\\{\mu_{n}\\}_{n\in\mathbb{N}}$ of a measure $\mu$ is usually not true under
the total variation (TV) topology on $\mathcal{M}(\mathbb{N}^{\infty})$,
except in some trivial cases.

The entropy is also inherited by the sequence of concentrating measures
$\\{\mu_{n}\\}_{n\in\mathbb{N}}$ from the original measure $\mu$, considering
Proposition 4.5 as well as the following result.

######  4.7 Lemma.

For a finite measure $\mu\in\mathcal{\hat{M}}(\mathbb{N}^{\infty})$ and its
concentrating measures $\\{\mu_{n}\\}_{n\in\mathbb{N}}$, we have

$\lim_{n\rightarrow\infty}h_{\mu_{n}}(\sigma)=h_{\mu}(\sigma)$

with respect to $(\mathbb{N}^{\infty},\mathcal{B}_{\mathbb{N}^{\infty}})$.

###### Proof.

We justify the result in case of $h_{\mu}(\sigma)=\infty$ and
$h_{\mu}(\sigma)<\infty$ respectively.

If

$h_{\mu}(\sigma)=\liminf_{k\rightarrow\infty}-\frac{1}{k}\sum_{\omega\in\mathbb{N}^{k}}\mu([\omega])\log\mu([\omega])=\infty$,

we have

$-\frac{1}{k}\sum_{\omega\in\mathbb{N}^{k}}\mu([\omega])\log\mu([\omega])=\infty$

for any $k\in\mathbb{N}$. Then for any $a>0$ and $k\in\mathbb{N}$, we can find
$N_{a,k}$ large enough, such that

$-\frac{1}{k}\sum_{\omega\in\mathbb{N}_{n}^{k}}\mu_{n}([\omega])\log\mu_{n}([\omega])>a$

for any $n>N_{a,k}$. This is enough to force
$\lim_{n\rightarrow\infty}h_{\mu_{n}}(\sigma)=h_{\mu}(\sigma)=\infty$ by
reduction to absurdity.

Now if

$h_{\mu}(\sigma)=\liminf_{k\rightarrow\infty}-\frac{1}{k}\sum_{\omega\in\mathbb{N}^{k}}\mu([\omega])\log\mu([\omega])<\infty$,

then for any small $\epsilon>0$, we can find $K_{\epsilon}$ large enough, such
that

$h_{\mu}(\sigma)\leq-\frac{1}{k}\sum_{\omega\in\mathbb{N}^{k}}\mu([\omega])\log\mu([\omega])<h_{\mu}(\sigma)+\epsilon$

for any $k>K_{\epsilon}$. Now for fixed $k>K_{\epsilon}$, since
$-\sum_{\omega\in\mathbb{N}^{k}}\mu([\omega])\log\mu([\omega])<\infty$, we can
find $N_{k,\epsilon}$ large enough such that

$\big{|}\sum_{\omega\in\mathbb{N}^{k}}\mu([\omega])\log\mu([\omega])-\sum_{\omega\in\mathbb{N}_{n}^{k}}\mu_{(}[\omega])\log\mu_{n}([\omega])\big{|}<\epsilon$

for any $n>N_{k,\epsilon}$. This is enough to force
$\lim_{n\rightarrow\infty}h_{\mu_{n}}(\sigma)=h_{\mu}(\sigma)$.

∎

######  4.8 Remark.

In fact Lemma 4.7 holds for any sequence of measures converging to the finite
measure $\mu$ under the setwise topology in $\mathcal{\hat{M}}(X)$, not only
for the sequence of its concentrating measures, for any topological ambient
space $X$. The version now is enough for our purpose in this work.

##  5\. Setwise approximation of projective measures on the attractors of
families of infinite PIFS

In this section we aim at proving Theorem 1.4. After the proof we formulate
some interesting applications of Theorem 1.4 in some circumstances. We first
show a result which links the upper and lower dimensions of the projective
measure $\nu$ on the attractor $J$ of a single PIFS.

######  5.1 Lemma.

Let

$S=\\{s_{i}:X\rightarrow X\\}_{i\in I}$

be a parabolic iterated function system with a countable index set $I$. For an
ergodic measure $\mu$ on $I^{\infty}$, let $\nu=\mu\circ\pi^{-1}$ be its
projective measure. Then

$dim_{*}\nu=dim^{*}\nu$.

###### Proof.

To prove the result, it suffices for us to prove

(5.1) $dim_{*}\nu\geq dim^{*}\nu.$

Let $J$ be the attractor of the PIFS. For a set $A\subset J$ with
$\nu(A)=\mu\circ\pi^{-1}(A)>0$, consider the set

$\pi\circ\sigma^{-1}\circ\pi^{-1}(A)=\pi(\cup_{i\in I}i\pi^{-1}(A))=\cup_{i\in
I}\pi(i\pi^{-1}(A))$.

Note that for any $i\in I$, due to
$\pi(\omega)=s_{\omega|_{n}}\circ\pi\circ\sigma^{n}(\omega)$ for any
$\omega\in I^{\infty}$, we have

$\pi(i\pi^{-1}(A))=s_{i}(A)$,

in which $i\pi^{-1}(A):=\\{\omega\in
I^{\infty}:\omega_{1}=i,\sigma(\omega)\in\pi^{-1}(A)\\}$. Since
$0<s_{i}^{\prime}(x)\leq 1$ for any $i\in I$, we have

$HD\big{(}\pi(i\pi^{-1}(A))\big{)}=HD(A)$

for any $i\in I$. Since $I$ is countable, this forces

$HD(\pi\circ\sigma^{-1}\circ\pi^{-1}(A))=HD(A)$.

