# On the dimension of limit sets on $\mathbb{P}(\mathbb{R}^{3})$ via
stationary measures: variational principles and applications

Yuxiang Jiao, Jialun Li, Wenyu Pan and Disheng Xu

###### Abstract

In this article, we establish the variational principle of the affinity
exponent of Borel Anosov representations. We also establish such a principle
of the Rauzy gasket. In [LPX23], they obtain a dimension formula of the
stationary measures on ${\mathbb{P}}({\mathbb{R}}^{3})$. Combined with our
result, it allows us to study the Hausdorff dimension of limit sets of Anosov
representations in $\mathrm{SL}_{3}(\mathbb{R})$ and the Rauzy gasket. It
yields the equality between the Hausdorff dimensions and the affinity
exponents in both settings.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Actions and random walks of $\mathrm{SL}_{n}({\mathbb{R}})$
    2. 2.2 Different notions of entropies
  3. 3 Free sub-semigroups in hyperbolic groups
    1. 3.1 Preliminaries on geometric group theory
    2. 3.2 An extension lemma
    3. 3.3 Proof of Proposition 3.1.
  4. 4 Variational principle for Anosov representations
    1. 4.1 The variational principle to positive linear functions on $\mathfrak{a}^{+}$
    2. 4.2 The variational principle of critical exponent
  5. 5 Hausdorff dimension of the Rauzy Gasket
    1. 5.1 Preliminaries and notation
    2. 5.2 The upper bound of the Hausdorff dimension
    3. 5.3 The lower bound of the Hausdorff dimension

##  1 Introduction

This article is the second part of [LPX23]. We provide different examples of
computing the Hausdorff dimension of limit sets on a projective space using
stationary measures.

First, we consider the limit sets of Anosov representations. For a finitely
generated hyperbolic group $\Gamma,$ let $|\cdot|$ be the word norm on
$\Gamma$ with respect to a fixed finite generating set. A homomorphism
$\rho:\Gamma\to\mathrm{SL}_{n}({\mathbb{R}})$ is called a Borel Anosov
representation if there exists $c>0$ such that for every $1\leqslant
p\leqslant n-1,$

$\frac{\sigma_{p}(\rho(\gamma))}{\sigma_{p+1}(\rho(\gamma))}\geqslant
ce^{c|\gamma|},\quad\forall\gamma\in\Gamma,$

where $\sigma_{p}(g)$ denotes the $p$-th maximal singular value of
$g\in\mathrm{SL}_{n}({\mathbb{R}}).$

The concept of Anosov representation was first introduced by Labourie in
[Lab06] to study Hitchin components of the representations of surface groups.
They are a generalization of convex cocompact representations in
$\mathrm{PSO}(n,1).$ Given a Borel Anosov representation $\rho,$ we consider
the action of $\rho(\Gamma)$ on the projective space
${\mathbb{P}}({\mathbb{R}}^{n})$. It always admits a limit set, denoted by
$L(\rho(\Gamma)),$ which is the closure of attracting fixed points of
$\rho(\gamma)$ on ${\mathbb{P}}({\mathbb{R}}^{n})$ for $\gamma\in\Gamma$.

To compute the Hausdorff dimension of the limit set of $\rho(\Gamma)$, the
difficulty lies in the fact that $\rho(\Gamma)$ usually acts on
${\mathbb{P}}({\mathbb{R}}^{n})$ non-conformally. For example, consider the
action of diagonal matrix $\mathrm{diag}(2,1,1/2)$ on
$\mathbb{P}(\mathbb{R}^{3})$. The stretch rates in different directions are
different. When computing the Hausdorff dimension of the set, one needs to
estimate the minimal number of balls to cover it. Naturally, in the non-
conformal setting, an efficient way is to arrange the balls according to the
stretch rate. Therefore, it leads us to consider the affinity exponent and we
expect the Hausdorff dimension of the limit set equals its affinity exponent.
The concept of affinity exponent was proposed by Falconer to study the
Hausdorff dimension of self-affine fractals [Fal88]. Later, Pozzetti-
Sambarino-Wienhard [PSW22] extended this concept to study Anosov
representations.

For $0\leqslant s\leqslant n-1,$ let
$\psi_{s}:\mathrm{SL}_{n}({\mathbb{R}})\to{\mathbb{R}}$ be given by

$\psi_{s}(g)\coloneqq\sum_{1\leqslant i\leqslant\lfloor
s\rfloor}(\log\sigma_{1}(g)-\log\sigma_{i+1}(g))+(s-\lfloor
s\rfloor)(\log\sigma_{1}(g)-\log\sigma_{\lfloor s\rfloor+2}(g)),\quad
g\in\mathrm{SL}_{n}({\mathbb{R}}).$

Then the affinity exponent of $\rho$ is given by

$s_{\mathrm{A}}(\rho)\coloneqq\sup\left\\{s:\sum_{\gamma\in\Gamma}\exp(-\psi_{s}(\rho(\gamma)))=\infty\right\\}.$
(1.1)

It is shown in [PSW22] that the affinity exponent is always an upper bound of
the Hausdorff dimension of the limit set. To obtain the equality between two
notions of dimensions, it remains to show a reversed inequality. An approach
to give a lower bound of the dimension of $L(\rho(\Gamma))$ is to consider
measures supported on it. A probability Borel measure $\mu$ on a metric space
is called exact dimensional if there exists $\alpha$ such that

$\lim_{r\to 0}\frac{\log\mu(B(x,r))}{\log
r}=\alpha,\quad\mu\mathrm{-a.e.}~{}x,$

and $\alpha$ is called the exact dimension of $\mu,$ which will be denoted by
$\dim\mu$. Due to a result by Young [You82], the Hausdorff dimension of a set
is bounded below by the exact dimension of measures (if exist) supported on
it. Therefore, our approach is to construct satationary measures supported on
the limit set whose exact dimensions approximate the affinity exponent.

Let us recall the definition of Lyapunov dimension of stationary measures,
which is the expected value of exact dimensions. Let $\nu$ be a finitely
supported probability measure on $\mathrm{SL}_{n}({\mathbb{R}})$ with the
Lyapunov spectrum
$\lambda(\nu)=\left\\{\lambda_{1}(\nu)\geqslant\cdots\geqslant\lambda_{n}(\nu)\right\\}.$
We denote $\chi_{i}(\nu)=\lambda_{1}(\nu)-\lambda_{i+1}(\nu)$ for $1\leqslant
i\leqslant n-1.$ Let $\mu$ be a $\nu$-stationary measure on
${\mathbb{P}}({\mathbb{R}}^{n}).$ The Furstenberg entropy is given by

$h_{\mathrm{F}}(\mu,\nu)=\int\log\frac{\mathrm{d}g\mu}{\mathrm{d}\mu}(\xi)\left(\frac{\mathrm{d}g\mu}{\mathrm{d}\mu}(\xi)\right)\mathrm{d}\nu(g)\mathrm{d}\mu(\xi).$
(1.2)

Assume further that $\chi_{n-1}(\nu)>0,$ then the Lyapunov dimension of $\mu$
is defined to be

$\dim_{\mathrm{LY}}\mu=d+\frac{h_{\mathrm{F}}(\mu,\nu)-(\chi_{1}(\nu)+\cdots+\chi_{d}(\nu))}{\chi_{d+1}(\nu)},$
(1.3)

where $d$ is the maximal integer such that
$\chi_{1}(\nu)+\cdots+\chi_{d}(\nu)\leqslant h_{\mathrm{F}}(\mu,\nu).$

To achieve the affinity exponent in our setting, we will approximate it by
Lyapunov dimensions of stationary measures. We call it a variational principle
of critical exponent. Such variational principle has been considered in
several different contexts to obtain a lower bound of the Hausdorff dimension
of dynamically invariant sets: Morris-Shmerkin [MS19] and Morris-Sert [MS23]
on self-affine IFSs on ${\mathbb{R}}^{n}$ and He-Jiao-Xu [HJX23] on
$\mathrm{Diff}(\SS^{1})$.

In conclusion, our approach can be roughly summarized by the following
inequalities

$s_{\mathrm{A}}(\rho)\stackrel{{\scriptstyle\text{(1)}}}{{\geqslant}}\dim
L(\rho(\Gamma))\stackrel{{\scriptstyle\text{(2)}}}{{\geqslant}}\sup_{\mu}\dim\mu\stackrel{{\scriptstyle\text{(3)}}}{{=}}\sup_{\mu}\dim_{\mathrm{LY}}\mu\stackrel{{\scriptstyle\text{(4)}}}{{\geqslant}}s_{\mathrm{A}}(\rho);$

here (1) is established in [PSW22], (2) is established in [You82], [Rap21] and
[LL23b], (3) is established in [HS17] and [LPX23] for special cases. We obtain
the following variational principle, which establishes (4). We say
$\rho:\Gamma\to\mathrm{SL}_{n}({\mathbb{R}})$ is Zariski dense if
$\rho(\Gamma)$ is Zariski dense in $\mathrm{SL}_{n}({\mathbb{R}}).$

######  Theorem 1.1.

Let $\rho:\Gamma\to\mathrm{SL}_{n}({\mathbb{R}})$ be a Zariski dense Borel
Anosov representation. For every $\epsilon>0,$ there exists a finitely
supported probability measure $\nu$ on $\rho(\Gamma)$ whose support generates
a Zariski dense subgroup such that the unique $\nu$-stationary measure $\mu$
satisfying

$\dim_{\mathrm{LY}}\mu\geqslant s_{\mathrm{A}}(\rho)-\epsilon.$

To obtain the approximations, we need to find a probability measure $\nu$ on
$\rho(\Gamma)$ with large entropy and controlled Lyapunov exponents. In our
setting, the Furstenberg entropy equals the random walk entropy (2.1), which
characterizes the freeness of the semigroup generated by
$\operatorname{\mathrm{supp}}\nu.$ We aim to find a finite subset of
$\rho(\Gamma)$ which freely generates a free semigroup, and consider the
uniform random walk on this subset.

At the core of the proof, we use a geometric group theoretic argument to
construct free semigroups, which is different from the one in the IFS setting.
Thanks to the hyperbolicity of the group, we can always approximate the group
by free semigroups, in the sense of the growth rate of groups. However, the
approximating semigroups should satisfy some additional conditions coming from
the dynamics. Our key argument, as presented in Sections 3 and 4, establishes
such a desired construction.

A direct application of Theorem 1.1 is computing the Hausdorff dimension of
limit sets of Anosov representations in $\mathrm{SL}_{3}({\mathbb{R}}).$ Let
us recall the dimension formula of stationary measures established in the
first part of our paper [LPX23].

######  Theorem 1.2 ([LPX23, Theorem 1.10]).

Let $\nu$ be a Zariski dense, finitely supported probability measure on
$\mathrm{SL}_{3}(\mathbb{R})$ that satisfies the exponential separation
condition, and $\mu$ be its Furstenberg measure on
$\mathbb{P}(\mathbb{R}^{3})$. Then we have $\dim\mu=\dim_{\mathrm{LY}}\mu.$

As a consequence, we can derive the Hausdorff dimension of limit sets, as
shown in [LPX23].

######  Theorem 1.3 ([LPX23, Theorem 1.3]).

Let $\Gamma$ be a hyperbolic group and
$\rho:\Gamma\rightarrow\mathrm{SL}_{3}(\mathbb{R})$ be an irreducible Anosov
representation, then the Hausdorff dimension of the limit set
$L(\rho(\Gamma))$ in $\mathbb{P}(\mathbb{R}^{3})$ equals the affinity exponent
$s_{\mathrm{A}}(\rho)$.

In a similar vein of Theorem 1.1, we also obtain a variational principle on
the flag variety, Proposition 4.4. An application is to obtain an estimation
of the Hausdorff dimension of limit sets of Borel Anosov representations on
the flag variety by Ledrappier-Lessa [LL23a] .

#### Rauzy gasket

Another example we consider is the Rauzy gasket which is a fractal set on
${\mathbb{P}}({\mathbb{R}}^{3})$ formed by projective actions of
$\mathrm{SL}_{3}({\mathbb{R}})$. We also establish the identity between its
Hausdorff dimension and its affinity exponent: we show the affinity exponent
is an upper bound of its Hausdorff dimension and a variational principle of
the affinity exponent.

Let $\Delta$ be the projectivization of $\\{(x,y,z):x,y,z\geqslant 0\\}$ in
${\mathbb{P}}({\mathbb{R}}^{3}).$ Let $\Gamma_{\mathscr{R}}$ be the semigroup
generated by

$A_{1}=\begin{pmatrix}1&1&1\\\ 0&1&0\\\ 0&0&1\end{pmatrix},\
A_{2}=\begin{pmatrix}1&0&0\\\ 1&1&1\\\ 0&0&1\end{pmatrix},\
A_{3}=\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 1&1&1\end{pmatrix},$

and we call it the Rauzy semigroup. Then as
$\Gamma_{\mathscr{R}}\subset\mathrm{SL}_{3}(\mathbb{R})$, the semigroup
$\Gamma_{\mathscr{R}}$ acts on $\mathbb{P}(\mathbb{R}^{3})$. Due to the choice
of $\Delta$, the semigroup $\Gamma_{\mathscr{R}}$ preserves $\Delta$. The
Rauzy gasket $X$ is the unique attractor of the Rauzy semigroup, which can be
defined formally as

$\bigcap_{n\rightarrow\infty}\bigcup_{i_{j}\in\\{1,2,3\\}}(A_{i_{1}}\cdots
A_{i_{n}}\Delta).$

The Rauzy gasket, depicted in Figure 1, was first introduced in 1991 by Arnoux
and Rauzy [AR91] in the context of interval exchange transformations. They
conjectured that the gasket has Lebesgue measure 0. Levitt [Lev93]
rediscovered the gasket in 1993 and confirmed the Arnoux-Rauzy conjecture, the
proof of which is essentially due to Yoccoz. Later, in the study of Novikov’s
problem, Dynnikov and De Leo [DD09] provided a numerical estimate of the
Hausdorff dimension of the Rauzy gasket. They suggested lower and upper bounds
are $1.7$ and $1.8$, respectively. Meanwhile, Arnoux asked whether the
Hausdorff dimension is less than or equal to $2$ [AS13]. Avila, Hubert, and
Skripchenko [AHS16] provided a positive answer to this question, and
Gutiérrez-Romo and Matheus in [GRM20] proved the lower bound is greater than
$1.19$. Recently, Pollicott and Sewell [PS21] used a renewal theoretical
argument to show that the Hausdorff dimension of the Rauzy gasket is less than
$1.7407$. See also [Fou20] for a weaker upper bound.

Figure 1: Figure in [AS13]

Let $\rho$ be the natural embedding of Rauzy semigroup $\Gamma_{\mathscr{R}}$
into $\mathrm{SL}_{3}({\mathbb{R}}).$ The affinity exponent
$s_{\mathrm{A}}(\Gamma_{\mathscr{R}})$ of the Rauzy gasket is defined as the
critical exponent of (1.1), which also works for the Rauzy semigroup. The
following theorem confirms a folk-lore conjecture of the Hausdorff dimension
of the Rauzy gasket.

######  Theorem 1.4.

The Hausdorff dimension of the Rauzy gasket $X$ is equal to its affinity
exponent $s_{\mathrm{A}}(\Gamma_{\mathscr{R}}).$

######  Remark 1.5.

Recently Natalia Jurga also obtained 1.4 independently.

Unlike Anosov representations, the Rauzy semigroup is non-uniformly
hyperbolic. The estimate of the upper bound of the Hausdorff dimension
requires the study of the points where the action lacks hyperbolicity. To
obtain a lower bound of the Hausdorff dimension, we also establish a
variational principle for the Rauzy semigroup. We build on the variational
principle for IFS setting [MS23]. Like Anosov representations, we aim to find
free semigroups. In this process, the difficulty lies in that non-uniform
hyperbolicity makes us lose control of word lengths. We make use of the prefix
argument to overcome the issue. This prefix argument also occurs in [HJX23].

#### Organization.

In Section 2, we discuss different notions of entropy. We establish a
geometric group theoretic lemma in Section 3 and the variational principle for
Anosov representations in Section 4. Section 5 is devoted to the study of the
Hausdorff dimension of the Rauzy gasket.

#### Acknowledgement.

We would like to thank Wenyuan Yang for carefully explaining the basic ideas
and arguments in [Yan19], which is useful for Section 3. We would like to
thank Cagri Sert, François Ledrappier, and Pablo Lessa for helpful
discussions. Part of the work was done in the conference “Beyond uniform
hyperbolicity” at the Banach Center in Będlewo, Poland, in 2023. We thank the
organizers and the hospitality of the center.

##  2 Preliminaries

###  2.1 Actions and random walks of $\mathrm{SL}_{n}({\mathbb{R}})$

Consider the $n$-dimensional Euclidean space $\mathbb{R}^{n}$ and denote by
$\|\cdot\|$ the Euclidean norm. By abuse of the notation, we denote by
$\|\cdot\|$ the norm on $\wedge^{2}\mathbb{R}^{n}$ induced by the one in
$\mathbb{R}^{n}$. Let ${\mathbb{P}}({\mathbb{R}}^{n})$ be the projective
space. The metric on ${\mathbb{P}}({\mathbb{R}}^{n})$ is given by

$d(\mathbb{R}v,\mathbb{R}w):=\frac{\|v\wedge w\|}{\|v\|\|w\|}\,\,\,\text{for
any}\,\,\,\mathbb{R}v,\mathbb{R}w\in\mathbb{P}(\mathbb{R}^{n}),$

which is bi-Lipschitz equivalent to the $\mathrm{SO}(n)$-invariant metric on
${\mathbb{P}}({\mathbb{R}}^{n}).$

Let
$\mathfrak{a}=\left\\{\lambda=\mathrm{diag}(\lambda_{1},\cdots,\lambda_{n}):\lambda_{i}\in{\mathbb{R}},\sum_{i}\lambda_{i}=0\right\\}$
be a Cartan algebra of $\mathfrak{sl}_{n}({\mathbb{R}})$ and
$\mathfrak{a}^{+}=\left\\{\lambda\in\mathfrak{a}:\lambda_{1}\geqslant\cdots\geqslant\lambda_{n}\right\\}$
be a positive Weyl chamber. Set $A^{+}=\exp\mathfrak{a}^{+}$ and
$K=\mathrm{SO}_{n}({\mathbb{R}}).$

For every $g\in\mathrm{SL}_{n}({\mathbb{R}}),$ it admits the Cartan
decomposition $g=\widetilde{k}_{g}a_{g}k_{g}\in KA^{+}K.$ Here,
$a_{g}=\mathrm{diag}(\sigma_{1}(g),\cdots,\sigma_{n}(g))$ where
$\sigma_{1}(g)\geqslant\cdots\geqslant\sigma_{n}(g)$ are singular values of
$g$ as the notation before. The Cartan projection of $g$ is defined to be
$\kappa(g)\coloneqq\mathrm{diag}(\log\sigma_{1}(g),\cdots\log\sigma_{n}(g))\in\mathfrak{a}^{+}.$
Let $e_{1},\cdots,e_{n}$ be the standard orthonormal basis of $\mathbb{R}^{n}$
and $E_{i}={\mathbb{R}}e_{i}\in{\mathbb{P}}({\mathbb{R}}^{n})$ be the
corresponding point in the projective space. We consider the following notions
as in our first paper [LPX23]:

  * •

$V_{g}^{+}\coloneqq\widetilde{k}_{g}E_{1}\in\mathbb{P}(\mathbb{R}^{n})$, which
is an attracting point of $g$;

  * •

$H_{g}^{-}\coloneqq k_{g}^{-1}(E_{2}\oplus\cdots\oplus
E_{n})\subset\mathbb{P}(\mathbb{R}^{d})$, which is a repelling hyperplane of
$g$

(if $\sigma_{1}>\sigma_{2}$, then $V_{g}^{+}$ and $H_{g}^{-}$ are uniquely
defined);

  * •

$b(g^{-},\epsilon)\coloneqq\\{x\in\mathbb{P}(\mathbb{R}^{n}):d(x,H_{g}^{-})>\epsilon\\}$
for any $\epsilon>0$;

  * •

$B(g^{+},\epsilon)\coloneqq\\{x\in\mathbb{P}(\mathbb{R}^{n}):d(x,V_{g}^{+})\leqslant\epsilon\\}$
for any $\epsilon>0$.

