# Multifractal Formalism from Large Deviations

Mirmukhsin Makhmudov  Mathematical Institute
Leiden University
Postbus 9512, 2300 RA Leiden, The Netherlands<EMAIL_ADDRESS>,  Evgeny Verbitskiy  Mathematical Institute
Leiden University
Postbus 9512, 2300 RA Leiden, The Netherlands  Bernoulli Institute
University of Groningen
PO Box 407, 9700 AK Groningen, The Netherlands<EMAIL_ADDRESS>and
Qian Xiao  Mathematical Institute
Leiden University
Postbus 9512, 2300 RA Leiden, The Netherlands  School of Mathematics
South China University of Technology
Guangzhou 510641, People’s Republic of China<EMAIL_ADDRESS>
###### Abstract.

It has often been observed that the Multifractal Formalism and the Large
Deviation Principles are intimately related. In fact, Multifractal Formalism
was heuristically derived using the Large Deviations ideas. In numerous
examples in which the multifractal results have been rigorously established,
the corresponding Large Deviation results are valid as well. Moreover, the
proofs of multifractal and large deviations are remarkably similar. The
natural question then is whether under which conditions multifractal formalism
can be deduced from the corresponding large deviations results. More
specifically, given a sequence of random variables
$\\{{X_{n}}\\}_{n\in\mathbb{N}}$, satisfying a Large Deviation Principle, what
can be said about the multifractal nature of the level sets
$K_{\alpha}=\\{\omega:\lim_{n}\frac{X_{n}(\omega)}{n}=\alpha\\}$. Under some
technical assumptions, we establish the upper and lower bounds for
multifractal spectra in terms of the large deviation rate functions, and show
that many known results of multifractal formalism are covered by our setup.

##  1\. Introduction

The concept of multifractal formalism (MF) was proposed in 1980’s by Parisi,
Frisch, Hentchel [Benzi-etc1984, Frisch-Parisi1983] in the context of the
study of turbulence. It has been suggested that natural local quantities have
multiple fluctuation scales. These possible fluctuations can be described
using a _singluarity spectrum_. The multifractal formalism states that the
singularity spectrum is dual, in the sense of Legendre transform, to some
integral ‘free energy’-type function of the system. The basic idea behind
multifractal formalism is underpinned (or motivated) by an assumption that a
certain probabilistic the Large Deviation Principle (LDP) holds for the
system.

The first rigorous mathematical results [Olsen1995, Olsen1996, Barreira-Pesin-
Schmeling1997] have been obtained in the early and mid-1990s (Riedli,
Falconer, Olsen, Pesin), and since the late 1990’s there was an explosion of
research in this area [Olsen2002, Olivier, Chazottes, Olivier1, Olivier2,
Olivier3, Fan1996, Fan3, Fan1, Fan4]. Multifractal formalism has been
rigorously established for very large classes of dynamical systems and a
plethora of local observables. However, a somewhat curious phenomenon has
occurred. Though the original motivation and first rigorous results relied
rather heavily on probabilistic methods of Large Deviations, in subsequent
works the link to Large Deviations has become somewhat less pronounced. In
fact, we are not aware of a single rigorous multifractal result without an
accompanying Large Deviations result. However, it is also clear that more
assumptions are required for the validity of multifractal formalism than for
the validity of Large Deviations Principles. For example, for multifractal
analysis, the phase space must be a metric space, while this is not really a
requirement for Large Deviations.

This brings us to the natural question: assuming that the local quantity of
interest is a pointwise limit of quantities, whose probabilistic behaviour is
governed by Large Deviations, what could be said about the corresponding
singularity spectrum and multifractal formalism?

Let us compare two old results, first LDP for Bernoulli random variables, and
second, what we would now call a multifractal result for frequencies of digits
in binary expansions. However, this result precedes the multifractal idea by
nearly 50 years.

######  Theorem 1.1 (LDP for coin-tossing).

Suppose $\left(X_{k}\right)$ are i.i.d. Bernoulli random variables with
$\mathbb{P}\left(X_{1}=0\right)=\mathbb{P}\left(X_{1}=1\right)=$
$\frac{1}{2}$. Set $S_{n}=\sum_{i=1}^{n}X_{i}$, then $\\{S_{n}/n\\}$ satisfies
LDP with the rate function $I(p)=\log 2-H(p)$, where $H(p)=-p\log
p-(1-p)\log(1-p)$ if $p\in[0,1]$, otherwise $H(p)=-\infty$. Namely,

$\lim_{n\to\infty}\frac{1}{n}\log{\mathbb{P}\Big{(}\frac{S_{n}}{n}\in(a,b)\Big{)}}=-\inf_{p\in(a,b)}I(p),\;\;\text{
for all }-\infty\leq a\leq b\leq\infty.$

The above theorem is one of the first Large Deviation results, and it is
useful to compare it with possibly the first multifractal result by
Besicovitch [Besicovitch1935] and Eggleston [Eggleston1949] in the 1930s and
the 1940s. This multifractal result was, first, proved by Besicovitch
[Besicovitch1935] for $N=2$ (binary shift), and later, extended by Eggleston
[Eggleston1949] for general $N$ ($N$ is the cardinality of the alphabet).

######  Theorem 1.2 (Besicovitch-Eggleston).

Let $\Omega=\\{0,1\\}^{\mathbb{Z}_{+}}$, then

$\operatorname{dim}_{H}\left\\{\omega\in\Omega:\,\lim_{n}\frac{1}{n}\sum_{j=0}^{n-1}\omega_{j}=p\right\\}=\frac{H(p)}{\log
2},\;\;\forall p\in[0,1],$

where $\dim_{H}$ denotes the Hausdorff dimension.

It is important to stress that not only rate functions in the above theorems
are related, but in fact, the proofs are also very similar.

Next to the Hausdorff dimension, there is another, but in some sense more
dynamical set characteristic, known as topological entropy, which is used
frequently to analyse non-compact non-invariant sets. However, in the setting
of symbolic dynamics $(\Omega:=(0,...,l-1)^{\mathbb{Z}_{+}},\sigma)$, one has
$h_{{\textsf{top}}}(\sigma,Z)=\log l\cdot\dim_{H}(Z)$ for all
$Z\subset\Omega$, in particular, $h_{{\textsf{top}}}(\sigma,\Omega)=\log l$,
where $\sigma:\Omega\rcirclearrowleft$ is the left shift map. Using the
topological entropy, the previous results can be summarized as

(1)
$\framebox{$h_{{\textsf{top}}}(\sigma,K_{p})=h_{{\textsf{top}}}(\sigma,\Omega)-I(p),\;\forall
p\in\mathbb{R}$},$

where $K_{p}:=\\{\omega\in\Omega:\lim_{n}S_{n}/n=p\\}$ and
$S_{n}(\omega):=\sum_{i=0}^{n-1}\omega_{i},\;\omega\in\Omega$.

###  1.1. Multifractal analysis of Birkhoff averages of Hölder continuous
functions

Most of the first results in the multifractal analysis of dynamical systems
can be reduced to the results in the multifractal analysis of Birkhoff
averages of certain potentials. For simplicity, suppose
$\Omega=\mathcal{A}^{\mathbb{Z}_{+}}$ is the set of all infinite sequences
$\omega=(\omega_{0},\omega_{1},\ldots)$ in a finite alphabet $\mathcal{A}$,
equipped with a metric $\rho$, generating the product topology. Denote by
$\sigma:\Omega\to\Omega$ the left shift on $\Omega$. The topological pressure
of a continuous function ${\varphi}:\Omega\to\mathbb{R}$ is defined as

$P({\varphi})=\lim_{n\to\infty}\frac{1}{n}\log\sum_{(a_{0},\ldots,a_{n-1})\in
A^{n}}\Bigl{(}\sup_{\omega\in[a_{0}^{n-1}]}\exp(S_{n}{\varphi}(\omega))\Bigr{)},\quad
S_{n}\varphi(\omega)=\sum_{k=0}^{n-1}{\varphi}(\sigma^{k}\omega),$

where $[a_{0}^{n-1}]:=\\{\omega\in\Omega:\omega_{i}=a_{i},\;i=0,...,n-1\\}$ is
the cylinder of the length $n$. It is well-known that the pressure function
$q\mapsto P_{q}({\varphi}):=P(q{\varphi})$, is convex. Moreover, by the
celebrated result of D.Ruelle the pressure function for a Hölder continuous
potential ${\varphi}$ is real-analytic.

It turns out that the real-analyticity of the pressure function immediately
gives us the Large Deviations Principle for ergodic averages
$X_{n}(\omega)=S_{n}{\varphi}(\omega)$. More specifically, one has the
following result111This is not the most general form of the Gartner-Ellis
theorem.

######  Theorem 1.3 (Gartner-Ellis).

Suppose $\\{X_{n}\\}$ is a sequence of real-valued random variables defined on
a common probability space $(\Omega,\mathcal{F},m)$. Assume that for every
$q\in\mathbb{R}$, the logarithmic moment generating function, defined as the
limit

$\Lambda(q)\triangleq\lim_{n\rightarrow\infty}\frac{1}{n}\log\mathbb{E}_{m}e^{qX_{n}}=\lim_{n\rightarrow\infty}\frac{1}{n}\log\int
e^{qX_{n}(\omega)}m(d\omega)$

exists and is finite. Denote by $\Lambda^{*}$ the Legendre transform (convex
dual) of $\Lambda$, defined by

$\Lambda^{*}(\alpha)=\sup_{q\in\mathbb{R}}\Bigl{(}\alpha
q-\Lambda(q)\Bigr{)}.$

Then

  * (a)

for any closed set $F\subset\mathbb{R}$,

$\limsup_{n\rightarrow\infty}\frac{1}{n}\log m\left(\left\\{\omega\in\Omega:\
\frac{1}{n}X_{n}(\omega)\in F\right\\}\right)\leq-\inf_{\alpha\in
F}\Lambda^{*}(\alpha);$

  * (b)

if, furthermore $\Lambda(q)$ is differentiable on $\mathbb{R}$, for any open
set $G\subset\mathbb{R}$,

$\liminf_{n\rightarrow\infty}\frac{1}{n}\log m\left(\left\\{\omega\in\Omega:\
\frac{1}{n}X_{n}(\omega)\in G\right\\}\right)\geq-\inf_{\alpha\in
G}\Lambda^{*}(\alpha).$

In order to apply the Gartner-Ellis theorem to ergodic averages
$X_{n}(\omega)=S_{n}{\varphi}(\omega)$ it is necessary to introduce the
reference measure $m$. A good choice would be the uniform  or the measure of
maximal entropy for $\Omega$. Namely, let $l=|\mathcal{A}|$ the number of
different letters in alphabet $\mathcal{A}$, and let
$m=\rho^{\mathbb{Z}+}_{N}$ be the product of uniform measures $\rho_{N}$ on
$\mathcal{A}$. Then one immediately concludes that

$\Lambda_{\varphi}(q)=P_{\varphi}(q)-\log l=P(q{\varphi})-\log l.$

Therefore, for Hölder continuous functions ${\varphi}$’s, the logarithmic
moment generating function $\Lambda_{\varphi}(q)$ is also real-analytic, and
hence the Gartner-Ellis theorem is applicable. The so-called large deviations
rate function is then

${\mathcal{I}}_{\varphi}(\alpha)=\Lambda^{*}_{\varphi}(\alpha)=\sup_{q\in\mathbb{R}}\Bigl{[}q\alpha-\Lambda_{\varphi}(q)\Bigr{]}=\sup_{q\in\mathbb{R}}\Bigl{[}q\alpha-
P_{\varphi}(q)+\log N\Bigr{]}=\log l+P^{*}_{\varphi}(\alpha).$

Note also that $h_{{\textsf{top}}}(\sigma,\Omega)=\log l$. Hence,

${\mathcal{I}}_{\varphi}(\alpha)=h_{\textsf{top}}(\sigma,\Omega)+P_{\varphi}^{*}(\alpha)$

######  Theorem 1.4 ([Barreira-Pesin-Schmeling1997]).

Suppose $\Omega=A^{\mathbb{Z}_{+}}$ and ${\varphi}:\Omega\to\mathbb{R}$ is
Hölder continuous. For $\alpha\in\mathbb{R}$, consider the set

$K_{\alpha}=\left\\{\omega\in\Omega:\
\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}{\varphi}(\sigma^{k}\omega)=\alpha\right\\}.$

Then there exist $\underline{\alpha},\overline{\alpha}\in\mathbb{R}$ such that

  * •

for every $\alpha\in(\underline{\alpha},\overline{\alpha})$,
$K_{\alpha}\neq\varnothing$ and

$\dim_{H}(K_{\alpha})=\frac{1}{\log
l}\inf_{q}\Bigl{(}P(q{\varphi})-q\alpha\Bigr{)}=-\frac{1}{\log
l}\sup_{q}\Bigl{(}q\alpha-P(q{\varphi})\Bigr{)}=-\frac{P^{*}_{\varphi}(\alpha)}{\log
l}.$

  * •

the multifractal spectrum

$\alpha\mapsto\mathcal{D}_{\varphi}(\alpha):=\dim_{H}(K_{\alpha})$

is a real-analytic function of $\alpha$ on
$(\underline{\alpha},\overline{\alpha})$.

Again, if we consider the topological entropy
$h_{{\textsf{top}}}(\sigma,\cdot)$ instead of the Hausdorff dimension
$\dim_{H}(\cdot)$, taking into account, that
$h_{{\textsf{top}}}(\sigma,\Omega)=\log l$, we obtain that the large
deviations rate function and the topological entropy of level sets of ergodic
averages are related

$h_{{\textsf{top}}}(\sigma,K_{\alpha})=h_{{\textsf{top}}}(\sigma,\Omega)-{\mathcal{I}}_{\varphi}(\alpha).$

It was observed in [Takens-Verbitskiy1999], that the result of Theorem 1.4 can
be extended to a much larger class of dynamical systems and observables
${\varphi}$ with the key property being that for every $q\in\mathbb{R}$, there
is a unique equilibrium state $\mu_{q}$ for $q{\varphi}$. To ensure
uniqueness, one has to make two types of assumptions: on the map
$f:\Omega\to\Omega$ and the potential (observable)
${\varphi}:\Omega\to\mathbb{R}$. For example, in [Takens-Verbitskiy1999] it
was assumed that

  * •

the map $f:\Omega\to\Omega$ is an _expansive_ homeomorphism with
_specification_ property

  * •

the continuous observable ${\varphi}:\Omega\to\mathbb{R}$ is in the _Bowen
class_ , i.e., for all ${\varepsilon}>0$,

$\sup_{d_{n}(\omega,\omega^{\prime})<{\varepsilon}}\Bigl{|}(S_{n}{\varphi})(\omega)-(S_{n}{\varphi})(\omega^{\prime})\Bigr{|}\leq
K({\varepsilon})<\infty$

where the supremum is taken over all $\omega,\omega^{\prime}$ such that

$d_{n}(\omega,\omega^{\prime})=\max_{k=0,\ldots,n-1}d(f^{k}(\omega),f^{k}(\omega^{\prime}))<{\varepsilon}.$

Under these conditions, the Gartner-Ellis theorem holds for ergodic sums
$X_{n}(\omega)=S_{n}{\varphi}(\omega)$ with a differentiable logarithmic
moment generating function $\Lambda(q)$, and the multifractal spectrum for
ergodic averages is given by the Legendre transform of the pressure function,
i.e., for all $\alpha\in(\underline{\alpha},\overline{\alpha})$,

$h_{\textsf{top}}(f,K_{\alpha})=\inf_{q}\Bigl{(}P(q{\varphi})-q\alpha\Bigr{)}=-{P^{*}_{\varphi}(\alpha)}.$

Furthermore, the logarithmic moment-generating function $\Lambda$ is related
to the pressure function $P_{{\varphi}}$ by
$\Lambda(q)=P_{{\varphi}}(q)-h_{{\textsf{top}}}(f,\Omega),\;q\in\mathbb{R}$.

###  1.2. Multifractal analysis of Birkhoff averages of continuous functions

If one considers simply continuous observables
${\varphi}:\Omega\to\mathbb{R}$, it is no longer true that there is a unique
equilibrium state $\mu_{q}$ for the potential $q{\varphi}$ for all
$q\in\mathbb{R}$. Therefore, a different approach is required.

In the case of symbolic systems ($\Omega=A^{\mathbb{Z}_{+}}$,
$\sigma:\Omega\to\Omega$ is the left-shift), the first results were obtained
independently by Fan and Feng [Fan1, Fan1996] and Olivier [Olivier1,
Olivier2]. The methods are quite different: Olivier relied on the density of
Hölder continuous function in the space of all continuous functions. Fan and
Feng constructed the so-called Moran fractal. The approach of Fan and Feng
turned out to be very suitable for generalization to abstract compact spaces.
In [Takens-Verbitskiy2003] the following variational principle for
multifractal spectra has been obtained:

######  Theorem 1.5 ([Takens-Verbitskiy2003]).

Suppose $T:\Omega\to\Omega$ is a continuous transformation of a compact metric
space $(\Omega,d)$ with the specification property and
${\varphi}:\Omega\to\mathbb{R}$ is a continuous observable. Then there exist
$\underline{\alpha},\overline{\alpha}\in\mathbb{R}$ such that for every
$\alpha\in(\underline{\alpha},\overline{\alpha})$,
$K_{\alpha}\neq\varnothing$, and

$h_{\textsf{top}}(T,K_{\alpha})=\sup_{\mu\in\mathcal{M}_{T}(\Omega)}\Bigl{\\{}h_{\mu}(T):\quad\int{\varphi}d\mu=\alpha\Bigr{\\}},$

where supremum is taken over $\mathcal{M}(\Omega,T)$ of all $T$-invariant
measures on $\Omega$, and $h_{\mu}(T)$ is the Kolmogorov–Sinai entropy of
$\mu$.