Successively we can show $HD(\pi\circ\sigma^{-n}\circ\pi^{-1}(A))=HD(A)$ for
any $n\in\mathbb{N}$, which forces

$HD(\cup_{n\in\mathbb{N}}\pi\circ\sigma^{-n}\circ\pi^{-1}(A))=HD(A)$.

Since $\mu$ is ergodic with respect to $\sigma$, then

$\begin{array}[]{ll}&\nu\big{(}\cup_{n\in\mathbb{N}}\pi\circ\sigma^{-n}\circ\pi^{-1}(A)\big{)}\\\
=&\mu\circ\pi^{-1}\big{(}\cup_{n\in\mathbb{N}}\pi\circ\sigma^{-n}\circ\pi^{-1}(A)\big{)}\\\
\geq&\mu\big{(}\cup_{n\in\mathbb{N}}\sigma^{-n}\circ\pi^{-1}(A)\big{)}\\\
=&1.\end{array}$

This implies the inequality (5.1).

∎

Lemma 5.1 may not be true for some invariant measure $\mu$ on the symbolic
space. Apply it to families of PIFS, we have the following result instantly.

######  5.2 Corollary.

Let

$\big{\\{}S^{\mathbb{t}}=\\{s_{i}^{\mathbb{t}}:X\rightarrow X\\}_{i\in
I}\big{\\}}_{\mathbb{t}\in U}$

be a family of parabolic iterated function systems with a countable index set
$I$. Let $\mu$ be an ergodic measure on $I^{\infty}$ and $\nu_{\mathbb{t}}$ be
the projective measure under $\pi_{\mathbb{t}}$ at time $\mathbb{t}$, then

$dim_{*}\nu_{\mathbb{t}}=dim^{*}\nu_{\mathbb{t}}$

for any $\mathbb{t}\in U$.

Note that the equality holds everywhere instead of Lebesgue _a.e._ with
respect to $\mathbb{t}\in U$. Now we consider setwisely approximating the
projective measure of $\mu$ by the sequence of projective measures of its
concentrating measures on the ambient space $X$. The following result is an
instant corollary of Proposition 4.5.

######  5.3 Corollary.

Let

$S=\\{s_{i}:X\rightarrow X\\}_{i\in\mathbb{N}}$

be an infinite parabolic iterated function system with its attractor $J$. For
a probability measure $\mu$ on the symbolic space $\mathbb{N}^{\infty}$ such
that $\mu(\cup_{n=1}^{\infty}\mathbb{N}_{n}^{\infty})=1$, let $\mu_{n}$ be its
$n$-th concentrating measure on $\mathbb{N}_{n}^{\infty}$ for any
$n\in\mathbb{N}$. Then their projections
$\\{\nu_{n}=\mu_{n}\circ\pi^{-1}\\}_{n\in\mathbb{N}}$ and
$\nu=\mu\circ\pi^{-1}$ satisfy

$\nu_{n}\stackrel{{\scriptstyle s}}{{\rightarrow}}\nu$

as $n\rightarrow\infty$ on $J$.

As to absolute continuity of the projective measures on the ambient space, we
have the following result.

######  5.4 Lemma.

Let

$S=\\{s_{i}:X\rightarrow X\\}_{i\in\mathbb{N}}$

be an infinite parabolic iterated function system with attractor $J$. For an
ergodic probability measure $\mu$ on the symbolic space $\mathbb{N}^{\infty}$,
with $\mu(\cup_{n=1}^{\infty}\mathbb{N}_{n}^{\infty})=1$ and its $n$-th
concentrating measure $\mu_{n}$ also being ergodic on
$\mathbb{N}_{n}^{\infty}$ for any $n\in\mathbb{N}$, consider their projections
$\\{\nu_{n}=\mu_{n}\circ\pi^{-1}\\}_{n\in\mathbb{N}}$ and
$\nu=\mu\circ\pi^{-1}$ on $J$. The sequence of measures
$\\{\nu_{n}\\}_{n\in\mathbb{N}}$ is absolutely continuous with respect to
$\mathfrak{L}^{1}$ if and only if $\nu$ is absolutely continuous with respect
to $\mathfrak{L}^{1}$.

###### Proof.

First note that all the projective measures $\\{\nu_{n}\\}_{n\in\mathbb{N}}$
and $\nu$ are of pure type since they are all projected from ergodic measures
on the symbolic spaces. By Corollary 5.3 we have

$\nu_{n}\stackrel{{\scriptstyle s}}{{\rightarrow}}\nu$

as $n\rightarrow\infty$ on $J$. Then the absolute continuity of the sequence
of measures $\\{\nu_{n}\\}_{n\in\mathbb{N}}$ and the absolute continuity of
$\nu$ with respect to $\mathfrak{L}^{1}$ is equivalent to each other in virtue
of Proposition 3.1 and Proposition 3.5. ∎

We will only use the result that absolute continuity of the sequence of
measures $\\{\nu_{n}\\}_{n\in\mathbb{N}}$ implies the absolute continuity of
$\nu$ with respect to $\mathfrak{L}^{1}$ in the following. This is true even
if $\\{\mu_{n}\\}_{n\in\mathbb{N}}$ are not ergodic according to Proposition
3.1. Equipped with all the above results, now we are well prepared to prove
Theorem 1.4.