We have the following useful lemma [BQ16, Lemma 14.2] for later use.

######  Lemma 2.1.

For any $g\in\mathrm{SL}_{n}(\mathbb{R})$ and
$V=\mathbb{R}v\in\mathbb{P}(\mathbb{R}^{n})$, we have

$\displaystyle d(V,H_{g}^{-})\leqslant\frac{\|gv\|}{\|g\|\|v\|}\leqslant
d(V,H_{g}^{-})+\frac{\sigma_{2}(g)}{\sigma_{1}(g)},\quad
d(gV,V_{g}^{+})d(V,H_{g}^{-})\leqslant\frac{\sigma_{2}(g)}{\sigma_{1}(g)}.$

Let $\nu$ be a finitely supported probability measure on
$\mathrm{SL}_{n}({\mathbb{R}}).$ It induces random walks on the projective
space and the flag variety. Recall the flag variety of ${\mathbb{R}}^{n}$ as

$\mathcal{F}({\mathbb{R}}^{n})\coloneqq\left\\{\,\xi=(\xi^{1}\subset\xi^{2}\subset\cdots\subset\xi^{i}\subset\cdots\subset\xi^{n-1}):\xi^{i}\text{
is a linear subspace of }{\mathbb{R}}^{n}\text{ of dimension }i\,\right\\}$

and that $\mathrm{SL}_{n}(\mathbb{R})$ acts on $\mathcal{F}({\mathbb{R}}^{n})$
canonically.

We denote $G_{\nu}$ to be the group generated by
$\operatorname{\mathrm{supp}}\nu.$ If we further assume that $G_{\nu}$ is
Zariski dense in $\mathrm{SL}_{n}({\mathbb{R}}).$ Then the Lyapunov spectrum
$\lambda(\nu)=\left\\{\lambda_{1}(\nu)\geqslant\cdots\geqslant\lambda_{n}(\nu)\right\\}$
is simple. Moreover, the random walks induced by $\nu$ on the projective space
${\mathbb{P}}({\mathbb{R}}^{n})$ and the flag variety
$\mathcal{F}({\mathbb{R}}^{n})$ both have a unique stationary measure ([Fur63,
GR85, GM89], see also [BQ16, Chapter 10]).

###  2.2 Different notions of entropies

The Furstenberg entropy is mysterious and might be difficult to compute. We
recall another notion of the entropy associated to the random walk. The random
walk entropy of $\nu$ is

$h_{\mathrm{RW}}(\nu)=\lim_{k\to\infty}\frac{1}{k}H(\nu^{*k}).$ (2.1)

######  Remark 2.2.

In [BHR19] and [Rap21], the notion of the entropy they used is the following:
for a discrete measure $\nu=\sum p_{i}\delta_{g_{i}}$, $H(\nu)\coloneqq
H(p)=-\sum p_{i}\log p_{i}$. This notion works well in many settings of IFSs.
For general semigroup actions, the random walk entropy $h_{\mathrm{RW}}(\nu)$
as in [HS17] is more precise. In particular, if
$\operatorname{\mathrm{supp}}\nu$ freely generates a free semigroup then
$H(\nu)=h_{\mathrm{RW}}(\nu)$. This kind of entropy was first studied by Avez
in [Ave72] to study the structure of the group action on the boundary.

To obtain a more calculable dimension formula, it is expected to show that the
Furstenberg entropy in 1.2 is equal to the random walk entropy. In the
following proposition we will see that for some concrete examples, we do have
this equality. We say a representation
$\rho:\Gamma\to\mathrm{SL}_{n}({\mathbb{R}})$ is Zariski dense if
$\rho(\Gamma)$ is Zariski dense.

######  Proposition 2.3.

Let $\Gamma$ be a hyperbolic group and
$\rho:\Gamma\rightarrow\mathrm{SL}_{n}(\mathbb{R})$ be a Zariski dense Borel
Anosov representation. Let $\nu$ be a finitely supported probability measure
on $\rho(\Gamma)$ such that $G_{\nu}$ is Zariski dense. Then the unique
$\nu$-stationary measure $\mu$ on $\mathbb{P}(\mathbb{R}^{n})$ satisfies

$h_{\mathrm{F}}(\mu,\nu)=h_{\mathrm{RW}}(\nu).$

######  Remark 2.4.

The Zariski density assumptions on $\rho$ and $G_{\nu}$ are both nonnecessary.
It is shown in [LL23a] that for every $\nu$ supported on $\rho(\Gamma)$ with
non-elementary $G_{\nu}$, the equality of entropies holds.

To show Propostion 2.3, we may consider the random walk on the flag variety.

######  Proposition 2.5.

Let $\nu$ be a probability measure on $\mathrm{SL}_{n}(\mathbb{R})$ such that
$G_{\nu}$ is a Zariski dense discrete subgroup. Let
$\mu_{\mathcal{F}}=\mu_{\mathcal{F}}(\nu)$ and $\mu=\mu(\nu)$ be the unique
$\nu$-stationary measure on $\mathcal{F}({\mathbb{R}}^{n})$ and
$\mathbb{P}(\mathbb{R}^{n})$ respectively. Then we have

$h_{\mathrm{RW}}(\nu)=h_{\mathrm{F}}(\mu_{\mathcal{F}},\nu)\geqslant
h_{\mathrm{F}}(\mu,\nu).$

###### Proof.

Proposition 2.5 follows from [Fur02, Theorem 2.31] (originally in [KV83,
Theorem 3.2], [Led85, Section 3.2]). In order to apply their result, we need
to use [Fur02, Theorem 2.21] (originally in [KV83], [Led85]) to obtain that
$(\operatorname{\mathrm{supp}}\mu_{\mathcal{F}},\mu_{\mathcal{F}})$ is the
Poisson boundary of $(G_{\nu},\nu)$. Then the Furstenberg entropy of the
Poisson boundary is exactly the random walk entropy for discrete $G_{\nu}$ due
to [Fur02, Theorem 2.31]. ∎

###### Proof of 2.3.

By Proposition 2.5 and that the image of an Anosov representation is discrete,
it remains to prove that
$h_{\mathrm{F}}(\mu_{\mathcal{F}},\nu)=h_{\mathrm{F}}(\mu,\nu)$.

Consider the canonical projection

$\pi:\mathcal{F}({\mathbb{R}}^{n})\to\mathbb{P}(\mathbb{R}^{n}),~{}\xi=(\xi^{1}\subset\xi^{2}\subset\cdots\subset\xi^{n-1})\mapsto\xi^{1}.$

Due to uniqueness of the Furstenberg measure on $\mathbb{P}(\mathbb{R}^{n})$,
we know that $\pi_{*}(\mu_{\mathcal{F}})=\mu$. By classical Rokhlin’s
disintegration theorem, we can define a desintegration $\\{\mu^{\xi}\\}$ of
the measure $\mu_{\mathcal{F}}$ over $\mu$, where $\mu^{\xi}$ is a well-
defined probabilty measure on $\pi^{-1}(\xi)$ for $\mu$ a.e. $\xi$.

Let $L_{\mathcal{F}}(\rho(\Gamma))$ and $L(\rho(\Gamma))$ be the limit sets on
$\mathcal{F}({\mathbb{R}}^{n})$ and $\mathbb{P}(\mathbb{R}^{n})$ (the closure
of attracting fixed points of proximal elements) respectively. [BQ16, Lemma
9.5] tells us that $L_{\mathcal{F}}(\rho(\Gamma))$ (resp. $L(\rho(\Gamma))$)
is the unique $\rho(\Gamma)$-minimal closed invariant subset on
$\mathcal{F}({\mathbb{R}}^{n})$ (resp. $\mathbb{P}(\mathbb{R}^{n})$). Hence
$\mu_{\mathcal{F}}$ (resp. $\mu$) is supported on the limit set
$L_{\mathcal{F}}(\rho(\Gamma))$ (resp. $L(\rho(\Gamma))$). Before continuing,
we need a lemma about the structure of the limit sets.

######  Lemma 2.6.

Let $\rho:\Gamma\rightarrow\mathrm{SL}_{n}(\mathbb{R})$ be a Borel Anosov
representation. Then the canonical projection
$\pi:L_{\mathcal{F}}(\rho(\Gamma))\rightarrow L(\rho(\Gamma))$ has a trivial
fibre.

###### Proof.

Because $\rho:\Gamma\rightarrow\mathrm{SL}_{n}({\mathbb{R}})$ is Borel Anosov,
we can apply [Can, Theorem 31.1]. Then there exists a continuous $\rho$
equivariant map $(\xi^{1},\xi^{\prime})$ from $\partial\Gamma$ to
$\mathbb{P}(\mathbb{R}^{n})\times\mathrm{Grass}(n-1,\mathbb{R}^{n})$, such
that $\xi^{1}$ satisfies the Cartan property in [Can, Chapter 30] and
$\xi^{1}(x)\subset\xi^{\prime}(x),$

$\xi^{1}(x)\oplus\xi^{\prime}(y)=\mathbb{R}^{n},\quad\forall x\neq y.$

It follows that $\xi^{1}$ is injective from $\partial\Gamma$ to
$\mathbb{P}(\mathbb{R}^{n})$. Moreover, the image of $\xi^{1}$ is exactly
$L(\rho(\Gamma)).$

From Borel Anosov property, $\xi^{1}$ can be extended to a limit map
$\xi:\partial\Gamma\rightarrow\mathcal{F}$ given by
$\xi(x)=(\xi^{1}(x)\subset\xi^{2}(x)\subset\cdots\subset\xi^{n-1}(x))$. 111The
existence of the limit map $\xi^{k}(x)$ is from $P_{k}$-Anosov and the
consistence condition $\xi^{k}(x)\subset\xi^{k+1}(x)$ can be deduced from the
Cartan property of the limit map (see for example Section 30 and 31 in [Can]),
that is $\xi^{k}(x)=\lim_{n\rightarrow\infty}U_{k}(\gamma_{n})$. Here
$\gamma_{n}$ is a sequence converges to the boundary point $x$ and
$U_{k}(\gamma_{n})=k_{\gamma_{n}}(E_{1}\oplus\cdots\oplus E_{k})$ from the
Cartan decomposition
$\gamma_{n}=k_{\gamma_{n}}a_{\gamma_{n}}k^{\prime}_{\gamma_{n}}$.  The image
of $\xi$ is $L_{\mathcal{F}}(\rho(\Gamma))\subset\mathcal{F}$. The map
$\xi^{1}$ being injective implies that the natural projection of
$L_{\mathcal{F}}(\rho(\Gamma))$ to $L(\rho(\Gamma))$ has trivial fiber. ∎

By Lemma 2.6, we know that the fiber is trivial for all $\xi\in
L(\rho(\Gamma))$. Hence we have the relation of relative measure-preserving of
$(\mathcal{F}({\mathbb{R}}^{n}),\mu_{\mathcal{F}})\rightarrow(\mathbb{P}(\mathbb{R}^{n}),\mu)$,
that is for $\nu$ a.e. $g$ and $\mu$ a.e. $\xi$

$g\mu^{\xi}=g\delta_{\pi^{-1}\xi}=\delta_{\pi^{-1}(g\xi)}=\mu^{g\xi}.$

By [Fur02, Proposition 2.25] (see also [KV83]), we obtain

$h_{\mathrm{F}}(\mu_{\mathcal{F}},\nu)=h_{\mathrm{F}}(\mu,\nu)$

Then by Proposition 2.5, the proof is complete. ∎

##  3 Free sub-semigroups in hyperbolic groups

This section is devoted to establish the geometric group theoretical
preparation for the later proof of variational principle. In this section,
$\Gamma$ is a finitely generated group with a fixed symmetric generating set
$\mathcal{S}$. For every $g\in\Gamma,$ let $|g|$ denote the word length of $g$
with respect to $\mathcal{S}.$ For a positive integer $L,$ let
$A(L)\coloneqq\left\\{g\in\Gamma:|g|=L\right\\}$ refer to an annulus. The
following is the main proposition of this section.

######  Proposition 3.1.

Let $\Gamma$ be a non-elementary torsion-free hyperbolic group and
$\rho:\Gamma\to\mathrm{SL}_{n}({\mathbb{R}})$ be a faithful representation.
Let ${\mathbf{G}}$ be the Zariski closure of $\rho(\Gamma)$ which is assumed
to be Zariski connected. Then there exists a finite subset $F\subset\Gamma$
with $\\#F\geqslant 3,$ constants $C_{1},C_{2},L_{0}>0$ and
$m\in{\mathbb{Z}}_{+}$ such that the following holds.

For every subset $S\subset A(L)$ for some $L\geqslant L_{0}$ there exists a
subset $S^{\prime}\subset S$ with $\\#S^{\prime}\geqslant C_{1}^{-1}\\#S$ and
$F^{\prime}\subset F$ with $\\#F^{\prime}=\\#F-2$ satisfying

  1. (1)

$\left\\{\rho(f)^{m}:f\in F^{\prime}\right\\}$ generates a semigroup whose
Zarski closure is ${\mathbf{G}}.$

  2. (2)

$\widetilde{S}\coloneqq\left\\{sf^{\varsigma}:s\in S^{\prime},f\in
F^{\prime},\varsigma=m,2m\right\\}\subset\Gamma$ freely generates a free
semigroup.

  3. (3)

For any sequence of elements
$\widetilde{s}_{1},\cdots,\widetilde{s}_{k}\in\widetilde{S},$ we have

$|\widetilde{s}_{1}\cdots\widetilde{s}_{k}|\geqslant\sum_{i=1}^{k}|\widetilde{s}_{i}|-kC_{2}.$
(3.1)

###  3.1 Preliminaries on geometric group theory

Recall that $\Gamma$ is a finitely generated group and $\mathcal{S}$ is a
fixed symmetric generating set. Let
$X=\operatorname{\mathrm{Cay}}(\Gamma,\mathcal{S})$ be the Cayley graph of
$\Gamma$ with respect to $\mathcal{S}.$ Endow $X$ with the graph metric
$d_{\mathcal{S}},$ which makes $X$ a proper geodesic space. We abbreviate
$d_{\mathcal{S}}$ to $d$ in this section. Then $\Gamma$ has a natural left
action on $(X,d)$ by isometries. Letting $o$ be the point in $X$ corresponding
to the identity in $\Gamma,$ we fix $o$ as the base point. Then the word
length $|g|$ is equal to $d(o,go).$

For every subset $Y\subset X$ and $r>0,$ we use $\mathcal{N}_{r}(Y)$ to denote
the $r$-neighborhood of $Y.$ For two subsets $Y_{1},Y_{2}\subset X,$ we use
$d_{\mathrm{H}}(Y_{1},Y_{2})$ to denote the Hausdorff distance between
$Y_{1},Y_{2}$ with respect to $d,$ given by

$d_{\mathrm{H}}(Y_{1},Y_{2})\coloneqq\inf\left\\{0<r\leqslant\infty:Y_{1}\subset\mathcal{N}_{r}(Y_{2})\text{
and }Y_{2}\subset\mathcal{N}_{r}(Y_{1})\right\\}.$

For a subset $Y\subset X$ and a point $x\in X,$ we denote $\pi_{Y}(x)$ to be
the projection of $x$ on $Y,$ that is $\pi_{Y}(x)\coloneqq\left\\{y\in
Y:d(x,y)=d(x,Y)\right\\}.$ For another subset $Z\subset X,$ the projection of
$Z$ on $Y$ is $\pi_{Y}(Z)\coloneqq\cup_{z\in Z}\pi_{Y}(z).$

For a rectifiable path $\alpha\subset X,$ we denote $\alpha_{-}$ and
$\alpha_{+}$ to be the initial and terminal points of $\alpha,$ respectively.
The length of $\alpha$ is denoted by $\ell(\alpha).$ For every pair $x,y\in
X,$ we denote $[x,y]$ to be a choice of a geodesic between $x$ and $y.$

A path $\alpha$ is called a $c$-quasi-geodesic for $c\geqslant 1$ if

$\ell(\beta)\leqslant c\cdot d(\beta_{-},\beta_{+})+c$

for every rectifiable subpath $\beta\subset\alpha.$ Morse lemma states that
every $c$-quasi-geodesic $\alpha$ is contained in the
$c^{\prime}$-neighborhood of $[\alpha_{-},\alpha_{+}]$ in a
$\delta$-hyperbolic space, where $c^{\prime}$ only depends on $c$ and
$\delta.$ A subset $Y\subset X$ is called $c$-quasi-convex for $c\geqslant 0$
if for every $x_{1},x_{2}\in Y,$ we have
$[x_{1},x_{2}]\subset\mathcal{N}_{c}(Y).$ A quasi-geodesic can also be
interpreted as a quasi-convex subset.

Recall that $\Gamma$ is a ($\delta$-)hyperbolic group if and only if $X$ is a
($\delta$-)hyperbolic space. That is, for every geodesic triangle in $X,$
every edge is contained in the $\delta$-neighborhood of the other two edges.
In the following of this section, we always assume that $\Gamma$ is a finitely
generated $\delta$-hyperbolic group. Every infinite order element $g\in\Gamma$
is loxodromic in the following sence

$\lim_{n\to+\infty}\frac{1}{n}|g^{n}|=\lim_{n\to+\infty}\frac{1}{n}d(o,g^{n}o)>0.$

In fact, for every loxodromic element $g,$ the map $n\mapsto g^{n}o$ is a
quasi-isometric embedding from ${\mathbb{Z}}$ to $(X,d).$ That is, there
exists $c_{1},c_{2},c_{3},c_{4}>0$ such that for every $m,n\in{\mathbb{Z}},$

$c_{1}|m-n|-c_{2}\leqslant d(g^{m}o,g^{n}o)\leqslant c_{3}|m-n|+c_{4}.$

For every loxodromic element $g\in\Gamma,$ the set

$E(g)\coloneqq\left\\{h\in\Gamma:d_{\mathrm{H}}(h\left\langle g\right\rangle
o,\left\langle g\right\rangle o)<\infty\right\\}$

is a subgroup of $\Gamma$ satisfying $[E(g):\left\langle
g\right\rangle]<\infty.$ We use $\operatorname{\mathrm{Ax}}(g)=E(g)o\subset X$
to denote the axis corresponding to $g,$ which is a quasi convex subset. We
also remark that if $g_{1}\in E(g_{2})$ is also loxodromic then
$E(g_{1})=E(g_{2}).$ Two loxodromic elements $g_{1},g_{2}$ are called
independent222This definition of independence is different with the one in
[Yan19]. But this is enough for our applications for hyperbolic groups.  if
$E(g_{1})\neq E(g_{2}),$ namely $E(g_{1})\cap E(g_{2})$ is finite. Moreover,
we have the following bounded intersection property.