For such systems, the Large Deviation Principle has been established in 1990
by Young [Young]. We will not recall the most general form of the result, but
reformulate it in a such form that the relation to the multifractal formalism
becomes apparent immediately.

######  Theorem 1.6 ([Young], Theorem 1).

Suppose $m$ is a not necessarily invariant Ahlfors-Bowen reference measure on
$\Omega$. Suppose $\varphi:\Omega\to\mathbb{R}$ is continuous. Then for all
$c\in\mathbb{R}$, one has

$\limsup_{n\rightarrow\infty}\frac{1}{n}\log
m\left\\{\frac{1}{n}S_{n}\varphi\geq
c\right\\}\leq\sup_{\mu\in\mathcal{M}_{T}(\Omega)}\left\\{h_{\mu}(T):\int\varphi
d\mu\geq c\right\\}-h_{{\textsf{top}}}(T,\Omega).$

If, furthermore, $T$ has a specification property, then for all
$c\in\mathbb{R}$, one has

$\liminf_{n\rightarrow\infty}\frac{1}{n}\log
m\left\\{\frac{1}{n}S_{n}\varphi>c\right\\}\geq\sup_{\mu\in\mathcal{M}_{T}(\Omega)}\left\\{h_{\mu}(T):\int\varphi
d\mu>c\right\\}-h_{{\textsf{top}}}(T,\Omega).$

This result allows us to say, that under the above conditions, the sequence of
random variables $\\{X_{n}(\omega)=(S_{n}{\varphi})(\omega)\\}$ satisfies the
Large Deviation Principle with the rate function

$I_{\varphi}(\alpha)=h_{{\textsf{top}}}(T,\Omega)-\sup_{\mu\in\mathcal{M}_{T}(\Omega)}\Bigl{\\{}h_{\mu}(T):\quad\int{\varphi}d\mu=\alpha\Bigr{\\}},$

As in the cases considered above, combining the results of Theorems 1.5 and
1.6, one concludes that for topological dynamical systems with specification
the large deviations rate function $I_{{\varphi}}$ and the multifractal
spectrum $\mathcal{E}_{{\varphi}}(\alpha):=h_{{\textsf{top}}}(T,K_{\alpha})$
are related by

$\mathcal{E}_{\varphi}(\alpha)=h_{\textsf{top}}(T,\Omega)-I_{\varphi}(\alpha).$

In fact, the methods developed by Young are instrumental in the proof of
Theorem 1.5 in [Takens-Verbitskiy2003].

###  1.3. Multifractal analysis beyond Birkhoff sums

A number of multifractal results have been obtained for the level sets of the
form

$K_{\alpha}=\Bigl{\\{}\omega\in\Omega:\,\lim_{n\to\infty}\frac{1}{n}X_{n}(\omega)=\alpha\Bigr{\\}},$

where $X_{n}$’s are not necessarily Birkhoff sums of the form
$S_{n}{\varphi}$, for some ${\varphi}$. For example, such _non-additive_
observable arise naturally in the multifractal analysis of pointwise Lyapunov
exponents for the products of matrices. Namely, suppose a map $M:\Omega\to
M_{d}(\mathbb{R})$ is given where $M_{d}(\mathbb{R})$ is the algebra of
$d\times d$ real matrices. We can associate to every $x\in\Omega$ and
$n\in\mathbb{N}$, a random variable
$X_{n}(x):=\log||M(T^{n-1}x)M(T^{n-2}x)\cdots M(x)||$ provided that
$||M(T^{n-1}x)M(T^{n-2}x)\cdots M(x)||\neq 0$, where $||\cdot||$ is a norm on
$M_{d}(\mathbb{R})$. Clearly, $\\{X_{n}\\}_{n\in\mathbb{N}}$ is not an
additive sequence if $d\geq 2$.

A non-additive sequence of potentials can also be encountered if one considers
weighted ergodic sums $\\{S_{n}^{w}f\\}$ of a continuous potential $f$ on the
space $\Omega$, i.e., $S_{n}^{w}f:=\sum_{i=0}^{n-1}w_{i}f\circ
T^{i},\;n\in\mathbb{Z}_{+}$ and $w_{n}\in\mathbb{R}$ are weights [Fan2021]. It
is easy to see that $\\{S_{n}^{w}f\\}$ is not additive unless all the weights
$w_{n},\;n\in\mathbb{Z}_{+}$ are the same.

Another example of non-additive sequences arises in studying pointwise
entropies of probability measures. Suppose $\mu$ is a fully supported
probability measure on $\Omega$. For a fixed $\delta>0$, define a sequence of
random variables as $X_{n}(x):=-\log\mu(B_{n}(x,\delta)),\;n\in\mathbb{N}$,
$x\in\Omega$, where
$B_{n}(x,\delta):=\\{y\in\Omega:d(T^{i}y,T^{i}x)<\delta,\;\;i=0,\dots,n-1\\}$,
$n\geq 1$.

We shall discuss these and other examples in greater detail in Section 7 and
show how our results can be applied in these cases.

##  2\. Preliminaries and Notations

In this section, we assume that we are given a compact metric space
$(\Omega,d)$ equipped with a continuous transformation $T:\Omega\to\Omega$. We
denote by $M(\Omega),M(\Omega,T)$ the sets of all Borel probability measures
and $T$-invariant Borel probability measures on $\Omega$, respectively. For
$x,y\in\Omega$ and $n\in\mathbb{N}$, the Bowen metric $d_{n}$ is defined as
$d_{n}(x,y):=\max_{0\leq i\leq n-1}d(T^{i}x,T^{i}y)$. By
$B_{n}(x,\varepsilon)$ we denote an open ball of radius $\varepsilon>0$ in the
metric $d_{n}$ centred at $x\in\Omega$, i.e.,
$B_{n}(x,\varepsilon):=\\{y\in\Omega:d_{n}(x,y)<\varepsilon\\}$.

###  2.1.  Topological Entropy

The notion of topological entropy of non-compact and non-invariant sets has
been introduced by Bowen [Bowen] in 1973. This paper uses an equivalent
definition of Bowen’s topological entropy given in [Pesin]. For
$Z\subset\Omega$, and all $t\in\mathbb{R},\;{\varepsilon}>0$, and
$N\in\mathbb{N}$, let

(2)
$m(Z,t,{\varepsilon},N):=\inf\left\\{\sum_{i=1}^{\infty}e^{-n_{i}t}:Z\subset\bigcup_{i=1}^{\infty}B_{n_{i}}(x_{i},{\varepsilon}),\;n_{i}\geq
N\right\\}.$

By the standard convention, $m(\emptyset,t,{\varepsilon},N)=0$, for all
$t,{\varepsilon},N$. Obviously, $m(Z,t,{\varepsilon},N)$ does not decrease on
$N$, thus we can define

$m(Z,t,{\varepsilon}):=\lim_{N\to\infty}m(Z,t,{\varepsilon},N).$

One can show [Pesin] that $m(\cdot,t,{\varepsilon})$ is an outer measure with
properties similar to those of $t$-dimensional Hausdorff outer measure. In
particular, there exists a critical value $t^{\prime}\in\mathbb{R}$ such that

$m(Z,t^{\prime},{\varepsilon})=\begin{cases}+\infty,\quad~{}~{}~{}\text{if}~{}t^{\prime}<t,\\\
0,\quad~{}~{}~{}~{}~{}\text{if}~{}t^{\prime}>t.\end{cases}$

We denote this critical value by $h_{{\textsf{top}}}(T,Z,{\varepsilon})$. Thus
$h_{{\textsf{top}}}(T,Z,{\varepsilon})=\inf\\{t\in\mathbb{R}:m(Z,t,{\varepsilon})=0\\}=\sup\\{t\in\mathbb{R}:m(Z,t,{\varepsilon})=+\infty\\}$.
Furthermore, since $h_{{\textsf{top}}}(T,Z,{\varepsilon})$ is monotonic in
${\varepsilon}$, we can define the topological entropy of $Z$ by

$h_{{\textsf{top}}}(T,Z):=\lim\limits_{{\varepsilon}\to
0+}h_{{\textsf{top}}}(T,Z,{\varepsilon}).$

It should be stressed that the set $Z$ is not assumed to be compact nor
$T$-invariant. Finally, by our convention,
$h_{{\textsf{top}}}(T,\emptyset)=-\infty$, and $h_{{\textsf{top}}}(T,Z)\geq 0$
for all non-empty subsets $Z\subset\Omega$. We summarize the basic properties
of the topological entropy in the following remark.

######  Remark 2.1 ([Pesin]).

  * (i)

Monotonicity: if $Z_{1}\subset Z_{2}$, then $h_{{\textsf{top}}}(T,Z_{1})\leq
h_{{\textsf{top}}}(T,Z_{2})$;

  * (ii)

Countable stability: if $Z=\bigcup_{n}Z_{n}$, then
$h_{{\textsf{top}}}(T,Z)=\sup_{n}h_{{\textsf{top}}}(T,Z_{n})$.

######  Remark 2.2.

The notions of topological entropy and Hausdorff dimension coincide if the
underlying space $\Omega$ is a symbolic space. Suppose
$\Omega=\\{1,2,\ldots,l\\}^{\mathbb{Z}_{+}}$ with integer $l\geq 2$, and the
metric on $\Omega$ is defined as

$d(x,y)=\frac{1}{l^{k}},\ \ \text{where}\ \ k=\min\\{i\geq 0:x_{i}\neq
y_{i}\\},$

then $Z\subseteq\Omega$, $\dim_{H}(Z)=\frac{h_{{\textsf{top}}}(T,Z)}{\log l}$.

###  2.2. Local (pointwise) entropies

Consider a Borel probability measure $\mu$ on the metric space $(\Omega,d)$,
and define the lower and upper pointwise entropies of $\mu$ at $x\in\Omega$ as

$\underline{h}_{\mu}(T,x):=\lim_{\varepsilon\to
0}\liminf_{n\to\infty}-\frac{1}{n}\log\mu(B_{n}(x,\varepsilon)),\;\;\;\overline{h}_{\mu}(T,x):=\lim_{\varepsilon\to
0}\limsup_{n\to\infty}-\frac{1}{n}\log\mu(B_{n}(x,\varepsilon)).$

Note that the limit in $\varepsilon$ exists due to the monotonicity. D.J. Feng
and W. Huang [Feng-Huang2012] extended the notion of measure-theoretic entropy
to non-invariant measures by motivating the result of M. Brin and A. Katok. In
fact, for a probability measure $\mu\in\mathcal{M}(\Omega)$, D.J. Feng and W.
Huang [Feng-Huang2012] defined the upper and lower measure-theoretic entropies
of $\mu$ relative to the transformation $T$ by

$\underline{h}_{\mu}(T):=\int_{\Omega}\underline{h}_{\mu}(T,x)\mu(dx)\text{
and }\overline{h}_{\mu}(T):=\int_{\Omega}\overline{h}_{\mu}(T,x)\mu(dx).$

Note that the above formulas are consistent with the classical notion of
measure-theoretic entropy due to the Brin-Katok theorem [Brin]. In fact, if
$\mu\in\mathcal{M}(\Omega,T)$, then
$\underline{h}_{\mu}(T)=\overline{h}_{\mu}(T)=h_{\mu}(T)$, where $h_{\mu}(T)$
is the measure-theoretic entropy of $\mu$ with respect to $T$. In [Feng-
Huang2012], the authors also extended the classical variational principle to
the non-invariant setting. More precisely, they showed for a compact, but not
necessarily invariant set $K\subseteq\Omega$ that

(3)
$h_{{\textsf{top}}}(T,K)=\sup\\{\underline{h}_{\mu}(T):\mu\in\mathcal{M}(\Omega),\;\mu(K)=1\\}.$

The next theorem is a dynamical analogue of the Mass distribution principle
and is very useful for the estimation of topological entropies of various
sets.

######  Theorem 2.1 ([Ma-Wen]).

Let $\mu$ be a Borel probability measure on $\Omega$, and $E$ be a Borel
subset of $\Omega$.

  * (1)

If $\underline{h}_{\mu}(T,x)\leq t$ for some $t<\infty$ and all $x\in E$, then
$h_{{\textsf{top}}}(T,E)\leq t$.

  * (2)

If $\mu(E)>0,$ and $\underline{h}_{\mu}(T,x)\geq t$ for some $t>0$ and all
$x\in E$, then $h_{{\textsf{top}}}(T,E)\geq t$.

###  2.3. Ahlfors-Bowen measures

######  Definition 2.1.

We say that a not necessarily invariant Borel probability measure $m$ on
$\Omega$ is Ahlfors-Bowen if there exists $h>0$ such that for every
${\varepsilon}>0$ there exists a finite positive constant, $C({\varepsilon})$
such that for every $n\geq 1$ and all $\omega\in\Omega$, one has

(4) $\frac{1}{C({\varepsilon})}e^{-nh}\leq
m\bigr{(}B_{n}(\omega,{\varepsilon})\bigr{)}\leq{C({\varepsilon})}e^{-nh}.$

It is easy to see from the definition that an Ahlfors-Bowen measure has a very
simple spectrum of local entropies. More specifically, $h_{m}(T,\omega)=h$ for
all $\omega\in\Omega$. Furthermore, one has the following simple lemma for the
constant $h$.

######  Lemma 2.1.

Suppose $m$ is Ahlfors-Bowen measure for $T$ on $\Omega$, then
$h=h_{\textsf{top}}(T,\Omega)$.

###### Proof.

Since $m$ is an Ahlfors-Bowen measure, for any $\omega\in\Omega$, one has
$h_{m}(T,\omega)=h$. Therefore, by Theorem 2.1, we immediately conclude that
$h_{{\textsf{top}}}(T,\Omega)=h$. ∎

###  2.4. Conditions on the underlying transformation

In applications, topological dynamical systems (TDS) usually have some sort of
expansivity and mixing properties. Therefore, there is a vast amount of
literature devoted to studying dynamical systems with these properties. We
also impose expansivity and mixing properties on our underlying transformation
to prove some of our results.

######  Definition 2.2.

A continuous transformation $T:\Omega\to\Omega$ is called expansive if there
exists a constant $\rho>0$ such that if

$d(T^{n}(x),T^{n}(y))<\rho\ \text{for all non-negative integer }n,\text{then}\
\;x=y.$

The maximal $\rho$ with such a property is called the expansive constant.

A slightly stronger version of topological mixing is topological exactness.
Topological exactness of the transformation $T$ is that for every
${\varepsilon}>0$ and $x\in\Omega$ there is
$N:=N(x,{\varepsilon})\in\mathbb{N}$ such that
$T^{N}(B(x,{\varepsilon}))=\Omega$. Basically, by using the compactness of the
underlying space $\Omega$, one can uniformise the constant $N$ with
$x\in\Omega$. A relatively stronger version of topological exactness which we
call strong topological exactness also gives uniform control over the Bowen
balls of the transformation.

######  Definition 2.3.

We call a continuous transformation $T:\Omega\to\Omega$ strongly topologically
exact if for any ${\varepsilon}>0$ there is a natural number
$M_{1}\in\mathbb{N}$ such that for all $n\in\mathbb{N}$ and $x\in\Omega$,

(5) $T^{n+M_{1}}(B_{n}(x,{\varepsilon}))=\Omega.$

Note that topologically mixing subshifts of finite type are strongly
topologically exact. P.Walters ([Walters2000]) showed that an expansive
strongly topologically exact transformation has a weak form of the
specification property, namely, one has the following theorem:

######  Theorem 2.2 ([Walters2000]).

Let $T:\Omega\to\Omega$ be an expansive strongly topologically exact
transformation, then $T$ satisfies the weak specification condition, i.e., for
any $\varepsilon>0$ there exists a natural number $M_{2}\in\mathbb{N}$ such
that for for all $x,x^{\prime}\in\Omega$ and every $n_{1},n_{2}\in\mathbb{N}$,
there exists $w\in\Omega$ with $d(T^{i}w,T^{i}x)<\varepsilon,0\leq i\leq
n_{1}-1$ and $d(T^{n_{1}+M_{2}+j}w,T^{j}x^{\prime})<\varepsilon,\;0\leq j\leq
n_{2}-1$. Note that the latter condition is equivalent to the following: for
every ${\varepsilon}>0$ there exists an integer $M_{2}>0$ such that for all
$x,x^{\prime}\in\Omega$ and $n_{1},n_{2}\in\mathbb{N}$, one has

(6) $B_{n_{1}}(x,{\varepsilon})\cap
T^{-(n_{1}+M_{2})}(B_{n_{2}}(x^{\prime},{\varepsilon}))\neq\emptyset.$

######  Remark 2.3.

If $T:\Omega\to\Omega$ is an expansive homeomorphism with weak specification
property, then the measure of maximal entropy for $T$ is Ahlfors-Bowen
[Walters2000, Theorem 4.6].