Proof of Theorem 1.4:

###### Proof.

First note that since
$\big{\\{}S^{\mathbb{t}}=\\{s_{i}^{\mathbb{t}}:X\rightarrow
X\\}_{i\in\mathbb{N}}\in\Gamma_{X}(\vartheta)\big{\\}}_{\mathbb{t}\in U}$,
there exist sequences of positive real numbers
$\\{0<\gamma_{n},u_{n}<1\\}_{n\in\mathbb{N}}$,
$\\{M_{n}>0\\}_{n\in\mathbb{N}}$ (it is possible that these sequences satisfy
$\sup\\{\gamma_{n}\\}_{n\in\mathbb{N}}=1$,
$\inf\\{u_{n}\\}_{n\in\mathbb{N}}=0$ and
$\sup\\{M_{n}\\}_{n\in\mathbb{N}}=\infty$) and a sequence of open
neighbourhoods $V_{n}$ of $v$ (it is also possible that
$\cap_{n=1}^{\infty}V_{n}=\\{v\\}$), such that its $n$-th family of truncates

$\big{\\{}S_{n}^{\mathbb{t}}=\\{s_{i}^{\mathbb{t}}:X\rightarrow
X\\}_{i\in\mathbb{N}_{n}}\in\Gamma_{X}(\vartheta,V_{n},\gamma_{n},u_{n},M_{n})\big{\\}}_{\mathbb{t}\in
U}$

for any fixed $n\in\mathbb{N}$. Denote their attractors by
$\\{J_{n,\mathbb{t}}\subset J_{\mathbb{t}}\\}_{n\in\mathbb{N},\mathbb{t}\in
U}$. Consider the projections of the concentrating measures

$\nu_{n,\mathbb{t}}=\mu_{n}\circ\pi_{\mathbb{t}}^{-1}$

on the limit sets for $\mathbb{t}\in U,n\in\mathbb{N}$. Since the sequence of
infinite random variables in $\mathbb{Y}$ is independent under the ergodic law
$\mu$, the concentrating measures $\\{\mu_{n}\\}_{n\in\mathbb{N}}$ are also
ergodic with respect to the shift map $\sigma$ in virtue of Lemma 4.3.
Moreover, according to Lemma 4.7, we have

$h_{\mu_{n}}(\sigma)>0$

for $n$ large enough. Now apply the [SSU1, Theorem 2.3 (i)] and Corollary 5.2
to the ergodic projections $\\{\nu_{n,\mathbb{t}}\\}_{\mathbb{t}\in U}$
originated from the family of finite PIFS

$\big{\\{}S_{n}^{\mathbb{t}}=\\{s_{i}^{\mathbb{t}}:X\rightarrow
X\\}_{i\in\mathbb{N}_{n}}\in\Gamma_{X}(\vartheta,V_{n},\gamma_{n},u_{n},M_{n})\big{\\}}_{\mathbb{t}\in
U}$

satisfying the continuity and transversality condition, we have

$dim_{*}\nu_{n,\mathbb{t}}=dim^{*}\nu_{n,\mathbb{t}}=\min\Big{\\{}\cfrac{h_{\mu_{n}}(\sigma)}{\lambda^{\mathbb{t}}_{\mu_{n}}(\sigma)},1\Big{\\}}$

for Lebesgue _a.e._ $\mathbb{t}\in U$ and any $n\in\mathbb{N}$ large enough.
In virtue of Corollary 5.3, we have

$\nu_{n,\mathbb{t}}\stackrel{{\scriptstyle s}}{{\rightarrow}}\nu_{\mathbb{t}}$

as $n\rightarrow\infty$ on $J$ for any $\mathbb{t}\in U$. Now apply the semi-
continuity result Theorem 2.3 to the setwisely convergent sequence of
projective measures $\\{\nu_{n,\mathbb{t}}\\}_{\mathbb{t}\in U}$, we have

(5.2)
$\begin{array}[]{ll}\limsup_{n\rightarrow\infty}\min\Big{\\{}\cfrac{h_{\mu_{n}}(\sigma)}{\lambda^{\mathbb{t}}_{\mu_{n}}(\sigma)},1\Big{\\}}=\limsup_{n\rightarrow\infty}dim_{*}\nu_{n,\mathbb{t}}\leq
dim_{*}\nu_{\mathbb{t}}\\\
=dim^{*}\nu_{\mathbb{t}}\leq\liminf_{n\rightarrow\infty}dim^{*}\nu_{n,\mathbb{t}}=\liminf_{n\rightarrow\infty}\min\Big{\\{}\cfrac{h_{\mu_{n}}(\sigma)}{\lambda^{\mathbb{t}}_{\mu_{n}}(\sigma)},1\Big{\\}}\end{array}$

for Lebesgue _a.e._ $\mathbb{t}\in U$ and any $n\in\mathbb{N}$ large enough.
Then Theorem 1.4 (i) is proved by letting $n\rightarrow\infty$ in (5.2), since
the Lebesgue measure of unions of countablly many Lebesgue null sets is still
null.

To prove Theorem 1.4 (ii), for fixed $n\in\mathbb{N}$, apply [SSU1, Theorem
2.3 (ii)] to the ergodic projections $\\{\nu_{n,\mathbb{t}}\\}_{\mathbb{t}\in
U}$ originated from the $n$-th truncated family of PIFS
$\\{S_{n}^{\mathbb{t}}\\}_{\mathbb{t}\in U}$ satisfying the continuity and
transversality condition, we see that $\nu_{n,\mathbb{t}}$ is absolutely
continuous for _a.e._ $\mathbb{t}\in\Big{\\{}\mathbb{t}\in
U:\cfrac{h_{\mu_{n}}(\sigma)}{\lambda^{\mathbb{t}}_{\mu_{n}}(\sigma)}>1\Big{\\}}$.
So in virtue of Lemma 5.4, $\nu_{\mathbb{t}}$ is absolutely continuous for
_a.e._ $\mathbb{t}$ in

$\cup_{\\{n_{j}\\}_{j=1}^{\infty}\mbox{ is an infinite subsequence of
}\\{n\\}_{n=1}^{\infty}}\cap_{j=1}^{\infty}\Big{\\{}\mathbb{t}\in
U:\cfrac{h_{\mu_{n_{j}}}(\sigma)}{\lambda^{\mathbb{t}}_{\mu_{n_{j}}}(\sigma)}>1\Big{\\}}:=U_{a}$.