######  Lemma 3.2.

Let
$\operatorname{\mathrm{Ax}}(g_{1}),\operatorname{\mathrm{Ax}}(g_{2})\subset X$
be different axes, then for every $r>0,$
$\mathcal{N}_{r}(\operatorname{\mathrm{Ax}}(g_{1}))\cap\mathcal{N}_{r}(\operatorname{\mathrm{Ax}}(g_{2}))$
is bounded.

###### Proof.

Since $d_{\mathrm{H}}(\operatorname{\mathrm{Ax}}(g_{i}),\left\langle
g_{i}\right\rangle o)<\infty,$ it suffices to show that
$\mathcal{N}_{r}(\left\langle g_{1}\right\rangle
o))\cap\mathcal{N}_{r}(\left\langle g_{2}\right\rangle o)$ is bounded.
Otherwise, there exists infinitely many pairs of integers $(n_{1},n_{2})$ such
that $d(g_{1}^{n_{1}}o,g_{2}^{n_{2}}o)\leqslant 2r.$ Then there exists
$(n_{1},n_{2})\neq(n_{1}^{\prime},n_{2}^{\prime})$ such that

$g_{2}^{-n_{2}}g_{1}^{n_{1}}=g_{2}^{-n_{2}^{\prime}}g_{1}^{n_{1}^{\prime}}.$

This implies that $g_{1}^{n_{1}^{\prime}-n_{1}}=g_{2}^{n_{2}^{\prime}-n_{2}}$,
which contradicts
$\operatorname{\mathrm{Ax}}(g_{1})\neq\operatorname{\mathrm{Ax}}(g_{2}).$ ∎

In the case of $\Gamma$ is torsion-free, every nontrivial element is
loxodromic. By a classification of virtually cyclic group [Hem04, Lemma 11.4],
$E(g)$ is cyclic for every nontrivial element $g\in\Gamma.$ Moreover, $g_{1}$
and $g_{2}$ are independent if and only if $E(g_{1})\cap
E(g_{2})=\left\\{\mathrm{id}\right\\}.$

###  3.2 An extension lemma

Now we give the main technical tool for showing Proposition 3.1. It is a
variant of [Yan19, Lemma 2.19], which states more generally for the group with
contracting elements. A direct proof for the case of hyperbolic groups is
given in this section, which is also inspired by the work of W. Yang.

For a subset $S\subset\Gamma,$ we say $S$ is $R$-separated for $R>0$ if
$\left\\{so:s\in S\right\\}$ is $R$-separated in $X.$

######  Proposition 3.3.

Let $\Gamma$ be a non-elementary hyperbolic group. Let $F\subset\Gamma$ be a
finite subset of pairwise independent loxodromic elements with $\\#F\geqslant
3.$ Then there exist $m_{0},R,C_{1},L_{0}>0$ such that the following holds.

For every $R$-separated subset $S\subset A(L)$ for some $L\geqslant L_{0}$
there exists a subset $S^{\prime}\subset S$ with $\\#S^{\prime}\geqslant
C_{1}^{-1}\\#S$ and $F^{\prime}\subset F$ with $\\#F^{\prime}=\\#F-2$ such
that for every $m\geqslant m_{0},$

$\widetilde{S}\coloneqq\left\\{sf^{\varsigma}:s\in S^{\prime},f\in
F^{\prime},\varsigma=m,2m\right\\}\subset\Gamma$

freely generates a free semigroup. Furthermore, there exists $C_{2}>0$ only
depends on $F$ such that for any sequence of elements
$\widetilde{s}_{1},\cdots,\widetilde{s}_{k}\in\widetilde{S},$ we have

$|\widetilde{s}_{1}\cdots\widetilde{s}_{k}|\geqslant\sum_{i=1}^{k}|\widetilde{s}_{i}|-kC_{2}.$
(3.2)

Before going through the proof, we need some preparation on hyperbolic groups.
Recall that $X$ is a $\delta$-hyperbolic space. Let $\left\langle
x,y\right\rangle_{z}=(d(x,z)+d(y,z)-d(x,y))/2$ be the Gromov product, then
$d(z,[x,y])-10\delta\leqslant\left\langle x,y\right\rangle_{z}\leqslant
d(z,[x,y]).$

######  Definition 3.4.

A $(\tau,D)$-chain is a sequence of points $x_{0},x_{1},\cdots,x_{n}\in X$
such that

  * •

$\left\langle x_{i-1},x_{i+1}\right\rangle_{x_{i}}\leqslant\tau$ for every
$1\leqslant i\leqslant n-1,$ and

  * •

$d(x_{i-1},x_{i})\geqslant D$ for every $1\leqslant i\leqslant n.$

By connecting two consecutive points in a $(\tau,D),$ we obtain a quasi-
geodesic. The following lemma shows such path is indeed a uniform quasi-
geodesic. See also [Gou22, Section 3B].

######  Lemma 3.5.

For every $\tau>0,$ there exists $D=D(\tau),c=c(\tau)>0$ such that for every
$(\tau,D)$-chain $x_{0},x_{1},\cdots,x_{n},$ the path
$\bigcup_{i=1}^{n}[x_{i-1},x_{i}]$ is a $c$-quasi-geodesic.

The following it a bounded projection property for different axes. A general
version for spaces with contracting elements can be found in [Yan19, Lemma
2.17].

######  Lemma 3.6.

Let $f_{1},f_{2}\in\Gamma$ be independent loxodromic elements. There exists
$\tau_{1}>0$ such that for every $g\in\Gamma,$ we have

$\min\left\\{d(o,\pi_{\operatorname{\mathrm{Ax}}(f_{1})}(go)),d(o,\pi_{\operatorname{\mathrm{Ax}}(f_{2})}(go))\right\\}\leqslant\tau_{1}.$

###### Proof.

Let $z_{i}\in\pi_{\operatorname{\mathrm{Ax}}(f_{i})}(go),$ we first show that
$[o,z_{i}][z_{i},go]$ is a $c_{1}$-quasi-geodesic for some $c_{1}>0$ only
depends on $\operatorname{\mathrm{Ax}}(f_{1})$ and
$\operatorname{\mathrm{Ax}}(f_{2}).$ Note that there exists $c_{2}>0$ such
that $\operatorname{\mathrm{Ax}}(f_{i})$ is $c_{2}$-quasi-convex. Fixing an
$i\in\left\\{1,2\right\\},$ for every $x\in[o,z_{i}]$ and $y\in[z_{i},go],$ we
have

$d(y,x)\geqslant
d(y,\operatorname{\mathrm{Ax}}(f_{i}))-c_{2}=d(y,z_{i})-c_{2}.$

Hence $d(x,z_{i})+d(z_{i},y)\leqslant d(x,y)+2d(z_{i},y)\leqslant
3d(x,y)+2c_{2}$ is a $(3+2c_{2})$-quasi-geodesic.

By Morse lemma, $[o,z_{i}]\subset\mathcal{N}_{c}([o,go])$ where
$c=c(c_{1})>0.$ If both $d(z_{1},o)$ and $d(z_{2},o)$ are larger than
$\tau_{1},$ we can choose $z_{i}^{\prime}\in[o,z_{i}]$ with
$d(o,z_{i}^{\prime})=\tau_{1}.$ Let $w_{i}\in[o,go]$ with
$d(w_{i},z_{i}^{\prime})\leqslant c,$ then $|d(w_{i},o)-\tau_{1}|\leqslant c.$
Hence
$w_{1},w_{2}\in\mathcal{N}_{3c}(\operatorname{\mathrm{Ax}}(f_{1}))\cap\mathcal{N}_{3c}(\operatorname{\mathrm{Ax}}(f_{2})).$
Since $d(w_{1},o)\geqslant\tau_{1}-c,$ this contradicts Lemma 3.2 for a
sufficiently large $\tau_{1}.$ ∎

######  Definition 3.7.

Let $g\in\Gamma$ and $\tau>0,$ we call a loxodromic element $f\in\Gamma$ is
$\tau$-contracting for $g$ if for every $h\in E(f),$ we have $\left\langle
go,ho\right\rangle_{o}\leqslant\tau$ and $\left\langle
g^{-1}o,ho\right\rangle_{o}\leqslant\tau.$

######  Lemma 3.8.

Let $F\subset\Gamma$ be a finite subset of pairwise independent loxodromic
elements with $\\#F\geqslant 3.$ There exists $\tau>0$ such that for every
$g\in\Gamma,$ there exists $F_{g}\subset F$ with $\\#F_{g}=\\#F-2$ such that
every element in $F_{g}$ is $\tau$-contracting for $g.$

###### Proof.

By the previous lemma, there exists $\tau_{1}>0$ such that for every
$f_{1},f_{2}\in F$ and $g\in\Gamma,$ we have
$\min\left\\{d(o,\pi_{\operatorname{\mathrm{Ax}}(f_{1})}(go)),d(o,\pi_{\operatorname{\mathrm{Ax}}(f_{2})}(go))\right\\}\leqslant\tau_{1}.$
Note that there exists $c_{1}>0$ such that for every $f\in F,$
$\operatorname{\mathrm{Ax}}(f)$ is $c_{1}$-quasi-convex. We take
$\tau=\tau_{1}+c_{1}+10\delta.$

For each $g\in\Gamma$ and $f\in F$ satisfying
$d(o,\pi_{\operatorname{\mathrm{Ax}}(f)}(go))\leqslant\tau_{1}.$ For every
$h\in E(f),$ we have
$[o,ho]\in\mathcal{N}_{c_{1}}(\operatorname{\mathrm{Ax}}(f)).$ Hence

$d(go,[o,ho])\geqslant d(go,\operatorname{\mathrm{Ax}}(f))-c_{1}\geqslant
d(go,o)-d(o,\pi_{\operatorname{\mathrm{Ax}}(f)}(go))-c_{1}\geqslant
d(o,go)-\tau_{1}-c_{1}.$

Then $\left\langle o,ho\right\rangle_{go}\geqslant
d(go,[o,ho])-10\delta\geqslant d(o,go)-\tau_{1}-c_{1}-10\delta=d(o,go)-\tau.$
We obtain

$\left\langle go,ho\right\rangle_{o}=d(o,go)-\left\langle
o,ho\right\rangle_{go}\leqslant\tau.$

To complete the proof, we apply the argument to both $g$ and $g^{-1}.$ By the
previous lemma, there are at least $\\#F-2$ elements in $F$ which are
$\tau$-contracting for $g.$ ∎

Now we are at the stage of proving Proposition 3.3.

###### Proof of Proposition 3.3..

We apply the previous lemma to $F$ and obtain a constant $\tau>0.$ Let
$C_{1}=\\#F(\\#F-1)/2$. By the pigeonhole principle, there exists
$F^{\prime}\subset F$ with $\\#F^{\prime}=\\#F-2$ such that

$S^{\prime}=\left\\{s\in S:F_{s}=F^{\prime}\right\\}$

has the cardinality at least $C_{1}^{-1}\\#S,$ where $F_{s}\subset F$ is the
subset given by the previous lemma consisting of $\tau$-contracting elements.

By Lemma 3.5, there exists $D=D(\tau)>0$ and $c_{1}=c_{1}(\tau)>0$ such that
every $(\tau,D)$-chain forms a $c_{1}$-quasi-geodesic. By Morse Lemma, every
$c$-quasi-geodesic $\alpha$ is contained in
$\mathcal{N}_{C}([\alpha_{-},\alpha_{+}])$ for some $C>0.$ Now we take
$C_{2}=2C,R=4C+1,$ $L_{0}=D+4C.$ The choice of $m_{0}$ will be given later.
Let $c_{2}>0$ such that every axis $\operatorname{\mathrm{Ax}}(f)$ is
$c_{2}$-quasi-convex for $f\in F.$ Now we show that $\widetilde{S}$ freely
generates a free semigroup.

Let $k\geqslant 1$ and $s_{i}\in S^{\prime},f_{i}\in
F^{\prime},\varsigma_{i}\in\left\\{m,2m\right\\}$ for $1\leqslant i\leqslant
k.$ We consider

$x_{i}=s_{1}f_{1}^{\varsigma_{1}}\cdots s_{i}f_{i}^{\varsigma_{i}}o,\quad
y_{i}=s_{1}f_{1}^{\varsigma_{1}}\cdots s_{i}o.$

We claim that $x_{0},y_{1},x_{1},\cdots,y_{k},x_{k}$ is a $(\tau,D)$-chain.
Assume that $m_{0}$ is large enough guaranteeing $|f^{m}|>D$ for every $f\in
F$ and $m\geqslant m_{0}.$ Since $|s_{i}|=L\geqslant D+4C$ and
$|f_{i}^{\varsigma_{i}}|\geqslant D$ for each $i,$ the second condition in
Definition 3.4 is verified. Besides, for each $i,$ we have

$\left\langle x_{i-1},x_{i}\right\rangle_{y_{i}}=\left\langle
s_{i}^{-1}o,f_{i}^{\varsigma_{i}}o\right\rangle_{o}\leqslant\tau,\quad\left\langle
y_{i},y_{i+1}\right\rangle_{x_{i}}=\left\langle
f_{i}^{-\varsigma_{i}}o,s_{i+1}o\right\rangle_{o}\leqslant\tau$

by the $\tau$-contracting property.

Hence $\alpha=[x_{0},y_{1}][y_{1},x_{1}]\cdots[x_{k-1},y_{k}][y_{k},x_{k}]$ is
a $c_{1}$-quasi geodesic, which is contained in the $C$-neighborhood of
$[\alpha_{-},\alpha_{+}]=[o,x_{k}].$ Then we have two estimates on the length
of $d(o,x_{k}).$ Firstly, since $|s_{i}|\geqslant D+4C$ and
$|f_{i}^{\varsigma_{i}}|\geqslant D,$ we have

$d(o,x_{k})\geqslant\sum_{i=1}^{k}(d(x_{i-1},y_{i})+d(y_{i},x_{i}))-4kC\geqslant
k(2D+2C)-4kC>0.$ (3.3)

This implies that $x_{k}\neq o.$ Besides, we have

$d(o,x_{k})\geqslant\sum_{i=1}^{k}d(x_{i-1},x_{i})-2kC=\sum_{i=1}^{k}d(x_{i-1},x_{i})-kC_{2},$

which gives the desired estimate (3.2).

#### Checking the freeness.

We consider two such sequences $s_{i},f_{i},\varsigma_{i}$ for $1\leqslant
i\leqslant k$ and $s_{j}^{\prime},f_{j}^{\prime},\varsigma_{j}^{\prime}$ for
$1\leqslant j\leqslant\ell.$ We get two sequences of points $x_{i},y_{i}$ and
$x_{j}^{\prime},y_{j}^{\prime}$ in $X.$ Assume that $x_{k}=x_{\ell}^{\prime},$
we are going to show that $k=\ell$ and
$s_{i}=s_{i}^{\prime},f_{i}=f_{i}^{\prime},$
$\varsigma_{i}=\varsigma_{i}^{\prime}.$ By an inductive argument, it suffices
to check for $i=1.$ Let $\alpha$ be a fixed geodesic connecting $o$ and
$x_{k}=x_{\ell}^{\prime}.$ Then there exist $z_{1},z_{1}^{\prime}\in\alpha$
such that $d(y_{1},z_{1}),d(y_{1}^{\prime},z_{1}^{\prime})\leqslant C.$ Since
$d(y_{1},o)=d(y_{1}^{\prime},o)=L,$ we have
$d(z_{1},z_{1}^{\prime})=|d(o,z_{1})-d(o,z_{1}^{\prime})|\leqslant 2C.$ This
implies that $d(y_{1},y_{1}^{\prime})\leqslant 4C.$ By the $R$-separation,
$y_{1}=y_{1}^{\prime}.$

Now we consider a geodesic $\beta$ connecting $y_{1}=y_{1}^{\prime}$ and
$x_{k}=x_{\ell}^{\prime}.$ Without loss of generality, we assume that
$d(y_{1},x_{1})\leqslant d(y_{1}^{\prime},x_{1}^{\prime}).$ Take a point
$w\in\pi_{\beta}(x_{1})$ then $d(x_{1},w)\leqslant C.$ Letting
$x^{\prime}\in[y_{1}^{\prime},x_{1}^{\prime}]$ with
$d(y_{1}^{\prime},x_{1}^{\prime})=d(x_{1},y_{1}),$ we have
$d(x^{\prime},w)\leqslant 3C$ since
$[y_{1}^{\prime},x_{1}^{\prime}]\in\mathcal{N}_{C}(\beta).$ This implies
$w\in\mathcal{N}_{3C}([y_{1},x_{1}])\cap\mathcal{N}_{3C}([y_{1}^{\prime},x_{1}^{\prime}]).$
By the quasi-convexity of axes, we have $[y_{1},x_{1}]\subset
s_{1}\mathcal{N}_{c_{2}}(\operatorname{\mathrm{Ax}}(f_{1}))$ and
$[y_{1}^{\prime},x_{1}^{\prime}]\subset
s_{1}\mathcal{N}_{c_{2}}(\operatorname{\mathrm{Ax}}(f_{1}^{\prime})).$
Therefore, both $y_{1}$ and $w$ are contained in
$s_{1}(\mathcal{N}_{3C+c_{2}}(\operatorname{\mathrm{Ax}}(f_{1}))\cap\mathcal{N}_{3C+c_{2}}(\operatorname{\mathrm{Ax}}(f_{1}^{\prime}))).$
Note that $d(y_{1},w)\geqslant
d(y_{1},x_{1})-d(x_{1},w)\geqslant|f_{1}^{\varsigma_{1}}|-C,$ we conclude that

$\operatorname{\mathrm{diam}}\mathcal{N}_{3C+c_{2}}(\operatorname{\mathrm{Ax}}(f_{1}))\cap\mathcal{N}_{3C+c_{2}}(\operatorname{\mathrm{Ax}}(f_{1}^{\prime}))\geqslant
d(y_{1},w)\geqslant|f_{1}^{\varsigma_{1}}|-C.$

By Lemma 3.2, the $(3C+c_{2})$-neighborhoods of axes of different elements in
$F$ have uniformly bounded intersections. We take $m_{0}$ sufficiently large
at beginning such that $|f^{m}|-C$ is strictly larger than diameters of all
such intersections for every $f\in F,m\geqslant m_{0}$, this forces
$f_{1}=f_{1}^{\prime}.$

Finally, we should show $\varsigma_{1}=\varsigma_{1}^{\prime}.$ Otherwise, we
assume that $\varsigma_{1}=m$ and $\varsigma_{1}^{\prime}=2m.$ In this case,
we have

$y_{2}=s_{1}f_{1}^{m}s_{2}o,\quad x_{1}=s_{1}f_{1}^{m}o,\quad
x_{1}^{\prime}=s_{1}f_{1}^{2m}o,\quad
y_{2}^{\prime}=s_{1}f_{1}^{2m}s_{2}^{\prime}o.$

By the $\tau$-contracting property, these points form a $(\tau,D)$-chain. This
leads to that

$x_{k},y_{k},\cdots,y_{2},x_{1},x_{1}^{\prime},y_{2}^{\prime},\cdots,y_{\ell}^{\prime},x_{\ell}^{\prime}.$

is also a $(\tau,D)$-chain. Applying the estimate in (3.3), this contradicts
$x_{k}=x_{\ell}^{\prime}.$ ∎

###  3.3 Proof of Proposition 3.1.

To prove Proposition 3.1, we need to find a finite subset consisting of
pairwise independent loxodromic elements satisfying the desired condition on
Zariski closures.