###  2.5. Class of random variables

We say that the sequence $\\{X_{n}\\}_{n\geq 1}$ of real-valued random
variables (observables) on $\Omega$ is subadditive if for every $m,n\geq 1$,

$X_{n+m}\leq X_{n}+X_{m}\circ T^{n}.$

We call a sequence $\\{X_{n}\\}_{n\geq 1}$ of random variables on $\Omega$
weakly almost additive (or simply, almost additive), if there are constants
$A_{n}\geq 0,\;n\in\mathbb{N}$ with $\lim_{n\to\infty}\frac{A_{n}}{n}=0$ such
that for all $x\in\Omega$ and $n,m\in\mathbb{N}$,

(7) $|X_{n+m}(x)-X_{n}(x)-X_{m}(T^{n}x)|\leq A_{n}.$

Recently, Cuneo [Cuneo2020, Theorem 1.2] showed that weakly almost additive
sequences of continuous functions can be approximated uniformly by Birkhoff’s
sums of continuous functions.

For a function $H:\Omega\to\mathbb{R}$ and $n\in\mathbb{N}$, $\delta>0$, we
denote the $(n,\delta)$-variation of the function $H$ by
$v_{n,\delta}(H):=\sup\limits_{d_{n}(y,z)\leq\delta}|H(y)-H(z)|$.

######  Definition 2.4.

A sequence $\\{X_{n}\\}$ of random variables on $\Omega$ satisfies the Bowen
condition, if there exists $\delta>0$ such that
$\sup\limits_{n\in\mathbb{N}}v_{n,\delta}(X_{n})<+\infty$. We say that the
sequence $\\{X_{n}\\}$ satisfies the weak Bowen condition if there exists
$\delta>0$ such that

(8) $\lim_{n\to\infty}\frac{v_{n,\delta}(X_{n})}{n}=0.$

It should be mentioned that the continuity of the random variables
$X_{n},\;n\geq 1$, is not assumed in the definition of the weak Bowen
condition, and neither the Bowen condition nor the weak Bowen condition
implies the continuity of each individual random variable $X_{n}$. However, if
a sequence of functions $\\{X_{n}\\}$ satisfies the weak Bowen condition, then
each function $X_{n}$ is bounded. Indeed, since $\Omega$ is compact, for every
$n$, there exists a finite cover $\\{B_{n}({\omega_{i},\delta/2})\\}_{i}$, and
the observable $X_{n}$ is a bounded function on each ball
$B_{n}({\omega_{i},\delta/2})$ in the cover.

######  Remark 2.4.

One can observe that if $T$ is expansive, then the sequence of ergodic sums of
any continuous potential $\psi:\Omega\to\mathbb{R}$ satisfies weak Bowen’s
condition. Namely, if $X_{n}=S_{n}\psi=\sum_{i=0}^{n-1}\psi\circ T^{i}$, then
$v_{n,\delta}(X_{n})\leq\sum_{i=1}^{n}v_{i,\delta}(\psi)$, and thus

$\lim_{n\to\infty}\frac{v_{n,\delta}(X_{n})}{n}\leq\lim_{n\to\infty}\frac{\sum_{i=1}^{n}v_{i,\delta}(\psi)}{n}=0$

by Stolz-Cesaro’s theorem, since $\psi:\Omega\to\mathbb{R}$ is continuous and
hence $\lim\limits_{n\to\infty}v_{n,\delta}(\psi)=0$, where positive number
$\delta$ is smaller than the expansive constant of $T$.

##  3\. Large Deviations

###  3.1. Some concepts from convex analysis

In this section, we consider functions with values in
$\overline{\mathbb{R}}:=\mathbb{R}\cup\\{\pm\infty\\}$. A convex function
${\varphi}:\mathbb{R}\to\overline{\mathbb{R}}$ is called a proper convex
function if ${\varphi}(t)>-\infty$ for all $t\in\mathbb{R}$ and
${\varphi}\nequiv+\infty$. The essential domain of the convex function
${\varphi}$ is
$\text{dom}({\varphi}):=\\{t\in\mathbb{R}:{\varphi}(t)<+\infty\\}$. Note that
$\text{dom}({\varphi})\subset\mathbb{R}$ is a convex set.

Suppose ${\varphi}:\mathbb{R}\to(-\infty,+\infty]$ is a proper convex function
and assume that ${\varphi}$ is finite at some $t\in\mathbb{R}$. A real number
$t^{*}$ is a subgradient of ${\varphi}$ at $t$ if for all $s\in\mathbb{R}$,
${\varphi}(t)+t^{*}(s-t)\leq{\varphi}(s)$ holds. Denote the set of all
subgradients of ${\varphi}$ at $t$ by $\partial{\varphi}(t)$. It is easy to
check that
$\partial{\varphi}(t)=\mathbb{R}\bigcap\Big{[}\sup_{s>0}\frac{{\varphi}(t)-{\varphi}(t-s)}{s},\;\inf_{s>0}\frac{{\varphi}(t+s)-{\varphi}(t)}{s}\Big{]}=\mathbb{R}\bigcap[{\varphi}^{\prime}_{-}(t),{\varphi}^{\prime}_{+}(t)]$,
where
${\varphi}^{\prime}_{-}(t),\;{\varphi}^{\prime}_{+}(t)\in\overline{\mathbb{R}}$
are left and right derivatives of ${\varphi}$ at $t$. We define the domain of
the multivalued mapping
$\partial{\varphi}:t\in\text{dom}({\varphi})\mapsto\partial{\varphi}(t)\subset\mathbb{R}$
by
$\text{dom}(\partial{\varphi}):=\\{t\in\text{dom}({\varphi}):\partial\varphi(t)\neq\emptyset\\}$.
Note that in general, $\text{dom}(\partial{\varphi})$ is not a convex set.

######  Definition 3.1.

A proper convex function ${\varphi}:\mathbb{R}\to\overline{\mathbb{R}}$ is
called essentially strictly convex if ${\varphi}$ is strictly convex on every
convex subset of $\text{dom}(\partial{\varphi})$. Note that a convex function
${\varphi}$ is called strictly convex in a convex set $D\subset\mathbb{R}$, if
for every $t_{1},t_{2}\in D$ with $t_{1}\neq t_{2}$ and $\lambda\in(0,1)$,
${\varphi}(\lambda
t_{1}+(1-\lambda)t_{2})<\lambda{\varphi}(t_{1})+(1-\lambda){\varphi}(t_{2})$.

It is a well-known fact that finite convex functions are very close to
differentiable functions. However, if it comes to extended convex functions,
then one needs to be more careful about the essential domain of the function.

######  Definition 3.2.

A proper convex function ${\varphi}:\mathbb{R}\to\overline{\mathbb{R}}$ is
essentially smooth if it satisfies the following conditions:

  * (a)

$\text{dom}({\varphi})^{\circ}\neq\emptyset$, where $A^{\circ}$ denotes the
interior of a set $A$;

  * (b)

${\varphi}$ is differentiable on $\text{dom}({\varphi})^{\circ}$;

  * (c)

${\varphi}$ is steep, namely,
$\lim_{n\rightarrow\infty}\left|{\varphi}\left(t_{n}\right)\right|=\infty$
whenever $\left\\{t_{n}\right\\}$ is a sequence in
$\text{dom}({\varphi})^{\circ}$ converging to a boundary point of
$\text{dom}({\varphi})^{\circ}$.

An extended, not necessarily convex, function
${\varphi}:\mathbb{R}\to\overline{\mathbb{R}}$ is lower semicontinuous if for
all $\lambda\in\mathbb{R}$, $\\{t\in\mathbb{R}:{\varphi}(t)\leq\lambda\\}$ is
a closed subset of $\mathbb{R}$. Equivalently, a function ${\varphi}$ is lower
semicontinuous if and only if $\liminf\limits_{x\to
x_{0}}{\varphi}(x)\geq{\varphi}(x_{0})$ for all $x_{0}\in\mathbb{R}$. Note
that a lower semicontinuous function achieves its minimum in any compact set.

The following theorem shows that essentially strictly convex and essentially
smooth convex functions are intimately related in terms of the Legendre
transform.

######  Theorem 3.1 ([Rockafellar-Book]).

A lower semicontinuous proper convex function ${\varphi}$ is essentially
strictly convex if and only if its Legendre transform,
${\varphi}^{*}(t):=\sup\limits_{q\in\mathbb{R}}\\{tq-{\varphi}(q)\\},\;t\in\mathbb{R}$,
is essentially smooth.

###  3.2. Large deviations

######  Definition 3.3.

An extended function $I:\mathbb{R}\to[0,+\infty]$ is called a  large
deviations rate function (or simply a rate function) if it is lower
semicontinuous. A rate function $I$ is called good if the sub-level sets,
$\\{t:I(t)\leq\lambda\\}\subset\mathbb{R}$ are compact for all
$\lambda\in\mathbb{R}$. For $E\subset\mathbb{R}$, $I(E)$ abbreviates
$\inf\limits_{t\in E}I(t)$ i.e., $I(E):=\inf\limits_{t\in E}I(t)$.

Note that a good rate function achieves its minimum in any closed set.

Suppose $(\Omega,\mathcal{F},\mu)$ is a probability space and
$X_{n}:\Omega\to\mathbb{R}$, $n\geq 1$, are random variables.

######  Definition 3.4.

The sequence of random variables $\\{\frac{1}{n}X_{n}\\}_{n=1}^{+\infty}$
satisfies the large deviations principle (LDP) with a rate function $I$ if for
all Borel sets $E\subset\mathbb{R}$,

$-I(E^{\circ})\leq\liminf_{n\rightarrow\infty}\frac{1}{n}\log\mu\Big{(}\Big{\\{}\omega:\
\frac{X_{n}(\omega)}{n}\in
E\Big{\\}}\Big{)}\leq\limsup_{n\rightarrow\infty}\frac{1}{n}\log\mu\Big{(}\Big{\\{}\omega:\
\frac{X_{n}(\omega)}{n}\in E\Big{\\}}\Big{)}\leq-I(\overline{E}),$

where $\overline{E}$, $E^{\circ}$ denote the closure and the interior of $E$,
respectively.

One of the most convenient ways to establish the validity of the LDP is
provided by the Gartner-Ellis theorem which we have stated in a partial case
in Theorem 1.3. Denote the normalised log-moment generating function of
$X_{n}$ by ${\varphi}_{n}$, i.e., for $t\in\mathbb{R}$

${\varphi}_{n}(t):=\frac{1}{n}\log\int\limits_{\Omega}e^{tX_{n}}d\mu.$

Note that ${\varphi}_{n}$ may take value $+\infty$. Define the log-moment
generating function of the sequence $X:=\\{X_{n}\\}$ by
${\varphi}_{X}(t):=\limsup\limits_{n\to\infty}{\varphi}_{n}(t),\;t\in\mathbb{R}$.
Then the Gartner-Ellis theorem states that if for all $t\in\mathbb{R}$, the
limit, $\lim\limits_{n\to\infty}{\varphi}_{n}(t)$ exists and is finite, and
the log-moment generating function ${\varphi}_{X}$ is essentially smooth, then
$\\{\frac{1}{n}X_{n}\\}_{n\in\mathbb{N}}$ satisfies the LDP with a good rate
function $I_{X}:={\varphi}_{X}^{*}$. An inverse statement to this in some
sense is the celebrated Varadhan’s lemma which we reformulate slightly for our
purposes below. One can find a general version of the theorem in [Dembo-
Zeitouni-Book, Theorem 4.3.1].

######  Theorem 3.2 ([Dembo-Zeitouni-Book]).

Suppose that $\\{\frac{1}{n}X_{n}\\}_{n=1}^{+\infty}$ satisfies the LDP with a
good rate function $I:\mathbb{R}\rightarrow[0,\infty]$, and

$\limsup_{n\rightarrow
0}\frac{1}{n}\log\int_{\Omega}e^{tX_{n}}d\mu={\varphi}_{X}(t)<\infty$

hold for all $t\in\mathbb{R}$. Then the limit $\lim\limits_{n\rightarrow
0}\frac{1}{n}\log\mathbb{E}_{\mu}\left[e^{tX_{n}}\right]$ exists and

${\varphi}_{X}(t)=\lim\limits_{n\rightarrow
0}\frac{1}{n}\log\int_{\Omega}e^{tX_{n}}d\mu=\sup_{q\in\mathbb{R}}\\{tq-I(q)\\}=I^{*}(t),\;\;\;\text{for
all }\;t\in\mathbb{R}.$

Sometimes, a weak version of LDP is more convenient for purposes, since it is
easier to establish than the full LDP.

######  Definition 3.5.

Let $(\Omega,\mathcal{F},\mu)$ be a probability space and
$X_{n}:\Omega\to\mathbb{R},\;\;n\geq 1$ be random variables. The sequence
$\\{\frac{1}{n}X_{n}\\}_{n\in\mathbb{N}}$ satisfies the weak Large Deviation
Principle (weak LDP) with a rate function $I$ if

  * 1)

for all compact set $F\Subset\mathbb{R}$,

$\limsup_{n\to\infty}\frac{1}{n}\log\mu\Big{(}\left\\{\frac{1}{n}X_{n}\in
F\right\\}\Big{)}\leq-I(F)$

  * 2)

for all open set $G\subset\mathbb{R}$,

$\liminf_{n\to\infty}\frac{1}{n}\log\mu\Big{(}\left\\{\frac{1}{n}X_{n}\in
G\right\\}\Big{)}\geq-I(G)$

Note that the only difference between LDP and weak LDP is in the upper large
deviation bound.

As one can see in the Gartner-Ellis theorem, the LDP upper bound requires
fewer conditions than the lower bound. However, one still needs to make rather
mild assumptions (for example, the existence of the limit log-moment
generating function as in Theorem 1.3) to obtain the upper bound.
Nevertheless, one can establish the upper weak large deviation bound without
any additional assumption about the sequence as the following folklore theorem
states.

######  Theorem 3.3.

Let $\\{X_{n}\\}$ be a sequence of random variables on a probability space
$(\Omega,\mathcal{F},\mu)$. Then the sequence
$\\{\frac{1}{n}X_{n}\\}_{n\in\mathbb{N}}$ satisfies the weak large deviations
upper bound with the rate function $I_{X}:={\varphi}_{X}^{*}$, i.e., for any
compact set $F\Subset\mathbb{R}$, the following holds

(9) $\limsup_{n\to\infty}\frac{1}{n}\log\mu\Big{(}\left\\{\frac{1}{n}X_{n}\in
F\right\\}\Big{)}\leq-I_{X}(F).$

##  4\. Main results

The main goal of this paper is to obtain the Multifractal spectra for the
sequence $\\{\frac{1}{n}X_{n}\\}$ under the assumption that the sequence
satisfies LDP. As a byproduct, we also obtain a formula that relates the
entropy spectra with the LDP rate function. Although we assume in our theorems
that the reference measure satisfies the Ahlfors-Bowen condition, one should
be able to generalise the results to a multifractal measure $\tau$ with the
smooth multifractal spectrum, i.e. $\alpha\mapsto
h_{{\textsf{top}}}(T,E_{\alpha})$ is a smooth function, where
$E_{\alpha}:=\\{\omega:h_{\tau}(\omega)=\alpha\\}$.

Assume that the ambient space $(\Omega,d)$ is compact, and we have a
continuous transformation $T:\Omega\to\Omega$ with finite topological entropy,
i.e., $h_{{\textsf{top}}}(T,\Omega)<\infty$. It is also assumed that a
sequence $X:=\\{X_{n}\\}$ of random variables on $\Omega$ and a reference
measure $\nu\in\mathcal{M}(\Omega)$ satisfying Ahlfors-Bowen condition with
the parameter $h=h_{{\textsf{top}}}(T,\Omega)$ are given. It should be
stressed that we are not assuming $\\{X_{n}:n\in\mathbb{N}\\}$ are continuous
functions, and $\nu$ is $T$-invariant. We denote, for $\alpha\in\mathbb{R}$,
the $\alpha$-level set of the sequence $\\{\frac{1}{n}X_{n}\\}$ by

$K_{\alpha}:=\\{x\in\Omega:\lim\limits_{n\to\infty}\frac{1}{n}X_{n}(x)=\alpha\\},$

and the domain of the multifractal spectrum by
$\mathcal{L}_{X}:=\\{\alpha\in\mathbb{R}:K_{\alpha}\neq\emptyset\\}$. We shall
study the entropy spectrum
$\mathcal{E}_{X}(\alpha):=h_{{\textsf{top}}}(T,K_{\alpha})$. In the statements
of our theorems, we shall assume some of the following main conditions:

  * (A1)

The sequence $\\{X_{n}\\}_{n}$ satisfies the weak Bowen condition;

  * (A2)

The sequence $\\{X_{n}\\}_{n}$ is weakly almost additive;

  * (A3)

$T:\Omega\to\Omega$ is an expansive strongly topologically exact
transformation.