It is easy to see that

$\Big{\\{}\mathbb{t}\in
U:\limsup_{n\rightarrow\infty}\cfrac{h_{\mu_{n}}(\sigma)}{\lambda^{\mathbb{t}}_{\mu_{n}}(\sigma)}>1\Big{\\}}\subset
U_{a}$.

This justifies Theorem 1.4 (ii). ∎

One can see from the above proof that the limit of the sequence

$\Big{\\{}\min\big{\\{}\cfrac{h_{\mu_{n}}(\sigma)}{\lambda^{\mathbb{t}}_{\mu_{n}}(\sigma)},1\big{\\}}\Big{\\}}_{n=1}^{\infty}$

always exists for Lebesgue _a.e._ $\mathbb{t}\in U$. More interesting results
are possible according to different _a.e._ limit behaviours of the sequence
$\Big{\\{}\cfrac{h_{\mu_{n}}(\sigma)}{\lambda^{\mathbb{t}}_{\mu_{n}}(\sigma)}\Big{\\}}_{\mathbb{t}\in
U,n\in\mathbb{N}}$.

In the following of the section we apply Theorem 1.4 to various families of
PIFS to deduce some interesting results. The first application is on the
Hausdorff dimension of the attractors of a family of infinite PIFS.

######  5.5 Corollary.

Let

$\big{\\{}S^{\mathbb{t}}=\\{s_{i}^{\mathbb{t}}:X\rightarrow
X\\}_{i\in\mathbb{N}}\in\Gamma_{X}(\vartheta)\big{\\}}_{\mathbb{t}\in U}$

be a family of infinite parabolic iterated function systems satisfying the
continuity and transversality condition with respect to the vector-time
parameter $\mathbb{t}\in U$. For an ergodic probability measure $\mu$ on the
symbolic space $\mathbb{N}^{\infty}$ with positive entropy $h_{\mu}(\sigma)$
and $\mu(\cup_{n=1}^{\infty}\mathbb{N}_{n}^{\infty})=1$, let $\mu_{n}$ be its
$n$-th concentrating measure for any $n\in\mathbb{N}$. If the sequence of
infinite random variables in $\mathbb{Y}$ under the law $\mu$ is independent,
then

  1. (i).

$HD(J_{\mathbb{t}})\geq\lim_{n\rightarrow\infty}\min\Big{\\{}\cfrac{h_{\mu_{n}}(\sigma)}{\lambda^{\mathbb{t}}_{\mu_{n}}(\sigma)},1\Big{\\}}$
for Lebesgue _a.e._ $\mathbb{t}\in U$.

  2. (ii).

$\mathfrak{L}^{1}(J_{\mathbb{t}})>0$ for Lebesgue _a.e._
$\mathbb{t}\in\Big{\\{}\mathbb{t}:\limsup_{n\rightarrow\infty}\big{\\{}\cfrac{h_{\mu_{n}}(\sigma)}{\lambda^{\mathbb{t}}_{\mu_{n}}(\sigma)}\big{\\}}>1\Big{\\}}$.

###### Proof.

These results follow directly from Theorem 1.4 and the fact that
$supp(\nu_{\mathbb{t}})\subset J_{\mathbb{t}}$ for any $\mathbb{t}\in U$. ∎

More interesting corollaries on the Hausdorff dimension of the attractors can
be formulated from Theorem 1.4, considering different limit behaviours of the
sequence
$\Big{\\{}\cfrac{h_{\mu_{n}}(\sigma)}{\lambda^{\mathbb{t}}_{\mu_{n}}(\sigma)}\Big{\\}}_{\mathbb{t}\in
U,n\in\mathbb{N}}$ and the fact that $supp(\nu_{\mathbb{t}})$ is always
contained in $J_{\mathbb{t}}$.

Now we apply Theorem 1.4 or Corollary 5.5 to some families of infinite PIFS
with _exploding_ measures $\mu$ on $\mathbb{N}^{\infty}$, that is, measures
$\mu$ satisfying

$h_{\mu}(\sigma)=\infty$.

There will be some stronger results available in due courses.

######  5.6 Corollary.

Let

$\big{\\{}S^{\mathbb{t}}=\\{s_{i}^{\mathbb{t}}:X\rightarrow
X\\}_{i\in\mathbb{N}}\in\Gamma_{X}(\vartheta,V,\gamma,u,M)\big{\\}}_{\mathbb{t}\in
U}$

be a family of infinite parabolic iterated function systems satisfying the
continuity and transversality condition with respect to the vector-time
parameter $\mathbb{t}\in U$ for some fixed open neighbourhood $V$ of $v$ and
some fixed parameters $0<\vartheta,\gamma,u<1,M>0$. For an ergodic exploding
probability measure $\mu$ on the symbolic space $\mathbb{N}^{\infty}$ with
$\mu(\cup_{n=1}^{\infty}\mathbb{N}_{n}^{\infty})=1$, if the sequence of
infinite random variables in $\mathbb{Y}$ under the law $\mu$ is independent,
then

  1. (i).

For Lebesgue _a.e._ $\mathbb{t}\in U$,
$dim_{*}\nu_{\mathbb{t}}=dim^{*}\nu_{\mathbb{t}}=1$,

  2. (ii).

$\nu_{\mathbb{t}}$ is absolutely continuous for Lebesgue _a.e._ $\mathbb{t}\in
U$,

in which $\nu_{\mathbb{t}}=\mu\circ\pi_{\mathbb{t}}^{-1}$ is the projective
measure at time $\mathbb{t}$.