######  Lemma 3.9.

Let $\Gamma$ be a non-elementary torsion-free hyperbolic group and
$\rho:\Gamma\to\mathrm{SL}_{n}({\mathbb{R}})$ be a faithful representation.
Let ${\mathbf{G}}$ be the Zariski closure of $\rho(\Gamma)$ and assume
${\mathbf{G}}$ is Zariski connected. For every $k\geqslant 0,$ there exists a
finite subset $F\subset\Gamma$ consisting of pairwise independent nontrivial
elements such that for every subset $F^{\prime}\subset F$ with
$\\#F^{\prime}\geqslant F-k,$ $\rho(F^{\prime})$ generates a semigroup whose
Zarski closure is ${\mathbf{G}}.$

###### Proof.

We will find $F=F_{k}$ for each $k\geqslant 0$ by an induction on $k.$

For the case of $k=0,$ applying [MS23, Lemma 3.6], we can find a finite subset
$F_{0}\subset\Gamma$ such that $\rho(F_{0})$ generates a semigroup whose
Zariski closure is ${\mathbf{G}}.$ Furthermore, we can assume that elements in
$F_{0}$ are pairwise independent by the following process. If $f_{1},f_{2}\in
F_{0}$ are not independent, then there exists $f\in\Gamma$ and
$m_{1},m_{2}\in{\mathbb{Z}}$ such that $f_{i}=f^{m_{i}}$ since $\Gamma$ is
torsion free. We can remove $f_{1},f_{2}$ and add $f$ in $F_{0}.$

Assume that $F_{k-1}$ is found. Recall that for every $f\in\Gamma,$ $E(f)$ is
a cyclic group containing all elements which are not independent of $f.$
Notice that Zariski closure of $\rho(E(f))$ is commutative. On the other hand,
applying Tits alternative for hyperbolic groups, $\Gamma$ contains a
nonabelian free subgroup. This implies that $\rho(E(f))$ is not Zariski dense
in ${\mathbf{G}}.$ By the connectivity of ${\mathbf{G}},$ the set
$S=\Gamma\setminus\bigcup_{f\in F_{k-1}}E(f)$ is Zariski dense in
${\mathbf{G}}.$ Applying [MS23, Lemma 3.6] and the argument for $k=0$ case, we
can find a finite subset $\widetilde{F}\subset S$ consisting of pairwise
independent elements such that $\rho(\widetilde{F})$ generates a semigroup
whose Zariski closure is ${\mathbf{G}}.$ We take
$F_{k}=F_{k-1}\cup\widetilde{F},$ which gives a desired construction for the
$k$-case. ∎

###### Proof of 3.1.

Applying Lemma 3.9 to the case of $k=2,$ we can find a finite subset
$F\subset\Gamma$ consisting of pairwise independent elements such that for
every $F^{\prime}\subset F$ with $\\#F^{\prime}=\\#F-2,$ $\rho(F^{\prime})$
generates a semigroup whose Zariski closure is ${\mathbf{G}}.$ We apply
Proposition 3.3 to $F.$ Let $m_{0}$ be the constant given by this proposition.
In order to find $m\geqslant m_{0}$ satisfying the first condition in the
proposition, we need the following lemma.

######  Lemma 3.10.

For every $g\in\mathrm{SL}_{n}(\mathbb{R}),$ there exists a positive integer
$\ell$ such that for every $m$ coprime with $\ell,$ the Zariski closure of
$\left\langle g^{m}\right\rangle$ is equal to the Zariski closure of
$\left\langle g\right\rangle.$

###### Proof.

Let ${\mathbf{H}}$ be the Zariski closure of $\left\langle g\right\rangle.$
Let $\ell$ be the number of connected components of ${\mathbf{H}}.$ Let
${\mathbf{H}}^{\prime}$ be the Zariski closure of $\left\langle
g^{m}\right\rangle$ where $m$ is coprime with $\ell.$ Then
${\mathbf{H}}^{\prime}$ is a finite index algebraic subgroup of ${\mathbf{H}}$
and hence a union of some connected components of ${\mathbf{H}}.$ Note that
the action of $\left\langle g\right\rangle$ given by left translation is
transitive among connected components of ${\mathbf{H}}$ and so does
$\left\langle g^{m}\right\rangle$, due to $m$ coprime with $\ell$. Hence
${\mathbf{H}}^{\prime}={\mathbf{H}}.$ ∎

We take $m\geqslant m_{0}$ to be a sufficiently large prime number such that
for every $f\in F,$ the Zariski closure of
$\left\langle\rho(f)^{m}\right\rangle$ equals to the Zariski closure of
$\left\langle\rho(f)\right\rangle.$ Then for every $F^{\prime}\subset F$ with
$\\#F^{\prime}=\\#F-2,$ the Zariski closure of the semigroup generated by
$\left\\{\rho(f)^{m}:f\in F^{\prime}\right\\}$ equals to that of $\rho(F),$
which is ${\mathbf{G}}.$ The first condition holds.

Note that for every finite subset $S\subset\Gamma,$ it contains an
$R$-separated subset of cardinality at least $(\\#\mathcal{S}+1)^{-R}\\#S,$
where $\mathcal{S}$ is the symmetric generating set of $\Gamma.$ Then the last
two conditions follow from Proposition 3.3 by enlarging $C_{1}$ suitably. ∎

##  4 Variational principle for Anosov representations

In this section, we will show a variational principle for dimensions of limit
sets of Borel Anosov representations. Let $\Gamma$ be a finitely generated
hyperbolic group with a fixed finite symmetric generating set
$\mathcal{S}\subset\Gamma.$ Let $\rho:\Gamma\to\mathrm{SL}_{n}({\mathbb{R}})$
be an irreducible Borel Anosov representation. Recall that
$\kappa(\rho(\gamma))=\mathrm{diag}(\log\sigma_{1}(\rho(\gamma)),\cdots,\log\sigma_{n}(\rho(\gamma)))$
is the Cartan projection of $\rho(\gamma).$

###  4.1 The variational principle to positive linear functions on
$\mathfrak{a}^{+}$

Recall the Cartan algebra
$\mathfrak{a}=\\{\lambda=\mathrm{diag}(\lambda_{1},\cdots,\lambda_{n}):\lambda_{i}\in\mathbb{R},\
\sum\lambda_{i}=0\\}$ and a positive Weyl chamber $\mathfrak{a}^{+}$ mentioned
in Section 2.1. Let $\psi$ be a linear function on $\mathfrak{a}$ which is
positive with respect to $\mathfrak{a}^{+}.$ Specifically, $\psi$ can be
expressed as

$\psi(\lambda)=\sum_{i=1}^{n-1}a_{i}\cdot\alpha_{i}(\lambda),$

where $a_{1},\cdots,a_{n-1}\geqslant 0$ are not all zero and
$\alpha_{i}(\lambda)=\lambda_{i}-\lambda_{i+1}$ are simple roots.

Recall that for a finitely supported probability measure $\nu$ on
$\mathrm{SL}_{n}({\mathbb{R}}),$ the Lyapunov spectrum of $\nu$ is
$\lambda(\nu)=(\lambda_{1}(\nu),\cdots,\lambda_{n}(\nu)).$ We also view
$\lambda(\nu)$ as an element in $\mathfrak{a}^{+}$ using the isomorphism
$\mathfrak{a}\cong{\mathbb{R}}^{n}$ and $\psi$ acts on $\lambda(\nu).$ Then
$\psi(\lambda(\nu))$ is a nonnegative number.

Throughout this subsection, we further assume that $\Gamma$ is torsion-free
and $\rho$ is faithful. Note that $\Gamma$ is non-cyclic and hence non-
elementary since $\rho$ is irreducible.

######  Proposition 4.1.

Let ${\mathbf{G}}$ be the Zariski closure of $\rho(\Gamma)$ which is assumed
to be Zariski connected. Let $\psi$ be a linear function as above. If the
series $\sum_{\gamma\in\Gamma}\exp(-\psi(\kappa(\rho(\gamma))))$ diverges,
then there exists $c>0$ such that the following holds. For every $\epsilon>0,$
there exists infinitely many positive integers $N$ with a finitely supported
probability measure $\nu$ on $\rho(\Gamma)$ such that

  * •

$G_{\nu}$ is Zariski dense in ${\mathbf{G}}.$

  * •

$\lambda_{p}(\nu)-\lambda_{p+1}(\nu)\geqslant cN$ for every $p=1,\cdots,n-1.$

  * •

$h_{\mathrm{RW}}(\nu)\geqslant(1-\epsilon)N$ and
$\psi(\lambda(\nu))\leqslant(1+\epsilon)N.$

We first recall an estimate on the lost of singular values under composition.
The following lemma is a direct consequence of combining Lemmas 2.5 and A.7 in
[BPS19].

######  Lemma 4.2.

Let $1\leqslant p\leqslant n.$ Given $c>0,$ then there exists $\delta>0$ such
that the following holds. Let
$(\gamma_{k})_{k\in{\mathbb{N}}}\in\mathcal{S}^{\mathbb{N}}$ satisfying for
every $\ell\leqslant m,$

$\frac{\sigma_{p}(\rho(\gamma_{\ell+1}\cdots\gamma_{m}))}{\sigma_{p+1}(\rho(\gamma_{\ell+1}\cdots\gamma_{m}))}\geqslant
c\cdot e^{c(m-\ell)}.$

Then for every $\ell\leqslant k\leqslant m,$ we have

$\sigma_{p}(\rho(\gamma_{\ell+1}\cdots\gamma_{m}))\geqslant\delta\cdot\sigma_{p}(\rho(\gamma_{\ell+1}\cdots\gamma_{k}))\sigma_{p}(\rho(\gamma_{k+1}\cdots\gamma_{m})),$
$\sigma_{p+1}(\rho(\gamma_{\ell+1}\cdots\gamma_{m}))\leqslant\delta^{-1}\cdot\sigma_{p+1}(\rho(\gamma_{\ell+1}\cdots\gamma_{k}))\sigma_{p+1}(\rho(\gamma_{k+1}\cdots\gamma_{m})).$

###### Proof of Proposition 4.1.

Applying Proposition 3.1, we obtain a finite subset $F\subset\Gamma$ with
$\\#F\geqslant 3,$ constants $C_{1},C_{2},L_{0}>0$ and a positive integer $m.$
For every $\epsilon>0$ sufficiently small, there are infinitely many integers
$N$ such that

$S_{1}=\left\\{\gamma\in\Gamma:\psi(\kappa(\rho(\gamma)))\leqslant N\right\\}$

has cardinality at least $e^{(1-\epsilon)N}$ due to the divergence of the
series. Since $\psi$ is positive, we have $\psi(\kappa(\rho(\gamma)))\geqslant
c_{1}(\log\sigma_{p}(\rho(\gamma))-\log\sigma_{p+1}(\rho(\gamma)))$ for some
$1\leqslant p\leqslant n-1$ and $c_{1}>0.$ Because $\rho$ is Borel Anosov,
there exists $c_{2}>0$ such that for $|\gamma|$ large enough,

$\psi(\kappa(\rho(\gamma)))\geqslant
c_{1}(\log\sigma_{p}(\rho(\gamma))-\log\sigma_{p+1}(\rho(\gamma)))\geqslant
c_{2}|\gamma|.$

Hence $S_{1}\subset\left\\{\gamma\in\Gamma:|\gamma|\leqslant
c_{2}^{-1}N\right\\}.$ Let $c_{3}=(2\\#\log\mathcal{S})^{-1},$ then

$\\#\left\\{\gamma\in\Gamma:|\gamma|\leqslant c_{3}N\right\\}\leqslant
2(\\#\mathcal{S})^{c_{3}N}\leqslant\frac{1}{2}e^{(1-\epsilon)N}$

providing $\epsilon$ small and $N$ large. Hence there exists $c_{3}N\leqslant
L\leqslant c_{2}^{-1}N$ such that

$S_{2}\coloneqq\left\\{\gamma\in S_{1}:|\gamma|=L\right\\}$

has cardinality at least $c_{2}(2N)^{-1}e^{(1-\epsilon)N}\geqslant
e^{(1-2\epsilon)N}$ assuming $N$ large. By Proposition 3.1 , there exists
$S_{3}\subset S_{2}$ and $F^{\prime}\subset F$ with $\\#S_{3}\geqslant
C_{1}^{-1}\\#S_{2}$ and $\\#F^{\prime}=\\#F-2$ such that

$\widetilde{S}\coloneqq\left\\{sf^{\varsigma}:s\in S_{3},f\in
F^{\prime},\varsigma=m,2m\right\\}$

freely generates a free semigroup. Assuming $N$ large, we have
$\\#\widetilde{S}\geqslant\\#S_{3}\geqslant e^{(1-3\epsilon)N}.$ Letting $\nu$
be the uniform measure on $\rho(\widetilde{S}),$ we now verify that this is a
desired construction.

  * •

Note that $\rho(f)^{m}=\rho(sf^{m})^{-1}\rho(sf^{2m})\in G_{\nu}$ for every
$f\in F^{\prime}.$ By the first condition in 3.1, we have $G_{\nu}$ is Zariski
dense in ${\mathbf{G}}.$

  * •

By the definition of Borel Anosov representations, we can take $c_{4}>0$ such
that for every $1\leqslant p\leqslant n-1$ and $|\gamma|$ large enough,
$\log\sigma_{p}(\rho(\gamma))-\log\sigma_{p+1}(\rho(\gamma))\geqslant
c_{4}|\gamma|$. Assuming $N$ is large enough only depends on $m,$ for every
$\widetilde{s}\in\widetilde{S}$ we have

$|\widetilde{s}|\geqslant c_{3}N-2m\max_{f\in F}|f|\geqslant
c_{3}^{\prime}N+C_{2}\quad\text{and}\quad|\widetilde{s}|\leqslant
c_{2}^{-1}N+2m\max_{f\in F}|f|\leqslant c_{2}^{\prime-1}N$

where $c_{2}^{\prime}=c_{2}/2,c_{3}^{\prime}=c_{3}/2$ and $C_{2}$ is the
constant given by Proposition 3.1. Then for every
$\widetilde{s}_{1},\cdots\widetilde{s}_{k}\in\widetilde{S},$ due to (3.1), we
have $|\widetilde{s}_{1}\cdots\widetilde{s}_{k}|\geqslant c_{3}^{\prime}kN.$
Hence for every $1\leqslant p\leqslant n-1,$ we have

$(\log\sigma_{p}-\log\sigma_{p+1})(\rho(\widetilde{s}_{1}\cdots\widetilde{s}_{k}))\geqslant
c_{4}c_{3}^{\prime}kN\geqslant
c_{2}^{\prime}c_{3}^{\prime}c_{4}\sum_{i=1}^{k}|\widetilde{s}_{i}|.$ (4.1)

Recall that $\operatorname{\mathrm{supp}}\nu=\widetilde{S}$. By the first
inequality in (4.1) we have

$\lambda_{p}(\nu)-\lambda_{p+1}(\nu)=\lim_{k\to\infty}\frac{1}{k}\int[\log\sigma_{p}(\rho(\gamma))-\log\sigma_{p+1}(\rho(\gamma))]\mathrm{d}\nu^{*k}(\gamma)\geqslant
c_{4}c_{3}^{\prime}N.$

Taking $c=c_{4}c_{3}^{\prime},$ we obtain the conclusion.

  * •

Since $\widetilde{S}$ freely generates a free semigroup and $\nu$ is the
uniform measure on $\widetilde{S},$ we have

$h_{\mathrm{RW}}(\nu)=\log\\#\widetilde{S}\geqslant(1-3\epsilon)N.$

Shrinking $\epsilon>0,$ we obtain the first estimate.

In order to estimate $\psi(\lambda(\nu)),$ we need an almost additivity
property of $\log\sigma_{p}.$ Recall the second inequality in (4.1) and the
constant cancellation property in (3.1), which verifies the condition of 4.2.
By applying Lemma 4.2, there exists $\delta>0$ only depending on $C_{2}$ and
$c_{2}^{\prime}c_{3}^{\prime}c_{4}$ such that

$\left|\log\sigma_{p}(\rho(\widetilde{s}_{1}\cdots\widetilde{s}_{k}))-\sum_{i=1}^{k}\log\sigma_{p}(\widetilde{s}_{i})\right|\leqslant-k\log\delta$

for every $\widetilde{s}_{1},\cdots,\widetilde{s}_{k}\in\widetilde{S}$ and
$1\leqslant p\leqslant n.$ Since $\psi$ is a linear function, we have

$\left|\psi(\kappa(\rho(\widetilde{s}_{1}\cdots\widetilde{s}_{k})))-\sum_{i=1}^{k}\psi(\kappa(\rho(\widetilde{s}_{i})))\right|\leqslant-
kC_{3}\log\delta,$

where $C_{3}>0$ only depends on $\psi.$ For each
$\widetilde{s}\in\widetilde{S},$ write $\widetilde{s}=sf^{\varsigma},$ where
$s\in S_{3}$ and $f\in F,\varsigma\in\left\\{m,2m\right\\}.$ Then there exists
$C_{4}>0$ only depending on $F$ and $m$ such that
$|\log\sigma_{p}(\rho\widetilde{s})-\log\sigma_{p}(\rho s)|\leqslant C_{4}$
for every $p.$ Hence $\psi(\kappa(\rho\widetilde{s}))\leqslant N+C_{3}C_{4}$
since $s\in S_{3}\subset S_{1}.$

Then for sufficiently large $N,$ we have

$\psi(\kappa(\rho(\widetilde{s}_{1}\cdots\widetilde{s}_{k})))\leqslant
kN+kC_{3}C_{4}-kC_{3}\log\delta\leqslant(1+\epsilon)kN.$

This implies that

$\psi(\lambda(\nu))=\lim_{k\to\infty}\frac{1}{k}\int\psi(\kappa(\rho(\gamma)))\mathrm{d}\nu^{*k}(\gamma)\leqslant(1+\epsilon)N.\qed$

###  4.2 The variational principle of critical exponent

This section is devoted to prove Theorem 1.1. We will also present a version
for flag varieties. In this section, we consider a Borel Anosov representation
$\rho:\Gamma\to\mathrm{SL}_{n}({\mathbb{R}})$ and let ${\mathbf{G}}$ be the
Zariski closure of $\rho(\Gamma).$ Recall the affinity exponent $s_{A}(\rho)$
given in (1.1), which is expressed as

$s_{\mathrm{A}}(\rho)\coloneqq\sup\left\\{s:\sum_{\gamma\in\Gamma}\exp(-\psi_{s}(\rho(\gamma)))=\infty\right\\}$

where

$\psi_{s}(g)\coloneqq\sum_{1\leqslant i\leqslant\lfloor
s\rfloor}(\log\sigma_{1}(g)-\log\sigma_{i+1}(g))+(s-\lfloor
s\rfloor)(\log\sigma_{1}(g)-\log\sigma_{\lfloor s\rfloor+2}(g)),\quad
g\in\mathrm{SL}_{n}({\mathbb{R}}).$

Note that $\psi_{s}(g)$ is a linear function on Cartan projection $\kappa(g).$
By abuse of the notation, we also denote $\psi_{s}$ to be the linear function
on $\mathfrak{a}$ satisfying $\psi_{s}(\kappa(g))=\psi_{s}(g).$ Then
$\psi_{s}$ is positive with respect to $\mathfrak{a}^{+}.$

Let $\mathscr{P}_{\mathrm{f.s.}}(\rho)$ be the family of finitely supported
probability measures on $\rho(\Gamma).$ Let
$\mathscr{P}_{\mathrm{f.s.}}^{\mathbf{G}}(\rho)$ be the family of
$\nu\in\mathscr{P}_{\mathrm{f.s.}}(\rho)$ satisfying $G_{\nu}$ is Zariski
dense in ${\mathbf{G}}.$ We will also consider
$\mathscr{P}_{\mathrm{f.s.}}^{{\mathbf{G}}^{0}}(\rho)$ where
${\mathbf{G}}^{0}$ denotes the identity (Zariski-)component of ${\mathbf{G}}.$
For $\nu\in\mathscr{P}_{\mathrm{f.s}}(\rho),$ let $\mu$ be an ergodic
stationary measure of $\nu$ on ${\mathbb{P}}({\mathbb{R}}^{d}).$ Recall the
Lyapunov dimension of $\mu$ in (1.3). Then Theorem 1.1 is interpreted as the
following.