We now present our first main result that estimates the entropy spectrum
$\mathcal{E}_{X}$ from above in terms of the LDP rate function.

######  Theorem A.

Assume (A1), and the sequence $\Big{\\{}\frac{1}{n}X_{n}\Big{\\}}$ satisfies
the weak large deviations upper bound with a rate function
$I_{X}:\mathbb{R}\to[0,+\infty]$, i.e. for all compact $F\Subset\mathbb{R}$
one has

(10) $\limsup_{n\to\infty}\frac{1}{n}\log\nu\Big{\\{}\frac{1}{n}X_{n}\in
F\Big{\\}}\leq-I_{X}(F),$

then for all $\alpha\in\mathcal{L}_{X}$, one has

(11) $\mathcal{E}_{X}(\alpha)\leq h_{{\textsf{top}}}(T,\Omega)-I_{X}(\alpha).$

Let ${\varphi}_{n}(q):=\frac{1}{n}\log\mathbb{E}_{\nu}e^{qX_{n}}$ and
${\varphi}_{X}:=\limsup\limits_{n\to\infty}{\varphi}_{n}$ be the log and log-
limit moment-generating functions of the random variable $X_{n}$,
respectively. Notice that the moment-generating functions ${\varphi}_{n}$
exist and are finite for all $n\in\mathbb{N}$ since the observables $X_{n}$
are bounded functions by the weak Bowen condition. The next result estimates
the spectrum $\mathcal{E}_{X}$ from above in terms of the log-moment
generating function ${\varphi}_{X}$ of $X$.

######  Theorem B.

Assume (A1), then for all $\alpha\in\mathcal{L}_{X}$

(12) $\mathcal{E}_{X}(\alpha)\leq
h_{{\textsf{top}}}(T,\Omega)-{\varphi}_{X}^{*}(\alpha),$

where ${\varphi}_{X}^{*}$ is the Legendre-Fenchel transform of
${\varphi}_{X}$.

######  Remark 4.1.

If one drops the condition (A1) in the above theorems, then the statements of
the theorems will be false. Assume $\Omega:=\\{0,1\\}^{\mathbb{Z}_{+}}$,
$T:\Omega\to\Omega$ is the left shift transformation, $x\in\Omega$ and
$\kappa\in\mathbb{R}\setminus\\{1\\}$. Set $X_{n}(\omega):=n$ if
$\omega_{i}=x_{i},\;i=0,1,\ldots,n^{2}-1$, and $X_{n}(\omega):=\kappa n$
otherwise. Then ${\varphi}_{X}(q)=\kappa q$ for every $q\in\mathbb{R}$, hence
by Theorem 1.3, the sequence $\\{X_{n}\\}_{n\in\mathbb{N}}$ satisfies LDP with
a good rate function ${\varphi}_{X}^{*}$. However, one has
$h_{{\textsf{top}}}(T,K_{1})=0>h_{{\textsf{top}}}(T,\Omega)-\varphi_{X}^{*}(1)=\log
2-\infty=-\infty$.

######  Remark 4.2.

We should note that one can prove the same statements as in Theorem A and
Theorem B with the following condition which is slightly weaker than the weak
Bowen condition: there exists $\delta>0$ such that

$\liminf\limits_{n\to\infty}\frac{v_{n,\delta}{(X_{n})}}{n}=0.$

One might expect equality in Theorem A and Theorem B, but it is easy to show
by examples that the conditions of these theorems are “too general” to prove
such equality. The subsequent theorem provides sufficient conditions for the
equality to hold in Theorem A and Theorem B.

######  Theorem C.

Assume the conditions (A1), (A2) and (A3). If the sequence
$\big{\\{}\frac{1}{n}X_{n}\big{\\}}_{n\in\mathbb{N}}$ satisfies LDP with an
essentially strictly convex good rate function $I_{X}$, then one has the
following:

  * (i)

There exists extended real numbers
$-\infty\leq\underline{\alpha}\leq\overline{\alpha}\leq+\infty$ such that
$(\underline{\alpha},\overline{\alpha})\subset\mathcal{L}_{X}\subset[\underline{\alpha},\overline{\alpha}]$.

  * (ii)

$\mathcal{E}_{X}(\alpha)=h_{{\textsf{top}}}(T,\Omega)-I_{X}(\alpha)$ holds for
all $\alpha\in(\underline{\alpha},\overline{\alpha})$.

Finally, it is well-known that a result of multifractal formalism for random
variables can be generalized straightforwardly to random vectors (see [Takens-
Verbitskiy2003]). In particular, one can directly extend the above theorems to
sequences $\\{\vec{X}_{n}\\}_{n\in N}$ of random vectors
$\vec{X}_{n}:\Omega\to\mathbb{R}^{d}$ with straightforward modification of the
conditions (A1)-(A3).

##  5\. Proofs of Theorem A and Theorem B: Upper bounds

###  5.1. A covering lemma

The Vitali covering lemma presented below plays a crucial role in establishing
upper bounds and is also instrumental in proving a key lemma in the subsequent
sections. Therefore, we shall state and prove it first for the sake of
completeness.

######  Lemma 5.1 (Vitali covering lemma).

Let $(\Xi,d)$ be a metric space, and suppose $\mathfrak{F}$ is a non-empty
finite collection of balls. Then there exists a disjoint subcollection
$\mathfrak{F}^{\prime}\subseteq\mathfrak{F}$, such that
$\bigcup\limits_{B\in\mathfrak{F}}B\subset\bigcup\limits_{B^{\prime}\in\mathfrak{F}^{\prime}}3B^{\prime}$,
where $3B^{\prime}$ denotes a ball with the same centre as the ball
$B^{\prime}$, but with three times the radius of $B^{\prime}$.

###### Proof.

We shall iteratively construct an ascending chain $\\{\mathfrak{F}_{s}\\}$ of
collections of pairwise disjoint balls contained in $\mathfrak{F}$ in the
following way: assume that in $s$-th step ($s\in\mathbb{N}$), we have
constructed such a subcollection $\mathfrak{F}_{s}$, ($\mathfrak{F}_{1}$ can
contain only one ball, therefore, such a disjoint collection of the balls
always exists) then we set $\mathfrak{F}_{s+1}=\mathfrak{F}_{s}$ if
$B^{\prime}\bigcap\bigcup\limits_{B\in\mathfrak{F}_{s}}B\neq\emptyset$ for
every $B^{\prime}\in\mathfrak{F}\setminus\mathfrak{F}_{s}$, otherwise we set
$\mathfrak{F}_{s+1}:=\mathfrak{F}_{s}\bigcup\\{B_{i_{s+1}}\\}$, where
$B_{i_{s+1}}$ is the ball having the largest radius among the balls non-
intersecting $\bigcup\limits_{B\in\mathfrak{F}_{s}}B$. Since $\mathfrak{F}$ is
finite, the constructed chain stabilizes at some $s^{\prime}\in\mathbb{N}$. We
claim that $\mathfrak{F}^{\prime}:=\mathfrak{F}_{s^{\prime}}$ is the desired
subcollection. Indeed, if $\mathfrak{F}\neq\mathfrak{F}_{s^{\prime}}$, then
for any $B^{\prime}\in\mathfrak{F}\setminus\mathfrak{F}_{s^{\prime}}$, by the
construction of $\mathfrak{F}_{s^{\prime}}$, we have
$B^{\prime}\bigcap\bigcup\limits_{B\in\mathfrak{F}_{s^{\prime}}}B\neq\emptyset$.
Thus there exists $B\in\mathfrak{F}_{s^{\prime}}$ such that $B^{\prime}\bigcap
B\neq\emptyset$, and hence $B^{\prime}\subset 3B$. ∎

###  5.2. Proofs of Theorem A and Theorem B

###### Proof of Theorem A.

First, we will show for any
$\alpha\in\mathcal{L}_{X}:=\\{\alpha\in\mathbb{R}:K_{\alpha}\neq\emptyset\\}$
that

(13) $I_{X}(\alpha)\leq h,$

where $h=h_{{\textsf{top}}}(T,\Omega)$ (c.f. Lemma 2.1). This, in particular,
implies that $\mathcal{L}_{X}\subset\text{dom}(I_{X})$.

Fix any decreasing sequence $\\{\delta_{m}\\}$ of positive numbers such that
$\delta_{m}\to 0$ as $m\to\infty$. For $\alpha\in\mathcal{L}_{X}$ set

(14)
$K_{\alpha,m}^{(n)}:=\\{w\in\Omega:-\delta_{m}<\frac{1}{n}X_{n}(w)-\alpha<\delta_{m}\\},\text{
and }Z_{m,j}:=\bigcap_{n\geq j}K_{\alpha,m}^{(n)}\bigcap K_{\alpha}.$

Then for every $m\in\mathbb{N}$, one has $K_{\alpha}=\bigcup_{j}Z_{m,j}$ .

We prove (13) by assuming the contradiction. Suppose that $I_{X}(\alpha)>h$
for some $\alpha\in\mathcal{L}_{X}$. Then there is $\Delta>0$ such that
$h<\Delta<I_{X}(\alpha)$. Take $\delta>0$, and consider the closed interval
$[\alpha-\delta,\alpha+\delta]$. Since $I_{X}$ is a lower semi-continuous
function, $I_{X}$ attains its minimum on $[\alpha-\delta,\alpha+\delta]$ at
some point $\alpha_{\delta}\in[\alpha-\delta,\alpha+\delta]$, i.e.,
$I_{X}(\alpha_{\delta})=I_{X}([\alpha-\delta,\alpha+\delta])$. Thus for all
$\delta>0$, $I_{X}(\alpha_{\delta})\leq I_{X}(\alpha)$, therefore,
$\limsup\limits_{\delta\to 0}I_{X}(\alpha_{\delta})\leq I_{X}(\alpha)$. But on
the other hand, $\liminf\limits_{\delta\to 0}I_{X}(\alpha_{\delta})\geq
I_{X}(\alpha)$ since $I_{X}$ is lower semi-continuous and
$\alpha_{\delta}\to\alpha$ as $\delta\to 0.$ Thus one can conclude that
$\lim\limits_{\delta\to 0}I_{X}([\alpha-\delta,\alpha+\delta])=I_{X}(\alpha)$,
therefore, there exists $\delta>0$ such that
$I_{X}([\alpha-\delta,\alpha+\delta])>\Delta$. Take any ${\varepsilon}>0$, and
fix it. Then since $\\{X_{n}\\}$ satisfies the weak Bowen condition, there
exists $N_{0}\in\mathbb{N}$ such that
$\frac{v_{n,{\varepsilon}}(X_{n})}{n}<\frac{\delta}{2}$ for all $n\geq N_{0}$.
Furthermore, since $\delta_{m}\to 0$, there is $m^{*}\in\mathbb{N}$ with
$\delta_{m^{*}}<\delta/2$. Finally, since
$\bigcup_{j}Z_{j,m^{*}}=K_{\alpha}\neq\emptyset$ and $\\{Z_{j,m^{*}}\\}_{j}$
is an ascending chain of the sets, there exists $z\in K_{\alpha}$ and
$j^{*}\in\mathbb{N}$ such that for all $j\geq j^{*}$, $z\in Z_{j,m^{*}}$.
Thus, for all $N\geq j^{*}$, one can readily check that

(15)
$B_{N}(z,{\varepsilon})\subset\Big{\\{}w\in\Omega:\frac{1}{N}X_{N}(w)\in\Big{(}\alpha-\delta_{m^{*}}-\frac{v_{N,{\varepsilon}}(X_{N})}{N},\alpha+\delta_{m^{*}}+\frac{v_{N,{\varepsilon}}(X_{N})}{N}\Big{)}\Big{\\}}.$

Therefore, for every $N\geq\max\\{j^{*},N_{0}\\}$ one has

(16)
$B_{N}(z,{\varepsilon})\subset\Big{\\{}\frac{1}{N}X_{N}\in\Big{[}\alpha-\delta,\alpha+\delta\Big{]}\Big{\\}}.$

Since $\nu$ is an Ahlfors-Bowen measure, for every
$N\geq\max\\{j^{*},N_{0}\\}$ one obtains

(17)
$\frac{1}{N}\log\nu\Big{\\{}\frac{1}{N}X_{N}\in[\alpha-\delta,\alpha+\delta]\Big{\\}}\geq\frac{1}{N}\nu(B_{N}(z,{\varepsilon}))\geq-h-\frac{\log
D({\varepsilon})}{N}.$

Then (10) leads

$-h\leq\limsup_{N\to\infty}\frac{1}{N}\log\nu\Big{\\{}\frac{1}{N}X_{N}\in[\alpha-\delta,\alpha+\delta]\Big{\\}}\leq-
I_{X}([\alpha-\delta,\alpha+\delta])<-\Delta.$

But this is a contradiction to $h<\Delta$, therefore, $I_{X}(\alpha)\leq h$
for every $\alpha\in\mathcal{L}_{X}$.

Now we prove the inequality (11). Let $\\{\delta_{m}\\}$ and $Z_{m,j}$ be as
above. Take any $\alpha\in\mathcal{L}_{X}$ and ${\varepsilon}>0$. Since
$I_{X}$ is a lower semi-continuous function, $\lim\limits_{\delta\to
0}I_{X}([a-\delta,a+\delta])=I_{X}(a)$ for all $a\in\mathbb{R}$. Since
$I_{X}(\alpha)<+\infty$ for $\alpha\in\mathcal{L}_{X}$, for arbitrary
$\sigma>0$, there exists $m^{\prime}\in\mathbb{N}$, and $\delta^{\prime}>0$
such that, for all $m\geq m^{\prime}$ and $\delta\leq\delta^{\prime}$

(18)
$\Big{|}I_{X}([\alpha-\delta_{m}-\delta;\alpha+\delta_{m}+\delta])-I_{X}(\alpha)\Big{|}<\sigma.$

Again $\lim_{n\to\infty}\frac{v_{n,{\varepsilon}}(X_{n})}{n}=0$, choose
sufficiently large $N_{1}\in\mathbb{N}$ such that
$\frac{v_{n,{\varepsilon}}(X_{n})}{n}<\delta^{\prime}$ for all $n\geq N_{1}$.
From (10) one can choose $N_{2}\in\mathbb{N}$ such that for all $n\geq N_{2}$

(19)
$\nu\Big{\\{}\frac{1}{n}X_{n}\in[\alpha-\delta_{m^{\prime}}-\delta^{\prime},\alpha+\delta_{m^{\prime}}+\delta^{\prime}]\Big{\\}}\leq
e^{n(-I_{X}([\alpha-\delta_{m^{\prime}}-\delta^{\prime},\alpha+\delta_{m^{\prime}}+\delta^{\prime}])+\sigma)}\leq
e^{n(-I_{X}(\alpha)+2\sigma)}.$

Now take any $j\in\mathbb{N}$ with $Z_{j,m^{\prime}}\neq\emptyset$, and for
all $N\geq\max\\{j,N_{1},N_{2}\\}$ consider the cover
$\\{B_{N}(z,{\varepsilon}/3)\\}_{z\in Z_{m^{\prime},j}}$ of
$Z_{m^{\prime},j}$. Clearly, it also covers $\overline{Z}_{m^{\prime},j}$,
therefore, by compactness, there is a finite subcover
$\\{B_{N}(z_{i},{\varepsilon}/3)\\}_{i}$ of $Z_{m^{\prime},j}$. Then by Vitali
covering lemma (Lemma 5.1) there exists a disjoint subcollection
$\\{B_{N}(z_{i_{k}},{\varepsilon}/3)\\}_{k}$ such that
$Z_{m^{\prime},j}\subset\cup_{i}B_{N}(z_{i},{\varepsilon}/3)\subset\cup_{k}B_{N}(z_{i_{k}},{\varepsilon}).$
Therefore, without loss of generality, one can assume that there is a finite
disjoint collection
$\mathcal{B}_{N}:=\\{B_{N}(z_{i},{\varepsilon}/3)\\}_{z_{i}\in
Z_{m^{\prime},j}}$ of the balls such that
$Z_{m^{\prime},j}\subset\bigcup_{i}B_{N}(z_{i},{\varepsilon})$. Since
$z_{i}\in Z_{m^{\prime},j}$, one can easily check for every $i$ that

(20)
$B_{N}(z_{i},{\varepsilon})\subset\Big{\\{}w\in\Omega:\frac{1}{N}X_{N}(w)\in\Big{(}\alpha-\delta_{m^{\prime}}-\frac{v_{N,{\varepsilon}}(X_{N})}{N},\alpha+\delta_{m^{\prime}}+\frac{v_{N,{\varepsilon}}(X_{N})}{N}\Big{)}\Big{\\}}.$

