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

Jialun Li, Wenyu Pan and Disheng Xu

###### Abstract

Let $\nu$ be a probability measure on $\mathrm{SL}_{3}(\mathbb{R})$ whose
support is finite and spans a Zariski dense subgroup. Let $\mu$ be the
associated stationary measure for the action on $\mathbb{P}(\mathbb{R}^{3})$.
Under the exponential separation condition on $\nu$, we prove that the
Hausdorff dimension of $\mu$ equals its Lyapunov dimension. This allows us to
study the limit sets of Anosov representations and Rauzy gasket. We show that
their Hausdorff dimensions are equal to the affinity exponents. In particular,
there is a dimension gap in the Barbot component.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Action of $\mathrm{GL}_{n}(\mathbb{R})$ on $\mathbb{P}(\mathbb{R}^{n})$
    2. 2.2 Projections in $\mathbb{R}^{3}$ and $\mathbb{P}(\mathbb{R}^{3})$
    3. 2.3 Decomposition of $\mathrm{SL}_{3}(\mathbb{R})$ for the composition maps $\pi_{V^{\perp}}g$
    4. 2.4 Linearize the projection of $L$-action
    5. 2.5 $q$-Adic partitions
    6. 2.6 Component measures and random measures
    7. 2.7 Regularity of stationary measures
    8. 2.8 Large deviation estimates
    9. 2.9 Coding, partitions of symbolic space and random words
    10. 2.10 Entropy
    11. 2.11 Entropy dimension of projection measure
  3. 3 Non-concentration on arithmetic sequences
  4. 4 Convolution and porosity
    1. 4.1 General results on entropy porosity
    2. 4.2 Decomposition of a general measure
    3. 4.3 Keeping porosity under projective transformation
    4. 4.4 Decomposition of random projection measures
    5. 4.5 Proof of 4.2
  5. 5 Entropy growth
    1. 5.1 Entropy growth under convolutions: Euclidean case
    2. 5.2 Convolution of measures on $\mathbb{P}(\mathbb{R}^{3})$ and measures on $L$: preparation
      1. 5.2.1 Linearization of projection of convolution measures
      2. 5.2.2 Decompose projection of convolution measures
      3. 5.2.3 General projection of convolution of measures
    3. 5.3 Entropy growth under convolutions: projective case
      1. 5.3.1 Find measures with small support that have positive entropy
      2. 5.3.2 Apply linearization
      3. 5.3.3 Replace measures by their good parts
      4. 5.3.4 Apply entropy growth of the Euclidean case
  6. 6 Exponential separation
  7. 7 Proofs of Theorem 1.10 and Theorem 1.11
    1. 7.1 Preparations: estimates for the entropy
    2. 7.2 Proof of Theorem 1.11
    3. 7.3 A Ledrappier-Young formula and the proof of 1.10 from 1.11.
  8. 8 Proof of 1.1 and 1.3
    1. 8.1 Proof of Theorem 1.3
    2. 8.2 Continuity and discontinuity of dimensions: proof of Theorem 1.1 and the continuity part of Theorem 1.3.
  9. A Auxiliary results
  10. B Spherical geometry
  11. C Proof of Lemma 5.6
  12. D Results in linear algebra
  13. E Fourier decay property of stationary measures on flag variety

##  1 Introduction

The Hausdorff dimension of the limit set on the visual boundary
$\partial\mathbb{H}^{3}$ of a discrete subgroup $\Gamma$ of
$\mathrm{Isom}_{+}(\mathbb{H}^{3})$ is a significant quantity of the group. It
has been extensively studied and is known to be related to the growth rate of
elements in $\Gamma$, the bottom of the eigenvalue of the Laplace operator and
the entropy of the geodesic flow on $\Gamma\backslash\mathbb{H}^{3}$, as
established by Sullivan in his works [Sul79], [Sul84]. The present paper is
concerned with the study of the Hausdorff dimension of limit sets in the real
projective plane $\mathbb{P}(\mathbb{R}^{3})$ of discrete subgroups of
$\mathrm{SL}_{3}(\mathbb{R})$. We usually denote by $\dim\cdot$ for the
Hausdorff dimension of a set and by $L(\cdot)$ for the limit set (see 1.2) of
projective actions of some discrete linear subgroups.

One major motivation in this paper is to generalize Bowen’s dimension rigidity
result [Bow79] to a different setting. Specifically, let $\Gamma$ be a surface
group, i.e. the fundamental group of a closed connected surface of genus at
least $2$, and let $\eta_{0}:\Gamma\rightarrow\mathrm{PSL}_{2}(\mathbb{R})$ be
a faithful representation onto a cocompact Fuchsian group of
$\mathrm{PSL}_{2}(\mathbb{R})$. We view $\mathrm{PSL}_{2}(\mathbb{R})$ as a
subgroup of $\mathrm{PSL}_{2}(\mathbb{C})$ and let $\iota_{0}$ be the natural
embedding of $\mathrm{PSL}_{2}(\mathbb{R})$ in $\mathrm{PSL}_{2}(\mathbb{C})$,
then $\iota_{0}\circ\eta_{0}$ is a representation of $\Gamma$ to
$\mathrm{PSL}_{2}(\mathbb{C})$. The limit set of
$\iota_{0}\circ\eta_{0}(\Gamma)$ is the smooth circle
$\mathbb{P}_{\mathbb{R}}^{1}$ in the sphere $\mathbb{P}_{\mathbb{C}}^{1}$. Let
$\rho:\Gamma\to\mathrm{PSL}_{2}(\mathbb{C})$ be a representation, which is a
small perturbation of $\iota_{0}\circ\eta_{0}.$ The limit set
$L(\rho(\Gamma))$ is known to be homeomorphic to a circle, which implies $\dim
L(\rho(\Gamma))\geqslant 1$. Bowen proved that $\dim L(\rho(\Gamma))=1$ if and
only if $\rho(\Gamma)$ is contained in a conjugate of
$\mathrm{SL}_{2}(\mathbb{R})$.111The actual result of Bowen is global, not
only for small perturbation. The version stated here is a weaker local
version.

##### Local dimension jump

We study a problem similar to that of Bowen: we consider the action of
$\mathrm{SL}_{3}(\mathbb{R})$ on $\mathbb{P}(\mathbb{R}^{3})$. A key
difference with Bowen’s setting is the non-conformality of the action, which
creates a lot of difficulties and also new phenomena. One of our main results
is a dimension jump phenomenon, which leads to a local dimension rigidity as
follows.

We consider $\mathrm{Hom}(\Gamma,\mathrm{SL}_{3}(\mathbb{R}))$, the set of
$\mathrm{SL}_{3}(\mathbb{R})$-representations of a surface group $\Gamma$, and
we deform representations here in a way similar to Bowen’s setting. More
precisely, recall that
$\eta_{0}:\Gamma\rightarrow\mathrm{PSL}_{2}(\mathbb{R})$ is a faithful
representation onto a cocompact Fuchsian group, and take an arbitrary lift
$\rho_{0}:\Gamma\rightarrow\mathrm{SL}_{2}(\mathbb{R})$ of $\eta_{0}$.222One
can do it by choosing the images of generators in the double cover
$\mathrm{SL}_{2}(\mathbb{R})$ of $\mathrm{PSL}_{2}(\mathbb{R})$. Denote by
$\iota$ the embedding from $\mathrm{SL}_{2}(\mathbb{R})$ to the upper left
corner of $\mathrm{SL}_{3}(\mathbb{R})$, then the composition
$\rho_{1}:=\iota\circ\rho_{0}$ is an element in
$\mathrm{Hom}(\Gamma,\mathrm{SL}_{3}({\mathbb{R}})).$ The action of
$\rho_{1}(\Gamma)$ preserves $\mathbb{P}(\mathbb{R}^{2})$ in
$\mathbb{P}(\mathbb{R}^{3})$, which is a smooth circle. Due to [Sul85], if
$\rho$ in $\mathrm{Hom}(\Gamma,\mathrm{SL}_{3}(\mathbb{R}))$ is sufficiently
close to $\rho_{1}$, i.e. its images on generators are close to that of
$\rho_{1}$, then $\rho(\Gamma)$ preserves a unique topological circle close to
$\mathbb{P}(\mathbb{R}^{2})$ in $\mathbb{P}(\mathbb{R}^{3})$, which turns out
to be the limit set $L(\rho(\Gamma))$ (see 1.2).

######  Theorem 1.1.

For every $\epsilon>0$, there exists a small neighborhood $O$ of $\rho_{1}$ in
$\mathrm{Hom}(\Gamma,\mathrm{SL}_{3}(\mathbb{R}))$ such that for any $\rho$ in
$O$ we have

  * •

either $\rho(\Gamma)$ acts reducibly, i.e. fixing a point or a projective line
in $\mathbb{P}(\mathbb{R}^{3})$;

  * •

or $\rho(\Gamma)$ is irreducible and

$|\dim L(\rho(\Gamma))-\frac{3}{2}|\leqslant\epsilon.$

In other words, once we perturb $\rho_{1}$ to any irreducible $\rho$, there is
a dimension jump with size about $1/2$, which does not occur in the conformal
case. A generic perturbation $\rho$ is irreducible. This is because reducible
representations form a proper subvariety in
$\mathrm{Hom}(\Gamma,\mathrm{SL}_{3}(\mathbb{R}))$ (cf. [BCLS15]).

If $\rho$ is reducible, the analysis becomes more subtle, and we have several
cases. If $\rho$ preserves a projective line in $\mathbb{P}(\mathbb{R}^{3})$,
then $L(\rho(\Gamma))$ is this projective line. Hence $\dim
L(\rho(\Gamma))=1$. If $\rho$ does not preserve a projective line, then
$L(\rho(\Gamma))$ is not Lipschitz [Bar10, Theorem 4.4]. And $L(\rho(\Gamma))$
is a graph-like set, and its dual representation $\rho^{\ast}$ preserves a
projective line which is actually $L(\rho^{\ast}(\Gamma))$. We hope to treat
this case in a forthcoming paper and conjecture that a similar dimension jump
phenomenon holds. We believe that one can distinguish different types of
$\rho$ in $O$ using the Hausdorff dimension of limit sets in the following
table. We conjecture that the entries with question marks are true.

Table 1: Hausdorff dimension of different types of representations

Reducibility type of $\rho\in O$ | $\dim L(\rho(\Gamma))$ | $\dim L(\rho^{\ast}(\Gamma))$
---|---|---
Semisimple | 1 | 1
point-irreducible | 1 |  $\approx 3/2$ (?)
line-irreducible |  $\approx 3/2$ (?) | $1$
Irreducible (in this paper) | $\approx 3/2$ | $\approx 3/2$

  * •

We follow the definition in [LLS21]. A reducible representation
$\rho:\Gamma\to\mathrm{SL}_{3}(\mathbb{R})$ is called point-irreducible if it
does not preserve any point in $\mathbb{P}(\mathbb{R}^{3})$; line-irreducible
if it does not preserve any projective line in $\mathbb{P}(\mathbb{R}^{3})$;
semisimple if it is a product of irreducible representations.

There are other settings where a similar dimension jump occurs. For example,
see [RS21] and references therein for results on graphs of Weierstrass-type
functions.

##### Anosov representations

The perturbed representation $\rho$ we considered in Theorem 1.1 is an example
of Anosov representations, which is a generalization of convex-cocompact
representations. The concept of Anosov representation was first introduced by
Labourie in [Lab06] to study Hitchin components of the representations of
surface groups. Here we use an equivalent definition which is due to [GW12],
[KLP18], [GGKW17], [BPS19], etc.

######  Definition 1.2.

Let $\Gamma$ be a hyperbolic group 333In this paper we always consider
finitely generated hyperbolic groups.and $\rho$ a representation from $\Gamma$
to $\mathrm{SL}_{3}(\mathbb{R})$. Then $\rho$ is called an Anosov
representation if there exist $C,c>0$ such that for any $\gamma\in\Gamma$, we
have

$\frac{\sigma_{1}(\rho(\gamma))}{\sigma_{2}(\rho(\gamma))}\geqslant\frac{1}{C}e^{c|\gamma|},$

where $\sigma_{i}(\rho(\gamma))$ are the singular values of $\rho(\gamma)$
with decreasing order and $|\gamma|$ is the word length with respect to a
fixed symmetric generating set. The limit set $L(\rho(\Gamma))$ of $\rho$ in
$\mathbb{P}(\mathbb{R}^{3})$ is defined to be the closure of the set of
attracting fixed points of elements in $\rho(\Gamma)$. We denote by
$\mathrm{HA}(\Gamma,\mathrm{SL}_{3}(\mathbb{R}))$ the subset of
$\mathrm{Hom}(\Gamma,\mathrm{SL}_{3}(\mathbb{R}))$ consisting of Anosov
representations.

If $\rho\in\mathrm{HA}(\Gamma,\mathrm{SL}_{3}(\mathbb{R}))$, then
$\rho(\Gamma)$ is discrete in $\mathrm{SL}_{3}(\mathbb{R})$. And any Anosov
representation is stable under small perturbation in
$\mathrm{Hom}(\Gamma,\mathrm{SL}_{3}(\mathbb{R}))$, hence
$\mathrm{HA}(\Gamma,\mathrm{SL}_{3}(\mathbb{R}))$ is open in
$\mathrm{Hom}(\Gamma,\mathrm{SL}_{3}(\mathbb{R}))$. It is not hard to check
that $\rho_{1}$ in Theorem 1.1 is an Anosov representation, and hence without
loss of generality, we can assume its small neighborhood
$O\subset\mathrm{HA}(\Gamma,\mathrm{SL}_{3}(\mathbb{R}))$.

##### Affinity exponent

The core of the proof of Theorem 1.1 is to show that if $\rho$ is an Anosov
representation, then $\dim L(\rho(\Gamma))$ equals its affinity exponent (or
affinity dimension).

The concept of affinity exponent of Anosov representations was introduced in
[PSW22], which is essentially a natural extension of the affinity exponent of
self-affine fractals due to Falconer [Fal88].

To motivate the definition, let us first recall the classical critical
exponent for Fuchsian groups and Kleinian groups. For a representation
$\rho:\Gamma\to\mathrm{PSL}(2,\mathbb{R})\,\,\,(\text{or}\,\,\,\mathrm{PSL}(2,\mathbb{C}))$,
the Poincaré series of the group $\rho(\Gamma)$ is defined by

$P_{\rho}(s)=\sum_{\gamma\in\Gamma}\Big{(}\frac{\sigma_{2}}{\sigma_{1}}\Big{)}^{s}(\rho(\gamma)).$

And the critical exponent $\delta(\rho(\Gamma))$ of $\rho(\Gamma)$ is defined
to be the critical exponent of series $P_{\rho}(s)$, i.e.

$\delta(\rho(\Gamma))={\sup}_{P_{\rho}(s)=\infty}s={\inf}_{P_{\rho(s)}<\infty}s.$

Sullivan [Sul79] established the beautiful equality between the critical
exponent, a dynamical invariant, and the Hausdorff dimension of the conic
limit set, a geometric invariant for a Kleinian group. It generalized the
earlier work by Patterson for Fuchsian groups [Pat76].

For a representation $\rho:\Gamma\to\mathrm{SL}_{3}(\mathbb{R})$, we can
similarly define a Poincaré series of the group $\rho(\Gamma)$ by

$P_{\rho}(s)=\left\\{\begin{aligned}
&\sum_{\gamma\in\Gamma}\Big{(}\frac{\sigma_{2}}{\sigma_{1}}\Big{)}^{s}(\rho(\gamma)),&0<s\leqslant
1;\\\
&\sum_{\gamma\in\Gamma}\Big{(}\frac{\sigma_{2}}{\sigma_{1}}\Big{)}(\rho(\gamma))\Big{(}\frac{\sigma_{3}}{\sigma_{1}}\Big{)}^{s-1}(\rho(\gamma)),&1<s\leqslant
2.\end{aligned}\right.$ (1.1)

We denote the critical exponent of $P_{\rho}(s)$ by $s_{\mathrm{A}}(\rho)$ and
call it the affinity exponent (or the affinity dimension). The affinity
exponent $s_{\mathrm{A}}(\rho)$ is always an upper bound of $\dim
L(\rho(\Gamma))\subset\mathbb{P}(\mathbb{R}^{3})$ if
$\rho:\Gamma\to\mathrm{SL}_{3}(\mathbb{R})$ is an Anosov representation
[PSW22]. In this paper, we show $s_{\mathrm{A}}(\rho)$ is also a lower bound
of $\dim L(\rho(\Gamma))$. Hence, we extend Sullivan’s theorem to Anosov
representations in $\mathrm{SL}_{3}(\mathbb{R})$.

######  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
$s_{\mathrm{A}}(\rho)$. Moreover, the exponent $s_{\mathrm{A}}(\rho)$ is
continuous on $\mathrm{HA}(\Gamma,\mathrm{SL}_{3}(\mathbb{R}))$.

The dimension jump phenomenon in Theorem 1.1, to some degree, is attributed to
the fact that the Hausdorff dimension of the limit set is not equal to the
affinity exponent when the representation preserves a plane. For higher
dimensional Lie groups, we can similarly define Anosov representations (cf.
[Can], etc.), associated limit sets, and affinity exponents. We pose the
following general question.

######  Question 1.4.

Let $\Gamma$ be a hyperbolic group. Which
$\rho\in\mathrm{HA}(\Gamma,\mathrm{SL}_{n}(\mathbb{R}))$ (can be reducible)
satisfies

$\dim L(\rho(\Gamma))=s_{\mathrm{A}}(\rho)?$

This question is in fact quite delicate even when one seeks a complete answer
for $\mathrm{SL}_{4}(\mathbb{R})$.

The continuity of the affinity exponent $s_{\mathrm{A}}(\rho)$ is essentially
due to [PS17] and [BCLS15]. For completeness, we provide a proof in Section
8.2. In the setting of Bowen’s result [Bow79], it is known that classical
critical exponents depend on representations analytically [Rue82]. It is
natural to ask the following question.

######  Question 1.5.

Does the affinity exponent $s_{\mathrm{A}}(\rho)$ depend on
$\rho\in\mathrm{HA}(\Gamma,\mathrm{SL}_{3}(\mathbb{R}))$ analytically?

When $\Gamma$ is a surface group, if
$\rho\in\mathrm{Hom}(\Gamma,\mathrm{SL}_{3}(\mathbb{R}))$ is in the Hitchin
component, that is, $\rho$ can be continuously deformed to a discrete and
faithful representation into
$\mathrm{SO}(2,1)(\cong\mathrm{SL}_{2}(\mathbb{R}))$; then the limit set
$L(\rho(\Gamma))$ is a $C^{1}$-circle [Lab06], and hence its Hausdorff
dimension is $1$. The representation $\rho_{1}$ in Theorem 1.1 is in the
Barbot component of $\mathrm{Hom}(\Gamma,\mathrm{SL}_{3}(\mathbb{R}))$. For an
irreducible Anosov representation in Barbot’s component, Barbot [Bar10] proved
that the limit set is not Lipschitz. Our result provides new knowledge for
Anosov representations in Barbot’s component. Bowen’s result is a global
result for all convex-cocompact representations. It is interesting to
investigate the following question.

######  Question 1.6.

Do all irreducible Anosov representations in Barbot’s component have an
affinity exponent greater than $1$?

In works like [Sul79], Patterson-Sullivan measures are central in computing
the Hausdorff dimension of limit sets. For Anosov representations, we also
have Patterson-Sullivan measures on the limit sets [Qui02], [PSW22]. Our
current computation of the Hausdorff dimension of limit sets does not use
these nice measures. It is interesting to explore the following question.