######  Proposition 4.3.

Let $\rho:\Gamma\to\mathrm{SL}_{n}({\mathbb{R}})$ be a Zariski dense Borel
Anosov representation (that is ${\mathbf{G}}=\mathrm{SL}_{n}({\mathbb{R}})$).
Then

$\displaystyle s_{A}(\rho)$
$\displaystyle\leqslant\sup\left\\{\dim_{\mathrm{LY}}\mu:\mu\text{ is the
unique stationary measure on ${\mathbb{P}}({\mathbb{R}}^{n})$ of some
}\nu\in\mathscr{P}_{\mathrm{f.s.}}^{\mathbf{G}}(\rho)\right\\}$

Besides, we also have a variational principle for dimensions on the flag
variety. We consider

$\psi_{F,s}(g)\coloneqq\inf\left\\{\sum_{1\leqslant i<j\leqslant
n}a_{ij}\log\frac{\sigma_{i}(g)}{\sigma_{j}(g)}:0\leqslant a_{ij}\leqslant
1,\sum_{1\leqslant i<j\leqslant n}a_{ij}=s\right\\}.$

Note that $\psi_{F,s}(g)$ is increasing with respect to $s.$ The affinity
exponent on the flag variety is given by

$s_{A,F}(\rho)\coloneqq\sup\left\\{s:\sum_{\gamma\in\Gamma}\exp(-\psi_{F,s}(\rho(\gamma)))=\infty\right\\}.$

For instance, in the case of $n=3,$ the function $\psi_{F,s}$ is given by

$\psi_{F,s}(g)=\left\\{\begin{aligned}
&s\min\left\\{\log\frac{\sigma_{1}(g)}{\sigma_{2}(g)},\log\frac{\sigma_{2}(g)}{\sigma_{3}(g)}\right\\},&s\leqslant
1;\\\
&(s-1)\left(\log\frac{\sigma_{1}(g)}{\sigma_{2}(g)}+\log\frac{\sigma_{2}(g)}{\sigma_{3}(g)}\right)+(2-s)\min\left\\{\log\frac{\sigma_{1}(g)}{\sigma_{2}(g)},\log\frac{\sigma_{2}(g)}{\sigma_{3}(g)}\right\\},&1<s\leqslant
2;\\\
&\log\frac{\sigma_{1}(g)}{\sigma_{2}(g)}+\log\frac{\sigma_{2}(g)}{\sigma_{3}(g)}+(s-2)\log\frac{\sigma_{1}(g)}{\sigma_{3}(g)},&2<s\leqslant
3.\end{aligned}\right.$

In general, $\psi_{F,s}(g)$ is always a minimum of finitely many linear
functions on $\kappa(g)$ which is positive with respect to $\mathfrak{a}^{+}.$
Specifically,
$\psi_{F,s}(g)=\min\left\\{\psi_{F,k}(\kappa(g)):k\in\mathcal{K}_{s}\right\\},$
where each $\psi_{F,k}(\kappa(g))$ is of the form

$\sum_{1\leqslant i<j\leqslant n}a_{ij}(\log\sigma_{i}(g)-\log\sigma_{j}(g))$

satisfying $0\leqslant a_{ij}\leqslant 1,\Sigma_{1\leqslant i<j\leqslant
n}a_{ij}=s$ and at most one of $a_{ij}$ is neither $0$ nor $1.$ For given $s$
and $n,$ there are at most finitely many such linear functionals. Hence
$\\#\mathcal{K}_{s}$ is finite.

For $\nu\in\mathscr{P}_{\mathrm{f.s}}(\rho),$ let $\mu$ be an ergodic
stationary measure of $\nu$ on $\mathcal{F}({\mathbb{R}}^{n}).$ Then the
Lyapunov dimension of $\mu$ is given by

$\dim_{\mathrm{LY}}\mu=\sup\left\\{\sum_{1\leqslant i<j\leqslant
n}d_{ij}:0\leqslant d_{ij}\leqslant 1,\sum_{1\leqslant i<j\leqslant
n}d_{ij}(\lambda_{i}(\nu)-\lambda_{j}(\nu))=h_{\mathrm{F}}(\mu,\nu)\right\\}.$

######  Proposition 4.4.

Let $\rho:\Gamma\to\mathrm{SL}_{n}({\mathbb{R}})$ be a Borel Anosov
representation. Then

$\displaystyle s_{A,F}(\rho)$
$\displaystyle\leqslant\sup\left\\{\dim_{\mathrm{LY}}\mu:\mu\text{ is an
ergodic stationary measure on $\mathcal{F}({\mathbb{R}}^{n})$ of some
}\nu\in\mathscr{P}_{\mathrm{f.s.}}^{{\mathbf{G}}^{0}}(\rho)\right\\}$

We will prove the case on flag varieties first. The case on the projective
space is very similar and we will only indicate where we need to modify in the
proof.

###### Proof of Proposition 4.4.

Since $\rho$ is Anosov, $\ker\rho$ is finite. Replacing $\Gamma$ by
$\Gamma/\ker\rho,$ we can assume that $\rho$ is faithful. Besides, by
Selberg’s lemma, $\rho(\Gamma)$ is virtually torsion-free. Replacing $\Gamma$
by a finite index torsion-free subgroup, we may assume that $\Gamma$ is
torsion free and the Zariski closure of $\rho(\Gamma)$ is Zariski connected.
This process does not affect the value of affinity exponent and the Zariski
closure of $\rho(\Gamma)$ is indeed the identity component of the original
one. Then the proposition is a direct consequence of the following lemma.

######  Lemma 4.5.

Let $\Gamma$ be a non-elementary torsion-free hyperbolic group and
$\rho:\Gamma\to\mathrm{SL}_{n}({\mathbb{R}})$ a faithful Borel Anosov
representation such that the Zariski closure ${\mathbf{G}}$ of $\rho(\Gamma)$
is Zariski connected. Let $s>0$ such that the series
$\sum_{\gamma\in\Gamma}\exp(-\psi_{F,s}(\rho(\gamma)))$ diverges. Then for
every $\epsilon>0$ sufficiently small, there exists
$\nu\in\mathscr{P}_{\mathrm{f.s.}}^{\mathbf{G}}(\rho)$ and an ergodic
$\nu$-stationary measure $\mu$ on $\mathcal{F}({\mathbb{R}}^{n})$ satisfying

$\dim_{\mathrm{LY}}\mu\geqslant s-\epsilon.$

###### Proof.

Recall that
$\psi_{F,s}(\rho(\gamma))=\min\left\\{\psi_{F,k}(\kappa(\rho(\gamma))):k\in\mathcal{K}_{s}\right\\}$
is a minimum of finitely many linear functions. Then

$\sum_{k\in\mathcal{K}_{s}}\sum_{\gamma\in\Gamma}\exp(-\psi_{F,k}(\kappa(\rho(\gamma))))\geqslant\sum_{\gamma\in\Gamma}\exp(-\psi_{F,s}(\rho(\gamma)))=\infty.$

This implies that there exists $k\in\mathcal{K}_{s}$ satisfying
$\sum_{\gamma\in\Gamma}\exp(-\psi_{F,k}(\rho(\gamma)))$ diverges. We fix a
such $\psi_{F,k}$ in latter discussions and assume that

$\psi_{F,k}(g)=\sum_{1\leqslant i<j\leqslant
n}a_{ij}(\log\sigma_{i}(g)-\log\sigma_{j}(g)),\quad\forall
g\in\mathrm{SL}_{n}({\mathbb{R}}).$

Now we apply Proposition 4.1 to $\psi_{F,k}.$ We obtain a constant $c>0$ and
there exists a positive integer $N$ and
$\nu\in\mathscr{P}_{\mathrm{f.s.}}(\rho)$ satisfying

  * •

$\operatorname{\mathrm{supp}}\nu$ generates a semigroup whose Zariski closure
is ${\mathbf{G}}.$

  * •

$\lambda_{p}(\nu)-\lambda_{p+1}(\nu)\geqslant cN$ for every $p=1,\cdots,n-1.$

  * •

$h_{\mathrm{RW}}(\nu)\geqslant(1-\frac{1}{2}c\epsilon)N$ and
$\psi_{F,k}(\lambda(\nu))\leqslant(1+\frac{1}{2}c\epsilon)N.$

Since $\nu$ has a simple Lyapunov spectrum, there exists a $\nu$-stationary
measure $\mu$ on $\mathcal{F}({\mathbb{R}}^{n})$ which corresponds to the
distribution of Oseledec’s splitting. Furthermore,
$(\operatorname{\mathrm{supp}}\mu,\mu)$ is the Poisson boundary for
$(\Gamma_{\nu},\nu)$ by [Fur02, Theorem 2.21], where $\Gamma_{\nu}$ is the
group generated by $\operatorname{\mathrm{supp}}\nu.$ By [Fur02, Theorem
2.31],
$h_{\mathrm{F}}(\mu,\nu)=h_{\mathrm{RW}}(\nu)\geqslant(1-\frac{1}{2}c\epsilon)N.$
Now we estimate $\dim_{\mathrm{LY}}\mu.$

Note that

$\psi_{F,k}(\lambda(\nu))=\sum_{1\leqslant i<j\leqslant
n}a_{ij}(\lambda_{i}(\nu)-\lambda_{j}(\nu))\leqslant(1+\frac{1}{2}c\epsilon)N.$

Assuming $a_{i_{0}j_{0}}>0$ for some $1\leqslant i_{0}<j_{0}\leqslant n,$ we
take

$a_{ij}^{\prime}=\left\\{\begin{aligned} &a_{ij}-\epsilon,&i=i_{0},j=j_{0};\\\
&a_{ij},&\text{otherwise}.\end{aligned}\right.$

Then

$\sum_{1\leqslant i<j\leqslant
n}a_{ij}^{\prime}(\lambda_{i}(\nu)-\lambda_{j}(\nu))\leqslant(1+\frac{1}{2}c\epsilon)N-\epsilon(\lambda_{i_{0}}(\nu)-\lambda_{j_{0}}(\nu))\leqslant(1-\frac{1}{2}c\epsilon)N\leqslant
h_{\mathrm{F}}(\mu,\nu).$

Hence

$\dim_{\mathrm{LY}}\mu\geqslant\sum_{1\leqslant i<j\leqslant
n}a_{ij}^{\prime}=\sum_{1\leqslant i<j\leqslant n}a_{ij}-\epsilon=s-\epsilon.$

We obtain the desired conclusion. ∎

∎

###### Proof of Proposition 4.3.

The only difference is to show the identity between the Furstenberg entropy
and the random walk entropy. Since we assume additionally that $\rho(\Gamma)$
is Zariski dense in $\mathrm{SL}_{n}({\mathbb{R}}),$ the equality of two
notions of entropies follows from Proposition 2.3. ∎

##  5 Hausdorff dimension of the Rauzy Gasket

We will verify the equality of the Haussdorff dimension of the Rauzy gasket
with its affinity exponent. To simplify notations, we abbreviate
$\Gamma_{\mathscr{R}}$ to $\Gamma$ in this section.

###  5.1 Preliminaries and notation

Recall that $\Delta$ is the projectivization of $\\{(x,y,z):x,y,z\geqslant
0\\}$ in ${\mathbb{P}}({\mathbb{R}}^{3}).$ There is a natural bijection
between $\Delta$ and the euclidean triangle
$\widetilde{\Delta}=\\{(x,y,z)\in\mathbb{R}^{3}:x+y+z=1,x,y,z\geqslant 0\\}$
by the projective map, which also preserves lines, hence triangles. The
euclidean distance $d_{E}$ on $\widetilde{\Delta}$ from $\mathbb{R}^{3}$ is
bi-Lipschitz equivalent to the projective distance $d$ coming from $\Delta$.
Since Lipschitz constants do not affect the statements of lemmas, we identify
$\Delta$ and $\widetilde{\Delta}$ in the following and we do not distinguish
the euclidean metric and the projective metric. Moreover, the area of
triangles in $\Delta$ will be understood as the area of the corresponding
triangle in $\widetilde{\Delta}$ in this section.

Recall that the Rauzy gasket $X\subset\mathbb{P}(\mathbb{R}^{3})$ is a
projective fractal set defined in the introduction. We may consider the
classical coding of $X$ by infinite words as the case of IFSs. Let
$\Lambda\coloneqq\left\\{1,2,3\right\\}$ be the set of symbols. We have the
following basic fact [AS13, Lemma 3].

######  Lemma 5.1.

For every $\mathbf{i}=(i_{1},i_{2},\cdots)\in\Lambda^{\mathbb{N}},$ we have
$\lim_{n\to\infty}\operatorname{\mathrm{diam}}A_{i_{1}}\cdots
A_{i_{n}}\Delta=0.$

This fact allows us to define the coding map

$\Phi:\Lambda^{\mathbb{N}}\to\Delta,\quad\mathbf{i}=(i_{1},i_{2},\cdots)\mapsto\cap_{n\in{\mathbb{N}}}A_{i_{1}}\cdots
A_{i_{n}}\Delta.$

Then the image of $\Phi$ is exactly the Rauzy Gasket $X.$

For any $\gamma\in\Gamma$ we denote by $\Delta_{\gamma}$ the image
$\gamma\cdot\Delta$. We also use $|\gamma|$ for the word length of $\gamma$
with respect to the standard generator set $\\{A_{i}\\}$ of $\Gamma$. For
later use, we consider the following notations. Recall that by freeness of
$\Gamma$, any $\gamma\in\Gamma$ can be decomposed uniquely as the following
$\gamma=A_{i_{1}}\cdots A_{i_{|\gamma|}}$. For any $n\leqslant|\gamma|$, we
denote by $\gamma_{n}\coloneqq A_{i_{1}}\cdots A_{i_{n}}$. We say that the
last $n$ digits of $\gamma$ are not the same for some $n\leqslant|\gamma|$, if
in the decomposition, $i_{|\gamma|-n+1},\dots,i_{|\gamma|}$ are not the same.
Recall $s_{\mathrm{A}}(\Gamma)$ defined in the introduction is the affinity
exponent of $\Gamma$.

We will also consider the transpose action of $\Gamma$. For any
$\gamma=A_{i_{1}}\cdots A_{i_{|\gamma|}}\in\Gamma$, the transpose action
$\gamma^{t}:\mathbb{P}(\mathbb{R}^{3})\to\mathbb{P}(\mathbb{R}^{3})$ of
$\gamma$ is defined by $\gamma^{t}:=A_{i_{|\gamma|}}^{t}\cdots A_{i_{1}}^{t}$,
where the transposes of $A_{i}$’s are

$A_{1}^{t}=\begin{pmatrix}1&0&0\\\ 1&1&0\\\ 1&0&1\end{pmatrix},\
A_{2}^{t}=\begin{pmatrix}1&1&0\\\ 0&1&0\\\ 0&1&1\end{pmatrix},\
A_{3}^{t}=\begin{pmatrix}1&0&1\\\ 0&1&1\\\ 0&0&1\end{pmatrix}.$

It is not hard to check that for any $\gamma\in\Gamma$, the transpose action
$\gamma^{t}$ preserves $\Delta$ since the entries of the matrix presentation
of $\gamma^{t}$ are all non-negative. For any $n\leqslant|\gamma|$, we denote
by $\gamma_{n}^{t}:=A_{i_{|\gamma|}}^{t}\cdots A_{i_{|\gamma|-n+1}}^{t}$.
Notice that $\gamma_{n}^{t}$ is not equal to $(\gamma_{n})^{t}$, it should be
identified as $(\gamma^{t})_{n}$.