###### Proof.

Note that since

(5.3) $\inf_{n\in\mathbb{N},x\in
X}\\{|(s_{n}^{\mathbb{t}})^{\prime}(x)|\\}\geq u$

for any $\mathbb{t}\in U$, we have

(5.4) $\lambda^{\mathbb{t}}_{\mu}(\sigma)\leq-\log u.$

Apply Theorem 1.4 to the projective measures
$\\{\nu_{\mathbb{t}}\\}_{\mathbb{t}\in U}$ originated from the family of
infinite parabolic iterated function systems

$\big{\\{}S^{\mathbb{t}}=\\{s_{i}^{\mathbb{t}}:X\rightarrow
X\\}_{i\in\mathbb{N}}\in\Gamma_{X}(\vartheta,V,\gamma,u,M)\big{\\}}_{\mathbb{t}\in
U}$

satisfying the continuity and transversality condition (note that
$\Gamma_{X}(\vartheta,V,\gamma,u,M)\subset\Gamma_{X}(\vartheta)$), the two
conclusions follow instantly in virtue of (5.4), $h_{\mu}(\sigma)=\infty$ and
Lemma 4.7.

∎

##  6\. Dimensional estimates of the exceptional parameters

This section is devoted to estimation on the upper bound of the local (global)
Hausdorff dimension of the exceptional parameters. We mean to prove Theorem
1.8 here. We start by showing several preceding results on the limit
behaviours of the (families of truncated) finite sub-systems of an (family of)
infinite PIFS.

######  6.1 Lemma.

Let

$S=\\{s_{i}:X\rightarrow X\\}_{i\in\mathbb{N}}\in\Gamma_{X}(\vartheta)$

be a PIFS. Let $\mu$ be a finite measure on $\mathbb{N}^{\infty}$ with its
concentrating measures $\\{\mu_{n}\\}_{n\in\mathbb{N}}$ on
$\\{\mathbb{N}_{n}^{\infty}\\}_{n\in\mathbb{N}}$ respectively. Considering the
sequence of Lyapunov exponents
$\\{\lambda_{\mu_{n}}(\sigma)\\}_{n=1}^{\infty}$ of the truncated systems
$\big{\\{}S_{n}=\\{s_{i}:X\rightarrow
X\\}_{i\in\mathbb{N}_{n}}\big{\\}}_{n\in\mathbb{N}}$, we have

$\lim_{n\rightarrow\infty}\lambda_{\mu_{n}}(\sigma)=\lambda_{\mu}(\sigma)$.

###### Proof.

First we represent $\lambda_{\mu}(\sigma)$ as

$\lambda_{\mu}(\sigma)=\sum_{i=1}^{\infty}-\int_{[i]}\log|s_{i}^{\prime}(\pi\circ\sigma(\omega))|d\mu(\omega)$.

Note that

(6.1)
$-\int_{[i]}\log|s_{i}^{\prime}(\pi\circ\sigma(\omega))|d\mu(\omega)<\infty$

for any $1\leq i<\infty$. We distinguish the case
$\lambda_{\mu}(\sigma)=\infty$ from the case $\lambda_{\mu}(\sigma)<\infty$.
If $\lambda_{\mu}(\sigma)=\infty$, we will claim that for any $a>0$, there
exists an integer $N_{a}$ large enough, such that for any $n>N_{a}$, we have

$\lambda_{\mu_{n}}(\sigma)>a$.

This means $\lim_{n\rightarrow\infty}\lambda_{\mu_{n}}(\sigma)=\infty$. To see
this, since $\lambda_{\mu}(\sigma)=\infty$, for any $a>0$, there exists
$N_{*}$ large enough, such that

$\sum_{i=1}^{N_{*}}-\int_{[i]}\log|s_{i}^{\prime}(\pi\circ\sigma(\omega))|d\mu(\omega)>2a$.

Now choose some small $0<\epsilon<a$, due to 6.1, for any $1\leq i\leq N_{*}$,
we can find an integer $N_{i}$ large enough, such that

$a_{i}:=\big{|}\int_{[i]}\log|s_{i}^{\prime}(\pi\circ\sigma(\omega))|d\mu(\omega)-\int_{[i]}\log|s_{i}^{\prime}(\pi\circ\sigma(\omega))|d\mu_{n}(\omega)\big{|}<\cfrac{\epsilon}{2^{i}}$

for any $n>N_{i}$. Now let $N_{a}=\max\\{N_{i}\\}_{i=1}^{N_{*}}$, we have

$\begin{array}[]{ll}&\sum_{i=1}^{N_{*}}-\int_{[i]}\log|s_{i}^{\prime}(\pi\circ\sigma(\omega))|d\mu_{n}(\omega)\\\
\geq&\sum_{i=1}^{N_{*}}-\int_{[i]}\log|s_{i}^{\prime}(\pi\circ\sigma(\omega))|d\mu(\omega)-\sum_{i=1}^{N_{*}}a_{i}\\\
\geq&2a-\sum_{i=1}^{N_{*}}\cfrac{\epsilon}{2^{i}}\\\ >&a\end{array}$

for any $n>N_{a}$, which justifies the claim.

The case $\lambda_{\mu}(\sigma)<\infty$ can be dealt with in a similar way,
which is left to the interested readers. ∎

Similar to Remark 4.8, lemma 6.1 holds for any sequence of measures converging
to the measure $\mu$ under the setwise topology on $\mathcal{\hat{M}}(X)$, not
only for the sequence of its concentrating measures, for any topological
ambient space $X$.