######  Question 1.7.

Is there a Patterson-Sullivan measure with the same Hausdorff dimension as the
limit set?

Related works on Hausdorff dimension of limit sets of Anosov representations
include [PSW22], [PSW21], [GMT19], [DK22], [KMO23]. In [Duf17], Dufloux
studied this problem for the non-conformal action of Schottky groups in
$\operatorname{PU}(1,n)$ on $\partial\mathbb{H}^{n}_{\mathbb{C}}$, the
boundary of the complex hyperbolic $n$-space. In [BHR19], Bárány-Hochman-
Rapaport studied self-affine contracting IFS on $\mathbb{R}^{2}$. Our proof
draws inspiration from their method. See also [HR21] and [Rap22] for further
development on this setting. There were a lot of more classical results in
this setting, we name a few, [Fal88], [HL95]. See [RS21] for the Hausdorff
dimension of the Weierstrass graph and [CP10] and [CPZ19] for more backgrounds
about non-conformal repellers.

##### Rauzy gasket

We also study the Rauzy gasket, a self-projective fractal set in
$\mathbb{P}(\mathbb{R}^{3})$, and we establish the identity between its
Hausdorff dimension and its affinity exponent.

Let $\Delta$ be the projectivization of $\\{(x,y,z):x,y,z\geqslant 0\\}$ in
${\mathbb{P}}({\mathbb{R}}^{3}).$ Let $\Gamma$ 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
$\Gamma\subset\mathrm{SL}_{3}(\mathbb{R})$, the semigroup $\Gamma$ acts on
$\mathbb{P}(\mathbb{R}^{3})$. Due to the choice of $\Delta$, the semigroup
$\Gamma$ preserves $\Delta$. The Rauzy gasket $X$ is the unique attractor of
the Rauzy semigroup, which can be defined formally as

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

For more background and history of the Rauzy gasket please see [DHS20] and
[JLPX].

Figure 1: Figure in [AS13]

The affinity exponent $s_{\mathrm{A}}(\Gamma)$ of the Rauzy gasket is the
critical exponent of the Poincaré series $P_{\Gamma}(s)$ defined in Eq. 1.1
using the Rauzy semigroup $\Gamma$. The following theorem confirms a folk-lore
conjecture of the Hausdorff dimension of Rauzy gasket.

######  Theorem 1.8.

Let $X,\Gamma$ be the Rauzy gasket and Rauzy semigroup respectively,
$s_{\mathrm{A}}(\Gamma)$ be the associated affinity exponent, then $\dim
X=s_{\mathrm{A}}(\Gamma)$.

######  Remark 1.9.

Recently Natalia Jurga also obtained 1.8 independently.

##### Hausdorff dimension and Lyapunov dimension of stationary measures

One key ingredient to prove 1.3 and 1.8 is to study the Hausdorff dimension
and Lyapunov dimension of stationary measures coming from random walks on
$\mathbb{P}(\mathbb{R}^{3})$, which is partially inspired by the methods in
[BHR19] and [HS17], which are developed from [Hoc14].

Recall that the Hausdorff dimension of a Borel probability measure $\mu$ on a
metric space $X$ is defined as

$\dim\mu:=\inf_{A\subset X,\mu(A)=1}\dim A.$

The Lyapunov dimension of measures is an analog of the affinity exponent of
sets. For our purpose, we only define it for stationary measures on
$\mathbb{P}(\mathbb{R}^{3})$ as follows.

A measure supported on $\mathrm{SL}_{3}(\mathbb{R})$ is called Zariski dense
if the group generated by $\operatorname{\mathrm{supp}}\nu$ is Zariski dense
in $\mathrm{SL}_{3}(\mathbb{R})$. Let $\nu$ be a finitely supported Zariski
dense probability measure on $\mathrm{SL}_{3}(\mathbb{R})$, and $\mu$ be the
unique $\nu$-stationary measure (Furstenberg measure) on
$\mathbb{P}(\mathbb{R}^{3})$ , that is, $\mu$ satisfies $\nu*\mu=\mu$, where
the convolution is induced by the projective action of
$\mathrm{SL}_{3}(\mathbb{R})$ on $\mathbb{P}(\mathbb{R}^{3})$. Let
$\lambda(\nu)=(\lambda_{1},\lambda_{2},\lambda_{3})$ be the Lyapunov vector of
$\nu$ with decreasing order and $\chi_{i}(\nu)=\lambda_{1}-\lambda_{i+1}$. As
$\nu$ is Zariski dense, we have $\chi_{2}(\nu)>\chi_{1}(\nu)>0$. Then the
Lyapunov dimension of $\mu$ can be defined as (following [KY79] and [DO80])

$\dim_{\mathrm{LY}}\mu=\left\\{\begin{aligned}
&\frac{h_{\mathrm{F}}(\mu,\nu)}{\chi_{1}(\nu)},&\text{ if
}h_{\mathrm{F}}(\mu,\nu)\leqslant\chi_{1}(\nu);\\\
&1+\frac{h_{\mathrm{F}}(\mu,\nu)-\chi_{1}(\nu)}{\chi_{2}(\nu)},&\text{
otherwise,}\end{aligned}\right.$

where $h_{\mathrm{F}}(\mu,\nu)$ is the Furstenberg entropy (see Eq. 2.81 for
definition).

In [LL23b] and [Rap21], the authors proved that
$\dim_{\mathrm{LY}}\mu\geqslant\dim\mu$. In this paper, we show the equality
actually holds under a mild separation assumption as in [HS17], [BHR19],
[Hoc14], etc. More precisely, a finitely supported measure $\nu$ on
$\mathrm{SL}_{3}(\mathbb{R})$ is called satisfying the exponential separation
condition, if there exist $C>0$ and $N>0$, for all $n\geqslant N$ and
$(g_{1},\cdots,g_{n}),\ (g_{1}^{\prime},\cdots,g_{n}^{\prime})$ in
$(\operatorname{\mathrm{supp}}\nu)^{\times n}$ with $g_{1}\cdots g_{n}\neq
g_{1}^{\prime}\cdots g_{n}^{\prime}$, then

$d(g_{1}\cdots g_{n},g_{1}^{\prime}\cdots g_{n}^{\prime})>C^{-n},$

where $d$ is a left $\mathrm{SL}_{3}(\mathbb{R})$-invariant and right
$\mathrm{SO}_{3}(\mathbb{R})$-invariant Riemannian metric on
$\mathrm{SL}_{3}(\mathbb{R})$. In particular, if
$\operatorname{\mathrm{supp}}\nu$ generates a discrete subgroup (for example,
$\nu$ is supported on an Anosov representation), then $\nu$ satisfies the
exponential separation condition.

######  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 stationary measure on
$\mathbb{P}(\mathbb{R}^{3})$. Then we have

$\dim\mu=\dim_{\mathrm{LY}}\mu.$

Ledrappier-Lessa [LL23b] proved 1.10 for measures $\nu$ supported on Hitchin
representations in $\mathrm{PSL}_{n}(\mathbb{R})$ for all $n\geqslant 2$. For
a Hitchin representation, the associated limit set and hence the support of
$\mu$ lies in a $C^{1+\beta}$-circle with $\beta>0$, which helps them to
obtain the dimension formula.

If we consider the action of $\mathrm{SL}_{2}(\mathbb{R})$ on
$\mathbb{P}(\mathbb{R}^{2})$, an early result is due to Ledrappier [Led83].
Later, Hochman-Solomayk advanced the knowledge in this setting [HS17]. In
particular, recall that the random walk entropy $h_{\mathrm{RW}}(\nu)$ of
$\nu$ is defined by

$h_{\mathrm{RW}}(\nu):=\lim_{n\rightarrow\infty}H(\nu^{*n})/n$ (1.2)

with $H(\nu^{*n})$ the Shannon entropy of a finite set. Hochman-Solomyak
obtained 1.10 by replacing the Furternberg entropy $h_{\mathrm{F}}(\mu,\nu)$
by the random walk entropy $h_{\mathrm{RW}}(\nu)$.

##### Dimension of the projection of stationary measures

Our strategy to show 1.10 is to combine a Ledrappier-Young formula in [Rap21]
and [LL23b], and a central result of dimension of projection measures which is
a projective version of that in [BHR19].

For any one-dimensional subspace $V\subset\mathbb{R}^{3}$, we consider the
orthogonal projection from $\mathbb{R}^{3}-V$ to $V^{\perp}$, which induces a
smooth projection (along projective lines passing through $\mathbb{P}(V)$)
from $\mathbb{P}(\mathbb{R}^{3})-\\{V\\}$ to $\mathbb{P}(V^{\perp})$. We
denote the smooth projection by $\pi_{V^{\perp}}$. For any probability measure
$\nu$ on $\mathrm{SL}_{3}(\mathbb{R})$, its inverse $\nu^{-}$ is defined by
$\nu^{-}(A):=\nu(\\{g^{-1}:g\in A\\})$ with a Borel set $A$. In particular,
$\nu$ is Zariski dense if and only if so is $\nu^{-}$.

Given $\nu$ a Zariski dense, finitely supported probability measure on
$\mathrm{SL}_{3}(\mathbb{R})$, let $\mu^{-}$ be the stationary measure of
$\nu^{-}$ on $\mathbb{P}(\mathbb{R}^{3})$. In [Rap21] and [LL23b], they obtain
a Ledrappier-Young formula, i.e. for $\mu^{-}$-a.e.
$V\in\mathbb{P}(\mathbb{R}^{3})$,

$\dim\mu=\dim\pi_{V^{\perp}}\mu+\frac{h_{\mathrm{F}}(\mu,\nu)-\chi_{1}(\nu)\dim\pi_{V^{\perp}}\mu}{\chi_{2}(\nu)};$
(1.3)

moreover, they established an upper bound:
$\dim\pi_{V^{\perp}}\mu\leqslant\min\\{1,h_{\mathrm{F}}(\mu,\nu)/\chi_{1}\\}$
(see Section 7.3 for more details).

In the following core theorem, we show that under the assumptions of Theorem
1.10, the upper bound above is actually also a lower bound. Then Theorem 1.10
is a direct corollary of the following theorem, (1.3) and the general fact
that $h_{\mathrm{F}}(\mu,\nu)\leqslant h_{\mathrm{RW}}(\nu)$.

######  Theorem 1.11.

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 stationary measure on
$\mathbb{P}(\mathbb{R}^{3})$. Then for $\mu^{-}$-a.e.
$V\in\mathbb{P}(\mathbb{R}^{3})$, we have

$\dim\pi_{V^{\perp}}\mu=\min\\{1,\frac{h_{\mathrm{RW}}(\nu)}{\chi_{1}}\\}.$

##### Convention

Throughout the paper, we use base-$q$ logarithm, where $q$ is a fixed integer
large enough that will be defined in Section 2.9.

##### Organization of the paper

After some preparations, the proof of 1.11 will occupy the most part the
paper, Section 3-Section 7. In Section 8, we will give the proof of 1.3 and
1.8 from 1.10 by admitting a variational principle proved in the second paper
[JLPX] joint with Yuxiang Jiao.

##### Acknowledgement

The authors would like to thank in particular Andres Sambarino for helping
discussions about the Barbot’s component, which motivates this work and the
observation of dimension jump. The authors would also like to thank François
Ledrappier for his courses on the Ledrappier-Young formula. We would like to
thank Cagri Sert, Federico Rodriguez-Hertz, 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.

### Notation

We summarize our main notation and conventions here.

$\log$ |  Logarithm with base $q$.
---|---
$q$ |  a fixed large integer defined in 2.30.
$V_{g}^{+}$ |  an attracting point in the projective space of $g\in\mathrm{SL}_{n}(\mathbb{R})$ defined in Lemma 2.4.
$H_{g}^{-}$ |  a repelling hyperplane in the projective space of $g\in\mathrm{SL}_{n}(\mathbb{R})$ defined in Lemma 2.4.
$b(g^{-},\epsilon),B(g^{+},\epsilon)$ |  repelling and attracting basins of $\mathbb{P}(\mathbb{R}^{n})$ defined in Lemma 2.4.
$E_{i},1\leqslant i\leqslant 3$ |  projectifications of standard basis of $\mathbb{R}^{3}$.
$\Pi_{V^{\perp}},\Pi_{V,W}$ |  linear projections defined in Definition 2.9.
$\Pi(V,W,W^{\prime})=\Pi_{V,W}|_{W^{\prime}}$ |  the restriction to $W^{\prime}$ of a linear projection $\Pi_{V,W}$ in Definition 2.9.
$\pi_{V^{\perp}},\pi_{V,W},\pi(V,W,W^{\prime})$ |  projective transformations induced by $\Pi_{V^{\perp}},\Pi_{V,W},\Pi(V,W,W^{\prime})$ in Definition 2.9.
$h_{V,g}$ |  the composition map $\pi_{V^{\perp}}\circ g|_{V^{\perp}}$ where $g\in\mathrm{SL}_{3}(\mathbb{R})$.
$U_{V}$, $L_{V}$ decomposition |  a decomposition of $\mathrm{SL}_{3}(\mathbb{R})$, where the parameter $V$ is an element in $\mathbb{P}(\mathbb{R}^{3})$.
$U$, $L$ decomposition |  $U_{V}$, $L_{V}$ decomposition with $V=E_{1}$.
$\mathrm{SL}_{2}^{\pm}(\mathbb{R})$ |  two by two matrices with determinant equal to $1$ or $-1$.
$\pi_{L_{V}}$ |  the projection from $U_{V}L_{V}$ to $L_{V}$.
$b(f_{\ell},r)$ |  the complement of the $r$-neighborhood of a hyperplane in $\mathbb{P}(\mathbb{R}^{3})$ 2.12.
$\mathcal{Q}_{n}$ |  $q$-adic decomposition of metric spaces as $\mathbb{R}$ and Lie groups.
${\mathbf{U}}(n),{\mathbf{I}}(n)$ |  random words defined in Section 2.9.
$g\mu$ |  the pushforward of $\mu$ under the action of $g$.
$\nu*\mu$ |  the convolution of $\nu$ on $\mathrm{SL}_{n}(\mathbb{R})$ and $\mu$ on $\mathbb{P}(\mathbb{R}^{n})$
$\theta*\tau$ |  the convolution of two measures on $\mathbb{R}$.
$[\theta.\mu]$ |  the projection $\pi_{E_{1}^{\perp}}$ of convolution of $\theta$ on $\mathrm{SL}_{3}(\mathbb{R})$ and $\mu$ on $\mathbb{P}(\mathbb{R}^{3})$. Section 5.2
$h_{\mathrm{F}}(\mu,\nu)$ |  the Furstenberg entropy defined in Eq. 2.81
$h_{\mathrm{RW}}(\nu)$ |  the random walk entropy of $\nu$ defined in Eq. 1.2.
$\sigma_{i}(g)$ |  Singular values of $g\in\mathrm{SL}_{n}(\mathbb{R})$.
$\chi_{i}(\nu)$ |  $\lambda_{1}(\nu)-\lambda_{i+1}(\nu)$, where $(\lambda_{1},\dots,\lambda_{n})$ are the Lyapunov exponents of $\nu$.
$\mathcal{F}=\mathcal{F}(\mathbb{R}^{3})$ |  Flag variety in $\mathbb{R}^{3}$, cf. Definition 3.3.
$C_{L}$ |  constant defined in Eq. 2.29.
$C_{p}$ |  constant defined in Eq. 2.47.
$\mathcal{C}$ |  subset of $\mathbb{P}(\mathbb{R}^{3})$ defined by $\mathbb{P}(\mathbb{R}^{3})-\mathbb{P}(E_{1}^{\perp})$ .

  * •

If $A$ and $B$ are two quantities, we write $A\ll B$ or $A=O(B)$ means that
there exists some constant $C>0$ (possibly depending on the ambient group and
the random walk $\nu$) such that $A\leqslant CB$.

  * •

We write $A\ll_{a}B$ and $A=O_{a}(B)$, if the constant $C$ depending on an
extra parameter $a$.

  * •

We $A\simeq B$, if $A\ll B$ and $B\ll A$. We write $A\simeq_{a}B$, if
$A\ll_{a}B$ and $B\ll_{a}A$.

##  2 Preliminaries

###  2.1 Action of $\mathrm{GL}_{n}(\mathbb{R})$ on
$\mathbb{P}(\mathbb{R}^{n})$

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
$\mathbb{R}^{n}$. Any element $g\in\mathrm{GL}_{n}(\mathbb{R})$ acts on
$\mathbb{R}^{n}$ and we denote the operator norm by $\|g\|$. Let
$e_{1},\cdots,e_{n}$ be the standard orthonormal basis of $\mathbb{R}^{n}$ and
$e_{1}^{*},\cdots,e_{n}^{*}$ be the dual basis of $(\mathbb{R}^{n})^{*}$. Let
$E_{i}=\mathbb{R}e_{i}$ be the corresponding element in
$\mathbb{P}(\mathbb{R}^{n})$ with $i=1,\cdots,n$ . Throughout the paper, we
always consider the following metric on the projective space
$\mathbb{P}(\mathbb{R}^{n})$ unless otherwise stated.

######  Definition 2.1.

The distance $d$ on $\mathbb{P}(\mathbb{R}^{n})$ is defined by

$d(\mathbb{R}v,\mathbb{R}w):=\frac{\|v\wedge w\|}{\|v\|\|w\|}\,\,\,\text{for
any}\,\,\,\mathbb{R}v,\mathbb{R}w\in\mathbb{P}(\mathbb{R}^{n}).$ (2.1)

When $A,B$ are two hyperplanes of $\mathbb{P}(\mathbb{R}^{n})$, we write
$d(A,B)$ for their Hausdorff distance.

######  Remark 2.2.

The metric $d$ is bi-Lipschitz equivalent to the standard
$\mathrm{SO}(n)$-invariant Riemannian metric $d_{R}$ on
$\mathbb{P}(\mathbb{R}^{n})$ (coming from the double cover of
$\mathbb{P}(\mathbb{R}^{n})$ by the unit sphere). More precisely, we have
$d(\mathbb{R}v,\mathbb{R}w)=\sin(d_{R}(\mathbb{R}v,\mathbb{R}w))$ for any
$\mathbb{R}v,\mathbb{R}w\in\mathbb{P}(\mathbb{R}^{n})$.

We will frequently consider the following subgroups of
$\mathrm{GL}_{n}(\mathbb{R})$:

$\displaystyle K:=$ $\displaystyle\mathrm{O}_{n}(\mathbb{R}),$ $\displaystyle
A:=$ $\displaystyle\\{a=\operatorname{diag}(a_{1},\ldots,a_{n}):a_{i}\neq
0\\},$ $\displaystyle A^{+}:=$
$\displaystyle\\{a=\operatorname{diag}(a_{1},\ldots,a_{n})\in
A:a_{1}\geqslant\ldots\geqslant a_{n}\\}.$

######  Definition 2.3.

For $g\in\mathrm{GL}_{n}(\mathbb{R})$, let
$\sigma_{1}(g)\geqslant\cdots\geqslant\sigma_{n}(g)$ be the singular values of
$g$ and $g=\tilde{k}_{g}a_{g}k_{g}\in KA^{+}K$ be the Cartan decomposition of
$g$ (singular value decomposition). Let $\epsilon>0$. Set

  * •

$\chi_{i}(g):=\log(\sigma_{1}(g)/\sigma_{i+1}(g)),1\leqslant i\leqslant n-1$;

  * •

$V_{g}^{+}:=\tilde{k}_{g}E_{1}\in\mathbb{P}(\mathbb{R}^{n})$, which is an
attracting point of $g$.