###  5.2 The upper bound of the Hausdorff dimension

The goal of this section is to show the upper bound of the Hausdorff dimension
of the Rauzy gasket, that is $\dim X\leqslant s_{\mathrm{A}}(\Gamma)$. The
following elementary lemma in linear algebra is the key observation, which
plays a crucial role in estimating the upper bound of $\dim X.$

Recall that $\\{e_{1},e_{2},e_{3}\\}$ is the standard orthonormal basis of
${\mathbb{R}}^{3}$ and
$E_{i}={\mathbb{R}}e_{i}\in{\mathbb{P}}({\mathbb{R}}^{3}).$

######  Lemma 5.2.

For every $n\in\mathbb{N}$, there exists $\epsilon_{n}>0$ such that for any
$\gamma\in\Gamma$, if the last $n$ digits of $\gamma$ are not the same, then
for $i=1,2,3$

$\|\gamma e_{i}\|\geqslant\epsilon_{n}\sigma_{1}(\gamma).$

###### Proof.

The idea to show the lemma is to consider the transpose action of $\Gamma$. We
list some basic facts on the action of $A_{i}^{t}$ on $\Delta.$

######  Lemma 5.3.

For every $i\neq j\in\left\\{1,2,3\right\\},$ the following holds:

  1. (1)

$A_{i}^{t}$ preserves $\Delta.$

  2. (2)

$A_{i}^{t}$ preserves $\Delta^{\prime},$ where $\Delta^{\prime}$ is the open
projective triangle in $\mathbb{P}(\mathbb{R}^{3})$ with vertices
$\begin{pmatrix}0\\\ 1\\\ 1\end{pmatrix},\begin{pmatrix}1\\\ 0\\\
1\end{pmatrix},\begin{pmatrix}1\\\ 1\\\ 0\end{pmatrix}.$

  3. (3)

$A_{i}^{t}E_{j}=E_{j}.$

  4. (4)

$A_{i}^{t}E_{i}=\begin{pmatrix}1\\\ 1\\\ 1\end{pmatrix}\in\Delta^{\prime}.$

Combining (2)(3)(4) in the lemma above, we obtain

######  Lemma 5.4.

For any $i\in\\{1,2,3\\}$, for any $\gamma=A_{i_{1}}\cdots
A_{i_{|\gamma|}}\in\Gamma$, if there exists some $i_{j}(\gamma)=i$, then
$\gamma^{t}E_{i}\in\Delta^{\prime}.$

We are back to the proof of Lemma 5.2. Let $\gamma\in\Gamma$ be an element
such that the last $n$ digits of $\gamma$ are not the same. Due to 2.1, we
have

$\|\gamma e_{i}\|\geqslant\|\gamma\|d(E_{i},H_{\gamma^{-}}).$ (5.1)

Recall the relation $(V_{(\gamma^{t})^{+}})^{\perp}=H_{\gamma^{-}}.$ So it is
sufficient to show the angle between $E_{i}$ and $V_{(\gamma^{t})^{+}}$ is
bounded away from $\pi/2$. Note that $\Delta$ has a special geometry property:
$E_{i}^{\perp}$ is the span $\left\\{E_{j}:j\neq i\right\\}$, which
corresponds to an edge of $\Delta.$ In order to show the angle between
$V_{(\gamma^{t})^{+}}$ and $E_{i}$ is bounded away from $\pi/2,$ it suffices
to show that $d(V_{(\gamma^{t})^{+}},\partial\Delta)$ is lower bounded by a
positive constant only depending on $n.$ To determine the position of
$V_{(\gamma^{t})^{+}},$ we use the fact that it is the attracting fixed point
of $\gamma^{t}\gamma$ on the projective plane. Hence
$V_{(\gamma^{t})^{+}}\in\gamma^{t}\gamma\Delta.$

######  Lemma 5.5.

$\gamma^{t}\gamma\Delta\subset\gamma_{n}^{t}\Delta\cap\overline{\Delta^{\prime}}.$

###### Proof.

Since $A_{i}$ and $A_{i}^{t}$ preserve $\Delta$ for $i=1,2,3,$ we obtain
$\gamma^{t}\gamma\Delta\subset\gamma_{n}^{t}\Delta.$ Now we show
$\gamma^{t}\gamma\Delta\subset\overline{\Delta^{\prime}}.$ There are two
possible cases. If all of $A_{1},A_{2},A_{3}$ occur in $\gamma,$ then
$\gamma^{t}E_{i}\in\Delta^{\prime}$ for each $i=1,2,3$ by Lemma 5.4. Therefore
$\gamma^{t}\gamma\Delta\subset\gamma^{t}\Delta\subset\Delta^{\prime}.$

Otherwise, there are at most two of $A_{i}$ occur in the $\gamma.$ Recall the
assumption that the last $n$ digits of $\gamma$ are not the same. Without loss
of generality, we can assume that both $A_{1}$ and $A_{2}$ occur in the last
$n$ digits of $\gamma.$ Now we consider the region

$\nabla_{z}\coloneqq\left\\{\begin{pmatrix}x\\\ y\\\
z\end{pmatrix}\in\Delta:z\leqslant x+y\right\\}.$

Then $A_{1},A_{2},A_{1}^{t},A_{2}^{t}$ preserve $\nabla_{z}.$ Moreover,
$A_{1}\Delta\subset\nabla_{z}$ and $A_{2}\Delta\subset\nabla_{z}.$ Since
$A_{3}$ does not occur in $\gamma,$ we have
$\gamma^{t}\gamma\Delta\subset\nabla_{z}.$ Finally, we notice that the
vertices of $\gamma_{n}^{t}\Delta$ satisfy
$\gamma_{n}^{t}E_{1},\gamma_{n}^{t}E_{2}\in\Delta^{\prime}$ and
$\gamma_{n}^{t}E_{3}=E_{3}$ by Lemma 5.4. This gives
$\gamma^{t}\gamma\Delta\subset\gamma_{n}^{t}\Delta\cap\nabla_{z}=\gamma_{n}^{t}\Delta\cap\overline{\Delta^{\prime}}.$
∎

Figure 2: The pattern of $\gamma^{t}_{n}\Delta\cap\overline{\Delta^{\prime}}$.

We complete the proof of Lemma 5.2. By an inductive argument on $n,$ we can
assume that there are exactly two of $\left\\{1,2,3\right\\}$ occur in
$i_{|\gamma|-n+1},\cdots i_{|\gamma|}.$ Without loss of generality, we assume
these two digits are $1$ and $2.$ By Lemma 5.4, the vertices of
$\gamma_{n}^{t}\Delta$ satisfy

$\gamma_{n}^{t}E_{1}\in\Delta^{\prime},\quad\gamma_{n}^{t}E_{2}\in\Delta^{\prime},\quad\gamma_{n}^{t}E_{3}=E_{3}.$

Notice that set $\gamma_{n}^{t}\Delta\cap\overline{\Delta^{\prime}}$ is a
closed quadrilateral (see Figure 2), which does not intersect the boundary of
$\Delta$. Thus for every such $\gamma_{n}$, we have
$d(\partial\Delta,\gamma_{n}^{t}\Delta\cap\overline{\Delta^{\prime}})>0$.
Since there are only finitely many such $\gamma_{n}$ for a given positive
integer $n,$ there exists $d_{n}>0$ only depending on $n$ such that

$d(\partial\Delta,\gamma_{n}^{t}\Delta\cap\overline{\Delta^{\prime}})>d_{n}$

for all such $\gamma_{n}^{t}$. Recalling
$V_{(\gamma^{t})^{+}}\in\gamma^{t}\gamma\Delta\subset\gamma_{n}^{t}\Delta\cap\Delta^{\prime},$
we have $d(V_{(\gamma^{t})^{+}},\partial\Delta)>d_{n}.$ ∎

The following lemma shows some basic estimates of the diameter and the area of
$\Delta_{\gamma}$.

######  Lemma 5.6.

There exists $C_{2}>1$ such that if the last $2$ digits of $\gamma$ are not
the same, then

  1. (1)

$\operatorname{\mathrm{diam}}(\Delta_{\gamma})\leqslant
C_{2}\cdot\frac{\sigma_{2}(\gamma)}{\sigma_{1}(\gamma)}$;

  2. (2)

$\mathrm{Area}(\Delta_{\gamma})\leqslant C_{2}\cdot\sigma_{1}(\gamma)^{-3}$.

###### Proof.

  1. (1)

It suffices to show $d(\gamma E_{i},\gamma E_{j})\leqslant
C_{2}\cdot\frac{\sigma_{2}(\gamma)}{\sigma_{1}(\gamma)}$. We have

$\displaystyle d(\gamma E_{i},\gamma E_{j})$ $\displaystyle=$
$\displaystyle\frac{\|\gamma e_{i}\wedge\gamma e_{j}\|}{\|\gamma
e_{i}\|\cdot\|\gamma
e_{j}\|}\leqslant\frac{\sigma_{1}(\gamma)\sigma_{2}(\gamma)\|e_{i}\wedge
e_{j}\|}{\|\gamma e_{i}\|\cdot\|\gamma e_{j}\|}$ $\displaystyle\leqslant$
$\displaystyle\frac{\sigma_{1}(\gamma)\sigma_{2}(\gamma)}{\epsilon_{2}^{2}\sigma_{1}(\gamma)^{2}}\text{
(by Lemma \ref{lem: crucial proj geo lem}) }$ $\displaystyle=$
$\displaystyle\epsilon_{2}^{-2}\cdot\frac{\sigma_{2}(\gamma)}{\sigma_{1}(\gamma)}.$

  2. (2)

We use the following elementary geometric fact: Let $x,y,z$ be three points in
$\mathbb{R}^{3}\setminus\\{o\\}$, then the area of the triangle formed by
$x,y,z$ (which we denote by $(xyz)$) is equal to

$\mathrm{Area}(xyz)=\frac{\|x\wedge y\wedge z\|}{2d_{E}(o,(xyz))},$

where $d_{E}(o,(xyz))$ is the distance from the origin $o$ to the two plane of
$(xyz)$. This is because the numerator gives the volume of the polyhedron of
$oxyz$.

For $\gamma\in\Gamma$, recall that the area of $\Delta_{\gamma}$ is understood
in the area of the corresponding subset of $\widetilde{\Delta}$ in the
euclidean space. Let $x_{\gamma},y_{\gamma},z_{\gamma}$ be the corresponding
points in $\widetilde{\Delta}$ of vertices of $\Delta_{\gamma}.$ Then

$\mathrm{Area}(x_{\gamma}y_{\gamma}z_{\gamma})=\frac{\|x_{\gamma}\wedge
y_{\gamma}\wedge
z_{\gamma}\|}{2d(o,(x_{\gamma}y_{\gamma}z_{\gamma}))}=\frac{\|x_{\gamma}\wedge
y_{\gamma}\wedge z_{\gamma}\|}{2d(o,\widetilde{\Delta})}.$

We know that

$x_{\gamma}=\gamma e_{1}/\omega(\gamma e_{1}),$

with $\omega(v)=v_{1}+v_{2}+v_{3}$, similarily for $y_{\gamma},z_{\gamma}$.
Therefore,

$\|x_{\gamma}\wedge y_{\gamma}\wedge z_{\gamma}\|=\|\gamma e_{1}\wedge\gamma
e_{2}\wedge\gamma e_{3}\|/\prod_{1\leqslant i\leqslant 3}\omega(\gamma e_{i})$

We actually have $\|\gamma e_{1}\wedge\gamma e_{2}\wedge\gamma
e_{3}\|=\|e_{1}\wedge e_{2}\wedge e_{3}\|=1$. Due to $\gamma E_{i}\in\Delta$,
we also have

$\omega(\gamma e_{i})\geqslant\|\gamma e_{i}\|$

Then by Lemma 5.2 we get the proof of (2).∎

The following geometric lemma is essentially proved in [PS21, Lemma 4.1].

######  Lemma 5.7.

For any $\delta>0$, there exists $c_{\delta}>0$ such that for any
$\gamma\in\Gamma$, there exists a finite open cover
$\\{D_{i}(\gamma):i=1,\cdots,k\\}$ of $\Delta_{\gamma}$ with
$\operatorname{\mathrm{diam}}D_{i}(\gamma)\leqslant\operatorname{\mathrm{diam}}\Delta_{\gamma}$
such that

$\sum_{i}\operatorname{\mathrm{diam}}^{1+\delta}D_{i}(\gamma)\leqslant
c_{\delta}\cdot\operatorname{\mathrm{diam}}^{1-\delta}\Delta_{\gamma}\cdot\mathrm{Area}^{\delta}\Delta_{\gamma}.$

###### Proof.

As in the proof of [PS21, Lemma 4.1], we know that every $\Delta_{\gamma}$ can
be covered by
$O(\operatorname{\mathrm{diam}}^{2}(\Delta_{\gamma})/\mathrm{Area}(\Delta_{\gamma}))$
disks $\\{D_{i}:i=1,\cdots,k\\}$ of diameter
$O(\mathrm{Area}(\Delta_{\gamma})/\operatorname{\mathrm{diam}}(\Delta_{\gamma}))$,
then we get the proof. ∎

We back to the proof of $\dim_{\mathrm{H}}(X)\leqslant
s_{\mathrm{A}}(\Gamma)$. Recall
$P_{\Gamma}(s)=\sum_{\gamma\in\Gamma}\varphi_{s}(\gamma)$ is the Poincaré
series of $\Gamma$, where $\varphi_{s}(\gamma)$ is defined by

$\varphi_{s}(\gamma)=\left\\{\begin{aligned}
&\Big{(}\frac{\sigma_{2}}{\sigma_{1}}\Big{)}^{s}(\gamma),&0<s\leqslant 1;\\\
&\Big{(}\frac{\sigma_{2}}{\sigma_{1}}\Big{)}(\gamma)\Big{(}\frac{\sigma_{3}}{\sigma_{1}}\Big{)}^{s-1}(\gamma),&1<s\leqslant
2.\end{aligned}\right.$

By definition of $s_{\mathrm{A}}$, it suffices to show if
$P_{\Gamma}(s)<\infty$ then for any $\epsilon>0$, $\dim X\leqslant
s+\epsilon$.

######  Definition 5.8.

For $x\in X$, we say $x$ is nice if every
$(i_{1}(x),i_{2}(x),\cdots)\in\Phi^{-1}(x)\in\Lambda^{\mathbb{N}}$ is not
ending by a single element in $\Lambda=\\{1,2,3\\}$.

We remark that if $x$ is nice, then $x$ is uniquely coding. This is because
the only possibility of $\Phi(\mathbf{i})=\Phi(\mathbf{i}^{\prime})$ is

$\mathbf{i}=(w,j_{1},j_{2},j_{2},\cdots)\text{ and
}\mathbf{i}^{\prime}=(w,j_{2},j_{1},j_{1},\cdots),$

where $w\in\Lambda^{N}$ for some $N\in{\mathbb{N}}$ and $j_{1}\neq
j_{2}\in\Lambda.$

Then the Rauzy Gasket $X$ can be decomposed as the set of nice points, which
we denote it by $X_{\mathrm{nice}}$, and a countable set. It suffices to show
the Hausdorff outer measure $H^{s+\epsilon}(X_{\mathrm{nice}})$ is $0$.

We consider

$\Gamma_{m}\coloneqq\left\\{\gamma\in\Gamma:\text{the last $2$ digits of
$\gamma$ are not the same and
}\operatorname{\mathrm{diam}}\Delta_{\gamma}\leqslant 1/m\right\\}.$

Now we construct two families of covers $\mathcal{U}_{m}$ and
$\mathcal{U}_{m}^{\prime}$ for $m\in\mathbb{N}$ as

$\mathcal{U}_{m}\coloneqq\left\\{D_{i}(\gamma):\gamma\in\Gamma_{m}\right\\},\quad\mathcal{U}_{m}^{\prime}\coloneqq\left\\{\Delta_{\gamma}:\gamma\in\Gamma_{m}\right\\},$

where $\\{D_{i}(\gamma)\\}$ is the finite open cover of $\Delta_{\gamma}$ we
obtained from Lemma 5.7. Let

$Y\coloneqq\bigcap_{m=1}^{\infty}\bigcup_{U\in\mathcal{U}_{m}}U\quad\text{and}\quad
Y^{\prime}\coloneqq\bigcap_{m=1}^{\infty}\bigcup_{U\in\mathcal{U}^{\prime}_{m}}U.$

Then the sequence of covers $\mathcal{U}_{m}$ (resp.
$\mathcal{U}_{m}^{\prime}$) is a family of Vitali covers of the set $Y$ (resp.
$Y^{\prime}$).333Recall that a Vitali cover $\mathcal{V}$ of a set $E$ is a
family of sets so that, for every $\delta>0$ and every $x\in E$, there is some
$U$ in the family $\mathcal{V}$ with $\operatorname{\mathrm{diam}}U<\delta$
and $x\in U$. Notice that $Y$ contains $Y^{\prime}$ by the construction of
$D_{i}(\gamma)$’s.

######  Lemma 5.9.

$Y\supset X_{\mathrm{nice}}$.

###### Proof.

We claim that for any $\mathbf{i}\in\\{1,2,3\\}^{\mathbb{N}}$ which is not
ending by a single element, the point $x=\Phi(\mathbf{i})$ is contained in
$Y$. Since $\mathbf{i}=(i_{1}(x),\cdots)$ is not ending by a single element in
$\\{1,2,3\\}$, there exists infinitely many $\ell$ such that
$i_{\ell-1}(x),i_{\ell}(x)$ are not the same (otherwise it will be ended by
only one element). Collect all such $\ell$ and consider all the elements
$\gamma=A_{i_{1}}\cdots A_{i_{\ell}}$. Then we get there are infinitely many
$\gamma\in\Gamma$ such that the last two digits of $\gamma$ are not the same
and $x\in\Delta_{\gamma}$. By 5.1,
$\operatorname{\mathrm{diam}}\Delta_{\gamma}\to 0$ as $\ell$ tending to
infinity. Therefore $x\in Y^{\prime}\subset Y$. ∎

As a consequence, it suffices to show the Hausdorff outer measure

$H^{s+\epsilon}(Y)=\lim_{\delta\to 0}H^{s+\epsilon}_{\delta}(Y)=0.$

Recall that $\operatorname{\mathrm{diam}}U\leqslant 1/m$ for every
$U\in\mathcal{U}_{m}.$ Then for $s\geqslant 1$, we have

$\displaystyle H^{s+\epsilon}(Y)$ $\displaystyle\leqslant$
$\displaystyle\limsup_{\delta\to
0}H^{s+\epsilon}_{\delta}(Y)~{}\leqslant~{}\limsup_{m\to\infty}\sum_{U\in\mathcal{U}_{m}}\operatorname{\mathrm{diam}}(U)^{s+\epsilon}$
$\displaystyle=$
$\displaystyle\limsup_{m\to\infty}\sum_{\gamma\in\Gamma_{m}}\sum_{i}\operatorname{\mathrm{diam}}(D_{i}(\gamma))^{s+\epsilon}$
$\displaystyle\leqslant$
$\displaystyle\limsup_{m\to\infty}c_{s+\epsilon-1}\sum_{\gamma\in\Gamma_{m}}\operatorname{\mathrm{diam}}^{2-s-\epsilon}\Delta_{\gamma}\cdot\mathrm{Area}^{s+\epsilon-1}\Delta_{\gamma}\qquad(\text{by
Lemma \ref{lem: 4.1 PS}})$ $\displaystyle\leqslant$
$\displaystyle\limsup_{m\to\infty}c_{s+\epsilon-1}C_{2}\sum_{\gamma\in\Gamma_{m}}\varphi_{s+\epsilon}(\gamma)\text{
\quad(by Lemma \ref{lem: proj geo lem} and the definition of $\varphi_{s}$}).$

Since $\sum\varphi_{s+\epsilon}(\gamma)<\infty$, the right-hand side of the
last inequality is $0$ when $m\to\infty$, which is exactly what we want. For
the case $s<1$ (we actually know $s\sim 1.72$ by numerical test), we do not
need to use Lemma 5.7 and consider $Y^{\prime}$ instead of $Y$ by the same
argument. We still get the proof.

###  5.3 The lower bound of the Hausdorff dimension

In this section, we will show that $\dim X\geqslant s_{\mathrm{A}}(\Gamma).$
Note that $X$ contains the boundary of $\Delta,$ which has the Hausdorff
dimension equal to $1.$ Therefore, we already have an a priori estimate that
$\dim X\geqslant 1$, which allows us to assume, without loss of generality,
that $s_{\mathrm{A}}(\Gamma)>1$. In other words, we assume the existence of a
value $s>1$ for which the series $\sum_{\gamma}\varphi_{s}(\gamma)$ diverges.
This technical assumption simplifies subsequent discussions. Recall that for
every exact dimensional probability measure $\mu$ supported on $X,$ we have
$\dim\mu\leqslant\dim X.$ The inequality $\dim X\geqslant
s_{\mathrm{A}}(\Gamma)$ is a direct consequence of the following variational
principle of the affinity exponent.

######  Lemma 5.10.

Let $s>1$ such that the series $\sum_{\gamma\in\Gamma}\varphi_{s}(\gamma)$
diverges, then for every $\epsilon>0,$ there exists a finitely supported
measure $\nu$ supported on $\Gamma$ and a $\nu$-stationary measure $\mu$
supported on $X$ satisfying $\dim\mu\geqslant s-\epsilon$.

Our idea is a stopping time argument which is partially inspired by the study
of the variational principle of iterated function systems [FH09] and self-
affine measures [MS23]. Specifically, our strategy is to combine our dimension
formula of stationary measures with a modification of the proof of Theorem 3.1
in [MS23]. The main difference of our proof from [MS23] is that due to the
presence of unipotent elements, the word lengths of elements may lose control.
So instead of considering the word length, we only consider $\kappa(\gamma).$
This is enough to estimate the Lyapunov exponents. Then we use a combinatorial
argument to make elements generating a free semigroup. This generates a large
enough random walk entropy. Therefore, we can find a stationary measure with a
large dimension approximating the affinity exponent. A similar argument to
address the issue of words with uncontrolled length also appears in [HJX23].