Thus by the choice of $N$, for all $i$,

(21)
$B_{N}(z_{i},{\varepsilon})\subset\Big{\\{}\frac{1}{N}X_{N}\in(\alpha-\delta_{m^{\prime}}-\delta^{\prime},\alpha+\delta_{m^{\prime}}+\delta^{\prime})\Big{\\}},$

and hence

(22)
$\bigcup_{i}B_{N}(z_{i},{\varepsilon})\subset\Big{\\{}\frac{1}{N}X_{N}\in(\alpha-\delta_{m^{\prime}}-\delta^{\prime},\alpha+\delta_{m^{\prime}}+\delta^{\prime})\Big{\\}}.$

Since $\mathcal{B}_{N}:=\\{B_{N}(z_{i},{\varepsilon}/3)\\}$ is a disjoint
collection of the balls, one can easily get the following from the above

(23)
$\sum_{i}\nu(B_{N}(z_{i},{\varepsilon}/3))\leq\nu\Big{(}\Big{\\{}\frac{1}{N}X_{N}\in(\alpha-\delta_{m^{\prime}}-\delta^{\prime},\alpha+\delta_{m^{\prime}}+\delta^{\prime})\Big{\\}}\Big{)}.$

Thus it follows from (19) that

(24) $\sum_{i}\nu(B_{N}(z_{i},{\varepsilon}/3))\leq
e^{N(-I_{X}(\alpha)+2\sigma)}.$

Since $\nu$ is an Ahlfors-Bowen measure, we can thus conclude that

(25) $\\#\mathcal{B}_{N}\leq
D({\varepsilon}/3)e^{N(h-I_{X}(\alpha)+2\sigma)}.$

Since $\\{B_{N}(z_{i},{\varepsilon})\\}_{i}$ is a cover of $Z_{m^{\prime},j}$,
one can immediately obtain from (2) that

(26) $m(Z_{m^{\prime},j},t,{\varepsilon},N)\leq\\#\mathcal{B}_{N}\cdot
e^{-Nt}.$

Then by combining the last two inequalities, one gets

(27) $m(Z_{m^{\prime},j},t,{\varepsilon},N)\leq
D({\varepsilon}/3)e^{-N(t-h+I_{X}(\alpha)-2\sigma)}$

Notice that $\sigma$ is not chosen yet, therefore, if $t>h-I_{X}(\alpha)$ then
one can choose $\sigma$ in the interval
$\Big{(}0,\frac{t-h+I_{X}(\alpha)}{2}\Big{)}$. Let $t>h-I_{X}(\alpha)$ be any
number. Then one can obtain from (27) that
$h_{{\textsf{top}}}(T,Z_{m^{\prime},j},{\varepsilon})\leq t$, and notice that
this inequality holds for any ${\varepsilon}>0$ and for all large
$j\in\mathbb{N}$. Therefore, $h_{{\textsf{top}}}(T,Z_{m^{\prime},j})\leq t$
for all sufficiently large $j$, thus by the countable stability of the
topological entropy, we can get $h_{{\textsf{top}}}(T,K_{\alpha})\leq t$ since
$\\{Z_{m^{\prime},j}\\}_{j}\uparrow K_{\alpha}$. Thus we can obtain the
desired result since $t$ is chosen arbitrarily in $(h-I_{X}(\alpha),\infty)$.
∎

###### Proof of Theorem B.

One can readily check that ${\varphi}_{X}^{*}$ is an extended convex lower-
semi-continuous function on $\mathbb{R}$. Indeed, the lower semi-continuity of
${\varphi}_{X}^{*}$ follows from the fact that it is the superior envelope of
the family $\\{u_{t}:t\in\mathbb{R}\\}$ of extended continuous functions
$\alpha\in\mathbb{R}\xrightarrow{u_{t}}\alpha
t-{\varphi}(t)\in\overline{\mathbb{R}}$. Furthermore, since
${\varphi}_{X}(0)=0$, one has ${\varphi}_{X}^{*}\geq 0$. Thus
${\varphi}_{X}^{*}$ is a rate function in the sense of Definition 3.3. By
Theorem 3.3, one has the weak large deviations upper bound for the sequence
$\Big{\\{}\frac{1}{n}X_{n}\Big{\\}}$ with the rate function
${\varphi}_{X}^{*}$, i.e., for all compact $F\Subset\mathbb{R}$, one has the
following

(28) $\limsup_{n\to\infty}\frac{1}{n}\log\nu\Big{\\{}\frac{1}{n}X_{n}\in
F\Big{\\}}\leq-\inf_{F}({\varphi}_{X}^{*})=:-{\varphi}_{X}^{*}(F).$

Thus one can immediately obtain the statement of the theorem from Theorem A.

∎

##  6\. Proof of Theorem C: Lower bound

The goal of this section is to prove Theorem C. For any real number $q$ and a
natural number $n$ denote

$\mathcal{Z}_{n}(q):=\int_{\Omega}e^{qX_{n}}d\nu.$

We will employ the Large Deviations technique. Namely, for every $q$, by using
Cramer’s exponential tilting method, we will construct a measure $\mu_{q}$
(Lemma 6.3) such that for every $\omega\in\Omega$, $n\in\mathbb{N}$

(29)
$\mu_{q}(B_{n}(\omega,{\varepsilon}))\leq\frac{e^{qX_{n}(\omega)}}{\mathcal{Z}_{n}(q)}\nu(B_{n}(\omega,{\varepsilon}))\times
K(n,q,{\varepsilon}),$

where the uniform ‘correction’ factor $K(n,q,{\varepsilon})>0$ is
subexponential in $n$.

Once (29) is established, we immediately obtain that for all
$\omega\in\Omega$, one has

$-\frac{1}{n}\log\mu_{q}(B_{n}(\omega,{\varepsilon}))\geq-\frac{qX_{n}(\omega)}{n}+\frac{1}{n}\log\mathcal{Z}_{n}(q)-\frac{1}{n}\log\nu(B_{n}(\omega,{\varepsilon}))+o(1).$

Therefore, for $\omega\in K_{\alpha}$ we then conclude

(30) ${\underline{h}}_{\mu_{q}}(T,\omega)=\lim_{{\varepsilon}\to
0}\liminf_{n\to\infty}-\frac{1}{n}\log\mu_{q}(B_{n}(\omega,{\varepsilon}))\geq
h_{{\textsf{top}}}(T,\Omega)-q\alpha+{\varphi}_{X}(q).$

Furthermore, for a given $\alpha$, we will identify $q$ such that
$\mu_{q}(K_{\alpha})=1$ (Lemma 6.7). As usual, $q=q_{\alpha}$ is such that
$\alpha={\varphi}^{\prime}(q_{\alpha})$. Therefore, by the entropy
distribution principle (Theorem 2.1), we can conclude that

(31) $h_{{\textsf{top}}}(T,K_{\alpha})\geq-
q_{\alpha}\alpha+{\varphi}_{X}(q_{\alpha})+h_{{\textsf{top}}}(T,\Omega)=h_{{\textsf{top}}}(T,\Omega)-{\varphi}_{X}^{*}(\alpha)=h_{{\textsf{top}}}(T,\Omega)-I_{X}(\alpha).$

After combining this with the upper bound, we obtain the claim of Theorem C.
Now, let us turn to the rigorous proof.

###  6.1. A fundamental inequality

In this subsection, we shall establish a compound inequality for the dynamical
systems which is introduced in Subsection 2.4. Recall that for any
${\varepsilon}>0$ there exists $M_{1}({\varepsilon})\in\mathbb{N}$ such that
for all $x\in\Omega$ and every $n\in\mathbb{N}$,
$T^{n+M_{1}}(B_{n}(x,{\varepsilon}))=\Omega$ since $T$ is strongly
topologically exact transformation, and there exists
$M_{2}({\varepsilon})\in\mathbb{N}$ such that for all $x,x^{\prime}\in\Omega$
and every $n_{1},n_{2}\in\mathbb{N}$, $B_{n_{1}}(x,{\varepsilon})\cap
T^{-(n_{1}+M_{2})}\big{(}B_{n_{2}}(x^{\prime},{\varepsilon})\big{)}\neq\emptyset$
since $T$ has the weak specification property (c.f. equations (5) and (6)).
Let $M_{1}({\varepsilon})$ and $M_{2}({\varepsilon})$ be the smallest integers
satisfying these properties, respectively. For ${\varepsilon}>0$, let

(32)
$M({\varepsilon}):=\max\\{M_{1}({\varepsilon}),M_{2}({\varepsilon}/2)\\}.$

One has the following lemma for the expansive dynamical systems with certain
mixing properties.

######  Lemma 6.1.

For any sufficiently small ${\varepsilon}$, there exists a positive constant
$C_{1}({\varepsilon})$ such that for any $n\in\mathbb{N}$ all $x\in\Omega$ and
for any measurable function $Y:\Omega\to[0,+\infty)$ one has

$\frac{C_{1}({\varepsilon})^{-1}}{\nu(B_{n}(x,\varepsilon))}\int_{B_{n}(x,\varepsilon)}Y\circ
T^{n+M}d\nu(y)\leq\int_{\Omega}Yd\nu\leq\frac{C_{1}({\varepsilon})}{\nu(B_{n}(x,\varepsilon))}\int_{B_{n}(x,\varepsilon)}Y\circ
T^{n+M}d\nu.$

######  Remark 6.1.

In the above lemma, ${\varepsilon}$ can be chosen in the interval $(0,\rho/2)$
where $\rho$ is the expansivity constant of the transformation $T$.

###### Proof.

We assume that $T$ is expansive. Let $\rho$ denote the expansivity constant of
$T$. Consider ${\varepsilon}\in(0,\rho/2)$, $n\in\mathbb{N}$, $x\in\Omega$ and
a measurable function $Y:\Omega\to[0,+\infty)$. Let $\nu_{x,n}$ be the
conditional probability measure on $B_{n}(x,{\varepsilon})$, i.e.,
$\nu_{x,n}(A):=\nu(A|B_{n}(x,{\varepsilon}))=\frac{\nu(A)}{\nu(B_{n}(x,\varepsilon))}$
for a Borel set $A\subset B_{n}(x,{\varepsilon})$. Set
$\tau_{x,n}:=\nu_{x,n}\circ T^{-(n+M)}$, then $\tau_{x,n}$ is always a
probability measure on $\Omega$ since
$T^{n+M}(B_{n}(x,{\varepsilon}))=\Omega$. Note that $\nu_{x,n}$ and
$\tau_{x,n}$ also depend on ${\varepsilon}$, but we omit this to avoid
cumbersome notations in our subsequent expressions. It is easy to check that

(33) $\displaystyle\int_{\Omega}Y(z)d\tau_{x,n}(z)=$
$\displaystyle\frac{1}{\nu(B_{n}(x,\varepsilon))}\int_{B_{n}(x,\varepsilon)}Y(T^{n+M}z)d\nu(z).$

Therefore, the claim of the lemma is equivalent to showing the existence of a
constant $C_{1}({\varepsilon})>0$ independent of $n$ and $x$ such that

(34)
$C_{1}({\varepsilon})^{-1}\int_{\Omega}Y(y)d\tau_{x,n}(y)\leq\int_{\Omega}Y(y)d\nu(y)\leq
C_{1}({\varepsilon})\int_{\Omega}Y(y)d\tau_{x,n}(y).$

We now present the proof of this statement in three steps.

Step 1: We start by showing that there exists a constant
$C_{2}({\varepsilon})>0$, dependent only on ${\varepsilon}$, such that for
every $k\in\mathbb{N}$ and $y\in\Omega$,

(35)
$C_{2}({\varepsilon})^{-1}\leq\frac{\tau_{x,n}(B_{k}(y,{\varepsilon}))}{\nu(B_{k}(y,{\varepsilon}))}\leq
C_{2}({\varepsilon}).$

Consider arbitrary $k\in\mathbb{N}$, $y\in\Omega$. By the choice of
$M:=M({\varepsilon})$, we have

$T^{-(n+M)}B_{k}(y,\varepsilon/2)\cap B_{n}(x,\varepsilon/2)\neq\emptyset.$

Then for any $z\in T^{-(n+M)}B_{k}(y,\varepsilon/2)\cap
B_{n}(x,\varepsilon/2)$, one has

$B_{n+M+k}(z,\varepsilon/2)\subset T^{-(n+M)}B_{k}(y,\varepsilon)\cap
B_{n}(x,\varepsilon).$

Since $\nu$ is an Ahlfors-Bowen measure (c.f. (4)), we can get

(36)
$D({\varepsilon}/2)^{-1}e^{-(n+M+k)h}\leq\nu(B_{n+M+k}(z,\varepsilon/2))\leq\nu(T^{-(n+M)}B_{k}(y,\varepsilon)\cap
B_{n}(x,\varepsilon)).$

Hence

(37)
$\displaystyle\begin{split}\tau_{x,n}(B_{k}(y,{\varepsilon}))=\frac{\nu(T^{-(n+M)}B_{k}(y,\varepsilon)\cap
B_{n}(x,\varepsilon))}{\nu(B_{n}(x,\varepsilon))}&\geq\frac{D({\varepsilon}/2)^{-1}e^{-(n+M+k)h}}{D({\varepsilon})e^{-nh}}\\\
&\geq
D(\varepsilon/2)^{-1}D(\varepsilon)^{-2}e^{-Mh}\nu(B_{k}(y,\varepsilon)).\end{split}$

In order to get an upper bound in (35), set
$B^{\prime}:=T^{-(n+M)}B_{k}(y,\varepsilon)\cap B_{n}(x,\varepsilon)$. Then
since $\\{B_{n+M+k}(z,2{\varepsilon}):z\in B^{\prime}\\}$ is a cover of
$\overline{B^{\prime}}$, by the the compactness of $\Omega$ and the Vitali
covering lemma (Lemma 5.1), there exists a finite disjoint subcollection
$\mathcal{G}:=\\{B_{n+M+k}(z_{i},2{\varepsilon})\\}_{z_{i}\in B^{\prime}}$
such that $\\{B_{n+M+k}(z_{i},6{\varepsilon})\\}_{z_{i}\in B^{\prime}}$ covers
$B^{\prime}$. Denote
$\mathscr{S}:=\\{T^{n}z_{i}:B_{n+M+k}(z_{i},2{\varepsilon})\in\mathcal{G}\\}$.
Because of the disjointness of $\mathcal{G}$ and by the definition of
$B^{\prime}$, it is easy to check that $\sharp\mathscr{S}=\sharp\mathcal{G}$
and $\mathscr{S}$ is a $(M,2{\varepsilon})$-separated set. Therefore, if we
denote the cardinality of a maximal $(M,2{\varepsilon})$-separated set in
$\Omega$ by $\mathfrak{s}(M,2{\varepsilon})$, then it is clear that
$\sharp\mathcal{G}=\sharp\mathscr{S}\leq\mathfrak{s}(M,2{\varepsilon})$. From
these and (4), it follows that

(38)
$\nu(B^{\prime})\leq\sum\limits_{i}\nu(B_{n+M+k}(z_{i},6{\varepsilon}))\leq
D(6{\varepsilon})\mathfrak{s}(M,2{\varepsilon})e^{-(n+M+k)h}$

This bound then implies

(39)
$\tau_{x,n}(B_{k}(y,{\varepsilon}))=\frac{\nu(B^{\prime})}{\nu(B_{n}(x,{\varepsilon}))}\leq\frac{D(6{\varepsilon})\mathfrak{s}(M,2{\varepsilon})e^{-(n+M+k)h}}{D({\varepsilon})^{-1}e^{-nh}}\leq
D(6{\varepsilon})D({\varepsilon})^{2}\mathfrak{s}(M,2{\varepsilon})\nu(B_{k}(y,{\varepsilon})).$

Combining (37), (39), the constant
$C_{2}({\varepsilon}):=\max\\{D(\varepsilon/2)D(\varepsilon)^{2}e^{Mh};D(6{\varepsilon})D({\varepsilon})^{2}\mathfrak{s}(M,2{\varepsilon})\\}$
satisfies the inequality (35).

Step 2: In this step, we shall prove (34) for continuous non-negative
functions: namely, if the constant
$C_{1}({\varepsilon}):=C_{2}({\varepsilon})D(2{\varepsilon})D({\varepsilon})$
and $\psi\in C(\Omega,\mathbb{R}_{+})$, then

(40) $C_{1}({\varepsilon})^{-1}\int_{\Omega}\psi
d\tau_{x,n}\leq\int_{\Omega}\psi d\nu\leq
C_{1}({\varepsilon})\int_{\Omega}\psi d\tau_{x,n}.$

Consider an arbitrary $N\in\mathbb{N}$ and let $E_{N}$ be a maximal
$(N,2{\varepsilon})$-separated set in $\Omega$. Thus
$\\{B_{N}(x,2{\varepsilon}):x\in E_{N}\\}$ is a cover of $\Omega$ and
$\\{B_{N}(x,{\varepsilon}):x\in E_{N}\\}$ are pairwise disjoint. By
successively applying (4) and (35) to these collections of the balls, one can
obtain

(41) $\int_{\Omega}\psi d\tau_{x,n}\geq
C_{2}({\varepsilon})^{-1}D(2{\varepsilon})^{-1}D({\varepsilon})^{-1}\int_{\Omega}\psi
d\nu-
C_{2}({\varepsilon})^{-1}(v_{N,2{\varepsilon}}(\psi)+D(2{\varepsilon})D({\varepsilon})v_{N,{\varepsilon}}(\psi)).$

Since $v_{N,2{\varepsilon}}(\psi)\to 0,v_{N,{\varepsilon}}(\psi)\to 0$ as
$N\to\infty$ (c.f. Remark 2.4), one obtains $\int_{\Omega}\psi d\tau_{x,n}\geq
C_{2}({\varepsilon})^{-1}D(2{\varepsilon})^{-1}D({\varepsilon})^{-1}\int_{\Omega}\psi
d\nu$. The opposite inequality is proved in exactly the same way.