  * •

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

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

  * •

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

  * •

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

The Cartan projection of $g$ is defined to be
$\kappa(g):=(\log\sigma_{1}(g),\cdots,\log\sigma_{n}(g))$.

We now state some basic contracting properties of the action of
$\mathrm{GL}_{n}(\mathbb{R})$ on $\mathbb{P}(\mathbb{R}^{n})$.

######  Lemma 2.4.

For any $g\in\mathrm{GL}_{n}(\mathbb{R})$ and any $\epsilon>0$, we have

  * •

$g$ acts on $b(g^{-},\epsilon)$ by contraction: for any $x\neq y$ in
$b(g^{-},\epsilon)$, we have
$\frac{d(g(x),g(y))}{d(x,y)}\leqslant\frac{\sigma_{2}}{\epsilon^{2}\sigma_{1}}$;

  * •

$g(b(g^{-},\epsilon))\subset
B(g^{+},\frac{\sigma_{2}}{\epsilon^{2}\sigma_{1}})$;

  * •

when $g\in\mathrm{SL}_{2}(\mathbb{R})$, we have
$\frac{\sigma_{1}(g)}{\sigma_{2}(g)}=\|g\|^{2}$.

Please see [BQ16] or [Li22, Lemma 2.11] for the proof.

We state another useful lemma [BQ16, Lemma 14.2 ] for later use.

######  Lemma 2.5.

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

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

######  Definition 2.6.

For a bi-Lipschitz map $f$ between two metric spaces $(X,d_{X})$ and
$(Y,d_{Y})$, we say $f$ scales by $u>0$ with distortion $C>1$ if for any
$x\neq x^{\prime}\in X$,

$\frac{1}{C}\leqslant\frac{d_{Y}(f(x),f(x^{\prime}))}{ud_{X}(x,x^{\prime})}\leqslant
C.$ (2.2)

We fix an identification between $\mathbb{P}(\mathbb{R}^{2})$ and
$\mathbb{R}/\mathbb{Z}$:

$\displaystyle\iota:\mathbb{P}(\mathbb{R}^{2})$
$\displaystyle\to\mathbb{R}/\mathbb{Z}$
$\displaystyle\mathbb{R}(\cos\theta,\sin\theta)$
$\displaystyle\mapsto\frac{\theta}{\pi}.$ (2.3)

When an element $g\in\mathrm{GL}_{2}(\mathbb{R})$ acts on
$\mathbb{P}(\mathbb{R}^{2})$, we view $g$ as a diffeomorphism of
$\mathbb{R}/\mathbb{Z}$ by conjugating it by $\iota$ and denote by
$|g^{\prime}x|$ and $|g^{\prime\prime}x|$ the $1$st and $2$nd derivatives of
$\iota g\iota^{-1}$ at $\iota x$. The following are estimates of
$|g^{\prime}x|$ and $|g^{\prime\prime}x|$, which can be found in [HS17, Page
826]. For later usage, we introduce $\mathrm{SL}_{2}^{\pm}(\mathbb{R})$, the
group of two by two matrices with determinant equal to $1$ or $-1$.

######  Lemma 2.7.

For $g\in\mathrm{SL}_{2}^{\pm}(\mathbb{R})$ and
$x\in\mathbb{P}(\mathbb{R}^{2})$, we have

$\|g\|^{-2}\leqslant|g^{\prime}x|\leqslant\|g\|^{2},\
|g^{\prime\prime}x|\leqslant 4\|g\|^{2}.$

######  Lemma 2.8.

For all $0<\epsilon<1/3$, the following holds. Let $g$ be an element in
$\mathrm{SL}_{2}^{\pm}(\mathbb{R})$. Then for action of $g$ on
$b(g^{-},\epsilon)\subset\mathbb{P}(\mathbb{R}^{2})$, it scales by
$\|g\|^{-2}$ with distortion $10\epsilon^{-2}$. Actually, we have a more
precise estimate: for any $x\in b(g^{-},\epsilon)$,

$\displaystyle-2\log\|g\|\leqslant\log|g^{\prime}(x)|$
$\displaystyle\leqslant-2\log\|g\|-2\log(\epsilon),$
$\displaystyle|g^{\prime\prime}(x)|$
$\displaystyle\leqslant\frac{10}{\epsilon^{3}\|g\|^{2}}.$

###  2.2 Projections in $\mathbb{R}^{3}$ and $\mathbb{P}(\mathbb{R}^{3})$

######  Definition 2.9.

For any line $V$ and any hyperplanes $W,W^{\prime}$ of $\mathbb{R}^{3}$ such
that $V\not\subset W,W^{\prime}$, and any $g\in\mathrm{SL}_{3}(\mathbb{R})$
such that $g^{-1}V\not\subset V^{\perp}$, we denote

  * •

the linear projection on $\mathbb{R}^{3}$ with kernel $V$ and image $W$ by
$\Pi_{V,W}$;

  * •

the orthogonal projection with kernel $V$ by
$\Pi_{V^{\perp}}(=\Pi_{V,V^{\perp}})$;

  * •

the linear projection from $W^{\prime}$ to $W$ along $V$ (i.e. with kernel
$V$) by $\Pi(V,W,W^{\prime})$, i.e.
$\Pi(V,W,W^{\prime})=\Pi_{V,W}|_{W^{\prime}}$;

  * •

the projective transformations associated to $\Pi_{V,W}$, $\Pi_{V^{\perp}}$,
$\Pi(V,W,W^{\prime})$ by $\pi_{V,W}$, $\pi_{V^{\perp}}$, $\pi(V,W,W^{\prime})$
respectively;

  * •

the composition map $\pi_{V^{\perp}}\circ g|_{V^{\perp}}$ by
$h_{V,g}:\mathbb{P}(V^{\perp})\to\mathbb{P}(V^{\perp})$.

One important observation is the following geometric lemma on a decomposition
of the map $\pi_{V^{\perp}}\circ g$, which enables us to change the direction
of projection and apply the ergodic theory on projections.

######  Lemma 2.10.

For any $g\in\mathrm{SL}_{3}(\mathbb{R})$, $V\in\mathbb{P}(\mathbb{R}^{3})$
such that $g^{-1}V\not\subset V^{\perp}$, we have:

$\displaystyle\pi_{V^{\perp}}\circ g$
$\displaystyle{}=h_{V,g}\circ\pi_{{g^{-1}V},V^{\perp}},$ (2.4)
$\displaystyle\pi_{V^{\perp}}\circ g$
$\displaystyle{}=h_{V,g}\circ\pi({g^{-1}V},V^{\perp},(g^{-1}V)^{\perp})\circ\pi_{(g^{-1}V)^{\perp}}.$
(2.5)

###### Proof.

It suffices to show the corresponding equations for linear maps in
$\mathbb{R}^{3}$. We have

$\displaystyle\Pi_{V^{\perp}}\circ g$ $\displaystyle=$
$\displaystyle(\Pi_{V^{\perp}}\circ
g)|_{V^{\perp}}\circ\Pi_{g^{-1}V,V^{\perp}}$ $\displaystyle=$
$\displaystyle(\Pi_{V^{\perp}}\circ
g)|_{V^{\perp}}\circ\Pi({g^{-1}V},V^{\perp},(g^{-1}V)^{\perp})\circ\Pi_{g^{-1}V,(g^{-1}V)^{\perp}}.$

The first equality holds because both $\Pi_{V^{\perp}}\circ g$ and
$(\Pi_{V^{\perp}}\circ g)|_{V^{\perp}}\circ\Pi_{g^{-1}V,V^{\perp}}$ are linear
maps from $\mathbb{R}^{3}$ to $V^{\perp}$. They have the same kernel which is
$g^{-1}V$, and their restrictions to $V^{\perp}$, a complement to $g^{-1}V$,
are the same.

The second equality is obtained similarly by analyzing the kernels and images
of the linear maps. ∎

###  2.3 Decomposition of $\mathrm{SL}_{3}(\mathbb{R})$ for the composition
maps $\pi_{V^{\perp}}g$

In this subsection, we introduce a decomposition of the Lie group
$\mathrm{SL}_{3}(\mathbb{R})$, which describes the structure of the
composition maps $\pi_{V^{\perp}}g$ with $V\in\mathbb{P}(\mathbb{R}^{3})$ and
$g\in\mathrm{SL}_{3}(\mathbb{R})$.

We start with the decomposition for $E_{1}\in\mathbb{P}(\mathbb{R}^{3})$. Let
$U,L$ be two closed Lie subgroups of $\mathrm{SL}_{3}(\mathbb{R})$ defined by

$\displaystyle U$ $\displaystyle{}:=\left\\{\begin{pmatrix}\lambda^{2}&x\ \
y\\\
0&\lambda^{-1}\mathrm{Id}_{2}\end{pmatrix}:\lambda\in\mathbb{R}^{+},x,y\in\mathbb{R}\right\\},$
$\displaystyle L$ $\displaystyle{}:=\left\\{\begin{pmatrix}\det h&0\\\
n&h\end{pmatrix}:n\in\mathbb{R}^{2},h\in\mathrm{SL}_{2}^{\pm}(\mathbb{R})\right\\}.$

The group $U$ is a solvable subgroup, and the group $L$ is isomorphic to
$\mathrm{SL}_{2}^{\pm}(\mathbb{R})\ltimes\mathbb{R}^{2}$.

For a general $V\in\mathbb{P}(\mathbb{R}^{3})$, we define $U_{V}:=k^{-1}Uk$
and $L_{V}:=k^{-1}Lk$, where $k$ is any matrix in
$\mathrm{SO}_{3}(\mathbb{R})$ satisfying $kV=E_{1}$. We also define the
projection

$\begin{split}\pi_{L_{V}}:U_{V}L_{V}&\to L_{V}\\\
u\ell&\mapsto\ell.\end{split}$ (2.6)

The following lemma shows that $U_{V},L_{V},\pi_{L_{V}}$ are well-defined and
explains the roles that $U_{V}$ and $L_{V}$ play in the composition maps
$\pi_{V^{\perp}}g$ with $g\in\mathrm{SL}_{3}(\mathbb{R})$.

######  Lemma 2.11.

For any $V\in\mathbb{P}(\mathbb{R}^{3})$ and any
$k\in\mathrm{SO}_{3}(\mathbb{R})$ such that $kV=E_{1}$. The followings hold.

  1. 1.

The groups $U_{V}$ and $L_{V}$ are well-defined, i.e. independent of the
choice of $k\in\mathrm{SO}_{3}(\mathbb{R})$.

  2. 2.

The product $U_{V}L_{V}$ equals the set

$\\{g\in\mathrm{SL}_{3}(\mathbb{R}):g^{-1}V\notin V^{\perp}\\},$

and it is Zariski open and Zariski dense in $\mathrm{SL}_{3}(\mathbb{R})$.

  3. 3.

The map $\pi_{L_{V}}$ is well-defined, i.e., if
$u\ell=u^{\prime}\ell^{\prime}$ in $U_{V}L_{V}$, then
$\pi_{L_{V}}(u\ell)=\pi_{L_{V}}(u^{\prime}\ell^{\prime})$.

  4. 4.

For the group $U_{V}$, we have

$\\{g\in\mathrm{SL}_{3}(\mathbb{R}):\pi_{V^{\perp}}g=\pi_{V^{\perp}}id\\}=U_{V}\cup
U_{V}k^{-1}\begin{pmatrix}1&0\\\ 0&-\mathrm{Id}_{2}\end{pmatrix}k.$ (2.7)

  5. 5.

For every $g\in U_{V}L_{V}$, write $g=k^{-1}u\ell k=k^{-1}u\begin{pmatrix}\det
h&0\\\ n&h\end{pmatrix}k$ with $u\in U$, $\ell\in L$, $n\in\mathbb{R}^{2}$ and
$h\in\mathrm{SL}_{2}^{\pm}(\mathbb{R})$. We have

$h_{V,g}=h$

as elements in $\mathrm{PGL}_{2}(\mathbb{R})$, where $h_{V,g}$ is given as in
2.9.

  6. 6.

Consider the map

$\displaystyle\Phi:L_{V}$ $\displaystyle\to\\{\text{maps
from}\,\,\mathbb{P}(\mathbb{R}^{3})\,\,\text{to}\,\,\mathbb{P}(V^{\perp})\\}$
$\displaystyle\ell$ $\displaystyle\mapsto\pi_{V^{\perp}}\ell.$

If $\ell\neq\ell^{\prime}\in L$ satisfy $\Phi(\ell)=\Phi(\ell^{\prime})$, then
$\ell^{\prime}\ell^{-1}=k^{-1}\begin{pmatrix}1&0\\\
0&-\mathrm{Id}_{2}\end{pmatrix}k$.

###### Proof.

  1. 1.

For any $V\in\mathbb{P}(\mathbb{R}^{3})$, let $k_{0}$ be a matrix in
$\mathrm{SO}_{3}(\mathbb{R})$ satisfying $kV=E_{1}$. We have

$\\{k\in\mathrm{SO}_{3}(\mathbb{R}):kV=E_{1}\\}=\left\\{\begin{pmatrix}1&0\\\
0&k_{1}\end{pmatrix}k_{0}:k_{1}\in\mathrm{SO}_{2}(\mathbb{R})\right\\}.$

Any matrix of the form $\begin{pmatrix}1&0\\\ 0&k_{1}\end{pmatrix}$ with
$k_{1}\in\operatorname{SO}_{2}(\mathbb{R})$ normalizes the groups $U$ and $L$.
This shows that $U_{V}$ and $L_{V}$ are well-defined.

With the first statement available, it suffices to prove the remaining
statements for the case $V=E_{1}$.

  2. 2.

We prove

$UL=\\{g\in\mathrm{SL}_{3}(\mathbb{R}):g^{-1}E_{1}\notin E_{1}^{\perp}\\}.$
(2.8)

The direction $\subset$ can be checked by a straightforward computation using
the definitions of $U$ and $L$.

For the direction $\supset$, given any $g\in\mathrm{SL}_{3}(\mathbb{R})$ such
that $g^{-1}E_{1}\notin E_{1}^{\perp}$, the entry in the first row and first
column of $g^{-1}$ is non-zero. So the 2 by 2 submatrix $g^{\prime}$ of $g$ in
the lower right corner is non-degenerate. We can write
$g^{\prime}=\lambda^{-1}h$ with $\lambda>0$ and
$h\in\mathrm{SL}_{2}^{\pm}(\mathbb{R})$, and hence we can find $u\in U$ and
$\ell\in L$ such that $g=u\ell$.

$UL$ is Zariski open and Zariski dense because (2.8) shows that it is the
complement of the proper Zariski closed subvariety
$\\{g\in\mathrm{SL}_{3}(\mathbb{R}):g^{-1}E_{1}\in E_{1}^{\perp}\\}$.

  3. 3.

Suppose $u\ell=u^{\prime}\ell^{\prime}$ in $UL$. Then
$\ell^{\prime}\ell^{-1}=u^{\prime-1}u\in U\cap L=\\{\mathrm{Id}_{3}\\}$.

  4. 4.

We can check that the group $U$ is contained in the group (2.7) by a
straightforward computation using the definition of $U$.

Given any $g\in\mathrm{SL}_{3}(\mathbb{R})$ such that
$\pi_{E_{1}^{\perp}}g=\pi_{E_{1}^{\perp}}\mathrm{id}$, we have $gE_{1}=E_{1}$.
So $g\in UL$ by (2.8) and we can write $g=u\ell$ with $u\in U$ and $\ell\in
L$. The relation $E_{1}=gE_{1}=u\ell E_{1}$ gives $\ell E_{1}=E_{1}$. As ha
result, we can write $\ell=\begin{pmatrix}\det h&0\\\ 0&h\end{pmatrix}$ with
$h\in\mathrm{SL}_{2}^{\pm}(\mathbb{R})$. So

$\pi_{E_{1}^{\perp}}\mathrm{id}=\pi_{E_{1}^{\perp}}g=\pi_{{E_{1}}^{\perp}}\begin{pmatrix}\det
h&0\\\ 0&h\end{pmatrix}=h.$

This implies $h=\mathrm{Id}_{2}$ or $-\mathrm{Id}_{2}$.

  5. 5.

The fifth statement holds because

$\displaystyle\pi_{E_{1}^{\perp}}g|_{E_{1}^{\perp}}=\pi_{E_{1}^{\perp}}\begin{pmatrix}\det
h&0\\\ n&h\end{pmatrix}\bigg{|}_{E_{1}^{\perp}}=h.$

  6. 6.

Since we have

$\pi_{E_{1}^{\perp}}(\ell^{\prime}\ell^{-1})=\pi_{E_{1}^{\perp}}(\ell^{\prime})\circ\ell^{-1}=\pi_{E_{1}^{\perp}}(\ell)\circ\ell^{-1}=\pi_{E_{1}^{\perp}}id,$

the sixth statement follows from the fourth statement.

∎

A.1 fixes an identification between $\mathbb{P}(V^{\perp})$ with
$\mathbb{P}(\mathbb{R}^{2})$ for
$V\in\mathcal{C}=\mathbb{P}(\mathbb{R}^{3})-\mathbb{P}(E_{1}^{\perp})$. This
identification provides a unique choice of $k\in\mathrm{SO}_{3}(\mathbb{R})$.

###  2.4 Linearize the projection of $L$-action

In this part, we consider the point $E_{1}\in\mathbb{P}(\mathbb{R}^{3})$ and
the corresponding $UL$-decomposition. We fix a left $L$-invariant and right
$\operatorname{SO}(2)$-invariant Riemannian metric $d$ on $L$. For $\ell\in L$
and $x\in\mathbb{P}(\mathbb{R}^{3})$, we introduce the following notation

$[\ell(x)]:=\pi_{E_{1}^{\perp}}\ell(x)\in\mathbb{P}(E_{1}^{\perp})\cong\mathbb{P}(\mathbb{R}^{2}).$

We will always write $\ell=\begin{pmatrix}\det h&0\\\ n&h\end{pmatrix}$ with
$n\in\mathbb{R}^{2}$ and $h\in\mathrm{SL}_{2}^{\pm}(\mathbb{R})$ as the matrix
representation of $\ell$. Then

$[\ell(x)]=\mathbb{R}(h(b,c)^{t}+an),$ (2.9)

for any $x=\mathbb{R}(a,b,c)^{t}\in\mathbb{P}(\mathbb{R}^{3})$.

The matrix $h$ acts on
$\mathbb{P}(E_{1}^{\perp})\cong\mathbb{P}(\mathbb{R}^{2})$. Let
$h=\tilde{k}_{h}a_{h}k_{h}\in KA^{+}K$ be the Cartan decomposition of $h$, and
let $H_{h}^{-}\in\mathbb{P}(E_{1}^{\perp})$ be its repelling point given as in
2.3.

Observing that by Eq. 2.4 and 2.11, for $\ell=\begin{pmatrix}\det h&0\\\
n&h\end{pmatrix}$, we have

$\pi_{{E_{1}}^{\perp}}\ell=h\circ\pi_{\ell^{-1}E_{1},E_{1}^{\perp}}.$ (2.10)

(As in Figure 3, for any $x$, the projection
$\pi_{\ell^{-1}E_{1},E_{1}^{\perp}}x$ can be viewed as the intersection of the
projective line joint by $x$ and $\pi_{(\ell^{-1}E_{1})^{\perp}}x$ with the
projective line in red $\mathbb{P}(E_{1}^{\perp})$, and we decompose
$\pi_{E_{1}^{\perp}}\ell$ as the composition of $h$ with the projection
$\pi_{\ell^{-1}E_{1},E_{1}^{\perp}}$.)