######  6.2 Lemma.

Let $G\subset U$ be a subset. Consider a sequences of real functions
$\\{f_{n}:G\rightarrow\mathbb{R}\\}_{n=1}^{\infty}$ such that

$f_{n}(\mathbb{t})\rightarrow f(\mathbb{t})$

as $n\rightarrow\infty$ for some $f(\mathbb{t}):G\rightarrow\mathbb{R}$ at any
$\mathbb{t}\in G$. Now if

$HD(\\{\mathbb{t}\in G:f_{n}(\mathbb{t})<0\\})\leq h_{n}$,

for a sequence of reals $\\{h_{n}\\}_{n=1}^{\infty}$, then we have

$HD(\\{\mathbb{t}\in
G:f(\mathbb{t})<0\\})\leq\limsup_{n\rightarrow\infty}h_{n}$.

###### Proof.

Note that since $\lim_{n\rightarrow\infty}f_{n}(\mathbb{t})\rightarrow
f(\mathbb{t})$ at any time $\mathbb{t}\in G$, then

$\\{\mathbb{t}\in
G:f(\mathbb{t})<0\\}\subset\cap_{k=1}^{\infty}\cup_{n=k}^{\infty}\\{\mathbb{t}\in
G:f_{n}(\mathbb{t})<0\\}$.

This means

$\begin{array}[]{ll}&HD(\\{\mathbb{t}\in G:f(\mathbb{t})<0\\})\\\
\leq&\inf\big{\\{}HD(\cup_{n=k}^{\infty}\\{\mathbb{t}\in
G:f_{n}(\mathbb{t})<0\\})\big{\\}}_{k=1}^{\infty}\\\
\leq&\inf\big{\\{}\sup\\{h_{n}\\}_{n=k}^{\infty}\big{\\}}_{k=1}^{\infty}\\\
=&\limsup_{n\rightarrow\infty}h_{n}.\end{array}$

∎

Be careful that Lemma 6.2 will not be true if the two $<$ are both substituted
by $\leq$ in it. Now we are well prepared to prove Theorem 1.8.

Proof of Theorem 1.8:

###### Proof.

Since $\mu$ is ergodic and $\mathbb{Y}$ is independent under the law $\mu$,
according to Lemma 4.3, consider the sequence of measures
$\\{\nu_{n,\mathbb{t}}\\}_{\mathbb{t}\in U,n\in\mathbb{N}}$ projected from the
ergodic concentrating measures $\\{\mu_{n}\\}_{n\in\mathbb{N}}$ of $\mu$. For
fixed $n\in\mathbb{N}$, as the family of $n$-th truncated finite PIFS

$\big{\\{}S_{n}^{\mathbb{t}}=\\{s_{i}^{\mathbb{t}}:X\rightarrow
X\\}_{i\in\mathbb{N}_{n}}\in\Gamma_{X}(\vartheta,V_{n},\gamma_{n},u_{n},M_{n})\big{\\}}_{\mathbb{t}\in
U}$

satisfying the continuity and strong transversality condition with respect to
the time parameter $\mathbb{t}\in U$ for some open neighbourhood $V_{n}$ of
$v$, some $0<\gamma_{n},u_{n}<1$ and $M_{n}>0$, apply [SSU1, Theorem 5.3] to
the family of truncated finite PIFS $S_{n}^{\mathbb{t}}$, we have

(6.2) $HD\Big{(}\big{\\{}\mathbb{t}\in
G:dim^{*}\nu_{n,\mathbb{t}}<\min\\{\frac{h_{\mu_{n}}(\sigma)}{\lambda_{\mu_{n}}^{\mathbb{t}}(\sigma)},\alpha\\}\big{\\}}\Big{)}\leq\min\Big{\\{}\sup_{\mathbb{t}\in
G}\frac{h_{\mu_{n}}(\sigma)}{\lambda_{\mu_{n}}^{\mathbb{t}}(\sigma)},\alpha\Big{\\}}+d-1$

for any $n\in\mathbb{N}$ and $\epsilon>0$. Note that

$dim^{*}\nu_{\mathbb{t}}=\lim_{n\rightarrow\infty}dim^{*}\nu_{n,\mathbb{t}}$

since $\lim_{n\rightarrow\infty}\nu_{n,\mathbb{t}}\stackrel{{\scriptstyle
s}}{{\rightarrow}}\nu_{\mathbb{t}}$ and

$\lim_{n\rightarrow\infty}\min\\{\cfrac{h_{\mu_{n}}(\sigma)}{\lambda_{\mu_{n}}^{\mathbb{t}}(\sigma)},\alpha\\}$

exists for any $\mathbb{t}\in G$ since
$\lim_{n\rightarrow\infty}\cfrac{h_{\mu_{n}}(\sigma)}{\lambda_{\mu_{n}}^{\mathbb{t}}(\sigma)}$
exists everywhere on $G$. Moreover,

$\lim_{n\rightarrow\infty}\Big{(}\min\big{\\{}\sup_{G}\cfrac{h_{\mu_{n}}(\sigma)}{\lambda_{\mu_{n}}^{\mathbb{t}}(\sigma)},\alpha\big{\\}}+d-1\Big{)}=K_{\alpha,G}$.

Apply Lemma 6.2 here, we get (6.3).

∎

In the non-exploding case, Theorem 1.8 degenerates into a more simple version
as following.

######  6.3 Corollary.

Let

$\big{\\{}S^{\mathbb{t}}=\\{s_{i}^{\mathbb{t}}:X\rightarrow
X\\}_{i\in\mathbb{N}}\in\Gamma_{X}(\vartheta)\big{\\}}_{\mathbb{t}\in U}$

be a family of infinite parabolic iterated function systems satisfying the
continuity and strong transversality condition with respect to the vector-time
parameter $\mathbb{t}\in U$. For an ergodic probability measure $\mu$ on the
symbolic space $\mathbb{N}^{\infty}$ satisfying
$\mu(\cup_{n=1}^{\infty}\mathbb{N}_{n}^{\infty})=1$ and
$0<h_{\mu}(\sigma),\lambda^{\mathbb{t}}_{\mu}(\sigma)<\infty$ at any time
$\mathbb{t}\in U$, if the sequence of infinite random variables in
$\mathbb{Y}$ under the law $\mu$ is independent, then

(6.3) $HD\Big{(}\Big{\\{}\mathbb{t}\in
G:dim^{*}\nu_{\mathbb{t}}<\min\big{\\{}\cfrac{h_{\mu}(\sigma)}{\lambda^{\mathbb{t}}_{\mu}(\sigma)},\alpha\big{\\}}\Big{\\}}\Big{)}\leq\min\Big{\\{}\sup_{\mathbb{t}\in
G}\cfrac{h_{\mu}(\sigma)}{\lambda^{\mathbb{t}}_{\mu}(\sigma)},\alpha\Big{\\}}+d-1$

for any $0<\alpha<1$ and $G\subset U$.