Step 3: Now, let $\tilde{Y}$ be any bounded non-negative measurable function
on $\Omega$. By Lusin’s theorem and Tietz’s extension theorem ([Ercan1997]),
one can approximate $\tilde{Y}$ by non-negative continuous functions
$\psi_{k}:\Omega\to\mathbb{R},\;\;k\in\mathbb{N}$ such that
$\int_{\Omega}\psi_{k}d\nu\to\int_{\Omega}\tilde{Y}d\nu$ and
$\int_{\Omega}\psi_{k}d\tau_{x,n}\to\int_{\Omega}\tilde{Y}d\tau_{x,n}$ as
$k\to\infty$. Hence it can be obtained from (40) that

(42)
$C_{1}({\varepsilon})^{-1}\int_{\Omega}\tilde{Y}d\tau_{x,n}\leq\int_{\Omega}\tilde{Y}d\nu\leq
C_{1}({\varepsilon})\int_{\Omega}\tilde{Y}d\tau_{x,n}.$

Now we conclude (34) from (42). Consider a non-negative measurable function
$Y$ on $\Omega$, and let $Y_{N}:=Y\cdot\mathds{1}_{Y\leq N}$ for
$N\in\mathbb{N}$. Then $Y_{N}$ is a bounded non-negative function, and
$\\{Y_{N}\\}$ converges monotonically to $Y$ as $N\to\infty$. Therefore, by
monotone convergence theorem and from (42), one gets (34). ∎

###  6.2. A ”tilted” measure and its relationship with the reference measure

For any real number $q\in\mathbb{R}$ and a natural number $n\in\mathbb{N}$,
define a probability measure $\mu_{n,q}$ by Cramer’s tilting method as

$d\mu_{n,q}:=\frac{e^{qX_{n}}}{\mathbb{E}_{\nu}e^{qX_{n}}}d\nu=\frac{e^{qX_{n}}}{\mathcal{Z}_{n}(q)}d\nu.$

For any $q\in\mathbb{R}$, consider the sequence $\\{\mu_{n,q}\\}_{n\geq 1}$ of
probability measures. Since $\mathcal{M}(\Omega)$ is a compact set in weak∗
topology, there is a convergent subsequence $\\{\mu_{N_{j},q}\\}_{j}$, and we
denote the weak${}^{*}-$limit of this subsequence by $\mu_{q}$. In this
subsection, we shall establish an important relation (c.f. Lemma 6.3) between
$\mu_{q}$ and $\nu$ under the conditions of Theorem C. First, we prove an
important lemma.

######  Lemma 6.2.

For all $n\in\mathbb{N}$, $q\in\mathbb{R}$, there exists a positive constant
$C_{4}(n,q)$ with $\lim\limits_{n\to\infty}\frac{\log C_{4}(n,q)}{n}=0$ such
that for any $m\in\mathbb{N}$,

(43) $\displaystyle
C_{4}(n,q)^{-1}\leq\frac{\mathcal{Z}_{n+M+m}(q)}{\mathcal{Z}_{n}(q)\mathcal{Z}_{m}(q)}\leq
C_{4}(n,q),$

where $M\in\mathbb{N}$ is as in (32).

###### Proof.

Fix a sufficiently small ${\varepsilon}$. Consider $n,m\in\mathbb{N}$,
$q\in\mathbb{R}$. Note that by the weak Bowen condition, each $X_{k}$ is a
bounded function. In what follows, $||X_{k}||$ denotes the supremum norm of a
function $X_{k}$. Since the sequence $X$ is almost additive, one has

(44) $\displaystyle\mathcal{Z}_{n+M+m}(q)$
$\displaystyle=\int_{\Omega}e^{qX_{n+M+m}(y)}d\nu(y)\leq
e^{|q|(A_{n}+A_{M}+||X_{M}||)}\int_{\Omega}e^{qX_{n}(y)}e^{qX_{m}(T^{n+M}y)}d\nu(y).$

Let $E$ be a maximal $(n,\varepsilon)$-separated set in $\Omega$. Then
$\bigcup\limits_{x\in E}B_{n}(x,\varepsilon)=\Omega$ and
$\\{B_{n}(x,{\varepsilon}/2)\\}_{x\in E}$ is a disjoint collection of Bowen
balls. Therefore,

(45)
$\displaystyle\begin{split}\int_{\Omega}e^{qX_{n}(y)}e^{qX_{m}(T^{n+M}y)}d\nu(y)&\leq\sum_{x\in
E}\int_{B_{n}(x,\varepsilon)}e^{qX_{n}(y)}e^{qX_{m}(T^{n+M}y)}d\nu(y)\\\ &\leq
e^{|q|v_{n,{\varepsilon}}(X_{n})}\sum_{x\in
E}e^{qX_{n}(x)}\int_{B_{n}(x,\varepsilon)}e^{qX_{m}(T^{n+M}y)}d\nu(y)\\\ &\leq
e^{|q|v_{n,{\varepsilon}}(X_{n})}C_{1}(\varepsilon)\mathcal{Z}_{m}(q)\sum_{x\in
E}e^{qX_{n}(x)}\nu(B_{n}(x,\varepsilon)),\end{split}$

where the second inequality holds since $\\{X_{n}\\}$ satisfies weak Bowen’s
condition and the last inequality holds by Lemma 6.1 applied to $Y=X_{m}$. Now
we estimate the sum $\sum\limits_{x\in
E}e^{qX_{n}(x)}\nu(B_{n}(x,\varepsilon))$ from above. One has

(46) $\displaystyle\begin{split}\sum_{x\in
E}e^{qX_{n}(x)}\nu(B_{n}(x,\varepsilon))&\leq\sum_{x\in
E}e^{qX_{n}(x)}D(\varepsilon/2)D(\varepsilon)\nu(B_{n}(x,\varepsilon/2))\\\
&\leq
D(\varepsilon/2)D(\varepsilon)e^{|q|v_{n,{\varepsilon}/2}(X_{n})}\sum_{x\in
E}\int_{B_{n}(x,\varepsilon/2)}e^{qX_{n}(y)}d\nu\\\ &\leq
D(\varepsilon/2)D(\varepsilon)e^{|q|v_{n,{\varepsilon}/2}(X_{n})}\mathcal{Z}_{n}(q).\end{split}$

After combining (44), (45) and (46)

(47) $\frac{\mathcal{Z}_{n+M+m}(q)}{\mathcal{Z}_{n}(q)\mathcal{Z}_{m}(q)}\leq
C_{5}(n,q),$

where
$C_{5}(n,q):=e^{|q|(A_{n}+A_{M}+||X_{M}||)}e^{|q|(v_{n,{\varepsilon}/2}(X_{n})+v_{n,{\varepsilon}}(X_{n}))}C_{1}(\varepsilon)D(\varepsilon/2)D(\varepsilon)$.

With similar reasoning, one gets the following inequalities

(48) $\displaystyle\mathcal{Z}_{n+M+m}(q)$ $\displaystyle\geq
e^{-|q|(A_{n}+A_{M}+||X_{M}||)}\int_{\Omega}e^{qX_{n}(y)}e^{qX_{m}(T^{n+M}y)}d\nu(y),$
(49) $\displaystyle\int_{\Omega}e^{qX_{n}(y)}e^{qX_{m}(T^{n+M}y)}d\nu(y)$
$\displaystyle\geq
e^{-|q|v_{n,{\varepsilon}/2}(X_{n})}C_{1}(\varepsilon/2)^{-1}\mathcal{Z}_{m}(q)\sum_{x\in
E}e^{qX_{n}(x)}\nu(B_{n}(x,\varepsilon/2)),$ (50) $\displaystyle\sum_{x\in
E}e^{qX_{n}(x)}\nu(B_{n}(x,\varepsilon/2))$ $\displaystyle\geq
e^{-|q|v_{n,{\varepsilon}}(X_{n})}D(\varepsilon/2)^{-1}D(\varepsilon)^{-1}\mathcal{Z}_{n}(q),$

and after combining these inequalities

(51) $\frac{\mathcal{Z}_{n+M+m}(q)}{\mathcal{Z}_{n}(q)\mathcal{Z}_{m}(q)}\geq
C_{6}(n,q)^{-1},$

where
$C_{6}(n,q):=e^{|q|(A_{n}+A_{M}+||X_{M}||)}e^{|q|(v_{n,{\varepsilon}/2}(X_{n})+v_{n,{\varepsilon}}(X_{n}))}C_{1}(\varepsilon/2)D(\varepsilon/2)D(\varepsilon)$.

From (47) and (51), it is clear that
$C_{4}(n,q):=\max\\{C_{5}(n,q),C_{6}(n,q)\\}$ satisfies the conditions of the
lemma. ∎

######  Lemma 6.3.

For any sufficiently small ${\varepsilon}>0$ and for any $n\in\mathbb{N}$,
$q\in\mathbb{R}$ there exists a constant $C_{7}(n,q,{\varepsilon})>0$ such
that $\lim\limits_{n\to\infty}\frac{\log C_{7}(n,q,{\varepsilon})}{n}=0$ and
for all $x\in\Omega$,

(52) $\mu_{q}(B_{n}(x,\varepsilon))\leq
C_{7}(n,q,{\varepsilon})\frac{e^{qX_{n}(x)}}{\mathcal{Z}_{n}(q)}\nu(B_{n}(x,\varepsilon)).$

###### Proof.

Consider again ${\varepsilon}\in(0,\rho/2)$, where $\rho$ is the expansivity
constant of the transformation $T$, and $x\in\Omega$, $n\in\mathbb{N}$,
$q\in\mathbb{R}$. By the Portmanteau theorem, we have

(53)
$\displaystyle\mu_{q}(B_{n}(x,\varepsilon))\leq\liminf_{j\to\infty}\int_{B_{n}(x,\varepsilon)}\frac{e^{qX_{N_{j}}(y)}}{\mathcal{Z}_{N_{j}}(q)}d\nu(y)$

Now we estimate the numerator
$\int_{B_{n}(x,\varepsilon)}e^{qX_{N_{j}}(y)}d\nu(y)$ and denominator
$\mathcal{Z}_{N_{j}}(q)$ separately. By Lemma 6.2, one has

(54) $\frac{1}{\mathcal{Z}_{n+M+N_{j}-n-M}(q)}\leq
C_{4}(n,q)\frac{1}{\mathcal{Z}_{n}(q)\mathcal{Z}_{N_{j}-n-M}(q)}.$

For the integral in the numerator, the following estimations are valid

(55)
$\displaystyle\begin{split}\int_{B_{n}(x,\varepsilon)}e^{qX_{n+M+N_{j}-n-M}(y)}d\nu(y)&\leq
e^{|q|(A_{n}+A_{M})}\int_{B_{n}(x,\varepsilon)}e^{qX_{n}(y)}e^{qX_{M}(T^{n}y)}e^{qX_{N_{j}-n-M}(T^{n+M}y)}d\nu(y)\\\
&\leq
e^{|q|(A_{n}+A_{M}+||X_{M}||)}\int_{B_{n}(x,\varepsilon)}e^{qX_{n}(y)}e^{qX_{N_{j}-n-M}(T^{n+M}y)}d\nu(y)\\\
&\leq
C_{8}(n,q,{\varepsilon})e^{qX_{n}(x)}\int_{B_{n}(x,\varepsilon)}e^{qX_{N_{j}-n-M}(T^{n+M}y)}d\nu(y),\end{split}$

where
$C_{8}(n,q,{\varepsilon}):=e^{|q|(A_{n}+A_{M}+||X_{M}||+v_{n,{\varepsilon}}(X_{n}))}$.
Note that

(56) $\int_{B_{n}(x,\varepsilon)}e^{qX_{N_{j}-n-M}(T^{n+M}y)}d\nu(y)\leq
C_{1}(\varepsilon)\mathcal{Z}_{N_{j}-n-M}(q)\nu(B_{n}(x,\varepsilon)).$

Thus it follows from (55) that

(57) $\int_{B_{n}(x,\varepsilon)}e^{qX_{N_{j}}(y)}d\nu(y)\leq
C_{1}({\varepsilon})C_{8}(n,q,{\varepsilon})e^{qX_{n}(x)}\nu(B_{n}(x,\varepsilon))\mathcal{Z}_{N_{j}-n-M}(q).$

Hence it easy to see from (53)-(54) that the constant
$C_{7}(n,q,{\varepsilon}):=C_{1}({\varepsilon})C_{4}(n,q)C_{8}(n,q,{\varepsilon})$
satisfies (52), and by Lemma 6.2 and the weak Bowen condition,
$\lim\limits_{n\to\infty}\frac{\log C_{7}(n,q,{\varepsilon})}{n}=0$. ∎

###  6.3. Differentiability of the log-moment generating function

######  Lemma 6.4.

For all $q\in\mathbb{R}$, the limit
$\lim\limits_{n\to\infty}\frac{1}{n}\log\mathcal{Z}_{n}(q)$ exists and finite.
Thus, ${\varphi}_{X}=\lim\limits_{n\to\infty}\frac{1}{n}\log\mathcal{Z}_{n}$
is a finite convex function. Furthermore, ${\varphi}_{X}=I_{X}^{*}$ and
${\varphi}_{X}^{*}=I_{X}$.

First, we state a slight generalization of the classical Fekete’s lemma which
is used to verify the existence and finiteness of the limit
$\lim\limits_{n\to\infty}\frac{1}{n}\log\mathcal{Z}_{n}(q)$.

######  Lemma 6.5.

[Fan1996] Let $\\{a_{n}\\}_{n\geq 0}$ be a sequence of real numbers. Assume
that there exists a natural number $N\in\mathbb{N}$ and real numbers $D$,
$\Delta_{n}\geq 0$, $n\in\mathbb{N}$ with
$\lim_{n\to\infty}\frac{\Delta_{n}}{n}=0$ such that for all
$m\in\mathbb{Z}_{+}$ and $n\geq N$

(58) $a_{n+m}\leq a_{n}+a_{m}+D+\Delta_{n}.$

Then the limit $\lim_{n\to\infty}\frac{a_{n}}{n}$ exists and belongs to
$\mathbb{R}\bigcup\\{-\infty\\}$.

###### Proof of Lemma 6.4.

By using Lemma 6.2, it can be shown for $q\in\mathbb{R}$, $n\geq M+1$ and any
$m\in\mathbb{Z}_{+}$ that

(59)
$\log\mathcal{Z}_{n+m}(q)\leq\log\mathcal{Z}_{n}(q)+\log\mathcal{Z}_{m}(q)+2\log
C_{4}(n-M,q).$

Thus by Lemma 6.5, the limit
$\lim\limits_{n\to\infty}\frac{1}{n}\log\mathcal{Z}_{n}(q)$ exists and
$\lim\limits_{n\to\infty}\frac{1}{n}\log\mathcal{Z}_{n}(q)<+\infty$. By
applying a similar argument to the sequence $\\{-\log\mathcal{Z}_{n}(q)\\}$,
one shows that
$\lim\limits_{n\to\infty}\frac{1}{n}\log\mathcal{Z}_{n}(q)>-\infty$.

Clearly, each $\log\mathcal{Z}_{n}$ is a convex function, thus ${\varphi}_{X}$
is also a convex function as the pointwise limit of convex functions.

Since for all $q\in\mathbb{R}$, ${\varphi}_{X}(q)<+\infty$, and
$\\{\frac{1}{n}X_{n}\\}_{n\in\mathbb{N}}$ satisfies LDP with the good rate
function $I_{X}$, we immediately obtain from Theorem 3.2 that
$I_{X}^{*}={\varphi}_{X}$. Note that $I_{X}={\varphi}_{X}^{*}$ also holds by
the Fenchel duality lemma ([Dembo-Zeitouni-Book]), since $I_{X}$ is convex and
lower semi-continuous. ∎

Now we show that ${\varphi}_{X}$ is a differentiable function. In fact, the
following holds true.

######  Lemma 6.6.

The function ${\varphi}_{X}$ is continuously differentiable on $\mathbb{R}$,
i.e., ${\varphi}_{X}\in C^{1}(\mathbb{R})$.