Figure 2: $\pi_{{E_{1}}^{\perp}}\ell=h\circ\pi_{\ell^{-1}E_{1},E_{1}^{\perp}}$

Figure 3: The $[\ell]$-attracting region

The contracting region of $\pi_{E_{1}^{\perp}}\ell$ should be away from the
projective plane spanned by $\ell^{-1}{E_{1}}$ and $H_{h}^{-}$. At the same
time, note that we have the formula for the linear map

$\displaystyle\Pi_{\ell^{-1}E_{1},E_{1}^{\perp}}:\mathbb{R}^{3}$
$\displaystyle\to\mathbb{R}^{2}$ $\displaystyle(a,b,c)^{t}$
$\displaystyle\mapsto(b,c)^{t}+ah^{-1}n.$

This motivates the following construction.

Let $f_{h}$ be a linear form on $E_{1}^{\perp}\cong\mathbb{R}^{2}$ with kernel
$H_{h}^{-}$ of norm $1$. For any non-zero vector $v\in\mathbb{R}^{2}$, we have

$|f_{h}(v)|=\|v\|d(\mathbb{R}v,H_{h}^{-}).$ (2.11)

######  Definition 2.12.

Let $\ell=\begin{pmatrix}\det h&0\\\ n&h\end{pmatrix}\in L$. We define the
linear form $f_{\ell}$ on $\mathbb{R}^{3}$ by

$f_{\ell}(v):=f_{h}(\Pi_{\ell^{-1}E_{1},E_{1}^{\perp}}(v))\,\,\,\text{for}\,\,\,v\in\mathbb{R}^{3}.$

The kernel of $f_{\ell}$ is the plane spanned by
$\ell^{-1}{E_{1}}=\mathbb{R}(1,-h^{-1}n)^{t}$ 444We abuse the notation here
and $(1,-h^{-1}n)$ is actually the row vector $(1,-(h^{-1}n)^{t})$. and
$H_{h}^{-}\subset E_{1}^{\perp}\subset\mathbb{R}^{3}$.

For $\epsilon>0$, we define an $[\ell]$-attracting region by

$b(f_{\ell},\epsilon):=\left\\{x=\mathbb{R}v\in\mathbb{P}(\mathbb{R}^{3}):|f_{\ell}(v)|\geqslant\epsilon\|f_{\ell}\|\|v\|\right\\}.$
(2.12)

Figure 3 shows the region $b(f_{l},\epsilon)$, here the arrows in blue means
$h$ is repelling near $H_{h}^{-}\in\mathbb{P}(E_{1}^{\perp})$.

We summarize the structure of the rest of this subsection: in 2.13, we prove
the contracting properties of $\pi_{E_{1}^{\perp}}\ell(\cdot)=[\ell(\cdot)]$
on the region $b(f_{\ell},\epsilon)$; 2.14, 2.16 and 2.17 collect the
continuity properties of $\ell$, $f_{\ell}$ and $b(f_{\ell},\epsilon)$; we
also need an estimate of the diameter of a set acted by the projection of
$L$-action (2.18); 2.20 is the linearization of the map
$\pi_{E_{1}^{\perp}}\ell$.

We start with some basic estimates, which will be frequently used. Note that
for any $\ell\in L$,

$d(\ell^{-1}E_{1},E_{1}^{\perp})=d(\mathbb{R}(1,-h^{-1}n)^{t},E_{1}^{\perp})=1/\|(1,-h^{-1}n)^{t}\|.$
(2.13)

Hence

$(1+\|h^{-1}n\|^{2})^{1/2}=1/d(\ell^{-1}E_{1},E_{1}^{\perp}).$ (2.14)

For $x=\mathbb{R}v$ with $v=(a,b,c)^{t}$ and $\|v\|=1$, we have

  * •

$\|\Pi_{\ell^{-1}E_{1},E_{1}^{\perp}}(v)\|\leqslant
1/d(\ell^{-1}E_{1},E_{1}^{\perp});$ (2.15)

  * •

$\|f_{\ell}\|\in[1,1/d(\ell^{-1}E_{1},E_{1}^{\perp})];$ (2.16)

  * •

if $x\in b(f_{\ell},\epsilon)$, then

$\displaystyle\|\Pi_{\ell^{-1}E_{1},E_{1}^{\perp}}(v)\|\geqslant\epsilon\|v\|,$
(2.17) $\displaystyle d(x,\ell^{-1}{E_{1}})\geqslant\epsilon
d(\ell^{-1}{E_{1}},{E_{1}}^{\perp}).$ (2.18)

Eq. 2.15 holds because

$\|\Pi_{\ell^{-1}E_{1},E_{1}^{\perp}}(v)\|=\|(b,c)^{t}+ah^{-1}n\|\leqslant\|(1,-h^{-1}n)^{t}\|.$

Eq. 2.17 holds because

$\|\Pi_{\ell^{-1}E_{1},E_{1}^{\perp}}(v)\|\geqslant|f_{h}(\Pi_{\ell^{-1}E_{1},E_{1}^{\perp}}(v))|\geqslant\epsilon\|f_{\ell}\|\|v\|\geqslant\epsilon\|v\|.$

Eq. 2.16 is obtained as follows. For any $v\in E_{1}^{\perp}$,
$f_{\ell}(v)=f_{h}(v)$. As the norm of $f_{h}$ is $1$, $\|f_{\ell}\|\geqslant
1$. Meanwhile, by definition, we have
$\|f_{\ell}\|\leqslant\|f_{h}\|\cdot\|\Pi_{\ell^{-1}E_{1},E_{1}^{\perp}}\|$.
So we use Eq. 2.15 to get the upper bound.

We obtain LABEL:equ:_x_psi-1_{E_1} as follows:

$\displaystyle d(x,\ell^{-1}{E_{1}})$
$\displaystyle=\frac{\|(a,b,c)^{t}\wedge(1,-h^{-1}n)^{t}\|}{\|(1,-h^{-1}n)^{t}\|}$
$\displaystyle\geqslant
d(\ell^{-1}{E_{1}},E_{1}^{\perp})\|(b,c)^{t}+ah^{-1}n\|=d(\ell^{-1}{E_{1}},E_{1}^{\perp})\|\Pi_{\ell^{-1}E_{1},E_{1}^{\perp}}(v)\|,$

where for the inequality we project the vector
$(a,b,c)^{t}\wedge(1,-h^{-1}n)^{t}$ to the subspace generated by $E_{1}\wedge
E_{2},E_{1}\wedge E_{3}$ in $\wedge^{2}\mathbb{R}^{3}$ and due to Eq. 2.13.

######  Lemma 2.13.

Let $C_{1},C_{2}>2$. Let $\ell=\begin{pmatrix}\det h&0\\\ n&h\end{pmatrix}\in
L$ be such that $d(\ell^{-1}{E_{1}},E_{1}^{\perp})>1/C_{1}$. Then for
$x=\mathbb{R}v$ with $\|v\|=1$ in the $[\ell]$-attracting region
$b(f_{\ell},1/C_{2})$, we have

$\displaystyle\|\Pi_{E_{1}^{\perp}}\ell(v)\|=\|h\circ\Pi_{\ell^{-1}E_{1},E_{1}^{\perp}}(v)\|\geqslant\|h\|/C_{2},$
(2.19) $\displaystyle
d(\pi_{\ell^{-1}E_{1},E_{1}^{\perp}}(x),H_{h}^{-})\geqslant 1/C_{1}C_{2}.$
(2.20)

Moreover, for the action of $\pi_{E_{1}^{\perp}}\ell$ on
$b(f_{\ell},1/C_{2})$, we have

$\frac{d([\ell(x)],[\ell(x^{\prime})])}{d(x,x^{\prime})}\leqslant
2C_{1}C_{2}^{2}/\|h\|^{2}$ (2.21)

for any two distinct points $x,x^{\prime}$ in $b(f_{\ell},1/C_{2})$, and the
inclusion

$[\ell(b(f_{\ell},1/C_{2}))]\subset B(h^{+},C_{1}C_{2}/\|h\|^{2}).$ (2.22)

###### Proof.

We write $w=\Pi_{\ell^{-1}E_{1},E_{1}^{\perp}}(v)$. Hence
$\mathbb{R}w=\pi_{\ell^{-1}E_{1},E_{1}^{\perp}}(x)$.

Eq. 2.19 holds because

$\|hw\|\geqslant\|h\|\|w\|d(\mathbb{R}w,H_{h}^{-})=\|h\||f_{h}(w)|=\|h\||f_{\ell}(v)|\geqslant\|h\|/C_{2}.$

Here the first inequality follows from 2.5; the second one is by Eq. 2.11; the
last inequality is due to $\|f_{\ell}\|\geqslant 1$ (Eq. 2.16) and the
definition of $b(f_{\ell},1/C_{2})$.

As $\|v\|=1$, by Eq. 2.15, we have

$\|w\|=\|\Pi_{\ell^{-1}E_{1},E_{1}^{\perp}}(v)\|\leqslant
1/d(\ell^{-1}E_{1},E_{1}^{\perp})<C_{1}.$ (2.23)

Eq. 2.20 holds because

$d(\mathbb{R}w,H_{h}^{-})=|f_{h}(w)|/\|w\|\geqslant 1/C_{1}C_{2}.$

For Eq. 2.22, note that Eq. 2.10 gives

$[\ell(x)]=h(\mathbb{R}w).$

With Eq. 2.20 available, we apply 2.4 to the action of $h$ and obtain the
inclusion relation.

It remains to prove Eq. 2.21. Take any point $x^{\prime}\neq x$ in
$b(f_{\ell},1/C_{2})$. Write
$x^{\prime}=\mathbb{R}(a^{\prime},b^{\prime},c^{\prime})^{t}$ with
$\|(a^{\prime},b^{\prime},c^{\prime})^{t}\|=1$. We have

$\displaystyle d([\ell x],[\ell x^{\prime}])=$
$\displaystyle\frac{\|(h(b,c)^{t}+an)\wedge(h(b^{\prime},c^{\prime})^{t}+a^{\prime}n)\|}{\|h(b,c)^{t}+an\|\|h(b^{\prime},c^{\prime})^{t}+a^{\prime}n\|}$
$\displaystyle\leqslant$
$\displaystyle\left(\frac{C_{2}}{\|h\|}\right)^{2}\|(h(b,c)^{t}+an)\wedge(h(b^{\prime},c^{\prime})^{t}+a^{\prime}n)\|,$
(2.24)

where we use Eq. 2.19 to obtain the inequality.

Note that

$d(x,x^{\prime})=\|(a,b,c)^{t}\wedge(a^{\prime},b^{\prime},c^{\prime})^{t}\|\geqslant\max\\{\|(b,c)^{t}\wedge(b^{\prime},c^{\prime})^{t}\|,\|a(b^{\prime},c^{\prime})^{t}-a^{\prime}(b,c)^{t}\|\\}.$
(2.25)

Then the numerator of Eq. 2.24 is equal to

$\displaystyle\|h(b,c)^{t}\wedge h(b^{\prime},c^{\prime})^{t}+n\wedge
h(a(b^{\prime},c^{\prime})^{t}-a^{\prime}(b,c)^{t})\|$ $\displaystyle=$
$\displaystyle\|(b,c)^{t}\wedge(b^{\prime},c^{\prime})^{t}+h^{-1}n\wedge(a(b^{\prime},c^{\prime})^{t}-a^{\prime}(b,c)^{t})\|$
$\displaystyle\leqslant$
$\displaystyle\|(b,c)^{t}\wedge(b^{\prime},c^{\prime})^{t}\|+\|h^{-1}n\|\cdot\|a(b^{\prime},c^{\prime})^{t}-a^{\prime}(b,c)^{t}\|\leqslant
2C_{1}d(x,x^{\prime}),$ (2.26)

where the second line is due to $h\in\mathrm{SL}_{2}^{\pm}(\mathbb{R})$ and
the third line is due to $\|h^{-1}n\|\leqslant
1/d(\ell^{-1}E_{1},E_{1}^{\perp})\leqslant C_{1}$ ( see Eq. 2.14). This
verifies Eq. 2.21. ∎

######  Lemma 2.14.

There exist $\epsilon_{1}>0$ and $C_{3}>1$ such that for
$\ell=\begin{pmatrix}\det h&0\\\
n&h\end{pmatrix}\,\,\,\text{and}\,\,\,\ell^{\prime}=\begin{pmatrix}\det
h^{\prime}&0\\\ n^{\prime}&h^{\prime}\end{pmatrix}$ in $L$ with
$d(\ell,\ell^{\prime})\leqslant\epsilon_{1}$, we have

$\displaystyle\|id-h^{-1}h^{\prime}\|\leqslant
C_{3}d(\ell,\ell^{\prime}),\,\,\|h^{-1}n^{\prime}-h^{-1}n\|\leqslant
C_{3}d(\ell,\ell^{\prime}),$
$\displaystyle\|h^{-1}n-h^{\prime-1}n^{\prime}\|\leqslant
C_{3}d(\ell,\ell^{\prime})/d(\ell^{-1}{E_{1}},E_{1}^{\perp}),$ (2.27)
$\displaystyle\left|\log\|h^{\prime}\|-\log\|h\|\right|\leqslant
C_{3}d(\ell,\ell^{\prime}),$ $\displaystyle
d(\ell^{-1}{E_{1}},(\ell^{\prime})^{-1}{E_{1}})\leqslant
C_{3}d(\ell,\ell^{\prime})/d(\ell^{-1}{E_{1}},E_{1}^{\perp}).$

Moreover, if $\ell,\ell^{\prime}$ satisfy
$d(\ell^{\prime},\ell)\leqslant\min\\{\epsilon_{1},1/C_{3}\\}$, then

$\displaystyle d((\ell^{\prime})^{-1}E_{1},E_{1}^{\perp})\geqslant
d(\ell^{-1}E_{1},E_{1}^{\perp})/2.$

###### Proof.

Since $\ell$ is near $\ell^{\prime}$, we have $\det h=\det h^{\prime}$. For
any $\ell,\ell^{\prime}\in L$, as $d$ is a left $L$-invariant metric on $L$,
we have

$d(\ell,\ell^{\prime})=d(id,\ell^{-1}\ell^{\prime})=d\left(id,\begin{pmatrix}1&0\\\
h^{-1}n^{\prime}-h^{-1}n&h^{-1}h^{\prime}\end{pmatrix}\right).$

The metric $d$ and the distance induced by the norm are locally bi-Lipschitz.
Hence, we obtain the two inequalities of the first line.

We also have

$\displaystyle\|h^{-1}n-h^{\prime-1}n^{\prime}\|=\|h^{-1}n-h^{-1}n^{\prime}+h^{-1}n^{\prime}-h^{\prime-1}n^{\prime}\|$
$\displaystyle\ll$ $\displaystyle
d(\ell,\ell^{\prime})+\|(h^{-1}h^{\prime}-id)h^{\prime-1}n^{\prime}\|\ll
d(\ell,\ell^{\prime})(1+\|h^{\prime-1}n^{\prime}\|).$

Using Eq. 2.14, we obtain Eq. 2.27.

For $\log\|h\|$, we may suppose $\|h\|\geqslant\|h^{\prime}\|$. Then we have

$\log\|h\|-\log\|h^{\prime}\|=\log\left(1+\frac{\|h\|-\|h^{\prime}\|}{\|h^{\prime}\|}\right)\leqslant\frac{\|h\|-\|h^{\prime}\|}{\|h^{\prime}\|}\leqslant\frac{\|h-h^{\prime}\|}{\|h^{\prime}\|}\leqslant{\|h^{\prime-1}h-id\|}.$

For $d(\ell^{-1}{E_{1}},\ell^{\prime-1}{E_{1}})$, by definition and Eq. 2.27,
we have

$\displaystyle
d(\ell^{-1}{E_{1}},\ell^{\prime-1}{E_{1}})=\frac{\|(1,-h^{-1}n)^{t}\wedge(1,-h^{\prime-1}n^{\prime})^{t}\|}{\|(1,-h^{-1}n)^{t}\|\|(1,-h^{\prime-1}n^{\prime})^{t}\|}$

The numerator is bounded above by

$\displaystyle\|h^{-1}n-h^{\prime-1}n^{\prime}\|+\|h^{-1}n\wedge
h^{\prime-1}n^{\prime}\|$ $\displaystyle=$
$\displaystyle\|h^{-1}n-h^{\prime-1}n^{\prime}\|+\|h^{-1}n\wedge(h^{-1}n-h^{\prime-1}n^{\prime})\|\leqslant(1+\|h^{-1}n\|)\|h^{-1}n-h^{\prime-1}n^{\prime}\|,$

which yields the estimate of $d(\ell^{-1}E_{1},\ell^{\prime-1}E_{1})$.

For $d((\ell^{\prime})^{-1}E_{1},E_{1}^{\perp})$, using Eq. 2.14 and Eq. 2.27,
we have

$\displaystyle 1/d((\ell^{\prime})^{-1}E_{1},E_{1}^{\perp})$
$\displaystyle=\|(1,-h^{\prime-1}n^{\prime})\|\leqslant\|(1,-h^{-1}n)\|+\|h^{\prime-1}n^{\prime}-h^{-1}n\|$
$\displaystyle\leqslant
1/d(\ell^{-1}E_{1},E_{1}^{\perp})+C_{3}d(\ell,\ell^{\prime})/d(\ell^{-1}E_{1},E_{1}^{\perp})\leqslant
2/d(\ell^{-1}E_{1},E_{1}^{\perp}).$

∎

We need a lemma about the continuity of Cartan decomposition. For any
$\epsilon>0$, let $\mathcal{O}_{\epsilon}$ be the neighborhood of the identity
in $\mathrm{SL}_{2}(\mathbb{R})$ with radius $\epsilon$. Recall that $A$ is
the diagonal subgroup and
$A^{+}=\\{\mathrm{diag}(a_{1},a_{1}^{-1}):a_{1}\geqslant 1\\}$. Let
$\widetilde{A}^{\delta}=\\{\mathrm{diag}(a_{1},a_{1}^{-1}):a_{1}>1+\delta\\}$.

######  Lemma 2.15 (Effective Cartan decomposition, [GOS10], Theorem 1.6).

Given $\delta>0$, there exist $l_{0},\epsilon_{2}>0$ such that the following
holds. For any $\epsilon<\epsilon_{2}$ and any $g=k_{1}ak_{2}\in
K\widetilde{A}^{\delta}K$, we have

$\mathcal{O}_{\epsilon}g\mathcal{O}_{\epsilon}\subset(\mathcal{O}_{l_{0}\epsilon}\cap
K)k_{1}(\mathcal{O}_{l_{0}\epsilon}\cap
A^{+})ak_{2}(\mathcal{O}_{l_{0}\epsilon}\cap K).$

Since we will mostly deal with $h\in\mathrm{SL}_{2}^{\pm}(\mathbb{R})$ with
$\|h\|$ large, we fix $\delta=\log 2$ and the corresponding constants
$l_{0},\epsilon_{2}$ in the last lemma.