In order to apply the dimension formula (Theorem 1.2), we need to show the
Zariski density of $\Gamma$ at first.

######  Lemma 5.11.

The semigroup $\Gamma$ is Zariski dense in $\mathrm{SL}_{3}(\mathbb{R})$.

###### Proof.

Let ${\mathbf{H}}$ be the Zariski closure of semigroup $\Gamma$ in
$\mathrm{SL}_{3}(\mathbb{R})$, which is a real algebraic group. Let
$\mathfrak{h}$ be the Lie algebra of the connected component of
${\mathbf{H}}$. By [Bor91, Page 106, Remark], the fact that $A_{1}$ is
unipotent implies that the one-parameter unipotent group
$\left\\{s\in\mathbb{R},\ \begin{pmatrix}1&s&s\\\ 0&1&0\\\
0&0&1\end{pmatrix}\right\\}$ is in ${\mathbf{H}}$ and
$X_{1}=\begin{pmatrix}0&1&1\\\ 0&0&0\\\ 0&0&0\end{pmatrix}$ is in
$\mathfrak{h}$. Similarly, the nilpotent elements
$X_{2}=\begin{pmatrix}0&0&0\\\ 1&0&1\\\ 0&0&0\end{pmatrix}$,
$X_{3}=\begin{pmatrix}0&0&0\\\ 0&0&0\\\ 1&1&0\end{pmatrix}$ are also in
$\mathfrak{h}$. Then we can play with these elements in the Lie algebra
$\mathfrak{h}$ and prove that they generate all $\mathfrak{sl}_{3}$. We have
$Y_{3}=[X_{1},X_{2}]=\begin{pmatrix}1&0&1\\\ 0&-1&-1\\\ 0&0&0\end{pmatrix}$,
$Y_{1}=[X_{2},X_{3}]=\begin{pmatrix}0&0&0\\\ 1&1&0\\\ -1&0&-1\end{pmatrix}$,
$Y_{2}=[X_{3},X_{1}]=\begin{pmatrix}-1&-1&0\\\ 0&0&0\\\ 0&1&1\end{pmatrix}$
also in $\mathfrak{h}$. Then
$[Y_{1},X_{1}]+Y_{2}+Y_{3}+2X_{1}=\begin{pmatrix}0&0&4\\\ 0&0&0\\\
0&0&0\end{pmatrix}$, $[Y_{2},X_{2}]+Y_{3}+Y_{1}+2X_{2}=\begin{pmatrix}0&0&0\\\
4&0&0\\\ 0&0&0\end{pmatrix}$ also in $\mathfrak{h}$. From these elements, we
can obtain the whole $\mathfrak{sl}(3,\mathbb{R})$ and then ${\mathbf{H}}$
must be the whole group $\mathrm{SL}_{3}(\mathbb{R})$. ∎

Recall that for $s>1,$ the linear function $\psi_{s}$ on $\mathfrak{a}$ is
given by

$\psi_{s}(\lambda)=(\lambda_{1}-\lambda_{2})+(s-1)(\lambda_{1}-\lambda_{3}),\quad\forall\lambda\in\mathfrak{a}.$

Now we construct a good set coming from the divergent series
$\sum_{\gamma}\varphi_{s}(\gamma)=\sum_{\gamma}\exp(-\psi_{s}(\kappa(\gamma))).$

######  Definition 5.12.

For $\beta>0,x\in\mathfrak{a}^{+}$ and $n\in{\mathbb{N}},$ a subset
$S\subset\Gamma$ is called $(n,\beta,x)$-approximate if $\\#S\geqslant
e^{(1-\beta)n}$ and for every $\gamma\in S,$
$\|\frac{1}{n}\kappa(\gamma)-x\|\leqslant\beta.$

######  Lemma 5.13.

For any $\beta>0$, there exists $x\in\mathfrak{a}^{+}$ satisfying
$|\psi_{s}(x)-1|\leqslant\beta$ and infinitely many $n\in{\mathbb{N}}$ such
that there exists an $(n,\beta,x)$-approximate subset $S\subset\Gamma.$

###### Proof.

Note that
$\psi_{s}(\kappa(\gamma))\geqslant(s-1)\log\sigma_{1}(\gamma)=(s-1)\log\left\|\gamma\right\|.$
Hence for every $t>0,$ the set
$\left\\{\gamma\in\Gamma:\psi_{s}(\kappa(\gamma))<t\right\\}$ is finite by the
discreteness of $\Gamma.$ Combining with the hypothesis that the series
$\sum_{\gamma\in\Gamma}\exp(-\psi_{s}(\kappa(\gamma)))$ diverges, there are
infinitely many $n$ such that

$\left\\{\gamma:\psi_{s}(\kappa(\gamma))\in[(1-\beta/10)n,(1+\beta/10)n]\right\\}$

contains at least $e^{(1-\beta/10)n}$ elements.

Since we have $\psi_{s}(\kappa(\gamma))\geqslant(s-1)\log\sigma_{1}(\gamma)$
and $\psi_{s}(\kappa(\gamma))\geqslant-(s-1)\log\sigma_{3}(\gamma),$ the set
$\psi_{s}^{-1}([1-\beta/10,1+\beta/10])\cap\mathfrak{a}^{+}$ is compact.
Therefore we can cover it by finitely many balls of radius $\beta/100.$ By a
pigeonhole principle, there exists a center of some ball, say
$x\in\mathfrak{a}^{+},$ such that for infinitely many $n,$

$\left\\{\gamma:|\frac{\psi_{s}(\kappa(\gamma))}{n}-1|\leqslant\frac{\beta}{10},~{}\|\frac{\kappa(\gamma)}{n}-x\|\leqslant\frac{\beta}{100}\right\\}$

contains at least $e^{(1-\beta)n}$ elements. Using the fact that
$|\psi_{s}(\kappa(\gamma))|\leqslant 10\|\kappa(\gamma)\|,$ we obtain
$|\psi_{s}(x)-1|\leqslant\beta,$ which is exactly what we want. ∎

We will modify an $(n,\beta,x)$-approximate subset to a set which freely
generates a free semigroup. Since $\Gamma$ is itself a free semigroup, a
subset of $\Gamma$ can freely generate a free semigroup by avoiding the
“prefix relations”. We consider the following concepts.

######  Definition 5.14.

  1. (1)

An element $j_{1}\in\Gamma$ is called starting with $j_{2}\in\Gamma$ if there
is $j_{3}\in\Gamma\setminus\\{\mathrm{id}\\}$ such that $j_{1}=j_{2}j_{3}$. We
also say $j_{2}$ is a prefix of $j_{1}$ if $j_{1}$ is starting with $j_{2}.$

  2. (2)

An element $j_{1}\in\Gamma$ is called ending with
$j_{2}\in\Gamma\setminus\\{\mathrm{id}\\}$ if there is $j_{3}\in\Gamma$ such
that $j_{1}=j_{3}j_{2}$.

  3. (3)

An element $j$ is called minimal in a subset $S$ of $\Gamma$ if there is no
element $j^{\prime}\in S$ such that $j$ is starting with $j^{\prime}$.

Within a set $S$, a minimal element of $S$ is never a prefix of another
minimal element. So the set of minimal elements $S_{\min}$ of $S$ will freely
generate a free semigroup. Moreover, the subset
$S_{\min}^{*\ell}\subset\Gamma$ satisfies that there is no pair of elements
such that one is a prefix of the other. In the following lemma, using
hyperbolicity and discreteness, we get a lower bound of minimal elements in a
set with approximately the same sizes of Cartan projections.

######  Lemma 5.15.

There exists $C>0$ such that the following hold. For every $\beta>0,$
$n\in{\mathbb{N}}$ and $x\in\mathfrak{a}^{+}.$ For any element $j$ in the set

$\mathcal{J}=\\{\gamma\in\Gamma:\
\|\frac{\kappa(\gamma)}{n}-x\|\leqslant\beta,~{}~{}\gamma\text{ is ending with
}A_{1}A_{2}A_{3}\\},$

there are at most $e^{C\beta n}$ elements in $\mathcal{J}$ which is starting
with $j.$

###### Proof.

For every $V\in\Delta$, let $v=\sum_{i}v_{i}e_{i}\in V$ be a unit vector with
$v_{1},v_{2},v_{3}\geqslant 0.$ Since all the entries of $\gamma e_{i}$ are
non-negative, we have

$\|\gamma v\|\geqslant\max_{i}\\{v_{i}\|\gamma
e_{i}\|\\}\geqslant\frac{1}{2}\min_{i}\\{\|\gamma e_{i}\|\\}.$

Take $n=2$ in 5.2, for any element $\gamma$ with last two digits different, we
have $\|\gamma e_{i}\|\geqslant\epsilon_{2}\sigma_{1}(\gamma)$, where
$\epsilon_{2}$ is defined in 5.2. Hence

$\|\gamma v\|\geqslant\frac{\epsilon_{2}}{2}\sigma_{1}(\gamma).$

Then for any $j,jj^{\prime\prime}\in\mathcal{J}$ with
$j^{\prime\prime}\in\Gamma\setminus\\{\mathrm{id}\\}$ and any unit $v\in
V\in\Delta$, since $j^{\prime\prime}V\in\Delta$ and $j,j^{\prime\prime}$ end
with $A_{1}A_{2}A_{3}$, we obtain

$\sigma_{1}(jj^{\prime\prime})\geqslant\|jj^{\prime\prime}v\|=\frac{\|j(j^{\prime\prime}v)\|}{\|j^{\prime\prime}v\|}\frac{\|j^{\prime\prime}v\|}{\|v\|}\geqslant\frac{\epsilon_{2}^{2}}{4}\sigma_{1}(j)\sigma_{1}(j^{\prime\prime}).$
(5.2)

Since $\|\frac{1}{n}\kappa(jj^{\prime\prime})-x\|\leqslant\beta$ and
$\|\frac{1}{n}\kappa(j)-x\|\leqslant\beta$, we have

$\log\sigma_{1}(jj^{\prime\prime})\leqslant 2\beta n+\log\sigma_{1}(j).$ (5.3)

Combining (5.2),(5.3) we get

$\sigma_{1}(j^{\prime\prime})\leqslant 4e^{2\beta n}/\epsilon_{2}^{2}.$

Since the semigroup $\Gamma$ is discrete in $\mathrm{SL}_{3}(\mathbb{R})$, up
to a constant the number

$\\#\\{\gamma\in\Gamma,\ \|\gamma\|\leqslant t\\}$

is bounded by the volume of the set $\\{g\in\mathrm{SL}_{3}(\mathbb{R}),\
\|g\|\leqslant t\\}$ , which grows at most polynomially on $t$. Then the
possible number of $j^{\prime\prime}$ is bounded by $q^{C\beta n}$ for some
constant $C>0.$ ∎

In order to estimate the Lyapunov exponents of the constructing measure, we
need to estimate the Cartan projection of products. Let us recall some
notions.

######  Definition 5.16.

  1. (1)

For an element $g\in\mathrm{SL}_{3}(\mathbb{R})$, we call it
$(r,\epsilon)$-loxodromic for $r,\epsilon>0$, if
$\sigma_{i}(g)/\sigma_{i+1}(g)\geqslant 1/\epsilon$ and

$d(V_{g}^{+},H^{-}_{g})>r,\
d(V_{g,\wedge^{2}\mathbb{R}^{3}}^{+},H^{-}_{g,\wedge^{2}\mathbb{R}^{3}})>r,$

where $\wedge^{2}\mathbb{R}^{3}$ is the wedge representation of
$\mathrm{SL}_{3}(\mathbb{R})$ and
$V^{+}_{g,\wedge^{2}\mathbb{R}^{3}}=\widetilde{k}_{g}(E_{1}\wedge E_{2})$ and
$H^{-}_{g,\wedge^{2}\mathbb{R}^{3}}=k_{g}^{-1}(E_{2}\wedge E_{3}\oplus
E_{3}\wedge E_{1})$ are the corresponding attracting point and repelling
hyperplane in the projective space $\mathbb{P}(\wedge^{2}\mathbb{R}^{3}),$
where $g=\widetilde{k}_{g}a_{g}k_{g}\in KA^{+}K$ is the Cartan decomposition.

  2. (2)

Let $F$ be a subset in $\mathrm{SL}_{3}(\mathbb{R})$. We call $F$ a
$(r,\epsilon)$-Schottky family if every element in $F$ is
$(r,\epsilon)$-loxodromic and for any pair $(g,h)$ in $F$, we have

$d(V_{g}^{+},H^{-}_{h})>6r,\
d(V_{g,\wedge^{2}\mathbb{R}^{3}}^{+},H^{-}_{h,\wedge^{2}\mathbb{R}^{3}})>6r.$

  3. (3)

A set $F$ of $\mathrm{SL}_{3}(\mathbb{R})$ is called $\eta$-narrow if for $g$
in $F$, the attracting points $V_{g}^{+}$ (resp.
$V_{g,\wedge^{2}\mathbb{R}^{3}}^{+}$) are within $\eta$-distance of one
another and the repelling hyperplanes $H_{g}^{-}$ (resp.
$H_{g,\wedge^{2}\mathbb{R}^{3}}^{-}$) are within $\eta$ Hausdorff distance of
one another.

  4. (4)

A set $F$ is $\eta$-narrow around $h$ if for $g$ in $F$, the attracting points
$V_{g}^{+}$ (resp. $V_{g,\wedge^{2}\mathbb{R}^{3}}^{+}$) are within
$\eta$-distance to $V_{h}^{+}$(resp. $V_{h,\wedge^{2}\mathbb{R}^{3}}^{+}$) and
the repelling hyperplanes $H_{g}^{-}$ (resp.
$H_{g,\wedge^{2}\mathbb{R}^{3}}^{-}$) are within $\eta$ Hausdorff distance to
$H_{h}^{-}$ (resp. $H_{h,\wedge^{2}\mathbb{R}^{3}}^{-}$).

These definitions and properties are originally due to Benoist [Ben97]. We
borrow them from [MS23, Corollary 2.16, 2.17]

######  Lemma 5.17.

  1. (1)

For $r>4\epsilon>0$, if $F$ is a $(r,\epsilon)$-Schottky family, then the
semigroup generated by $F$ is a $(r/2,2\epsilon)$-Schottky family.

  2. (2)

Let $E$ be a $\eta$-narrow collection of $(r,\epsilon)$-loxodromic elements
with $r>4\max\\{\epsilon,\eta\\}$. Then, $E$ is a $(r/4,\epsilon)$-Schottky
family.

  3. (3)

If $F$ is a $(r,\epsilon)$-Schottky family, then there exists $C_{r}>0$ only
depending on $r$ such that for any $g_{1},\cdots,g_{\ell}$ in $F$, we have

$\|\kappa(g_{1}\cdots g_{\ell})-\sum_{1\leqslant
i\leqslant\ell}\kappa(g_{i})\|\leqslant\ell\cdot C_{r}.$

Now we state the main construction, which gives a good set to support a
desired random walk.

######  Lemma 5.18.

For every $\beta>0.$ There exists $N\in\mathbb{N}$,
$x\in\mathfrak{a}^{+},\mathcal{J}\subset\Gamma,$ such that

  1. (1)

$|\psi_{s}(x)-1|\leqslant\beta$.

  2. (2)

The semigroup generated by $\mathcal{J}$ is Zariski dense.

  3. (3)

For every $k\in\mathbb{N}$ and $j\in\mathcal{J}^{*k}:=\\{j_{1}\cdots
j_{k}:j_{i}\in\mathcal{J}\\}$, $\|\frac{1}{kN}\kappa(j)-x\|\leqslant 10\beta$.

  4. (4)

The set $\mathcal{J}$ contains a subset $\mathcal{J}_{1}$ satisfies
$\\#\mathcal{J}_{1}\geqslant e^{(1-10C\beta)N}$ and no element in
$\mathcal{J}_{1}$ is a prefix of another one, where $C>0$ is the constant
given by Lemma 5.15.

###### Proof.

Fix an $x\in\mathfrak{a}^{+}$ be given by Lemma 5.13. Let $S$ be a
$(n,\beta,x)$-approximate subset for some sufficiently large $n.$ Now we
construct the set $\mathcal{J}$ by modifying $S$.

#### Step 1.

For every $\gamma\in S$, we add $A_{1}A_{2}A_{3}$ at the end and denote the
new set by $W_{2}$. Since $\|\kappa(\gamma
A_{1}A_{2}A_{3})-\kappa(\gamma)\|\leqslant\|\kappa(A_{1}A_{2}A_{3})\|$, the
set $W_{2}$ is $(n,2\beta,x)$-approximate for $n$ large enough.

#### Step 2.

We apply a theorem of Abels-Margulis-Soifer [AMS95], see also [MS23, Theorem
3.2].

######  Theorem 5.19 (Abels-Margulis-Soifer).

Let $G$ be a Zariski-connected real reductive group and $\Gamma$ be a Zariski-
dense subsemigroup. Then there exists $0<r=r(\Gamma)$ such that for all
$0<\epsilon\leqslant r$, there exists a finite subset
$F=F(r,\epsilon,\Gamma)\subset\Gamma$ with the property that for every $g\in
G$, there exists $f\in F$ such that $fg$ is $(r,\epsilon)$-loxodromic in $G$.

We fix $r_{0},\epsilon_{0}>0$ sufficiently small comparing to $\beta$ with
$r_{0}>100\epsilon_{0}.$ By 5.19, we could find a finite subset
$F_{1}=F(r_{0},\epsilon_{0},\Gamma)$ of $\Gamma.$ Therefore for every element
$\gamma\in W_{2}$, there exists $f\in F_{1}$ such that $f\gamma$ is
$(r_{0},\epsilon_{0})$-loxodromic. By the pigeonhole principle, we can find an
$f\in F_{1}$ such that for at least $(\\#F_{1})^{-1}$ proportion of $\gamma$
in $W_{2}$, the product $f\gamma$ is $(r_{0},\epsilon_{0})$-loxodromic. Fix
this $f\in F_{1}$ and let

$W_{3}=\left\\{f\gamma:\gamma\in W_{2},f\gamma\text{ is
$(r_{0},\epsilon_{0})$-loxodromic }\right\\}.$

Then $W_{3}$ is $(n,3\beta,x)$-approximate assuming $n$ large enough.

#### Step 3.

By compactness, we can cover $\prod_{1\leqslant i\leqslant
2}\mathbb{P}(V_{i})\times\mathbb{P}(V_{i}^{*})$ with $O(\epsilon_{0}^{-8})$
balls of radius $\epsilon_{0}$, where $V_{1}=\mathbb{R}^{3}$ and
$V_{2}=\wedge^{2}\mathbb{R}^{3}$. By the pigeonhole principle, there exists a
subset $W_{4}\subset W_{3}$, such that
$\\#W_{4}\gg\epsilon_{0}^{8}\cdot\\#W_{3}$ and $W_{4}$ is an
$\epsilon_{0}$-narrow set of $(r_{0},\epsilon_{0})$-loxodromic elements. For
$n$ sufficiently large comparing to $\epsilon_{0},$ $W_{4}$ is
$(n,4\beta,x)$-approximate.