###### Proof.

By Lemma 6.4, we have that $I_{X}^{*}={\varphi}_{X}$. Thus since $I_{X}$ is an
essentially strictly convex rate function, ${\varphi}_{X}$ is an essentially
smooth function by Theorem 3.1. Hence since
$\text{dom}({\varphi}_{X})=\mathbb{R}$ by Lemma 6.4, we have that
${\varphi}_{X}$ is a differentiable function on $\mathbb{R}$. Note that
${\varphi}_{X}$ is a convex function, therefore, ${\varphi}_{X}\in
C^{1}(\mathbb{R})$ by Darboux’s intermediate value theorem. ∎

By the monotonicity of ${\varphi}_{X}^{\prime}$, we define
$\underline{\alpha}:=\lim_{t\to-\infty}{\varphi}_{X}^{\prime}(t)$ and
$\overline{\alpha}:=\lim_{t\to+\infty}{\varphi}_{X}^{\prime}(t)$. Then by
convexity of ${\varphi}_{X}$, it is easy to verify the following limits

(60)
$\underline{\alpha}=\lim\limits_{q\to-\infty}\frac{{\varphi}_{X}(q)}{q},\text{
and }\overline{\alpha}=\lim\limits_{q\to+\infty}\frac{{\varphi}_{X}(q)}{q}.$

###  6.4.  A measure supported on the level set and a proof of Theorem C

######  Lemma 6.7.

Under the conditions of Theorem C, for every
$\alpha\in(\underline{\alpha},\overline{\alpha})$, there exists
$q\in\mathbb{R}$ such that $\mu_{q}(K_{\alpha})=1$. In particular,
$(\underline{\alpha},\overline{\alpha})\subset\mathcal{L}_{X}$.

The proof of the lemma relies on the following well-known result in the theory
of convex functions.

######  Proposition 6.1 ([Ellis-Book, Rockafellar-Book]).

For any convex function $\Lambda$ defined on $\mathbb{R}$, we have
$(1)$ $\beta q\leq\Lambda^{\ast}(\beta)+\Lambda(q)\ \ \text{for any
}\;q,\beta\in\mathbb{R}$,
$(2)$ if $\Lambda$ is differentiable at $q$ then $\beta
q=\Lambda^{*}(\beta)+\Lambda(q)\Longleftrightarrow\beta=\Lambda^{\prime}(q)$.

######  Proof of Lemma 6.7.

Note that the for every $q\in\mathbb{R}$, the limit
$\lim\limits_{n\to\infty}\frac{1}{n}\log\mathbb{E}_{\nu}e^{qX_{n}}$ exists by
Lemma 6.4. One ([Fan2021]) can prove that the following limit

$C_{q}(\beta):=\lim_{n\to\infty}\frac{1}{n}\log\mathbb{E}_{\mu_{q}}e^{\beta
X_{n}},$

also exists, and
$C_{q}(\beta)={\varphi}_{X}(q+\beta)-{\varphi}_{X}(q),\;\forall\beta,q\in\mathbb{R}$.
Then we have $C_{q}^{\prime}(\beta)={\varphi}_{X}^{\prime}(q+\beta)$ and
$C_{q}(0)=0$, and thus $C_{q}^{*}(\gamma)\geq 0$ for any
$\gamma\in\mathbb{R}$. Moreover, $C_{q}^{\ast}(\alpha)=0$ if and only if
$\alpha=C_{q}^{\prime}(0)={\varphi}_{X}^{\prime}(q)$. The following two
observations show that $\mu_{q}$ is supported on the level set $K_{\alpha}$:

  * •

For any nonempty closed set $F\subset\mathbb{R}$ such that $\alpha\notin F$ ,
we have $\eta_{F}:=\inf\limits_{\gamma\in F}C_{q}^{*}(\gamma)>0$.

  * •

Since the limit $\lim_{n\to\infty}\frac{1}{n}\log\mathbb{E}_{\mu_{q}}e^{\beta
X_{n}}$ exists for any $\beta\in\mathbb{R}$, by the first part of Theorem 1.3,
for any closed set $F$, we have,

$\limsup_{n\to\infty}\frac{1}{n}\log\mu_{q}(\\{\omega\in\Omega:\frac{1}{n}X_{n}(\omega)\in
F\\})\leq-\eta_{F}<0.$

In fact, for $m\in\mathbb{N}$, consider
$F_{m}:=\mathbb{R}\setminus(\alpha-\frac{1}{m},\alpha+\frac{1}{m})$. From the
above observations and the Borel-Cantelli lemma, one has for every
$m\in\mathbb{N}$ that

(61) $\mu_{q}\Big{(}\bigcap_{j\in\mathbb{N}}\bigcup_{n\geq
j}\Big{\\{}\frac{1}{n}X_{n}\in F_{m}\Big{\\}}\Big{)}=0.$

Therefore,

(62)
$\mu_{q}(K_{\alpha})=\lim\limits_{m\to\infty}\mu_{q}\Big{(}\bigcup_{j\in\mathbb{N}}\bigcap_{n\geq
j}\Big{\\{}\frac{1}{n}X_{n}\in F_{m}^{c}\Big{\\}}\Big{)}=1.$

∎

###### Proof of Theorem C.

The first claim of Theorem C:
$(\underline{\alpha},\overline{\alpha})\subset\mathcal{L}_{X}$ is proven in
Lemma 6.7. Now we show that
$\mathcal{L}_{X}\subset[\underline{\alpha},\overline{\alpha}]$. Take any
$\alpha\in\mathcal{L}_{X}$, then by Theorem B, ${\varphi}_{X}^{*}(\alpha)\leq
h=h_{{\textsf{top}}}(T,\Omega)$. Thus by Young’s inequality (the first
statement of Proposition 6.1), one has for all $q\in\mathbb{R}$,

(63) $\alpha q\leq{\varphi}_{X}^{*}(\alpha)+{\varphi}_{X}(q)\leq
h+{\varphi}_{X}(q).$

Hence for $q<0$,

(64) $\alpha\geq\frac{h}{q}+\frac{{\varphi}_{X}(q)}{q},\text{ and for
}q>0,\;\;\alpha\leq\frac{h}{q}+\frac{{\varphi}_{X}(q)}{q}.$

Furthermore, from (60) and (64), for $q<0$, we conclude that
$\alpha\geq\lim\limits_{q\to-\infty}\big{(}\frac{h}{q}+\frac{{\varphi}_{X}(q)}{q}\big{)}=\underline{\alpha}$,
and for $q>0$, we conclude that
$\alpha\leq\lim\limits_{q\to+\infty}\big{(}\frac{h}{q}+\frac{{\varphi}_{X}(q)}{q}\big{)}=\overline{\alpha}$.

The second claim of Theorem C: By combining Lemma 6.3, Lemma 6.7, Proposition
6.1 and Theorem 2.1, one concludes the proof of the second claim (c.f.
(29)-(31)).

∎

##  7\. Examples and Discussion

In this section, we shall discuss some of the known results of multifractal
formalism, and show how they can be treated by the results of this paper.

###  7.1. Lyapunov exponents for products of matrices.

We shall adopt the setup in ([Feng2003],[Feng2009], [Feng-Lau2002]). Consider
a map $M:\Omega\to M_{d}(\mathbb{R})$, where
$\Omega:=\\{1,...,q\\}^{\mathbb{Z}_{+}}$ is a full shift space with the shift
transformation $T$, and $M_{d}(\mathbb{R})$ is the algebra of $d\times d$ real
matrices. For $n\in\mathbb{N}$, $x\in\Omega$, set
$M_{n}(x):=M(T^{n-1}x)\;M(T^{n-2}x)\cdots M(x)$. We impose the following two
conditions on the map $M$:

  * (C1):

$M_{n}(x)\neq O$ holds for all $n\in\mathbb{N}$ and $x\in\Omega$, where $O$ is
the zero matrix;

  * (C2):

$M$ depends only on finitely many coordinates, i.e., $M$ is locally constant.

Note that condition (C1) is automatically satisfied if, for example, the map
$M$ takes values in the set $GL_{d}(\mathbb{R})$ of irreducible matrices, or
the set $M_{d}^{+}(\mathbb{R})$ of $d\times d$ positive matrices. In
[Feng2009], the authors imposed an irreducibility condition which also implies
the condition (C1). Under the condition (C1), one can define the (upper)
Lyapunov exponent of the cocycle $(T,M)$ as: for $x\in\Omega$,

$\lambda^{+}_{M}(x):=\lim_{n\to\infty}\frac{1}{n}\log||M_{n}(x)||,$

provided the limit exists, where $||\cdot||$ is a matrix norm. Let
$K_{\alpha}:=\\{x\in\Omega:\lambda^{+}_{M}(x)=\alpha\\}$, and
$\mathcal{L}_{M}:=\\{\alpha\in\mathbb{R}:K_{\alpha}\neq\emptyset\\}$. We
denote $X_{n}(x):=\log||M_{n}(x)||$ for all $n\in\mathbb{N}$, $x\in\Omega$.
Then it is clear that $X_{n}$ is a locally constant function since $M$ is only
dependent on finitely many coordinates, therefore, the sequence $\\{X_{n}\\}$
satisfies the Bowen condition. In [Feng-Lau2002], the authors introduced the
pressure function $P_{M}$ of the cocycle $(T,M)$ by

$P_{M}(t):=\lim_{n\to\infty}\frac{1}{n}\log\sum_{\omega\in\Omega_{n}}\sup_{x\in[\omega]}||M(T^{n-1}x)\;M(T^{n-1}x)\cdots
M(x)||^{t},\ \ \forall~{}t\in\mathbb{R},$

where $\Omega_{n}:=\\{1,...,q\\}^{n}$, $n\in\mathbb{N}$. Note that the above
limit exists and

(65) $P_{M}(t)=\log q+{\varphi}_{X}(t),\;t\in\mathbb{R},$

where ${\varphi}_{X}$ is the log moment generating function of the sequence
$\\{X_{n}\\}$, i.e.,
${\varphi}_{X}(t)=\lim_{n\to\infty}\frac{1}{n}\log\int_{\Omega}\exp{(tX_{n})}d\nu,\;\;t\in\mathbb{R}$,
and $\nu$ is the uniform Bernoulli measure on $\Omega$. Then by applying
Theorem B, one can obtain that

(66) $h_{{\textsf{top}}}(T,K_{\alpha})\leq\log
q-{\varphi}_{X}^{*}(\alpha)=\inf\limits_{q\in\mathbb{R}}\\{-\alpha
q+P_{M}(q)\\}$

for all $\alpha\in\mathcal{L}_{M}$. Note that the last inequality generalizes
the upper bound of Theorem 1.1 in [Feng2009]. In the same theorem, the authors
also proved the lower bound under much stronger conditions than the conditions
we have imposed above. In this setup, we can not treat the lower bound with
Theorem C, since the sequence $\\{X_{n}\\}$ is not almost additive in general.
However, if the map $M$ takes values in the set $M_{d}^{+}(\mathbb{R})$ of
$d\times d$ positive matrices, then it was proven in [Feng-Lau2002],
[Feng2003] that the sequence $\\{X_{n}\\}$ is in fact almost additive.
Furthermore, in this case, D.J. Feng has proven in [Feng2003] that the
pressure function $P_{M}$ is differentiable, thus ${\varphi}_{X}$ is also
differentiable by (65). Then, in this particular setup, one can apply Theorem
C and obtain the equality in (66).

###  7.2. Ratios of the Birkhoff ergodic averages

In large deviation theory, the contraction principle describes how the large
deviation rate function evolves if the corresponding sequence of random
elements (probability measures, random variables) is transformed by a
continuous map. More specifically, if $\mathcal{X}$ and $\mathcal{Y}$ are
Hausdorff topological spaces and a family $\\{\mu_{n}\\}$ of probability
measures on $\mathcal{X}$ satisfies the LDP with the rate function $I$,
$F:\mathcal{X}\to\mathcal{Y}$ is a continuous map, then
$\tau_{n}:=\mu_{n}\circ F^{-1}$ satisfies LDP with the rate function
$J(y):=\inf_{x:\;F(x)=y}I(x)$. For example, suppose
$X_{n}^{(1)}=S_{n}{\varphi}$, $X_{n}^{(2)}=S_{n}\psi$, where ${\varphi},\psi$
are continuous functions on a compact metric space $\Omega$ and $\psi$ is
strictly positive. Suppose $\nu$ is a probability measure such that the
(vector) sequence $X=(X_{n}^{(1)},X_{n}^{(2)})$ satisfies the LDP with the
rate function $I_{X}(\alpha,\beta)$. Then a new sequence of random variables

$Z_{n}:=\frac{X_{n}^{(1)}}{X_{n}^{(2)}}=\frac{S_{n}{\varphi}}{S_{n}\psi},\;\;n\geq
1,$

satisfies LDP with the rate function
$J_{Z}(\gamma)=\inf_{\alpha/\beta=\gamma}I_{X}(\alpha,\beta)$. Thus we can
easily obtain many non-trivial observables satisfying LDP. In case,
$X_{n}^{(1)}=S_{n}{\varphi}$, $X_{n}^{(2)}=S_{n}\psi$, $\psi>0$, one easily
shows that $\\{nZ_{n}\\}$ satisfies the weak Bowen condition, therefore,
Theorem A applies, and hence $h_{{\textsf{top}}}(T,K_{\gamma}^{Z})\leq
h_{{\textsf{top}}}(T,\Omega)-\inf_{\alpha/\beta=\gamma}I_{X}(\alpha,\beta)=h_{{\textsf{top}}}(T,\Omega)-J_{Z}(\gamma)$.
In this case, one can also easily obtain a lower bound as well. Indeed, for
any $\alpha,\beta$ such that

$K_{\alpha,\beta}^{X}:=\\{\omega\in\Omega:\frac{1}{n}(S_{n}{\varphi})(\omega)\to\alpha,\;\frac{1}{n}(S_{n}\psi)(\omega)\to\beta\\},$

we have that $K^{X}_{\alpha,\beta}\subseteq K^{Z}_{\alpha/\beta}$. Therefore,
$h_{{\textsf{top}}}(T,K^{Z}_{\gamma})\geq
h_{{\textsf{top}}}(T,K^{X}_{\alpha,\beta})$, where $\gamma=\alpha/\beta$. Note
that Theorem C has been established for random variables, but the same
argument works for random vectors as well. (There are also other results for
the level sets of the form $K^{X}_{\alpha,\beta}$, c.f., [Takens-
Verbitskiy2003].) Hence $h_{{\textsf{top}}}(T,K_{\gamma}^{Z})\geq
h_{{\textsf{top}}}(T,\Omega)-I_{X}(\alpha,\beta),$ and thus

$\displaystyle h_{{\textsf{top}}}(T,K_{\gamma}^{Z})$
$\displaystyle\geq\sup_{\alpha/\beta=\gamma}[h_{{\textsf{top}}}(T,\Omega)-I_{X}(\alpha,\beta)]$
$\displaystyle=h_{{\textsf{top}}}(T,\Omega)-\inf_{\alpha/\beta=\gamma}I_{X}(\alpha,\beta)$
$\displaystyle=h_{{\textsf{top}}}(T,\Omega)-J_{Z}(\gamma).$

Therefore, combining two inequalities conclude that
$h_{{\textsf{top}}}(T,K_{\gamma}^{Z})=h_{{\textsf{top}}}(T,\Omega)-J_{Z}(\gamma)$.

###  7.3. Almost additive sequences over a subshift of finite type.

In [Bomfim-Varandas2015], the authors considered almost additive sequences of
continuous potentials and they obtained an expression for the multifractal
spectra of the level sets of the sequence in terms of a rate function. In this
subsection, we shall discuss that the setup considered in [Bomfim-
Varandas2015] is covered by Theorem C. Let us now recall the result of
[Bomfim-Varandas2015] using the notation of the present paper.