######  Lemma 2.16.

There exists $C_{4}>1$ such that the following holds. Let
$\ell=\begin{pmatrix}\det h&0\\\ n&h\end{pmatrix}\in L$ such that $\|h\|>2$.
The linear form $f_{\ell}$ is continuous with respect to $\ell$: if
$d(\ell,\ell^{\prime})\leqslant\min\\{\epsilon_{1},\epsilon_{2}/C_{3}\\}$,
then

$\|f_{\ell}-f_{\ell^{\prime}}\|\leqslant
C_{4}d(\ell,\ell^{\prime})/d(\ell^{-1}{E_{1}},E_{1}^{\perp}).$

###### Proof.

Write $\ell^{\prime}=\begin{pmatrix}\det h^{\prime}&0\\\
n^{\prime}&h^{\prime}\end{pmatrix}\in L$. When
$d(\ell,\ell^{\prime})\leqslant\epsilon_{1}$, 2.14 gives that

$\|\mathrm{id}-h^{-1}h^{\prime}\|<C_{3}d(\ell,\ell^{\prime})\leqslant\epsilon_{2}.$

Let $h=\tilde{k}_{h}a_{h}k_{h}$ and
$h^{\prime}=\tilde{k}_{h^{\prime}}a_{h^{\prime}}k_{h^{\prime}}$ be the Cartan
decomposition of $h$ and $h^{\prime}$ respectively. Then 2.15 gives

$d(k_{h},k_{h^{\prime}})\leqslant l_{0}d(h,h^{\prime}).$

Combining these two inequalities, we obtain

$\|f_{h}-f_{h^{\prime}}\|=\|e_{1}^{*}\circ k_{h}-e_{1}^{*}\circ
k_{h^{\prime}}\|\leqslant l_{0}d(h,h^{\prime})\ll d(\ell,\ell^{\prime}).$
(2.28)

For any $x=\mathbb{R}(a,b,c)^{t}$ with $\|(a,b,c)^{t}\|=1$, by 2.12, we obtain

$\displaystyle|f_{\ell}((a,b,c)^{t})-f_{\ell^{\prime}}((a,b,c)^{t})|=|f_{h}((b,c)^{t}+ah^{-1}n)-f_{h^{\prime}}((b,c)^{t}+a(h^{\prime})^{-1}n^{\prime})|$
$\displaystyle\leqslant$
$\displaystyle\|f_{h}-f_{h^{\prime}}\|\cdot\|(b,c)^{t}+ah^{-1}n\|+|f_{h^{\prime}}(ah^{-1}n-a(h^{\prime})^{-1}n^{\prime})|.$

To get the stated inequality, we use Eq. 2.28 and Eq. 2.15 to estimate the
first term, and Eq. 2.27 for the second term. ∎

We fix a large constant $C_{L}>0$ such that

$C_{L}>\max\\{1/\epsilon_{1},C_{3},C_{3}/\epsilon_{2},4C_{4}\\}$ (2.29)

where the constants $\epsilon_{1},\epsilon_{2},C_{3}$ and $C_{4}$ are given as
in 2.14, 2.15 and 2.16.

######  Lemma 2.17.

Fix any $C_{1},C_{2}>2$. Let $\ell=\begin{pmatrix}\det h&0\\\
n&h\end{pmatrix}\in L$ be such that $\|h\|>2$. If
$d(\ell^{-1}{E_{1}},{E_{1}}^{\perp})>1/C_{1}$, then for any
$\ell^{\prime}=\begin{pmatrix}\det h^{\prime}&0\\\
n^{\prime}&h^{\prime}\end{pmatrix}\in B(\ell,1/C_{L}C_{1}C_{2})\subset L$, we
have

$\displaystyle\frac{1}{2}\|h\|\leqslant\|h^{\prime}\|$ $\displaystyle\leqslant
2\|h\|,$ $\displaystyle b(f_{\ell},1/C_{2})$ $\displaystyle\subset
b(f_{\ell^{\prime}},1/2C_{2}).$ $\displaystyle
d((\ell^{\prime})^{-1}E_{1},E_{1}^{\perp})$ $\displaystyle>1/(2C_{1})$

###### Proof.

The first line follows from 2.14.

For the second line, due to 2.16 and $\|f_{\ell}\|\geqslant 1$ (Eq. 2.16), we
have

$\left|\|f_{\ell}\|-\|f_{\ell^{\prime}}\|\right|\leqslant
1/4C_{2}\leqslant\|f_{\ell}\|/4C_{2}.$

Hence $\|f_{\ell}\|\geqslant\|f_{\ell^{\prime}}\|/(1+1/4C_{2})$. Again, due to
2.16, for any $\mathbb{R}v\in b(f_{\ell},1/C_{2})$ with $\|v\|=1$, we have

$|f_{\ell^{\prime}}(v)|\geqslant|f_{\ell}(v)|-1/4C_{2}\geqslant\|f_{\ell}\|/C_{2}-1/4C_{2}\geqslant\|f_{\ell^{\prime}}\|/(C_{2}+1/4)-\|f_{\ell^{\prime}}\|/4C_{2}\geqslant\|f_{\ell^{\prime}}\|/2C_{2}.$

The third line follows from 2.14. ∎

For sets $E\subset L$ and $F\subset\mathbb{P}(\mathbb{R}^{3})$, let
$[E.F]=\\{[\ell(x)],\ \ell\in E,\ x\in F\\}$ be a subset in
$\mathbb{P}(\mathbb{R}^{2})$.

######  Lemma 2.18.

Fix any $C_{1},C_{2}>2$. Let $E$ be a subset in $L$ such that any
$\ell=\begin{pmatrix}\det h&0\\\ n&h\end{pmatrix}\in E$ satifies $\|h\|>2$ and
$\operatorname{\mathrm{diam}}E<1/C_{L}C_{1}C_{2}$. Suppose there exists
$\ell\in E$ such that $d(\ell^{-1}{E_{1}},E_{1}^{\perp})>1/C_{1}$. Let $F$ be
a subset in $[\ell]$-attracting region
$b(f_{\ell},1/C_{2})\subset\mathbb{P}(\mathbb{R}^{3})$. Then

$\operatorname{\mathrm{diam}}([E.F])\leqslant
16C_{L}C_{1}C_{2}^{2}(\operatorname{\mathrm{diam}}E+\operatorname{\mathrm{diam}}F)\|h\|^{-2}.$

###### Proof.

For any $\ell\in E$ and $x,x^{\prime}\in F$, 2.13 gives $d([\ell x],[\ell
x^{\prime}])\leqslant 2C_{1}C_{2}^{2}d(x,x^{\prime})\|h\|^{-2}$.

Let $\ell,\ell^{\prime}\in E$ and $x\in F$. Write
$\ell^{\prime}=\begin{pmatrix}\det h^{\prime}&0\\\
n^{\prime}&h^{\prime}\end{pmatrix}$ and $x=\mathbb{R}(a,b,c)^{t}$ with
$\|(a,b,c)^{t}\|=1$. We have

$d([\ell
x],[\ell^{\prime}x])=\frac{\|(h(b,c)^{t}+an)\wedge(h^{\prime}(b,c)^{t}+an^{\prime})\|}{\|h(b,c)^{t}+an\|\|h^{\prime}(b,c)^{t}+an^{\prime}\|}.$
(2.30)

Due to $x\in b(f_{\ell},1/C_{2})$, we obtain from 2.13 that

$\|h(b,c)^{t}+an\|\geqslant\|h\|/C_{2}.$

2.17 gives that $x\in b(f_{\ell^{\prime}},1/2C_{2})$ and

$\|h^{\prime}(b,c)^{t}+an^{\prime}\|\geqslant\|h^{\prime}\|/2C_{2}\geqslant\|h\|/4C_{2}.$

For the numerator of Eq. 2.30, we have

$\displaystyle\|(h(b,c)^{t}+an)\wedge(h^{\prime}(b,c)^{t}+an^{\prime})\|$
$\displaystyle=$ $\displaystyle\|(b,c)^{t}\wedge
h^{-1}h^{\prime}(b,c)^{t}+a^{2}h^{-1}n\wedge h^{-1}n^{\prime}+ah^{-1}n\wedge
h^{-1}h^{\prime}(b,c)^{t}-ah^{-1}n^{\prime}\wedge(b,c)^{t}\|$
$\displaystyle\leqslant$ $\displaystyle\|(b,c)^{t}\wedge
h^{-1}h^{\prime}(b,c)^{t}\|+\|a^{2}h^{-1}n\wedge
h^{-1}n^{\prime}\|+\|ah^{-1}n\wedge
h^{-1}h^{\prime}(b,c)^{t}-ah^{-1}n^{\prime}\wedge(b,c)^{t}\|.$

We use 2.14 to estimate the first two terms and get an upper bound
$2C_{L}d(\ell,\ell^{\prime})$. For the last term, we also use 2.14

$\displaystyle\|ah^{-1}n\wedge
h^{-1}h^{\prime}(b,c)^{t}-ah^{-1}n\wedge(b,c)^{t}\|$
$\displaystyle\leqslant\|h^{-1}n\wedge(h^{-1}h^{\prime}-id)(b,c)^{t}+(h^{-1}n-h^{-1}n^{\prime})\wedge(b,c)^{t}\|$
$\displaystyle\leqslant
C_{1}C_{L}d(\ell^{\prime},\ell)+C_{L}d(\ell^{\prime},\ell).$

Collecting all the terms, we obtain the lemma. ∎

Let $\ell=\begin{pmatrix}\det h&0\\\ n&h\end{pmatrix}\in L$. We are interested
in the map

$[\ell(\cdot)]=\pi_{E_{1}^{\perp}}\ell(\cdot)=h\circ\pi(\ell^{-1}E_{1},E_{1}^{\perp},(\ell^{-1}E_{1})^{\perp})\circ\pi_{(\ell^{-1}E_{1})^{\perp}}(\cdot).$
(2.31)

Now we are ready to explain how to linearize the map $[\ell(\cdot)]$ with
$\ell\in L$. We will approximate it by a composition of translation and
scaling map.

######  Definition 2.19.

For any $t\in\mathbb{R}$, we define the scaling map
$S_{t}:\mathbb{R}\to\mathbb{R}$ by $S_{t}x=q^{t}x$.

We have the identifications
$\mathbb{P}(V^{\perp})\cong\mathbb{P}(\mathbb{R}^{2})$ for
$V\in\mathbb{P}(\mathbb{R}^{3})$ in A.1 and
$\iota:\mathbb{P}(\mathbb{R}^{2})\cong\mathbb{R}/\mathbb{Z}$ (Section 2.1). In
the followings, for any $\star,\bullet\in\mathbb{P}(V^{\perp})$,
$d(\star,\bullet)$ means to view $\star,\bullet$ as elements in
$\mathbb{P}(\mathbb{R}^{2})$ and $d$ is the distance given as in 2.1;
$\star-\bullet$ means to view $\star,\bullet$ as elements in
$\mathbb{R}/\mathbb{Z}$ and $|\star-\bullet|$ is to compute their distance
using the rotational invariant distance on $\mathbb{R}/\mathbb{Z}$. We have
$d(\star,\bullet)=\sin(\pi|\star-\bullet|)$.

######  Lemma 2.20.

Fix any $C_{1},C_{2}>20$. Let $\ell_{0}=\begin{pmatrix}\det h_{0}&0\\\
n_{0}&h_{0}\end{pmatrix}\in L$ be such that $\|h_{0}\|>2$ and
$d(\ell_{0}^{-1}E_{1},E_{1}^{\perp})>1/C_{1}$. Then for any $\ell\in
B(\ell_{0},1/C_{L}C_{1}^{2}C_{2}^{2})\subset L$, and $x,x_{0}$ in the
$[\ell_{0}]$-attracting region $b(f_{\ell_{0}},1/C_{2})$ with
$d(x,x_{0})\leqslant(1/C_{1}C_{2})^{10}$, there exists a scaling map
$S_{-t(\ell_{0},x_{0})}:\mathbb{R}\to\mathbb{R}$ with

$\big{|}t(\ell_{0},x_{0})-2\log\|h_{0}\|\big{|}\leqslant 4\log(C_{1}C_{2})$
(2.32)

such that

$\displaystyle\big{|}[\ell(x)]-[\ell(x_{0})]-S_{-t(\ell_{0},x_{0})}\left(\pi_{(\ell_{0}^{-1}{E_{1}})^{\perp}}x-\pi_{(\ell_{0}^{-1}{E_{1}})^{\perp}}x_{0}\right)\big{|}$
(2.33) $\displaystyle\leqslant$ $\displaystyle
C_{L}(C_{2}C_{1})^{9}(d(\ell,\ell_{0})+d(x,x_{0}))d(x,x_{0})/\|h_{0}\|^{2}.$

######  Remark 2.21.

Let us explain the notions in Eq. 2.33. Let $\iota^{\prime}$ be the
identification between $\mathbb{R}/\mathbb{Z}$ and
$[-\frac{1}{2},\frac{1}{2})$. Let
$\pi_{\mathbb{R}/\mathbb{Z}}:\mathbb{R}\to\mathbb{R}/\mathbb{Z}$ be the
quotient map. It will be clear from the proof that
$\left|\pi_{(\ell_{0}^{-1}{E_{1}})^{\perp}}x-\pi_{(\ell_{0}^{-1}{E_{1}})^{\perp}}x_{0}\right|<\frac{1}{2}$.
By abusing the notation, we denote $\pi_{\mathbb{R}/\mathbb{Z}}\circ
S_{-t(\ell_{0},x_{0})}\circ\iota^{\prime}\left(\pi_{(\ell_{0}^{-1}E_{1})^{\perp}}x-\pi_{(\ell_{0}^{-1}E_{1})^{\perp}}x_{0}\right)$
by
$S_{-t(\ell_{0},x_{0})}\left(\pi_{(\ell_{0}^{-1}E_{1})^{\perp}}x-\pi_{(\ell_{0}^{-1}E_{1})^{\perp}}x_{0}\right)$.
Eq. 2.33 is to subtract elements in $\mathbb{R}/\mathbb{Z}$.

###### Proof.

Step 1: Approximate $[\ell_{0}(x)]-[\ell_{0}(x_{0})]$. We have

$\displaystyle[\ell_{0}(x)]-[\ell_{0}(x_{0})]=$ $\displaystyle
h_{0}\circ\pi(\ell_{0}^{-1}E_{1},E_{1}^{\perp},(\ell_{0}^{-1}E_{1})^{\perp})\circ\pi_{(\ell_{0}^{-1}{E_{1}})^{\perp}}x$
$\displaystyle-
h_{0}\circ\pi(\ell_{0}^{-1}E_{1},E_{1}^{\perp},(\ell_{0}^{-1}E_{1})^{\perp})\circ\pi_{(\ell_{0}^{-1}{E_{1}})^{\perp}}x_{0}.$

We will use a basic inequality in analysis to approximate the difference
gradually. For a $C^{2}$-map $g$ on $\mathbb{R}$ and two points $z<y$ in
$\mathbb{R}$, we have

$|gz-
gy-g^{\prime}y(z-y)|\leqslant|z-y|^{2}\sup\\{|g^{\prime\prime}w|:z\leqslant
w\leqslant y\\}.$ (2.34)

We want to apply Eq. 2.34 to

$\displaystyle g:=$ $\displaystyle h_{0},$ $\displaystyle z:=$
$\displaystyle\pi(\ell_{0}^{-1}E_{1},E_{1}^{\perp},(\ell_{0}^{-1}E_{1})^{\perp})\circ\pi_{(\ell_{0}^{-1}{E_{1}})^{\perp}}(x)=\pi_{\ell_{0}^{-1}E_{1},E_{1}^{\perp}}(x),$
$\displaystyle y:=$
$\displaystyle\pi(\ell_{0}^{-1}E_{1},E_{1}^{\perp},(\ell_{0}^{-1}E_{1})^{\perp})\circ\pi_{(\ell_{0}^{-1}{E_{1}})^{\perp}}(x_{0})=\pi_{\ell_{0}^{-1}E_{1},E_{1}^{\perp}}(x_{0}).$

Due to $x,x_{0}\in b(f_{\ell_{0}},1/C_{2})$, Eq. 2.20 in 2.13 implies that

$y,z\in b(h_{0}^{-},1/C_{1}C_{2}).$ (2.35)

Due to 2.8, we know

$|\log h_{0}^{\prime}(y)+2\log\|h_{0}\||\leqslant 2\log(C_{1}C_{2}).$ (2.36)

Using 2.7 and Eq. 2.21, we obtain

$\begin{split}d(z,y)&=\frac{d(h_{0}^{-1}[\ell_{0}(x_{0})],h_{0}^{-1}[\ell_{0}(x)])}{d([\ell_{0}(x_{0})],[\ell_{0}(x)])}\frac{d([\ell_{0}(x_{0})],[\ell_{0}(x)])}{d(x_{0},x)}d(x,x_{0})\\\
&\leqslant\|h_{0}\|^{2}\frac{2C_{1}C_{2}^{2}}{\|h_{0}\|^{2}}d(x,x_{0})=2C_{1}C_{2}^{2}d(x,x_{0}).\end{split}$
(2.37)

We have $[z,y]\subset b(h_{0}^{-},1/C_{1}C_{2})$ due to Eq. 2.35, Eq. 2.37 and
the hypothesis that $d(x,x_{0})\leqslant 1/(C_{1}C_{2})^{10}$. Hence, by
locally identifying $\mathbb{P}(\ell_{0}^{-1}E_{1})$ with $\mathbb{R}$, we can
use Eq. 2.34 and the estimate for $h_{0}^{\prime\prime}$ (2.8) to obtain

$\displaystyle\big{|}[\ell_{0}(x)]-[\ell_{0}(x_{0})]-h_{0}^{\prime}(y)(z-y)\big{|}\leqslant|z-y|^{2}\sup\\{|h_{0}^{\prime\prime}w|:z\leqslant
w\leqslant y\\}$ $\displaystyle\leqslant$
$\displaystyle|z-y|^{2}10(C_{1}C_{2})^{3}/\|h_{0}\|^{2}\leqslant
40C_{1}^{5}C_{2}^{7}d(x,x_{0})^{2}/\|h_{0}\|^{2}.$ (2.38)

Then we let

$\displaystyle\tilde{g}:=$
$\displaystyle\pi(\ell_{0}^{-1}E_{1},E_{1}^{\perp},(\ell_{0}^{-1}E_{1})^{\perp}),$
$\displaystyle z^{\prime}:=$
$\displaystyle\pi_{(\ell_{0}^{-1}{E_{1}})^{\perp}}x,$ $\displaystyle
y^{\prime}:=$ $\displaystyle\pi_{(\ell_{0}^{-1}{E_{1}})^{\perp}}x_{0}.$

We use Eq. 2.34 to approximate $\tilde{g}(z^{\prime})-\tilde{g}(y^{\prime})$.
Due to B.1 and 2.7, we have for any $w\in\mathbb{P}(E_{1}^{\perp})$,

$|\tilde{g}^{\prime\prime}(w)|\leqslant
4/d(\ell_{0}^{-1}E_{1},E_{1}^{\perp})\leqslant 4C_{1},\
1/C_{1}\leqslant|\tilde{g}^{\prime}(w)|\leqslant C_{1}.$ (2.39)

We apply LABEL:equ:_x_psi-1_{E_1} to $x,x_{0}$ and then B.2(2) yields

$d(z^{\prime},y^{\prime})=d(\pi_{(\ell_{0}^{-1}{E_{1}})^{\perp}}x,\pi_{(\ell_{0}^{-1}{E_{1}})^{\perp}}x_{0})\leqslant
C_{1}C_{2}d(x,x_{0}).$ (2.40)

Therefore, we use Eq. 2.34 to obtain

$|z-y-\tilde{g}^{\prime}(y^{\prime})(z^{\prime}-y^{\prime})|\leqslant|z^{\prime}-y^{\prime}|^{2}(4C_{1})\leqslant
4C_{1}^{3}C_{2}^{2}d(x,x_{0})^{2}.$ (2.41)

Combining with Section 2.4 and Eq. 2.36, we obtain

$|[\ell_{0}(x)]-[\ell_{0}(x_{0})]-S_{-t(\ell_{0},x_{0})}(\pi_{(\ell_{0}^{-1}{E_{1}})^{\perp}}x-\pi_{(\ell_{0}^{-1}{E_{1}})^{\perp}}x_{0})|\leqslant
44(C_{1}C_{2})^{7}d(x,x_{0})^{2}/\|h\|^{2}.$ (2.42)

Here $S_{-t(\ell_{0},x_{0})}=h_{0}^{\prime}(y)\tilde{g}^{\prime}(y^{\prime})$
is the contracting action with contracting rate of $h_{0}\circ\tilde{g}$ at
$y_{0}:=\pi_{(\ell_{0}^{-1}{E_{1}})^{\perp}}x_{0}$, and we obtain the estimate
of $t(\ell_{0},x_{0})$ (Eq. 2.32) using Eq. 2.36 and (2.39).