###### Proof.

Under the assumption
$0<h_{\mu}(\sigma),\lambda^{\mathbb{t}}_{\mu}(\sigma)<\infty$ at any time
$\mathbb{t}\in U$, considering Lemma 4.7 and Lemma 6.1, the result follows
instantly from Theorem 1.8.

∎

In the exploding case, that is, either $h_{\mu}(\sigma)=\infty$ or
$\lambda^{\mathbb{t}}_{\mu}(\sigma)=\infty$ for some $\mathbb{t}\in U$, the
result depends heavily on the asymptotic behaviour of the sequence of
concentrating measures of $\mu$, however, some estimations on the Hausdorff
dimension of some form of exceptional parameters may still be possible upon
reforming Lemma 6.2.

Note that most of the notions and results in this work apply to families of
_hyperbolic iterated function systems_ , which are families of iterated
function systems constituted by contractive hyperbolic maps only.

## References

  * [Ber] V. Berthé, Symbolic dynamics and representations, Les cours du C.I.R.M. Vol. 5 no 1 (2017) 1-16, Cours $n^{o}$ I.
  * [BHP] R. Basu, J. Hermon and Y. Peres, Characterization of cutoff for reversible Markov chains, Ann. Probab. Volume 45, Number 3 (2017), 1448-1487.
  * [BRS] B. Bárány, M. Rams and K. Simon. On the dimension of self-affine sets and measures with overlaps. Proceedings of the American Mathematical Society, 144(10):4427-4440, 2016.
  * [BV1] E. Breuillard and P. Varjú, On the dimension of Bernoulli convolutions, Ann. Probab. Volume 47, Number 4 (2019), 2582-2617.
  * [BV2] E. Breuillard and P. Varjú, Entropy of Bernoulli convolutions and uniform exponential growth for linear groups, Journal d’Analyse Mathématique volume 140, pages 443-481(2020).
  * [Doo] J. Doob, Measure Theory, Graduate Texts in Mathematics, 143, Springer-Verlag New York, 1994.
  * [Fal1] K. Falconer, The Hausdorff dimension of self-affine fractals, Math. Proc. Cambridge Philos. Soc. 103, 1988, 339-350.
  * [Fal2] K. Falconer, Fractal Geometry, Mathematical Foundations and Applications, 3rd Edition, John Wiley & Sons, 2014.
  * [FKZ] E. Feinberg, P. Kasyanov and M. Zgurovsky, Convergence of probability measures and Markov decision models with incomplete information, Proceedings of the Steklov Institute of Mathematics, December 2014, Volume 287, Issue 1, pp 96-117.
  * [Fur] H. Furstenberg, Intersections of Cantor sets and transversality of semigroups, In Problems in analysis (Sympos. Salomon Bochner, Princeton Univ., Princeton, N.J., 1969), pages 41-59. Princeton Univ. Press, Princeton, N.J., 1970. 3, 4, 10.
  * [GR] J. K. Ghosh and R. V. Ramamoorthi, Bayesian Nonparametrics, Springer, New York, 2003.
  * [Hai] M. Hairer, Ergodic Properties of Markov Processes, Lecture given at The University of Warwick in Spring 2006, http://www.hairer.org/notes/Markov.pdf, 2018.
  * [Hay] N. Haydn, The Central Limit Theorem for uniformly strong mixing measures, Stochastics and Dynamics, Vol. 12, No. 4 (2012).
  * [HL] O. Hernandez-Lerma and J. Lasserre, Markov Chains and Invariant Probabilities, Progress in Mathematics, Birkhäuser Basel, 2003.
  * [Hoc1] M. Hochman, On self-similar sets with overlaps and inverse theorems for entropy, Ann. of Math. (2) 180, 773-822, 2014.
  * [Hoc2] M. Hochman, Lectures on dynamical systems and entropy, http://math.huji.ac.il/~mhochman/courses/dynamics2014/notes.5.pdf, June 27, 2014.
  * [Hoc3] M. Hochman, Self similar sets, entropy and additive combinatorics, Geometry and Analysis of Fractals, Volume 88, 2014, pp 225-252.
  * [Hoc4] M. Hochman, Some problems on the boundary of fractal geometry and additive combinatorics, Recent Developments in Fractals and Related Fields, pp 129-174, Editors: J. Barral and S. Seuret, Conference on Fractals and Related Fields III, île de Porquerolles, France, 2015.
  * [HS] M. Hochman and P. Shmerkin, Local entropy averages and projections of fractal measures, Ann. of Math. (2) 175 1001-1059, 2012.
  * [Hut] J. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713-747.
  * [JW] B. Jessen and A. Wintner, Distribution functions and the Riemann zeta function, Trans. Amer. Math. Soc., 38(1):48-88, 1935.
  * [Kau] R. Kaufman, On Hausdorff dimension of projections, Mathematika 15 (1968), 153-155.
  * [Las] J. Lasserre, On the setwise convergence of sequences of measures, Journal of Applied Mathematics and Stochastic Analysis, 10:2 (1997), 131-136.
  * [LPS] E. Lindenstrauss, Y. Peres and W. Schlag, Bernoulli convolutions and an intermediate value theorem for entropy of K-partitions, J. d’Analyse Math 87 (2002), 337-367.