######  Theorem 7.1 ([Bomfim-Varandas2015]).

Let $(\Omega,T)$ be a topologically mixing subshift of finite type. Assume
that $X:=\\{X_{n}\\}_{n\in\mathbb{N}}$ is an almost additive sequence
satisfying Bowen’s condition, and $X$ is not cohomologous to a constant, i.e.,
$\frac{X_{n}}{n}$ does not uniformly converge to a constant as $n\to\infty$.
Furthermore, assume that
$\lim_{n\to\infty}\frac{1}{n}\int_{\Omega}X_{n}d\nu=\inf_{n\in\mathbb{N}}\frac{1}{n}\int_{\Omega}X_{n}d\nu=0$,
where $\nu$ is the unique measure of maximal entropy for $T$. Then

$h_{{\textsf{top}}}(T,K_{\alpha})=h_{{\textsf{top}}}(T,\Omega)-{\varphi}_{X}^{*}(\alpha).$

It is clear that the Bowen condition implies the weak Bowen condition, and a
mixing subshift of finite type satisfies the condition (A3) in Theorem C. In
Theorem C, it is also assumed that the sequence
$\\{\frac{1}{n}X_{n}\\}_{n\in\mathbb{N}}$ satisfies LDP with an essentially
strictly convex good rate function. Note that if the moment generating
function ${\varphi}_{X}$ of the sequence $X$ is finite and differentiable on
$\mathbb{R}$, then by Theorem 1.3 $\\{\frac{1}{n}X_{n}\\}_{n\in\mathbb{N}}$
satisfies LDP with a good rate function ${\varphi}_{X}^{*}$. Furthermore, by
Theorem 3.1, ${\varphi}_{X}^{*}$ is also an essentially strictly convex
function. Below, we show that the conditions of Theorem 7.1 imply
differentiability of the moment-generating function ${\varphi}_{X}$. In
[Bomfim-Varandas2015], it is observed that

(67) ${\varphi}_{X}(t)=P_{top}(T,tX)-h_{{\textsf{top}}}(T,\Omega),\text{ for
all }t\in\mathbb{R},$

where $P_{top}(T,tX)$ is the topological pressure (for the concept, see
[Bomfim-Varandas2015], [Barreira2006], [Feng-Huang2010], [Zhao-Zhang-Cao2011]
) of $T$ of the almost additive sequence $X=\\{X_{n}\\}$, and
${\varphi}_{X}(t)=\lim\limits_{n\to\infty}\frac{1}{n}\log\int\limits_{\Omega}e^{tX_{n}(x)}\nu(dx)$
as usual. In [Barreira-Doutor2009], by generalising the well-known result in
[Takens-Verbitskiy1999] to almost additive setting, the authors proved that
$t\mapsto P_{top}(T,tX)$ is differentiable at $0$ if the almost additive
sequence $X$ has unique equilibrium state (for the concept, see [Bomfim-
Varandas2015], [Barreira2006], [Feng-Huang2010], [Zhao-Zhang-Cao2011]), and
showed

$\frac{d}{dt}P_{top}(T,tX)\big{\rvert}_{t=t_{0}}=\lim_{n\to\infty}\int_{\Omega}\frac{X_{n}}{n}d\mu_{t_{0}X},$

where $\mu_{t_{0}X}$ is the unique equilibrium state of
$t_{0}X=\\{t_{0}X_{n}\\}$. In 2006, L. Barreira proved that if an almost
additive sequence $X$ satisfies Bowen’s condition, then it has a unique
equilibrium state [Barreira2006], therefore, from the above one can conclude
that the pressure function $t\in\mathbb{R}\mapsto P_{top}(T,tX)$ is
differentiable everywhere on $\mathbb{R}$. Then from (67), the log-moment
generating function ${\varphi}_{X}$ is also differentiable on $\mathbb{R}$,
and hence, the conditions of Theorem 7.1 imply the conditions of Theorem C.

###  7.4. Multifractal Formalism in the absence of strict convexity of the
rate function.

The Manneville–Pomeau map is one of the well-known examples of a non-uniformly
expanding interval map. For a given $s\in(0,1)$, the Manneville–Pomeau map
$T:[0,1]\to[0,1]$ is defined as

$T(x):=x+x^{1+s}\mod 1.$

It should be noted that $T$ is topologically conjugate to a one-sided full
shift, therefore, it has the specification property. It is also easy to check
that $T$ is strongly topologically exact and expansive. The upper Lyapunov
exponent $\lambda_{T}$ is defined as

$\lambda_{T}(x)=\lim\limits_{n\to\infty}\frac{1}{n}\log(T^{n})^{\prime}(x)$

provided that the limit exists at a point $x\in[0,1]$. Set

$K_{\alpha}:=\\{x\in\Omega:\lambda_{T}(x)=\alpha\\},\;\;\;\;\mathcal{L}_{T}:=\\{\alpha\in\mathbb{R}:K_{\alpha}\neq\emptyset\\}.$

Define, $\psi(x):=\log T^{\prime}(x)=\log(1+(1+s)x^{s}),\;x\in[0,1]$. Then it
is easy to notice that the Lyapunov exponent $\lambda_{T}$ exists at a point
$x$ if and only if the limit $\lim\limits_{n\to\infty}\frac{1}{n}S_{n}\psi(x)$
converges, and
$\lim\limits_{n\to\infty}\frac{1}{n}S_{n}\psi(x)=\lambda_{T}(x)$. It was shown
in [Prellberg-Slawny1992], [Urbanski1996] that the pressure function
$P_{\psi}(q):=P(T,q\psi),\;q\in\mathbb{R}$ exhibits a first order phase
transition at the point $q=-1$, in a sense that $P_{\psi}$ is not
differentiable at $q=-1$, furthermore, it is strictly convex positive
differentiable function in $(-1,\infty)$, and $P_{\psi}(q)=0$ for all $q<-1$.
Since $T$ is expansive and satisfies the specification, there is a unique
measure of maximal entropy $\nu\in\mathcal{M}([0,1])$ for $T$, and it also
becomes the Ahlfors-Bowen measure ([Walters2000]). Define $X_{n}:=S_{n}\psi$
for all $n\in\mathbb{N}$, and set
${\varphi}_{X}(q):=\limsup\limits_{n\to\infty}\frac{1}{n}\log\int_{0}^{1}e^{qX_{n}}d\nu,\;q\in\mathbb{R}$.
Then one can easily check that
${\varphi}_{X}(q)=h_{{\textsf{top}}}(T,[0,1])+P_{\psi}(q)$ for all
$q\in\mathbb{R}$. Thus since $\\{X_{n}\\}$ satisfies the weak Bowen condition
one obtains from Theorem B that for all $\alpha\in\mathcal{L}_{T}$,

(68) $h_{{\textsf{top}}}(T,K_{\alpha})\leq
h_{{\textsf{top}}}(T,[0,1])-{\varphi}_{X}^{*}(\alpha)=-P_{\psi}^{*}(\alpha)=\inf\limits_{q\in\mathbb{R}}\\{P(T,q\psi)-\alpha
q\\}.$

Now we discuss the lower bound. As have we already mentioned, ${\varphi}_{X}$
is not differentiable everywhere; otherwise, the sequence
$\big{\\{}\frac{1}{n}X_{n}\big{\\}}$ satisfies LDP with an essentially
strictly convex good rate function. However, Lemma 6.3 and Lemma 6.7 are still
applicable if
$\alpha\in\\{0\\}\cup((P_{\psi})^{\prime}_{+}(-1),\;\;\overline{\alpha})$,
where $(P_{\psi})^{\prime}_{+}(-1)>0$ is the right derivative of $P_{\psi}$ at
the point $-1$, and
$\overline{\alpha}:=\lim\limits_{q\to\infty}P^{\prime}_{\psi}(q)$. Thus one
can obtain from these lemmas that

(69)
$h_{{\textsf{top}}}(T,K_{\alpha})=h_{{\textsf{top}}}(T,[0,1])-{\varphi}_{X}^{*}(\alpha)=-P_{\psi}^{*}(\alpha)=\inf\limits_{q\in\mathbb{R}}\\{P(T,q\psi)-\alpha
q\\},$

holds for all
$\alpha\in\\{0\\}\cup((P_{\psi})^{\prime}_{+}(-1),\;\;\overline{\alpha})$.

We should note that in [Takens-Verbitskiy2003], the above equality was proven
for all $\alpha\in(0,\overline{\alpha})$ by a different method.

###  7.5. Local or pointwise entropies.

Recall that the upper and the lower local entropies of an invariant measure
$\mu$ at a point $x\in\Omega$ are defined as

$\overline{h}_{\mu}(T,x):=\lim_{{\varepsilon}\to
0+}\limsup_{n\to\infty}-\frac{1}{n}\log\mu(B_{n}(x,{\varepsilon})),\;\;\underline{h}_{\mu}(T,x):=\lim_{{\varepsilon}\to
0+}\liminf_{n\to\infty}-\frac{1}{n}\log\mu(B_{n}(x,{\varepsilon})).$

If $\overline{h}_{\mu}(T,x)=\underline{h}_{\mu}(T,x)$, we say that the local
entropy of the measure $\mu$ at $x$ exists and denote by $h_{\mu}(T,x)$ the
common value. Finally, consider the $\alpha$-level set $K^{(\mu)}_{\alpha}$ of
the local entropies

$K^{(\mu)}_{\alpha}:=\\{x\in\Omega:h_{\mu}(T,x)=\alpha\\},$

and the entropy spectrum
$\mathcal{E}_{\mu}(\alpha):=h_{{\textsf{top}}}(T,K^{(\mu)}_{\alpha})$.

In [Takens-Verbitskiy2000], the authors introduced the notion of weak entropy
doubling condition

$C_{n}({\varepsilon})=\sup_{x}\frac{\mu(B_{n}(x,{\varepsilon}))}{\mu(B_{n}(x,{\varepsilon}/2))}<\infty,\text{
and }\lim_{n}\frac{1}{n}\log C_{n}({\varepsilon})=0.$

Note that the weak entropy doubling condition implies the weak Bowen condition
for observables $X_{n}:\Omega\to\mathbb{R}$ given by

$X_{n}(x)=-\log\mu(B_{n}(x,{\varepsilon})).$

Indeed, it is easy to show that $v_{n,{\varepsilon}}(X_{n})\leq\log
C_{n}(2{\varepsilon})$.

In [Takens-Verbitskiy2000] it was shown that if $\mu$ is fully supported
measures satisfying the (weak) entropy doubling condition, then for any
$\alpha$ one has:

$\mathcal{E}_{\mu}(\alpha):=h_{{\textsf{top}}}(T,K^{(\mu)}_{\alpha})\leq
H^{*}_{\mu}(\alpha),$

where $H^{*}_{\mu}(\alpha)$ is the Legendre transform of the correlation
entropy function

$H_{\mu}(q)=\lim_{{\varepsilon}\to 0}\liminf_{n\to\infty}-\frac{1}{n}\log
I_{\mu}(q,n,{\varepsilon}),\quad
I_{\mu}(q,n,{\varepsilon})=\begin{cases}\int\mu(B_{n}(x,{\varepsilon}))^{q-1}d\mu,&\text{if
}q\neq 1,\\\ \int\log\mu(B_{n}(x,{\varepsilon}))d\mu,&\text{if }q=1.\\\
\end{cases}$

Note the similarity between this result and the upper bound (Theorem A) –
under very similar mild regularity assumptions one obtains an upper bound on
the multifractal spectrum.

In order to obtain a lower bound on the multifractal spectrum of local
entropies one needs additional assumptions on the measure $\mu$. For example,
$\mu$ is the so-called Bowen-Gibbs measure:

$\frac{1}{C({\varepsilon})}\leq\frac{\mu(B_{n}(x,{\varepsilon}))}{\exp(\sum_{k=0}^{n-1}{\varphi}(T^{k}x))}\leq
C({\varepsilon}).$

Then the multifractal analysis of level sets of local entropies is reduced to
the analysis of level sets of Birkhoff’s averages
$K_{\alpha}=\\{\omega:\frac{1}{n}\sum_{k=0}^{n-1}{\varphi}(T^{k}\omega)\to\alpha\\}$,
see [Takens-Verbitskiy2000].

###  7.6. Weighted ergodic averages.

Fan ([Fan2021]) recently studied multifractal spectra of a sequence
$\\{X_{n}\\}$ in the form $X_{n}:=\sum_{i=0}^{n-1}f_{i}\circ T^{i}$, where
$f_{i},\;\;i\in\mathbb{Z}_{+}$ are continuous functions on the shift space
$\Omega:=\mathcal{A}^{\mathbb{Z}_{+}}$, where $\mathcal{A}$ is a finite
alphabet with $\\#\mathcal{A}=q$. A motivation to study this kind of sequence
has been the weighted ergodic averages $S^{w}_{n}f=\sum_{i=0}^{n-1}w_{i}f\circ
T^{i},\;n\in\mathbb{Z}_{+}$ of a potential $f\in C(\Omega)$, where
$\\{w_{i}:i\in\mathbb{Z}_{+}\\}\subset\mathbb{R}$ are some fixed weights. This
is indeed a special case of the former sequence if one sets $f_{i}=w_{i}f$ for
all $i\in\mathbb{Z}_{+}$. To find the dimension spectra of the sequence
$\\{X_{n}\\}$ in terms of the partition function, Fan made two assumptions:

$\displaystyle\text{(H1):}\;\;\;\text{ The limit
}{\varphi}(t)=\lim_{n\to\infty}\frac{1}{n}\log\int_{\Omega}\exp{(tX_{n})}d\mu\text{
exists for all }t\in\mathbb{R},$
$\displaystyle\text{(H2):}\;\;\;\sup_{n\in\mathbb{Z}_{+}}\sup_{x_{0}^{n-1}=y_{0}^{n-1}}\sum_{i=0}^{n-1}|f_{i}(T^{i}(x))-f_{i}(T^{i}(y))|<+\infty.$

Recall that in the shift space, for all $Z\subset\Omega$,
$h_{{\textsf{top}}}(T,Z)=\log q\cdot\dim_{H}(Z)$. Under the assumptions
(H1)-(H2), Fan obtained the following theorem reformulated in terms of
topological entropies:

######  Theorem 7.2 ([Fan2021] ).

If $\lambda\geq 0$,

$h_{{\textsf{top}}}(T,\Omega)-{\varphi}^{*}(\max\partial{\varphi}(\lambda))\leq
h_{{\textsf{top}}}(T,K_{\partial{\varphi}(\lambda)})\leq
h_{{\textsf{top}}}(T,\Omega)-{\varphi}^{*}(\min\partial{\varphi}(\lambda)),$

where $\partial{\varphi}(\lambda)$ is the set of all subgradients of
${\varphi}$ at $\lambda$, and

$K_{[a;b]}=\Big{\\{}\omega\in\Omega:a\leq\liminf_{n\to\infty}\frac{X_{n}(\omega)}{n}\leq\limsup_{n\to\infty}\frac{X_{n}(\omega)}{n}\leq
b\Big{\\}}\text{ for }\;a,b\in\mathbb{R}.$

If $\lambda<0$, we have similar estimates but we have to exchange the roles of
$\min\partial{\varphi}(\lambda)$ and $\max\partial{\varphi}(\lambda)$. In
particular, if ${\varphi}$ is differentiable at $\lambda$, then for the level
set $K_{\alpha}$, $\alpha={\varphi}^{\prime}(\lambda)$, one has

(70)
$h_{{\textsf{top}}}(T,K_{\alpha})=h_{{\textsf{top}}}(T,\Omega)-{\varphi}^{*}(\alpha).$

Now we should note that condition (H2) implies the Bowen condition, and thus
the weak Bowen condition as well. In fact, the condition (H2) is strictly
stronger than the Bowen condition as the following example shows.

######  Example 7.1.

Let $\mathcal{A}=\\{0,1\\}$, and
$a_{n}=\frac{(-1)^{n}}{n^{\gamma}},\;\gamma\in(1,2],\;n\in\mathbb{N}$. Define
$f(\omega):=\sum_{n\in\mathbb{N}}a_{n}\omega_{n}$ for $\omega\in\Omega$, then
$f\in C(\Omega)$. It is easy to notice that

  * 1)

$\sup_{n\in\mathbb{N}}\sum_{j=1}^{\infty}(|a_{j}|+...+|a_{j+n-1}|)=\sum_{j\in\mathbb{N}}j|a_{j}|=+\infty$;

  * 2)

$\sup_{n\in\mathbb{N}}\sum_{j=1}^{\infty}|a_{j}+...+a_{j+n-1}|<+\infty$.

Thus

(71)
$\sup_{n\in\mathbb{N}}\sup_{\omega_{0}^{n-1}=\bar{\omega}_{0}^{n-1}}\sum_{i=0}^{n-1}|(f\circ
T^{i})(\omega)-(f\circ T^{i})(\bar{\omega})|=+\infty,$

but $\sup\limits_{n\in\mathbb{N}}\;v_{n}(f_{n})<+\infty$, where
$f_{n}:=\sum_{i=0}^{n-1}f\circ T^{i}$.

Since the sequence $\\{X_{n}\\}$ satisfies the weak Bowen condition, Theorem B
generalizes the inequality $h_{{\textsf{top}}}(T,K_{\alpha})\leq
h_{{\textsf{top}}}(T,\Omega)-{\varphi}^{*}(\alpha)$ which is a part of (70)
since it relaxes both assumptions (H1) and (H2).

It should be stressed that Theorem C is not applicable in this setup since the
sequence $\\{X_{n}\\}_{n\in\mathbb{N}}$ is not almost additive in general.

##  8\. Final remarks

A) The results of this paper can be extended to the case that the reference
measure is a Bowen-Gibbs measure, i.e., there exists potential $\zeta\in
C(\Omega,\mathbb{R})$ such that for every ${\varepsilon}>0$ there exists a
constant $C({\varepsilon})\geq 1$ satisfying for all $n\in\mathbb{N}$ and
$x\in\Omega$ that

(72)
$C({\varepsilon})^{-1}\exp(\zeta_{n}(x))\leq\nu(B_{n}(x,{\varepsilon}))\leq
C({\varepsilon})\exp(\zeta_{n}(x)).$

It is easy to see that the Ahlfors-Bowen measures satisfy the above property
(72) with a constant potential $\zeta\equiv const$. To extend the results for
such measures, instead of the topological entropy, one can consider another
dimension characteristics of the sets related to the potential $\zeta$. We
note that the Pesin-Pitskel topological pressure [Pesin-Pitskel1984]
associated with the potential $\zeta$ is suitable for this purpose.

B) It would be interesting if one could obtain an analogue of the above
results for the Hausdorff dimension instead of the topological entropy.
However, the Hausdorff dimension might be more difficult to treat as it is not
related to the dynamics of the underlying transformation.

Acknowledgements.

The authors are grateful to Aihua Fan for his valuable discussions. Qian Xiao
is particularly grateful to the China Scholarship Council and Leiden
University for their support, which made her visit to Leiden University
possible.

## References