Before proceeding to the general case, we would like to point out a corollary
of Step 1: the sign of $[\ell_{0}(x)]-[\ell_{0}(x_{0})]$ is the same with the
sign of $z^{\prime}-y^{\prime}$ as elements in $[-\frac{1}{2},\frac{1}{2})$.
The reason is as follows. Note that $\tilde{g}$ scales
$\mathbb{P}(\mathbb{R}^{2})$ by $1$ with distortion $C_{1}$ (2.4). As
$d(z^{\prime},y^{\prime})$ is small (Eq. 2.40), we can deduce that the sign of
$z-y$ is the same as that of $z^{\prime}-y^{\prime}$. As
$[\ell_{0}(x)]=h_{0}(z)$, $[\ell_{0}(x_{0})]=h_{0}(y)$, and $h_{0}$ contracts
the minor arc $[z,y]$, we obtain the corollary.

Step 2: We need to distinguish two cases.

Case 1: Suppose that $\ell^{-1}{E_{1}}$ and $\ell_{0}^{-1}{E_{1}}$ are in the
same side of the great circle passing through $x,x_{0}$,555Take
representatives $u,u_{0}\in\mathbb{S}^{2}$ of $\ell^{-1}{E_{1}}$ and
$\ell_{0}^{-1}{E_{1}}$ with $d_{R}(u_{0},u)<\pi/2$. The two points
$\ell^{-1}{E_{1}}$ and $\ell_{0}^{-1}{E_{1}}$ are in the same side of a great
circle actually means $u,u_{0}$ are in the same side. In the figure, we write
$\ell^{-1}{E_{1}},\ell_{0}^{-1}E_{1}$ instead of $u,u_{0}$. see Fig. 5. We
write $\ell_{0}=\begin{pmatrix}\det h_{0}&0\\\ n_{0}&h_{0}\end{pmatrix}$,
$\ell=\begin{pmatrix}\det h&0\\\ n&h\end{pmatrix}$,
$x_{0}=\mathbb{R}(a_{0},b_{0},c_{0})^{t}$, $x=\mathbb{R}(a,b,c)^{t}$. We have

$\displaystyle d([\ell(x)],[\ell(x_{0})])-d([\ell_{0}(x)],[\ell_{0}(x_{0})])=$
$\displaystyle\frac{\|((b,c)^{t}\wedge(b_{0},c_{0})^{t}+h^{-1}n\wedge(a(b_{0},c_{0})^{t}-a_{0}(b,c)^{t})\|}{\|h(b,c)^{t}+an\|\|h(b_{0},c_{0})^{t}+a_{0}n\|}$
$\displaystyle-\frac{\|((b,c)^{t}\wedge(b_{0},c_{0})^{t}+h_{0}^{-1}n_{0}\wedge(a(b_{0},c_{0})^{t}-a_{0}(b,c)^{t})\|}{\|h_{0}(b,c)^{t}+an_{0}\|\|h_{0}(b_{0},c_{0})^{t}+a_{0}n_{0}\|}.$

The idea is to compare the difference term by term using the formula

$\displaystyle\frac{A}{BC}-\frac{A_{0}}{B_{0}C_{0}}=\frac{AB_{0}C_{0}-A_{0}BC}{BCB_{0}C_{0}}=\frac{(A-A_{0})B_{0}C_{0}+A_{0}(B_{0}-B)C_{0}+A_{0}B(C_{0}-C)}{BCB_{0}C_{0}},$

with

$\displaystyle
A=\|((b,c)^{t}\wedge(b_{0},c_{0})^{t}+h^{-1}n\wedge(a(b_{0},c_{0})^{t}-a_{0}(b,c)^{t})\|,$
$\displaystyle B=\|h(b,c)^{t}+an\|,\,\,\,C=\|h(b_{0},c_{0})^{t}+a_{0}n\|$

and $A_{0},B_{0},C_{0}$ the corresponding one with $h,n$ replaced by
$h_{0},n_{0}$. From Eq. 2.14, we obtain

$A_{0}\leqslant(1+\|h_{0}^{-1}n_{0}\|)d(x,x_{0})\leqslant
2d(x,x_{0})/d(\ell_{0}^{-1}E_{1},E_{1}^{\perp})\leqslant 2C_{1}d(x,x_{0}).$

Similarly, we have $A_{0}\leqslant 4C_{1}d(x,x_{0})$. From 2.17, we can apply
2.13 to obtain all possible pairs between $\ell,\ell_{0}$ and $x,x_{0}$. Due
to Eq. 2.19, we obtain

$B,B_{0},C,C_{0}\geqslant\frac{1}{4C_{2}}\|h_{0}\|.$

Then due to $\|h(b,c)^{t}+an\|\leqslant\|h\|(1+\|h^{-1}n\|)$, using Eq. 2.14
we obtain

$B,B_{0},C,C_{0}\leqslant 4C_{1}\|h_{0}\|.$

For the differences, by triangle inequality, Eq. 2.27 and Eq. 2.25, we have

$|A-A_{0}|\leqslant\|(h^{-1}n-h_{0}^{-1}n_{0})\wedge(a(b_{0},c_{0})^{t}-a_{0}(b,c)^{t})\|\leqslant\frac{C_{L}d(\ell,\ell_{0})d(x,x_{0})}{d(\ell_{0}^{-1}E_{1},E_{1}^{\perp})}\leqslant
C_{L}C_{1}d(\ell,\ell_{0})d(x,x_{0}).$

Similarly, using triangle inequality and 2.14

$|B-B_{0}|,|C-C_{0}|\leqslant\|h_{0}\|(\|id-
h_{0}^{-1}h\|+\|h^{-1}_{0}n-h^{-1}_{0}n_{0}\|)\leqslant
2C_{L}\|h_{0}\|d(\ell,\ell_{0}).$

In the end, we obtain from these computations that

$|d([\ell(x)],[\ell(x_{0})])-d([\ell_{0}(x)],[\ell_{0}(x_{0})])|\leqslant
2^{16}C_{L}(C_{1}C_{2})^{4}d(x,x_{0})d(\ell,\ell_{0})/\|h_{0}\|^{2}.$ (2.43)

By the position of $\ell^{-1}{E_{1}},\ell_{0}^{-1}{E_{1}},x,x_{0}$ on
$\mathbb{P}(\mathbb{R}^{3})$ (see Fig. 5) and Eq. 2.40,
$\pi_{\ell^{-1}E_{1}^{\perp}}x-\pi_{\ell^{-1}E_{1}^{\perp}}x_{0}$ and
$\pi_{\ell_{0}^{-1}E_{1}^{\perp}}x-\pi_{\ell_{0}^{-1}E_{1}^{\perp}}x_{0}$ have
the same sign as elements in $[-1/2,1/2)$ . The corollary of Step 1 holds for
both $[\ell(x)]-[\ell(x_{0})]$ and $[\ell_{0}(x)]-[\ell_{0}(x_{0})]$. Then
they have the same sign. We have the relation

$\left|([\ell(x)]-[\ell(x_{0})])-([\ell_{0}(x)]-[\ell_{0}(x_{0})])\right|\leqslant\big{|}|[\ell(x)]-[\ell(x_{0})]|-|[\ell_{0}(x)]-[\ell_{0}(x_{0})]|\big{|}.$
(2.44)

Due to $d([\ell(x)],[\ell(x_{0})])\leqslant 1/2$ and $|a-b|\leqslant
2|\sin(a)-\sin(b)|$ for $a,b\in[0,\pi/3]$, combined with Eq. 2.43, we obtain

$|([\ell(x)]-[\ell(x_{0})])-([\ell_{0}(x)]-[\ell_{0}(x_{0})])|\leqslant
2^{17}C_{L}(C_{1}C_{2})^{4}d(x,x_{0})d(\ell,\ell_{0})/\|h_{0}\|^{2}.$ (2.45)

Combined with Eq. 2.42 for both $\ell$ and $\ell_{0}$, we obtain the lemma due
to $C_{1},C_{2}$ large.

Figure 4: Case 1

Figure 5: Case 2

Case 2: Suppose $\ell^{-1}{E_{1}}$ and $\ell_{0}^{-1}{E_{1}}$ are in different
sides of the great circle passing through $x,x_{0}$. By spherical trigonometry
of the triangle $(x,\ell^{-1}E_{1},\ell_{0}^{-1}E_{1})$, we have

$\sin\angle_{x}(\ell^{-1}{E_{1}},\ell_{0}^{-1}{E_{1}})=\frac{\sin\angle_{\ell^{-1}{E_{1}}}(x_{,}\ell_{0}^{-1}{E_{1}})}{d(x,\ell_{0}^{-1}{E_{1}})}d(\ell^{-1}{E_{1}},\ell_{0}^{-1}{E_{1}})\leqslant\frac{d(\ell^{-1}{E_{1}},\ell_{0}^{-1}{E_{1}})}{d(x,\ell_{0}^{-1}{E_{1}})}.$

Due to 2.14 and LABEL:equ:_x_psi-1_{E_1}, we have

$\sin\angle_{x}(\ell^{-1}{E_{1}},\ell_{0}^{-1}{E_{1}})\leqslant\frac{C_{L}d(\ell,\ell_{0})/d(\ell_{0}^{-1}{E_{1}},{E_{1}}^{\perp})}{d(\ell_{0}^{-1}{E_{1}},{E_{1}}^{\perp})/C_{2}}\leqslant
C_{L}C_{1}^{2}C_{2}d(\ell,\ell_{0})\leqslant 1/C_{2}.$ (2.46)

Again by spherical trigonometry of the triangle $(x,x_{0},\ell^{-1}E_{1})$, we
have

$d(\pi_{(\ell^{-1}{E_{1}})^{\perp}}x,\pi_{(\ell^{-1}{E_{1}})^{\perp}}x_{0})=\sin\angle_{\ell^{-1}E_{1}}(x,x_{0})=d(x,x_{0})\frac{\sin\angle_{x}(\ell^{-1}{E_{1}},x_{0})}{d(x_{0},\ell^{-1}{E_{1}})}.$

Due to the hypothesis of case 2, we obtain
$\sin\angle_{x}(\ell^{-1}{E_{1}},x_{0})\leqslant\sin\angle_{x}(\ell^{-1}{E_{1}},\ell_{0}^{-1}{E_{1}})$
(the angle $\angle_{x}(\ell^{-1}{E_{1}},\ell_{0}^{-1}{E_{1}})$ must be acute
due to $d(x,\ell^{-1}E_{1})\geqslant d(\ell^{-1}E_{1},\ell_{0}^{-1}E_{1})$).
Combined with Eq. 2.46 and LABEL:equ:_x_psi-1_{E_1}, we obtain

$d(\pi_{(\ell^{-1}{E_{1}})^{\perp}}x,\pi_{(\ell^{-1}{E_{1}})^{\perp}}x_{0})\leqslant
d(x,x_{0})\frac{\sin\angle_{x}(\ell^{-1}{E_{1}},\ell_{0}^{-1}E_{1})}{d(x_{0},\ell^{-1}{E_{1}})}\leqslant
C_{L}C_{1}^{3}C_{2}^{2}d(\ell,\ell_{0})d(x,x_{0}).$

The same inequality also holds with $\ell$ replaced by $\ell_{0}$. Combining
with Eq. 2.42 for $\ell$ and the triangle inequality, using the fact that
$\theta\leqslant 2\sin\theta$ for any $\theta\in[0,\pi/2]$, we obtain

$\displaystyle\left|[\ell(x)]-[\ell(x_{0})]-S_{-t(\ell_{0},x_{0})}\left(\pi_{(\ell_{0}^{-1}{E_{1}})^{\perp}}x-\pi_{(\ell_{0}^{-1}{E_{1}})^{\perp}}x_{0}\right)\right|$
$\displaystyle\leqslant$
$\displaystyle\left|[\ell(x)]-[\ell(x_{0})]-S_{-t(\ell,x_{0})}\left(\pi_{(\ell^{-1}{E_{1}})^{\perp}}x-\pi_{(\ell^{-1}{E_{1}})^{\perp}}x_{0}\right)\right|+2q^{-t(\ell,x_{0})}d(\pi_{(\ell^{-1}{E_{1}})^{\perp}}x,\pi_{(\ell^{-1}{E_{1}})^{\perp}}x_{0})$
$\displaystyle+$ $\displaystyle
2q^{-t(\ell_{0},x_{0})}d(\pi_{(\ell_{0}^{-1}{E_{1}})^{\perp}}x,\pi_{(\ell_{0}^{-1}{E_{1}})^{\perp}}x_{0})$
$\displaystyle\leqslant$ $\displaystyle
44(C_{1}C_{2})^{7}d(x,x_{0})^{2}/\|h\|^{2}+(q^{-t(\ell,x_{0})}+q^{-t(\ell_{0},x_{0})})C_{L}C_{1}^{3}C_{2}^{2}d(\ell,\ell_{0})d(x,x_{0})$
$\displaystyle\leqslant$ $\displaystyle
C_{L}(C_{1}C_{2})^{7}(2d(\ell,\ell_{0})+88d(x,x_{0}))d(x,x_{0})/\|h_{0}\|^{2}.\qed$

###  2.5 $q$-Adic partitions

The $q$-adic level-$n$ partition for $n\in\mathbb{N}$ of $[0,1)$ is defined by

$\mathcal{Q}_{n}=\left\\{[\frac{k}{q^{n}},\frac{k+1}{q^{n}}):0\leqslant
k<q^{n},\ k\in\mathbb{Z}\right\\}.$

Using the canonical identification between $[0,1)$ and the circle
$\mathbb{P}(\mathbb{R}^{2})$, we can regard $\mathcal{Q}_{n}$ as a partition
of $\mathbb{P}(\mathbb{R}^{2})$. We write $\mathcal{Q}_{t}=\mathcal{Q}_{[t]}$
when $t\in\mathbb{R}^{+}$ is not an integer.

We also need similar partitions of the group $L$ defined in Section 2.3. By
Theorem 2.1 of [KRS12], for $q$ large enough, there exists a collection of
Borel sets $\\{Q_{n,i}\subset L:n\in\mathbb{N},i\in\mathbb{N}\\}$, having the
following properties:

  1. 1.

$L=\cup_{i\in\mathbb{N}}Q_{n,i}$ for every $n\in\mathbb{N}$;

  2. 2.

$Q_{n,i}\cap Q_{m,j}=\emptyset$ or $Q_{n,i}\subset Q_{m,j}$ for
$n,m\in\mathbb{N}$ with $n\geqslant m$ and any $i,j\in\mathbb{N}$;

  3. 3.

There exists a constant $C_{p}>1$ (independent of large $q$) such that for
every $n\in\mathbb{N}$ and $i\in\mathbb{N}$ there exists $\ell\in Q_{n,i}$
with

$B(\ell,C_{p}^{-1}q^{-n})\subset Q_{n,i}\subset B(\ell,q^{-n}),$ (2.47)

where $B(\ell,q^{-n})$ is measured using the left $L$-invariant metric on $L$.

For each $n\in\mathbb{N}$, denote by $\mathcal{Q}_{n}^{L}$ the partition
$\\{Q_{n,i}:i\in\mathbb{N}\\}$ of $L$. We will usually omit the superscript
when there is no ambiguity about which space to partition. Fix a left-
invariant Haar measure on $L$. In view of (2.47), there exists a constant
$C>1$ independent of $n$ such that

$1/C\leqslant q^{5n}\times\,\,\,\text{measure of any atom
in}\,\,\,\mathcal{Q}_{n}^{L}\leqslant C.$

Hence there exists a constant $C^{\prime}>0$ such that for every $n$ and any
atom $Q\in\mathcal{Q}_{n}^{L}$,

$\\#\\{Q^{\prime}\in\mathcal{Q}_{n+1}^{L}:Q^{\prime}\subset Q\\}\leqslant
C^{\prime}.$

###  2.6 Component measures and random measures

For a probability measure $\theta$ on a measure space $X$ and a measurable
subset $Y$ of $X$, define the normalized restriction measure by

$\theta_{Y}=\begin{cases}\frac{1}{\theta(Y)}{\theta|_{Y}},\,\,\,&\text{if}\,\,\,\theta(Y)>0,\\\
0,\,\,\,&\text{otherwise}.\end{cases}$

For a probability measure $\theta$ on $\mathbb{P}(\mathbb{R}^{2})$ (or $L$)
equipped with a partition $\mathcal{Q}_{n}$, we write $\mathcal{Q}_{n}(x)$ for
the unique partition element containing $x$. We define measure valued random
variables $\theta_{x,n}$ such that $\theta_{x,n}=\theta_{\mathcal{Q}_{n}(x)}$
with probability $\theta(\mathcal{Q}_{n}(x))$. For integers $n_{2}\geqslant
n_{1}$ and an event $\mathcal{U}$, we write

$\mathbb{P}_{n_{1}\leqslant i\leqslant
n_{2}}(\theta_{x,i}\in\mathcal{U})=\frac{1}{n_{2}-n_{1}+1}\sum_{n=n_{1}}^{n_{2}}\mathbb{P}(\theta_{x,n}\in\mathcal{U}).$

###  2.7 Regularity of stationary measures

Let $\Lambda$ be a finite set and let
$\nu=\sum_{i\in\Lambda}p_{i}\delta_{g_{i}}$ be a finitely supported
probability measure on $\mathrm{SL}_{3}(\mathbb{R})$ such that the group
generated by the support of $\nu$,
$\langle\operatorname{\mathrm{supp}}\nu\rangle$, is Zariski dense in
$\mathrm{SL}_{3}(\mathbb{R})$. We write $\nu^{*n}$ for the convolution, that
is, $\nu^{*n}$ is the law of the product $g_{1}\cdots g_{n}$, with i.i.d.
$g_{i}$ following the law of $\nu$.

Let $\mu$ be the unique associated stationary measure on
$\mathbb{P}(\mathbb{R}^{3})$ ([Fur73]), that is to say, we have

$\mu=\nu*\mu:=\sum_{i\in\Lambda}p_{i}(g_{i})_{*}\mu,$

where $(g_{i})_{*}\mu$ is the pushforward measure defined by
$(g_{i})_{*}\mu(E)=\mu(g_{i}^{-1}E)$ for any measurable set
$E\subset\mathbb{P}(\mathbb{R}^{3})$.