  * [LY] T. Linder and S. Yüksel, Optimization and convergence of observation channels in stochastic control, SIAM J. Control Optim. Vol. 50, No. 2, pp. 864-887.
  * [Ma] L. Ma, Continuity of measure-dimension mappings, arXiv:2105.05491 [math.DS], 2021.
  * [MMR] P. Mattila, M. Morán and J. M. Rey, Dimension of a measure, Studia Math. 142 (2000), no. 3, 219-233.
  * [PS1] Y. Peres and B. Solomyak, Problems on self-simiar and self-affine sets, an update, in ‘Fractals and Stochastics II, Proceedings of the Greifswald conference’ (Aug 1998), (2000), 95-106.
  * [PS2] Y. Peres and B. Solomyak, Absolute continuity of Bernoulli convolutions, a simple proof, Math. Research Letters 3:2 (1996), 231-239.
  * [PS3] Y. Peres and B. Solomyak, Self-similar measures and intersections of Cantor sets, Trans. Amer. Math. Soc. 350, no. 10 (1998), 4065-4087.
  * [PolS] M. Pollicott and K. Simon, The Hausdorff dimension of $\lambda$-expansions with deleted digits, Trans. Amer. Math. Soc. 347 (1995), 967-983.
  * [PSS] Y. Peres, K. Simon and B. Solomyak, Absolute continuity for random iterated function systems with overlaps, Journal of the London Math. Soc. (2) 74 (2006), 739-756.
  * [Shi] P. Shields, The theory of Bernoulli Shifts, Web Edition 1.01, https://www.impan.pl/~gutman/The%20Theory%20of%20Bernoulli%20Shifts.pdf.
  * [Shm1] P. Shmerkin, On the exceptional set for absolute continuity of Bernoulli convolutions, Geom. Funct. Anal. 24 no. 3(2014),946-958.
  * [Shm2] P. Shmerkin, A modified multifractal formalism for a class of self-similar measures with overlap, Asian J. Math., 9(3):323-348, 2005.
  * [Shm3] P. Shmerkin, Overlapping self-affine sets, Indiana Univ. Math. J., 55(4):1291-1331, 2006\.
  * [Shm4] P. Shmerkin, Projections of self-similar and related fractals: a survey of recent developments, Fractal Geometry and Stochastics V, pages 53-74. Editors: Christoph Bandt, Kenneth J. Falconer, and Martina Zähle, Springer, 2015.
  * [SimS1] K. Simon and B. Solomyak, Hausdorff dimension for horseshoes in $\mathbb{R}^{3}$, Ergodic Th. and Dynam. Sys. 19 (1999), 1343-1363.
  * [SimS2] K. Simon and B. Solomyak, On the dimension of self-similar sets, Fractals 10 (2002), 59-65.
  * [Sol1] B. Solomyak, On the random series $\sum\pm\lambda^{i}$ (an Erdös problem), Annals of Math. 142 (1995), 611-625.
  * [Sol2] B. Solomyak, Measure and dimension for some fractal families, Math. Proc. Cambridge Phil. Soc. 124, no. 3 (1998), 531-546.
  * [Sol3] B. Solomyak, Notes on Bernoulli convolutions, In Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot. Part 1. Proc. Sympos. Pure Math. 72 (2004) 207-230. Amer. Math. Soc., Providence, RI.
  * [SS1] P. Shmerkin and B. Solomyak, Absolute continuity of self-similar measures, their projections and convolutions, Transactions of the American Math. Society 368 (2016), 5125-5151.
  * [SS2] P. Shmerkin and B. Solomyak, Absolute continuity of complex Bernoulli convolutions, Math. Proc. Cambridge Philos. Soc. 161 (2016), no. 3, 435-453.
  * [SSS] S. Saglietti, P. Shmerkin and B. Solomyak, Absolute continuity of non-homogeneous self-similar measures, Adv. Math., 335:60-110, 2018.
  * [SSU1] K. Simon, B. Solomyak and M. Urbański, Invariant Measures for Parabolic IFS with Overlaps and Random Continued Fractions, Transactions of the American Mathematical Society, Vol. 353, No. 12 (Dec., 2001), pp. 5145-5164.
  * [SSU2] K. Simon, B. Solomyak and M. Urbański, Hausdorff dimension of limit sets for parabolic IFS with overlaps, Pacific J. Math. 201 (2001), 441-478.
  * [Tao] T. Tao, An introduction to measure theory, Graduate Studies in Mathematics, Volume: 126, American Math. Society, 2011.
  * [Tsi] B. Tsirelson, Lecture notes for Probability for mathematicians, Part A: Independence, 3, Infinite independent sequences, http://www.math.tau.ac.il/~tsirel/Courses/ProbForMath/A3.pdf.
  * [Var1] P. P. Varjú, Recent progress on Bernoulli convolutions, In European Congress of Mathematics: Berlin, July 18-22, 2016. Eur. Math. Soc., Zurich.
  * [Var2] P. P. Varjú, Absolute continuity of Bernoulli convolutions for algebraic parameters, J. Amer. Math. Soc. 32 (2019) 351-397.
  * [Var3] P. P. Varjú, On the dimension of Bernoulli convolutions for all transcendental parameters, Annals of Mathematics 189 (2019), 1001-1011.
  * [Wal] P. Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, 79. Springer, New York-Berlin, 1982.
  * [You] L. Young, Dimension, entropy and Lyapunov exponents, Ergodic Theory Dynam. Systems 2(1982), no. 1, 109-124.