#### Step 4.

Before making the next modification, we recall the following lemma from
[Ben97] and [MS23].

######  Lemma 5.20 ([MS23, Lemma 3.4]).

There exists $r_{1}>0$ depending only on $\Gamma$ such that the following
hold. For every loxodromic element $g\in G$ and $\epsilon>0,\eta>0$ there
exists a Zariski dense $(r_{1},\epsilon)$-Schottky subgroup of $\Gamma$ which
is $\eta$-narrow around $g.$

Since $r_{1}$ is determined by $\Gamma,$ we can assume at first that
$r_{0}<r_{1}.$ Now we fix an element $g\in W_{4}.$ Then we can find a Zariski
dense $(r_{0},\epsilon_{0})$-Schottky subgroup $\Gamma_{1}$ of $\Gamma$ which
is $\epsilon_{0}$-narrow around $g.$ By the proof of $k=0$ in Lemma 3.9 (see
also [MS23, Lemma 3.6]), we can find a finite subset
$\\{\theta_{i}:i=1,\cdots,p\\}\subset\Gamma_{1},$ which generates Zariski
dense sub-semigroup. Let
$W^{\prime}=W_{4}\cup\left\\{\theta_{i}:i=1,\cdots,p\right\\},$ which consists
of $(r_{0},\epsilon_{0})$-loxodromic elements. Moreover, every element in
$W^{\prime}$ is $\epsilon_{0}$-narrow around $g.$ Hence $W^{\prime}$ is
$2\epsilon_{0}$-narrow. Therefore, $W^{\prime}$ is a
$(r_{0}/4,\epsilon_{0})$-Schottky family and the semigroup it generates is a
$(r_{0}/8,2\epsilon_{0})$-Shottky family, by Lemma 5.17.

Take an $m\in{\mathbb{N}}$ large enough depending on $\theta_{i}$, $\beta$ and
$\epsilon_{0}$. Let

$W_{5}\coloneqq W_{4}^{*m}\cup\\{\theta_{i}g^{m},\ i=1,\cdots,p\\}$

Now we verify that the set $\mathcal{J}\coloneqq W_{5}$ satisfies the
condition for $N=nm.$

  1. (1)

This is because $x$ is given by Lemma 5.13.

  2. (2)

Let ${\mathbf{H}}$ be the Zariski closure of the semigroup generated by
$W_{5},$ which is an algebraic subgroup of $\mathrm{SL}_{3}({\mathbb{R}}).$
Note that $g^{m}\in W_{4}^{*m}\subset W_{5}$ and $\theta_{i}g^{m}\in W_{5}$
for every $i.$ We obtain $\theta_{i}\in{\mathbf{H}}.$ Since
$\left\\{\theta_{i}:i=1,\cdots,p\right\\}$ generates a Zariski dense subgroup,
we have ${\mathbf{H}}=\mathrm{SL}_{3}({\mathbb{R}}).$

  3. (3)

Recall that the semigroup generated by $W^{\prime}$ is a
$(r,\epsilon)$-Schottky family, where $r=r_{0}/8$ and
$\epsilon=2\epsilon_{0}.$ Also recall that $W_{4}$ is
$(n,4\beta,x)$-approximate. By 5.17, for every $h\in W_{4}^{*m},$ we have

$\|\frac{\kappa(h)}{N}-x\|=\|\frac{\kappa(h)}{nm}-x\|\leqslant\frac{C_{r}}{n}+4\beta.$

For $h\in\\{\theta_{i}g^{m},i=1,\dots,p\\}$, we have

$\|\frac{\kappa(h)}{N}-x\|=\|\frac{\kappa(h)}{nm}-x\|\leqslant\frac{C_{r}}{n}+4\beta+\frac{\left\|\kappa(\theta_{i})\right\|}{nm}.$

By taking $m$ large enough depending on $\theta_{i},\beta$ and then taking $n$
large enough, we assume that
$C_{r}/n+\left\|\kappa(\theta_{i})\right\|/(nm)<\beta.$ Therefore, the set
$\mathcal{J}=W_{5}$ is $(N,5\beta,x)$-approximate. Since the semigroup
generated by $W_{5}$ is an $(r,\epsilon)$-Schottky family, for every
$j\in\mathcal{J}^{*k},$ we have

$\|\frac{\kappa(j)}{kN}-x\|\leqslant\frac{C_{r}}{N}+5\beta\leqslant 10\beta.$

  4. (4)

We take minimal elements $(W_{4})_{\min}$ in $W_{4}$. Let
$\mathcal{J}_{1}=((W_{4})_{\min})^{*m}.$ Then there is no element in
$\mathcal{J}_{1}$ which is a prefix of another element. Note that $W_{4}$ is
$(n,4\beta,x)$-approximate and every element in $W_{4}$ is ending with
$A_{1}A_{2}A_{3}.$ By 5.15, we have

$\\#\mathcal{J}_{1}=(\\#(W_{4})_{\min})^{m}\geqslant(\\#W_{4}/e^{4C\beta
n})^{m}\geqslant e^{(1-4\beta-4C\beta)nm}\geqslant e^{(1-10C\beta)N}.\qed$

Finally, we will construct the random walk and estimate the dimension of the
stationary measure. This part is to complete the proof of Lemma 5.10.

Let $\mathcal{J},\mathcal{J}_{1}$ be the sets given by Lemma 5.18. We take
$\nu=(1-\beta)\nu_{1}+\beta\nu_{2},$ where $\nu_{1}$ is the uniform measure on
$\mathcal{J}_{1}$ and $\nu_{2}$ is the uniform measure on
$\mathcal{J}\setminus\mathcal{J}_{1}$. Then by 5.18 the support of $\nu$
generates a Zariski dense subgroup in $\mathrm{SL}_{3}(\mathbb{R})$ and the
associated Lyapunov vector is close to $Nx.$ Hence, by 5.18(3),

$|\frac{\psi_{s}(\lambda(\nu))}{N}-1|\leqslant|\psi_{s}(x)-1|+5\cdot\|\frac{\lambda(\nu)}{N}-x\|\leqslant
100\beta.$ (5.4)

Now we should estimate the random walk entropy of $\nu.$ Firstly, note that
$\mathcal{J}_{1}$ freely generates a free semigroup, we have

$h_{\mathrm{RW}}(\nu_{1})=H(\nu_{1})\geqslant(1-10C\beta)N.$

Moreover, since the support of $\nu_{1}$ satisfies that no element is a prefix
of another one, by certain “continuity" property of the random walk entropy
and freeness of $\Gamma$, we have

######  Lemma 5.21.

Let $\nu,\nu_{1},\nu_{2}$ be probability measures which supported on $\Gamma$
such that $\nu=(1-\beta)\nu_{1}+\beta\nu_{2}$. If the support of $\nu_{1}$ is
a minimal set (i.e. no element is a prefix of another), then

$h_{\mathrm{RW}}(\nu)\geqslant(1-\beta)h_{\mathrm{RW}}(\nu_{1}).$ (5.5)

###### Proof.

By the concavity of the entropy function $H$,

$H(\nu_{1}^{*\ell_{1}}*\nu_{2}^{*j_{1}}*\nu_{1}^{*\ell_{2}}*\cdots*\nu_{2}^{*j_{k}})\geqslant\int
H(\nu_{1}^{*\ell_{1}}*\delta_{g_{1}}*\nu_{1}^{*\ell_{2}}*\cdots*\delta_{g_{k}})\mathrm{d}\nu_{2}^{*j_{1}}(g_{1})\cdots\mathrm{d}\nu_{2}^{*j_{k}}(g_{k}).$

We claim that
$H(\nu_{1}^{*\ell_{1}}*\delta_{g_{1}}*\nu_{1}^{*\ell_{2}}*\cdots*\delta_{g_{k}})=H(\nu_{1}^{*(\ell_{1}+\cdots+\ell_{k})})$.
It suffices to show that all the elements in the support of LHS are distinct.
Otherwise suppose that two elements $(h_{1},\cdots,h_{k})$ and
$(h_{1}^{\prime},\cdots,h_{k}^{\prime})$ in
$\operatorname{\mathrm{supp}}\nu_{1}^{*\ell_{1}}\times\cdots\times\operatorname{\mathrm{supp}}\nu_{1}^{*\ell_{k}}$
satisfy $h_{1}g_{1}\cdots h_{k}g_{k}=h_{1}^{\prime}g_{1}\cdots
h_{k}^{\prime}g_{k}$. Since no element in
$\operatorname{\mathrm{supp}}\nu_{1}$ is a prefix of another one, so does
$\operatorname{\mathrm{supp}}\nu_{1}^{*\ell_{1}}$. Therefore by freeness of
$\Gamma$, we obtain that $h_{1}=h_{1}^{\prime}$. By induction we get
$h_{2}=h_{2}^{\prime},\cdots,h_{k}=h_{k}^{\prime}$. Therefore

$H(\nu_{1}^{*\ell_{1}}*\nu_{2}^{*j_{1}}*\nu_{1}^{*\ell_{2}}*\cdots*\nu_{2}^{*j_{k}})\geqslant\int
H(\nu_{1}^{*(\ell_{1}+\cdots+\ell_{k})})\mathrm{d}\nu_{2}^{*j_{1}}(g_{1})\cdots
d\nu_{2}^{*j_{k}}(g_{k})=H(\nu_{1}^{*(\ell_{1}+\cdots+\ell_{k})}).$

Combining with the concavity of $H$, we get

$\displaystyle h_{\mathrm{RW}}((1-\beta)\nu_{1}+\beta\nu_{2})$
$\displaystyle=$
$\displaystyle\lim_{\ell\rightarrow\infty}\frac{1}{\ell}H(((1-\beta)\nu_{1}+\beta\nu_{2})^{*\ell})$
$\displaystyle\geqslant$
$\displaystyle\lim_{\ell\rightarrow\infty}\frac{1}{\ell}\sum_{j}\binom{l}{j}(1-\beta)^{\ell-j}\beta^{j}H(\nu_{1}^{*(\ell-j)})$
$\displaystyle=$
$\displaystyle\lim_{\ell\rightarrow\infty}\frac{1}{\ell}\sum_{j}\binom{\ell}{j}(1-\beta)^{\ell-j}\beta^{j}(\ell-j)h_{\mathrm{RW}}(\nu_{1})$
$\displaystyle\geqslant$
$\displaystyle\lim_{\ell\rightarrow\infty}(1-\beta)\sum_{j}\binom{\ell-1}{j}(1-\beta)^{\ell-j-1}\beta^{j}h_{\mathrm{RW}}(\nu_{1})$
$\displaystyle=$ $\displaystyle(1-\beta)h_{\mathrm{RW}}(\nu_{1}).$

∎

As a consequence
$h_{\mathrm{RW}}(\nu)\geqslant(1-\beta)h_{\mathrm{RW}}(\nu_{1})\geqslant(1-\beta)(1-10C\beta)N$.
Moreover, the group generated by $\operatorname{\mathrm{supp}}\nu$ is Zariski
dense in $\mathrm{SL}_{3}(\mathbb{R})$ and satisfies exponential separation
property (actually discreteness). Let $\mu$ be the unique $\nu$-stationary
measure. By the dimension formula, i.e. Theorem 1.2, we have
$\dim\mu=\dim_{\mathrm{LY}}\mu.$

We admit the fact that $h_{\mathrm{F}}(\mu,\nu)=h_{\mathrm{RW}}(\nu)$ and
postpone the proof to the end of this section. We give an estimate on
$\dim_{\mathrm{LY}}\mu.$ Since $\psi_{s}(\lambda(\nu))\geqslant(1-100\beta)N,$
we obtain

$s(\lambda_{1}(\nu)-\lambda_{3}(\nu))\geqslant\psi_{s}(\lambda(\nu))\geqslant(1-100\beta)N.$

Then $\lambda_{1}(\nu)-\lambda_{3}(\nu)\geqslant\frac{1}{2s}N$ assuming
$\beta$ small enough. For a given $0<\epsilon<s-1,$ we have

$\psi_{s-\epsilon}(\lambda(\nu))=\psi_{s}(\lambda(\nu))-\epsilon(\lambda_{1}(\nu)-\lambda_{3}(\nu))\leqslant(1+100\beta)N-\frac{\epsilon}{2s}N.$

Now we take $\beta>0$ sufficiently small comparing to $\epsilon,$ we obtain

$\psi_{s-\epsilon}(\lambda(\nu))\leqslant(1+100\beta)N-\frac{\epsilon}{2s}N\leqslant(1-\beta)(1-10C\beta)N\leqslant
h_{\mathrm{RW}}(\nu).$

Therefore, $\dim\mu=\dim_{\mathrm{LY}}\mu\geqslant s-\epsilon.$

To complete the proof, it remains to show the identity between the Furstenberg
entropy and the random walk entropy.

######  Lemma 5.22.

Let $\nu$ be a finitely supported probability measure on $\Gamma$ such that
$G_{\nu}$ is Zariski dense. Let $\mu$ be the unique $\nu$-stationary measure
on ${\mathbb{P}}({\mathbb{R}}^{3}),$ then
$h_{\mathrm{F}}(\mu,\nu)=h_{\mathrm{RW}}(\nu).$

###### Proof.

As discussions after Definition 5.8, $X\setminus X_{\mathrm{nice}}$ is a
countable set and hence a $\mu$-null set. We consider the space
$B=\mathrm{SL}_{3}({\mathbb{R}})^{\times{\mathbb{N}}}$ endowing with the
probability measure $\nu^{{\mathbb{N}}}.$ For almost every
$b=(b_{1},b_{2},\cdots)\in B,$ we consider the Furstenberg boundary $\mu_{b}$
given by the weak* limit

$\lim_{n\to\infty}(b_{1}b_{2}\cdots b_{n-1})_{*}\mu,$

which is a Dirac measure [BQ16, Lemma 4.5] and denoted by $\delta_{\xi(b)}.$
Because $\mu=\int\delta_{\xi(b)}\mathrm{d}\nu^{{\mathbb{N}}}(b)$ and
$\mu(X\setminus X_{\mathrm{nice}})=0,$ there exists a conull set
$\Omega\subset B$ such that $\xi(b)\in X_{\mathrm{nice}}$ for every
$b\in\Omega.$

Note that $\nu$ also induces a random walk on the flag variety
$\mathcal{F}=\mathcal{F}({\mathbb{R}}^{3})$ with a unique stationary measure
$\nu_{\mathcal{F}}.$ We can also consider the Furstenberg boundary on the flag
variety. Then for almost every $b\in B,$ we can associate a Dirac measure
$(\mu_{\mathcal{F}})_{b}=\delta_{\xi_{\mathcal{F}}(b)}.$ Denote
$\xi_{\mathcal{F}}(b)=(\xi(b),\xi_{2}(b))\in\mathcal{F}({\mathbb{R}}^{3}),$
where $\xi_{2}(b)$ is a two dimensional subspace in ${\mathbb{R}}^{3}.$ Then
for every sequence of positive numbers $(\chi_{n}),$ the image of every
nonzero limit of $(\chi_{n}b_{0}b_{1}\cdots b_{n-1})$ in
$\mathrm{End}(\wedge^{2}\mathbb{R}^{3})$ is $\xi_{2}(b).$ We aim to show that
$\xi_{\mathcal{F}}(b)$ is uniquely determined by $\xi(b)$ for every
$b\in\Omega.$

Let $b\neq b^{\prime}\in\Omega,$ then $\xi(b)=\xi(b^{\prime})\in
X_{\mathrm{nice}}.$ Recall that every point in $X_{\mathrm{nice}}$ is uniquely
coding by an element in $\left\\{1,2,3\right\\}^{\mathbb{N}}.$ We know that
$b^{\prime}=(b_{1}^{\prime},b_{2}^{\prime},\cdots)$ and
$b=(b_{1},b_{2},\cdots)$ are different partitions of a same infinite word
$A_{i_{1}}A_{i_{2}}\cdots A_{i_{n}}\cdots$. Since $\nu$ is finitely supported,
for every $n\geqslant 0$ we can find $m_{n}\geqslant 0$ such that

$|b_{1}b_{2}\cdots b_{n}|\leqslant|b_{1}^{\prime}b_{2}^{\prime}\cdots
b_{m_{n}}^{\prime}|\leqslant|b_{1}b_{2}\cdots b_{n}|+N_{\nu},$

where $|\cdot|$ denotes the word length in the semigroup $\Gamma$ with respect
to the free generating set $\left\\{A_{1},A_{2},A_{3}\right\\},$ where
$N_{\nu}$ is a constant only depending on $\nu.$

Passing to a subsequence, we can assume that $(b_{1}^{\prime}\cdots
b_{m_{n}}^{\prime})^{-1}b_{1}\cdots b_{n}$ is a fixed element in
$\mathrm{SL}_{3}({\mathbb{R}}).$ Taking $\chi_{n}=\|b_{1}b_{2}\cdots
b_{n}\|^{-1}$ and passing to a subsequence if necessary, we also assume that
$(\chi_{n}b_{1}b_{2}\cdots b_{n})$ admits the nonzero limit in
$\mathrm{End}(\wedge^{2}\mathbb{R}^{3}).$ Meanwhile,
$(\chi_{n}b_{1}^{\prime}\cdots b_{m_{n}}^{\prime})$ converges to this limit
composing with an element in $\mathrm{SL}_{3}({\mathbb{R}}).$ Therefore, these
two limits has the same image. By our discussions on $\xi_{2},$ we obtain
$\xi_{2}(b^{\prime})=\xi_{2}(b)$ and hence

$\xi_{\mathcal{F}}(b)=\xi_{\mathcal{F}}(b^{\prime}).$ (5.6)

Now we consider a conditional measure at a $\mu$-full measure set
$\xi(\Omega)\subset{\mathbb{P}}({\mathbb{R}}^{3})$. For every
$x\in\xi(\Omega)$, choose an element $b\in\Omega$ with
$\xi(b)=x\in\mathbb{P}(\mathbb{R}^{3})$ and let

$\mu^{x}=\delta_{\xi_{\mathcal{F}}(b)}.$

Due to Eq. 5.6, we have

$\mu_{\mathcal{F}}=\int_{\Omega}\delta_{\xi_{\mathcal{F}}(b)}\mathrm{d}\nu^{\mathbb{N}}(b)=\int_{\mathbb{P}(\mathbb{R}^{3})}\mu^{x}\mathrm{d}\mu(x).$

So the family of measures $\mu^{x}$ is actually the disintegration of
$\mu_{\mathcal{F}}$ with respect to the natural projection
$\pi:\mathcal{F}\to{\mathbb{P}}({\mathbb{R}}^{3}).$ Hence we obtain the
trivial-fiber property. Now we can apply a same argument as in the end of the
proof 2.3 by using relative measure preserving property. We obtain the desired
equality between the Furstenberg entropy and the random walk entropy. ∎

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Yuxiang Jiao. Peking University, No.5 Yiheyuan Road, Haidian District,
Beijing, China.
email<EMAIL_ADDRESS>
Jialun Li. CNRS-Centre de Mathématiques Laurent Schwartz, École Polytechnique,
Palaiseau, France.
email<EMAIL_ADDRESS>

Wenyu Pan. University of Toronto, 40 St. George St., Toronto, ON, M5S 2E4,
Canada.
email<EMAIL_ADDRESS>

Disheng Xu. Great Bay University, Songshanhu International Community,
Dongguan, Guangdong, 523000, China.
email<EMAIL_ADDRESS>