There are $3$ Lyapunov exponents
$\lambda_{1}\geqslant\lambda_{2}\geqslant\lambda_{3}$ of $\nu$ such that for
any $i=1,2,3$,

$\lambda_{i}=\lim_{n\to\infty}\int\frac{1}{n}\log\sigma_{i}(g_{i_{1}}\cdots
g_{i_{n}})d\nu(g_{i_{1}})\cdots d\nu(g_{i_{n}}).$

Since $\langle\operatorname{\mathrm{supp}}\nu\rangle$ is Zariski dense, we
have $\lambda_{1}>\lambda_{2}>\lambda_{3}$ (see for example [GR85], [GM89] ).
Set $\chi_{i}=\lambda_{1}-\lambda_{i+1}$ for $i=1,2$, and
$\lambda(\nu)=(\lambda_{1},\lambda_{2},\lambda_{3})$ to be the Lyapunov
vector.

We will also consider the measure $\nu^{-}$ on $\mathrm{SL}_{3}(\mathbb{R})$
defined by $\nu^{-}(g)=\nu(g^{-1})$. Let $\mu^{-}$ be the unique associated
stationary measure.

The results stated in this section and Section 2.8 hold under a more general
assumption that $\operatorname{\mathrm{supp}}\nu$ generates a Zariski dense
subgroup and that $\nu$ is of finite exponential moment. But for the interest
of the current paper, we will focus on the case when $\nu$ is finitely
supported.

Classical references of products of random matrices include [Fur63] and
[BL85]. We refer the reader to [BQ16]. We recall an equidistribution result
for $\mu$, $\mu^{-}$ ([GR85], [BL85, Chapter III, Theorem 4.3])

######  Lemma 2.22.

For any $x\in\mathbb{P}(\mathbb{R}^{3})$, as $n\rightarrow\infty$ we have

$\nu^{*n}*\delta_{x}\rightarrow\mu,\
(\nu^{-})^{*n}*\delta_{x}\rightarrow\mu^{-},$

and the convergence is uniform with respect to $x$.

We have the Guivarc’h regularity of stationary measures ( [Gui90], [BQ16, Prop
14.1])

######  Lemma 2.23.

There exist $C>0,\beta>0$ such that for any $n\geqslant 1$ and hyperplane $W$
in $\mathbb{P}(\mathbb{R}^{3})$

$\mu\\{x\in\mathbb{P}(\mathbb{R}^{3}):\ d(W,x)\leqslant q^{-n}\\}\leqslant
Cq^{-\beta n}.$

In particular, for any $V\in\mathbb{P}(\mathbb{R}^{3})$ and
$x\in\mathbb{P}(\mathbb{R}^{2})$

$(\pi_{V^{\perp}}\mu)\left(B(x,q^{-n})\right)\leqslant\mu\\{y:d((x,V),y)\leqslant
q^{-n}\\}\leqslant Cq^{-\beta n},$

where $(x,V)$ is the projective plane spanned by $x$ and $V$.

As a corollary, we have the following result.

######  Lemma 2.24.

Let $C_{1}>1$. For any $g\in\mathrm{SL}_{3}(\mathbb{R})$ and
$V\in\mathbb{P}(\mathbb{R}^{3})$ satisfy $d(g^{-1}V,V^{\perp})>1/C_{1},$ we
have for any for any $x\in\mathbb{P}(V^{\perp})$ and $r>0$,

$(\pi_{g^{-1}V,V^{\perp}}\mu)(B(x,r))\leqslant CC_{1}^{\beta}r^{\beta}.\,\,\,$

where $C,\beta$ are defined in Lemma 2.23.

###### Proof.

It follows from the proof of Lemma 2.10 that

$\pi_{g^{-1}V,V^{\perp}}=\pi(g^{-1}V,V^{\perp},(g^{-1}V)^{\perp})\circ\pi_{(g^{-1}V)^{\perp}}.$

2.23 holds for $\pi_{(g^{-1}V)^{\perp}}\mu$. Moreover, B.1 states that
$\pi(g^{-1}V,V^{\perp},(g^{-1}V)^{\perp})$ scales by $1$ with distortion
$C_{1}$. These imply the lemma. ∎

More generally, the Hölder regularity of stationary measures also holds for
irreducible representations ([Gui90], [BQ16, Proposition 10.1]). Let $p_{2}$
be the map from $\mathbb{P}(\mathbb{R}^{3})$ to the projective space of the
second symmetric power $\mathbb{P}(Sym^{2}(\mathbb{R}^{3}))$, defined by
sending $\mathbb{R}v$ to $\mathbb{R}v\otimes v$. It is
$\mathrm{SL}_{3}(\mathbb{R})$-equivariant. Let $(p_{2})_{*}\mu$ be the
pushfoward measure on $\mathbb{P}(Sym^{2}(\mathbb{R}^{3}))$. Then
$(p_{2})_{*}\mu$ is $\nu$-stationary and we have

######  Lemma 2.25.

There exist $C>0,\beta>0$ such that for any $n\geqslant 1$ and hyperplane $W$
in $\mathbb{P}(Sym^{2}(\mathbb{R}^{3}))$, we have

$(p_{2})_{*}\mu\\{x\in\mathbb{P}(Sym^{2}\mathbb{R}^{3}):\ d(W,x)\leqslant
q^{-n}\\}\leqslant Cq^{-\beta n}.$

###  2.8 Large deviation estimates

We first introduce several LDP (Large Deviation Principle) estimates for
random walks on $\mathrm{SL}_{3}(\mathbb{R})$. Set
$G=\mathrm{SL}_{3}(\mathbb{R})$ in this subsection. The following lemma can be
found in [LP82] and Proposition 14.3, Theorem 13.17 (iii), Theorem 13.11 (iii)
in [BQ16]. Recall that $\nu$ is finitely supported and Zariski dense and $\mu$
is the $\nu$-stationary measure on $\mathbb{P}(\mathbb{R}^{3})$.

######  Lemma 2.26.

For any $c>0$, there exist $C>0$ and $\epsilon>0$ such that for any
$n\in\mathbb{N}$, we have for any hyperplane $W$ in
$\mathbb{P}(\mathbb{R}^{3})$ and any
$V=\mathbb{R}v\in\mathbb{P}(\mathbb{R}^{3})$ with $\|v\|=1$,

$\displaystyle\nu^{*n}\\{g\in G:\ d(gV,W)\leqslant q^{-cn}\\}\leqslant
Cq^{-\epsilon n},$ (2.48) $\displaystyle\nu^{*n}\\{g\in G:\
d(V_{g}^{+},W)\leqslant q^{-cn}\\}\leqslant Cq^{-\epsilon n},$ (2.49)
$\displaystyle\nu^{*n}\\{g\in G:\ d(H_{g}^{-},V)\leqslant q^{-cn}\\}\leqslant
Cq^{-\epsilon n},$ (2.50) $\displaystyle\nu^{*n}\\{g\in G:\
d(V_{g}^{+},gV)\geqslant q^{-(\chi_{1}-c)n}\\}\leqslant Cq^{-\epsilon n},$
(2.51) $\displaystyle\nu^{*n}\\{g\in G:\
\|\frac{1}{n}\kappa(g)-\lambda(\nu)\|\geqslant c\\}\leqslant Cq^{-\epsilon
n},$ (2.52) $\displaystyle\nu^{*n}\\{g\in G:\
\frac{1}{n}|\log\|gv\|-\log\sigma_{1}(g)|\geqslant c\\}\leqslant Cq^{-\epsilon
n}.$ (2.53)

As a corollary, we have the following lemma as in [BQ16, Lemma 14.11].

######  Lemma 2.27.

For any $c>0$, there exist $C>0$ and $\epsilon>0$ such that for any
$m\leqslant n$, we have for any hyperplane $W$ in $\mathbb{P}(\mathbb{R}^{3})$
and any $V\in\mathbb{P}(\mathbb{R}^{3})$,

$\displaystyle\nu^{*n}\\{g\in G:\ d(gV,W)\leqslant q^{-cm}\\}\leqslant
Cq^{-\epsilon m},$ (2.54) $\displaystyle\nu^{*n}\\{g\in G:\
d(V_{g}^{+},W)\leqslant q^{-cm}\\}\leqslant Cq^{-\epsilon m},$ (2.55)
$\displaystyle\nu^{*n}\\{g\in G:\ d(V,H_{g}^{-})\leqslant q^{-cm}\\}\leqslant
Cq^{-\epsilon m}.$ (2.56)

###### Proof.

For Eq. 2.54, when $m=n$, this is Eq. 2.48. Since

$\nu^{*n}\\{g\in G:d(gV,W)\leqslant q^{-cm}\\}=\int_{G}\nu^{*m}\\{g\in
G:d(ghV,W)\leqslant q^{-cm}\\}d\nu^{*(n-m)}(h),$

then case $n\geqslant m$ follows.

For Eq. 2.55, we use triangle inequality

$d(V_{g}^{+},W)\geqslant d(gV,W)-d(V_{g}^{+},gV).$

Therefore

$\displaystyle\nu^{*n}\\{g\in G:\ d(V_{g}^{+},W)\leqslant q^{-cm}\\}$
$\displaystyle\leqslant$ $\displaystyle\nu^{*n}\\{g\in G:\ d(gV,W)\leqslant
2q^{-cm}\\}+\nu^{*n}\\{g\in G:\ d(V_{g}^{+},gV)\geqslant q^{-cm}\\}.$

We may assume $c<\chi_{1}/2$. Then $q^{-cm}\leqslant q^{-(\chi_{1}-c)n}$ and
the second term above can be estimated using Eq. 2.51. The first term can be
estimated using Eq. 2.54. Hence, we obtain Eq. 2.55.

For Eq. 2.56, due to $H_{g}^{-}=(V_{(g^{t})}^{+})^{\perp}$, it is equivalent
to prove

$\nu^{*n}\\{g\in G:\ d(V_{(g^{t})}^{+},W)\leqslant q^{-cm}\\}\leqslant
Cq^{-\epsilon m},$ (2.57)

with $W=V^{\perp}$, which is similar to Eq. 2.55. Let $\nu^{t}$ be the
probability measure on $G$ defined by $\nu^{t}(g)=\nu(g^{t})$. It is finitely
generated and Zariski dense. Eq. 2.57 holds as we have Eq. 2.55 for $\nu^{t}$.
∎

Next we use LDP of $\mathrm{SL}_{3}(\mathbb{R})$ to obtain LDP for the
$\mathrm{SL}_{2}(\mathbb{R})$-part in the $UL$-decomposition. It will allow us
to reduce the $\mathrm{SL}_{3}(\mathbb{R})$-action on
$\mathbb{P}(\mathbb{R}^{3})$ to the induced
$\mathrm{SL}_{2}(\mathbb{R})$-action on $\mathbb{P}(\mathbb{R}^{2})$ and
obtain the convergence of entropy for the induced action.

Recall that we have the projection map $\pi_{L}:UL\to L$ (see Eq. 2.6). In the
following lemma, it is important that it holds for all $n\gg\log C$, which
will be very useful in proving porosity in Section 4.

######  Lemma 2.28.

There exists $\beta>0$ such that the following holds. For any $C>1$, there
exists $N_{C}=O(\log C)>0$ such that we have for any $n>N_{C}$ and
$V\in\mathbb{P}(\mathbb{R}^{3})$

$\displaystyle\nu^{*n}\\{g\in G:\ d(g^{-1}V,V^{\perp})\leqslant 1/C\\}$
$\displaystyle\leqslant C^{-\beta},$ (2.58) $\displaystyle\nu^{*n}\\{g\in G:\
|\chi_{1}(h)-\chi_{1}(g)|\geqslant\log C\\}$ $\displaystyle\leqslant
C^{-\beta},$ (2.59)

where $h\in\mathrm{SL}_{2}^{\pm}(\mathbb{R})$ comes from the
$UL$-decomposition of $g$: $\pi_{L}(g)=\begin{pmatrix}\det h&0\\\
n&h\end{pmatrix}$ with $n\in\mathbb{R}^{2}$.

###### Proof.

Applying Eq. 2.54 to the measure $\nu^{-}$ and $W=V^{\perp}$, we obtain Eq.
2.58.

For any $g\in UL$, write $g=\begin{pmatrix}\lambda^{-2}&x&y\\\ 0&\lambda&0\\\
0&0&\lambda\end{pmatrix}\begin{pmatrix}\det h&0\\\ n&h\end{pmatrix}$. We
define $\tilde{h}:=\Pi_{E_{1}^{\perp}}g|_{E_{1}^{\perp}}=\lambda
h\in\mathrm{GL}(E_{1}^{\perp})$. We have
$q^{\chi_{1}(h)}=\|\tilde{h}\|^{2}/|\det\tilde{h}|.$ The determinant is given
by

$\begin{split}|\det\tilde{h}|=\lambda^{2}=|\langle
g^{-1}e_{1},e_{1}\rangle|=d(g^{-1}E_{1},E_{1}^{\perp})\|g^{-1}e_{1}\|.\end{split}$

The norm of $\|\tilde{h}v\|$ for $v\in E_{1}^{\perp}$ is given by

$\|\tilde{h}v\|=\|\Pi_{E_{1}^{\perp}}gv\|=\|(gv)\wedge
e_{1}\|=d(g\mathbb{R}v,E_{1})\|gv\|.$ (2.60)

Hence

$\|g\|d(gE_{2},E_{1})d(E_{2},H_{g}^{-})\leqslant
d(gE_{2},E_{1})\|ge_{2}\|\leqslant\|\tilde{h}\|\leqslant\|g\|.$ (2.61)

Therefore,

$\frac{(d(gE_{2},E_{1})d(E_{2},H_{g}^{-1})\|g\|)^{2}}{d(g^{-1}E_{1},E_{1}^{\perp})\|g^{-1}e_{1}\|}\leqslant
q^{\chi_{1}(h)}=\frac{\|\tilde{h}\|^{2}}{|\det{h}|}\leqslant\frac{\|g\|^{2}}{d(g^{-1}E_{1},E_{1}^{\perp})\|g^{-1}e_{1}\|}.$

It follows from 2.5 that

$d(E_{1},H^{-}_{g^{-1}})\leqslant\frac{\|g^{-1}e_{1}\|}{\|g^{-1}\|}\leqslant
1.$ (2.62)

We also have
$\|g\|^{2}/\|g^{-1}\|=\sigma_{1}(g)^{2}/\sigma_{1}(g^{-1})=\sigma_{1}(g)^{2}\sigma_{3}(g)=\sigma_{1}(g)/\sigma_{2}(g)=q^{\chi_{1}(g)}$.
Therefore

$q^{\chi_{1}(g)}(d(gE_{2},E_{1})d(E_{2},H_{g}^{-1}))^{2}\leqslant
q^{\chi_{1}(h)}\leqslant
q^{\chi_{1}(g)}\frac{1}{d(g^{-1}E_{1},E_{1}^{\perp})d(E_{1},H_{g^{-1}}^{-})}.$
(2.63)

Consider the elements $g\in G$ satisfying all the following inequalities:

  1. (i)

$d(E_{1},H^{-}_{g^{-1}})\geqslant q^{-C/8}$;

  2. (ii)

$d(g^{-1}E_{1},E_{1}^{\perp})\geqslant q^{-3C/4}$;

  3. (iii)

$d(E_{2},H_{g}^{-})\geqslant q^{-C/8}$;

  4. (iv)

$d(gE_{2},E_{1})\geqslant q^{-3C/8}$.

It follows from Eq. 2.54 and Eq. 2.56, that $\nu^{*n}$-measure of such $g$ is
greater than $1-C^{-\beta}$ for some $\beta>0$. Using (i), (ii) to Eq. 2.63,
we have $q^{\chi_{1}(h)-\chi_{1}(g)}\leqslant q^{C}$; using (iii), (iv) to Eq.
2.63, we have $q^{\chi_{1}(g)-\chi_{1}(h)}\leqslant q^{C}$. This finishes the
proof of Eq. 2.59. ∎

We need more LDP estimates of $h$.

######  Lemma 2.29.

For any $c>0$, there exist $C>0$ and $\epsilon>0$ such that for any
$x\in\mathbb{P}(E_{1}^{\perp})$ and $n\in\mathbb{N}$, we have

$\displaystyle\nu^{*n}\\{g\in G:\ \frac{1}{n}|\chi_{1}(h)-\chi_{1}n|\geqslant
c\\}$ $\displaystyle\leqslant Cq^{-\epsilon n},$ (2.64)
$\displaystyle\nu^{*n}\\{g\in G:\ d(H_{h}^{-},x)\leqslant q^{-cn}\\}$
$\displaystyle\leqslant Cq^{-\epsilon n},$ (2.65)
$\displaystyle\nu^{*n}\\{g\in G:\ d(V^{+}_{h},hx)\geqslant
q^{(-\chi_{1}+c)n}\\}$ $\displaystyle\leqslant Cq^{-\epsilon n},$ (2.66)

where $h\in\mathrm{SL}_{2}^{\pm}(\mathbb{R})$ comes from the
$UL$-decomposition of $g$: $\pi_{L}(g)=\begin{pmatrix}\det h&0\\\
n&h\end{pmatrix}$ with $n\in\mathbb{R}^{2}$.

###### Proof.

For Eq. 2.64, we compare $\chi_{1}(h)$ with $\chi_{1}(g)$ using Eq. 2.59. Then
use Eq. 2.52 for the LDP of $\chi_{1}(g)$.

For any $g\in UL$, write $g=\begin{pmatrix}\lambda^{-2}&x&y\\\ 0&\lambda&0\\\
0&0&\lambda\end{pmatrix}\begin{pmatrix}\det h&0\\\ n&h\end{pmatrix}$. We
define $\tilde{h}:=\Pi_{E_{1}^{\perp}}g|_{E_{1}^{\perp}}=\lambda
h\in\mathrm{GL}(E_{1}^{\perp})$. Take a unit vector $v$ in $x$. Due to 2.5,
that is

$d(H^{-}_{h},x)\geqslant\frac{\|\tilde{h}v\|}{\|\tilde{h}\|}-q^{-\chi_{1}(h)}.$

To obtain Eq. 2.65, it suffices to estimate the $\nu^{*n}$-measure of $g$ such
that

  1. (i)

either its corresponding $\tilde{h}$ satisfies $\|\tilde{h}v\|\leqslant
4q^{-cn}\|\tilde{h}\|$,

  2. (ii)

or $\chi_{1}(h)\leqslant 2cn$.

We can bound the $\nu^{*n}$-measure of $g$ satisfying (ii) by applying Eq.
2.64 to another sufficiently small constant $c$. For (i), by Eq. 2.61,

$\|\tilde{h}\|\leqslant\sigma_{1}(g).$

So due to Eq. 2.60, we have

$d(g\mathbb{R}v,E_{1})\frac{\|gv\|}{\sigma_{1}(g)}\leqslant\frac{\|\tilde{h}v\|}{\|\tilde{h}\|}.$

We use Eq. 2.48 and Eq. 2.53 to estimate the first term and the second term on
the left, respectively. This yields the estimate of the $\nu^{*n}$-measure of
$g$ satisfying (i), and hence we obtain Eq. 2.65.

Using 2.5, we have

$d(V^{+}_{h},hx)\leqslant\frac{q^{-\chi_{1}(h)}}{d(x,H^{-}_{h})}.$

Then Eq. 2.66 follows from Eq. 2.64 and Eq. 2.65. ∎

At the end of the subsection, we introduce two kinds of good subsets in $L$:

  * •

for any $n\in\mathbb{N}$ and $c>0$, set

$L(n,c):=\\{\ell\in L:\ d(\ell^{-1}E_{1},E_{1}^{\perp})\geqslant q^{-cn},\
\chi_{1}(h)\in[\chi_{1}-c,\chi_{1}+c]n\\};$ (2.67)

  * •

for any $t\geqslant 1$ and $C_{1}>0$, set

$L(t,C_{1}):=\\{\ell\in L:\ d(\ell^{-1}E_{1},E_{1}^{\perp})>1/C_{1},\ \|h\|\in
q^{t/2}[1/C_{p},C_{p}]\\},$ (2.68)

where $C_{p}$ is defined as in Eq. 2.47, and $h$ is the
$\mathrm{SL}_{2}^{\pm}(\mathbb{R})$ matrix of $\ell$: we have
$\ell=\begin{pmatrix}\det h&0\\\ n&h\end{pmatrix}$ with $n\in\mathbb{R}^{2}$.
For $\ell\in L(t,C_{1})$, we have $\|h\|\geqslant 2$. Moreover, it follows
from Eq. 2.58 and Eq. 2.64 that

$(\pi_{L}\nu^{*n})(L(n,c)^{c})\leqslant Cq^{-\epsilon(c)n}.$ (2.69)

###  2.9 Coding, partitions of symbolic space and random words

We consider the general coding scheme of the random walk. Consider the
symbolic space $\Lambda^{\mathbb{N}}$, the set of one-sided infinite words
over $\Lambda$. Let $\Lambda^{\ast}$ be the set of finite words over $\Lambda$
and for each $n\in\mathbb{N}$, $\Lambda^{n}$ be the set of words of length
$n$. For a word $\mathbf{i}=(i_{1},...,i_{n})\in\Lambda^{\ast}$, we define
$g_{\mathbf{i}}:=g_{i_{1}}\circ\cdots\circ g_{i_{n}}$. Similarly, define
$p_{\mathbf{i}}:=p_{i_{1}}\cdots p_{i_{n}}$.

The space $\Lambda^{\mathbb{N}}$ comes with the natural partitions into
level-$n$ cylinder sets. However, it will be more convenient to consider
general partitions into cylinders of varying lengths.

##### The partition $\Psi_{n}^{q}$ and the definition of $q$.

We consider a collection of words

$\Psi_{n}^{q}=\\{(i_{1},\cdots,i_{m})\in\Lambda^{\ast}:\chi_{1}(g_{i_{1}\ldots
i_{m}})\geqslant n>\chi_{1}(g_{i_{1}\ldots i_{j}})\ \text{ for all }j<m\\}.$

Different from the setting in [BHR19], the exponent $\chi_{1}$ here is not
monotone with respect to the length of the word $m$. However, note that the
measure $\nu$ has compact support, and the Cartan projection is subadditive
(see for example [Kas08, Lemma 2.3]). There exists a constant $c_{0}>0$,
depending only on the support of $\nu$, such that for every $n\in\mathbb{N}$
and for every $\mathbf{i}\in\Psi_{n}^{q}$,

$c_{0}q^{-n}\leqslant q^{-\chi_{1}(g_{\mathbf{i}})}\leqslant q^{-n}.$ (2.70)

If we take $q>1/c_{0}$, then for any $n\neq k$, we have

$\Psi^{q}_{n}\cap\Psi^{q}_{k}=\emptyset.$ (2.71)

By the definition of $\Psi_{n}^{q}$, no word in is a prefix of another word.
And the law of large numbers holds for $\nu$([Fur63]): we have for
$\nu^{\otimes\mathbb{N}}$-a.e. $\mathbf{i}\in\Lambda^{\mathbb{N}}$,

$\frac{\chi_{1}(g_{i_{1}\ldots i_{m}})}{m}\rightarrow\chi_{1}>0.$ (2.72)

Therefore, the set of $\mathbf{i}\in\Lambda^{\mathbb{N}}$ such that
$\chi_{1}(g_{i_{1}\ldots i_{m}})<n$ for all $m$ is of measure zero. Consider
the map $\Psi_{n}^{q}:\Lambda^{\mathbb{N}}\rightarrow\Lambda^{*}$ sending
$\mathbf{i}$ to its first $(m+1)$ coordinates with $m$ the least nonnegative
integer such that $\chi_{1}(g_{i_{1}\ldots i_{m}})\geqslant n$. It is defined
$\nu^{\otimes\mathbb{N}}$\- almost everywhere, and it gives a measurable
partition of $\Lambda^{\mathbb{N}}$.

######  Definition 2.30.

From now on, we fix the choice of an integer $q$ with
$q>\max\\{1/c_{0},C_{L},4C_{p}^{2}\\}$ (see Eq. 2.29 and Eq. 2.47), and the
logarithm is in base $q$, $\log=\log_{q}$.

######  Definition 2.31 (Random words $\mathbf{U}(n)$, $\mathbf{I}(n)$).

For any given $n\in\mathbb{N}$, we introduce the random words
${\mathbf{I}}(n)$ and ${\mathbf{U}}(n)$ taking values from subsets coming from
measurable partitions of $\Lambda^{\ast}$:

  * •

${\mathbf{I}}(n)$ is the random word taking values in
$\Psi_{n}(=\Psi_{n}^{q})$ according to the probability
$\nu^{\otimes\mathbb{N}}.$ i.e.

$\mathbb{P}({\mathbf{I}}(n)=\mathbf{i})=\begin{cases}p_{\mathbf{i}},\text{ if
}\mathbf{i}\in\Psi_{n}\\\ 0,\text{ otherwise}.\end{cases}$

  * •

${\mathbf{U}}(n)$ is the random word taking values from $\Lambda^{n}$
according to the probability $\nu^{\otimes n}$.

For each $n\in\mathbb{N}$ and for any measurable set $\mathcal{U}\subset G$,
we have

$\nu^{*n}\\{g\in
G:g\in\mathcal{U}\\}=\mathbb{P}_{i=n}\\{{\mathbf{U}}(i):g_{{\mathbf{U}}(i)}\in\mathcal{U}\\}.$
(2.73)

Due to the stationarity of the measure $\mu$ and that ${\mathbf{I}}(n)$ is
defined by stopping time, using the martingale property, we have for any
$n\in\mathbb{N}$

$\mu=\mathbb{E}_{j=n}(g_{{\mathbf{I}}(j)}\mu).$ (2.74)

The following lemma is the relation between ${\mathbf{U}}(i)$ and
${\mathbf{I}}(i)$.

######  Lemma 2.32.

There exist $C_{I}>1$ and $\beta>0$ such that the following holds. Let
$\mathcal{U}\subset\Lambda^{\ast}$ be a set of words. Suppose that for some
$\epsilon>0$ and $n(\epsilon)\in\mathbb{N}$, we have for every $n\geqslant
n(\epsilon)$,

$\mathbb{P}_{1\leqslant i\leqslant
n}\\{{\mathbf{U}}(i)\in\mathcal{U}\\}>1-\epsilon.$

Then for every $n\geqslant n(\epsilon)$,

$\mathbb{P}_{1\leqslant i\leqslant
n}\\{{\mathbf{I}}(i)\in\mathcal{U}\\}>1-C_{I}\epsilon/2-O(q^{-\beta n}).$

###### Proof.

The subtlety of the proof lies in that $\chi_{1}(g_{i_{1}\ldots i_{m}})$ is
not monotone with respect to the length of the word $m$.

For any $n\geqslant n(\epsilon)$, we want to bound

$\mathbb{P}_{1\leqslant i\leqslant n}\\{{\mathbf{I}}(i)\in\mathcal{U}^{c}\\}.$

Set $C^{\prime}>2(1-\log c_{0})/\chi_{1}$ and
$\mathcal{B}:=\\{\mathbf{i}\in\Psi_{i}:1\leqslant i\leqslant n\\}$. We have
$\Psi_{i}\cap\Psi_{j}=\emptyset$ for any $1\leqslant i<j\leqslant C^{\prime}n$
(Eq. 2.71).

We consider two cases. For any $\mathbf{i}\in\mathcal{B}$ with the word length
$l(\mathbf{i})\leqslant C^{\prime}n$, by definition,
$\mathbf{i}\in{\mathbf{U}}(j)$ for some $j\leqslant C^{\prime}n$. So we have

$\mathbb{P}_{1\leqslant i\leqslant
n}({\mathbf{I}}(i):{\mathbf{I}}(i)\in\mathcal{U}^{c},l({\mathbf{I}}(i))\leqslant
C^{\prime}n)\leqslant C^{\prime}\mathbb{P}_{1\leqslant i\leqslant
C^{\prime}n}({\mathbf{U}}(i):{\mathbf{U}}(i)\in\mathcal{U}^{c})\leqslant
C^{\prime}\epsilon.$ (2.75)

Consider any $\mathbf{i}\in\mathcal{B}$ with $l(\mathbf{i})>C^{\prime}n$. Take
$\epsilon_{1}=\min\\{\epsilon,\chi_{1}/2\\}$. It follows from Eq. 2.70 that

$\chi_{1}(g_{\mathbf{i}})\leqslant n-\log c_{0}\leqslant
C^{\prime}n(\chi_{1}-\epsilon_{1})\leqslant
l(\mathbf{i})(\chi_{1}-\epsilon_{1}).$

As $\mathbf{i}\in U(j)$ for some $j>C^{\prime}n$, the large deviation estimate
Eq. 2.52 yields

$\displaystyle\mathbb{P}_{1\leqslant i\leqslant
n}\\{{\mathbf{I}}(i):l({\mathbf{I}}(i))>C^{\prime}n\\}\leqslant$
$\displaystyle\frac{1}{n}\sum_{j>C^{\prime}n}\mathbb{P}\\{{\mathbf{U}}(j):\chi_{1}(g_{{\mathbf{U}}(j)})\leqslant
l({\mathbf{U}}(j))(\chi_{1}-\epsilon_{1})\\}$ $\displaystyle\leqslant$
$\displaystyle\frac{1}{n}\sum_{j\geqslant
C^{\prime}n}Cq^{-\beta(\epsilon_{1})j}.$ (2.76)

Combining Eqs. 2.75 and 2.9, we finish the proof of the lemma. ∎

###  2.10 Entropy

For a probability measure $\mu$ on a metric space $X$ and any countable
measurable partitions $\mathcal{E},\mathcal{E}^{\prime}$ of $X$, define the
entropy and conditional entropy by

$\displaystyle H(\mu,\mathcal{E}):=$
$\displaystyle\sum_{E\in\mathcal{E}}-\mu(E)\log\mu(E),$ $\displaystyle
H(\mu,\mathcal{E}^{\prime}|\mathcal{E}):=$ $\displaystyle
H(\mu,\mathcal{E}^{\prime}\vee\mathcal{E})-H(\mu,\mathcal{E}),$

where $\mathcal{E}^{\prime}\vee\mathcal{E}$ denotes the common refinement of
$\mathcal{E}^{\prime}$ and $\mathcal{E}$.

We collect some basic properties of entropy.

######  Lemma 2.33.

Let $\mathcal{E}$ and $\mathcal{F}$ be two measurable partitions of $X$. If
each atom in $\mathcal{E}$ intersects at most $k$ atoms in $\mathcal{F}$, then

$H(\mu,\mathcal{F})\leqslant H(\mu,\mathcal{E})+O(\log k).$

Suppose $X=\mathbb{P}(\mathbb{R}^{2})$ or $L$ and $\mathcal{Q}_{n}$ the
$q$-adic partition on $X$. This lemma implies that for any $m,n\in\mathbb{N}$,

$|H(\mu,\mathcal{Q}_{n})-H(\mu,\mathcal{Q}_{m})|=O(|m-n|).$ (2.77)

Let $f$ and $g$ be two continuous maps on $X$, such that
$\|f-g\|_{\infty}\leqslant q^{-n}$. Denote by $f\mu$ and $g\mu$ the
pushforward measures. Then we have

$|H(f\mu,\mathcal{Q}_{n})-H(g\mu,\mathcal{Q}_{n})|=O(1)$ (2.78)

(see [BHR19, (2.14)]).

######  Lemma 2.34 (Lemma 4.3 in [HS17]).

Suppose $f$ scales $\operatorname{\mathrm{supp}}\mu$ by $u>0$ with distortion
$C>1$ (see 2.6 for definition). Then

$H(f\mu,\mathcal{Q}_{n-\log u})=H(\mu,\mathcal{Q}_{n})+O(\log C).$

The following is about the concavity of entropy. Let $\delta\in[0,1]$, and
$\mu_{1},\mu_{2}$ be probability measures on $X$ and $\mathcal{Q}$ be a
measurable partition. We have

$\begin{split}\delta
H(\mu_{1},\mathcal{Q})+(1-\delta)H(\mu_{2},\mathcal{Q})&\leqslant
H(\delta\mu_{1}+(1-\delta)\mu_{2},\mathcal{Q})\\\ &\leqslant\delta
H(\mu_{1},\mathcal{Q})+(1-\delta)H(\mu_{2},\mathcal{Q})+H(\delta),\end{split}$
(2.79)

where $H(\delta):=-\delta\log(\delta)-(1-\delta)\log(1-\delta)$. The quantity
$H(\delta)$ is small if $\delta$ is sufficiently close to $0$ or $1$.

######  Lemma 2.35.

For any $k,n\in\mathbb{N}$, any $\epsilon_{1},\epsilon_{2}\in(0,1)$ and any
probability measures $\tau,\tau_{1},\tau_{2}$ on $X$ that satisfy
$\tau=(1-\epsilon_{2})\tau_{1}+\epsilon_{2}\tau_{2}$ and
$\operatorname{\mathrm{diam}}(\operatorname{\mathrm{supp}}\tau_{1})=O(q^{-k+\epsilon_{1}n})$,
we have

$\displaystyle\frac{1}{n}H\left(\tau,\mathcal{Q}_{n+k}|\mathcal{Q}_{k}\right)\geqslant$
$\displaystyle\frac{1}{n}H(\tau,\mathcal{Q}_{n+k})-\left(\epsilon_{2}+\frac{H(\epsilon_{2})}{n}+O(\epsilon_{1})\right),$
$\displaystyle\frac{1}{n}H\left(\tau,\mathcal{Q}_{n+k}|\mathcal{Q}_{k}\right)\geqslant$
$\displaystyle\frac{1}{n}H(\tau_{1},\mathcal{Q}_{n+k})-(\epsilon_{2}+O(\epsilon_{1})).$

###### Proof.

Note that the support of the measure $\tau_{1}$ intersects at most
$O(q^{\epsilon_{1}n})$ atoms of $\mathcal{Q}_{k}$. By Eq. 2.77 and Eq. 2.79,
we have

$\displaystyle\frac{1}{n}H\left(\tau,\mathcal{Q}_{n+k}|\mathcal{Q}_{k}\right)\geqslant(1-\epsilon_{2})\frac{1}{n}H(\tau_{1},\mathcal{Q}_{n+k}|\mathcal{Q}_{k})$
$\displaystyle\geqslant$
$\displaystyle(1-\epsilon_{2})\frac{1}{n}H(\tau_{1},\mathcal{Q}_{n+k})-O(\epsilon_{1})\geqslant\frac{1}{n}H(\tau,\mathcal{Q}_{n+k})-(\epsilon_{2}+H(\epsilon_{2})/n+O(\epsilon_{1})).$

For the second one, we have

$\displaystyle\frac{1}{n}H\left(\tau,\mathcal{Q}_{n+k}|\mathcal{Q}_{k}\right)\geqslant(1-\epsilon_{2})\frac{1}{n}H(\tau_{1},\mathcal{Q}_{n+k}|\mathcal{Q}_{k})$
$\displaystyle\geqslant$
$\displaystyle\frac{1}{n}H(\tau_{1},\mathcal{Q}_{n+k})-O(\epsilon_{1})-\epsilon_{2}\frac{1}{n}H(\tau_{1},\mathcal{Q}_{n+k}|\mathcal{Q}_{k})\geqslant\frac{1}{n}H(\tau_{1},\mathcal{Q}_{n+k})-(\epsilon_{2}+O(\epsilon_{1})).\qed$

###  2.11 Entropy dimension of projection measure

A probability measure $\mu$ on a metric space $X$ is called exact dimensional
if for $\mu$-a.e. $x$, we have that

$\lim_{r\rightarrow 0}\frac{\log\mu(B(x,r))}{\log r}$

exists and equals a constant independent of $x$. This constant is called the
exact dimension of the measure $\mu$. Young [You82] proved that if $\mu$ is
exact dimensional, then its exact dimension coincides with $\dim\mu$, the
Hausdorff dimension of $\mu$, which we defined in the Introduction. Hence, we
also use $\dim\mu$ to denote the exact dimension of $\mu$.

In the same paper [You82], Young proved a fundamental property of exact
dimensional measures (see also [FLR02]).

######  Lemma 2.36.

Let $\mu$ be an exact dimensional probability measure on $\mathbb{R}$. Then

$\dim\mu=\lim_{n\rightarrow\infty}\frac{1}{n}H(\mu,\mathcal{Q}_{n}).$

This lemma enables us to use the entropy to compute the dimension of the
measure $\mu$. The limit on the right-hand side is usually called the entropy
dimension.

Let $\mu$ be the $\nu$-stationary measure on $\mathbb{P}(\mathbb{R}^{3})$
given as in Section 2.7. In [LL23b] and [Rap21], it is proved that for
$\mu^{-}$-a.e. $V\in\mathbb{P}(\mathbb{R}^{3})$ the projection measure
$\pi_{V^{\perp}}\mu$ is exact dimensional. Thus we have

######  Lemma 2.37.

For $\mu^{-}$-a.e. $V\in\mathbb{P}(\mathbb{R}^{3})$, we have

$\dim\pi_{V^{\perp}}\mu=\lim_{n}\frac{1}{n}H(\pi_{V^{\perp}}\mu,\mathcal{Q}_{n}),$

and they are of the same value.

See [LL23b, Corollary 6.5] and [Rap21, Theorem 1.3] for the proof.

The following lemma is an application of “pseudo-continuity" property of
fixed-scale entropy.

######  Lemma 2.38.

For any $\epsilon>0$, any $m\geqslant M(\epsilon)$ large enough, any
$n\geqslant N(\epsilon,m)$, we have

$\inf_{V\in\mathbb{P}(\mathbb{R}^{3})}\mathbb{P}\left\\{{\mathbf{U}}(n):\alpha-\epsilon\leqslant\frac{1}{m}H(\pi_{(g^{-1}_{\mathbf{U}(n)}V)^{\perp}}\mu,\mathcal{Q}_{m})\leqslant\alpha+\epsilon\right\\}>1-\epsilon.$
(2.80)

###### Proof.

By 2.37, for $\mu^{-}$-almost every $V$, $\pi_{V^{\perp}}\mu$ has entropy
dimension $\alpha$. We have the equidistribution of random walks (2.22), i.e.
$\mathbb{E}(\delta_{{A}_{\mathbf{U}(n)}^{-1}V})$ weakly converges to $\mu^{-}$
as $n$ goes to infinity and the convergence is uniform with respect to $V$.
Then (2.80) can be shown immediately if the scale-$m$ entropy were continuous
as a function of measure. Since it is not, we need the following argument.

We can replace the entropy $\frac{1}{m}H(\cdot,\mathcal{Q}_{m})$ by a
continuous variant $F_{m}(\cdot)$ at the cost of $O(1/m)$, where $F_{m}$ is
continuous in the weak∗ topology. For a possible choice of $F_{m}$, see the
end of Section 2.8 in [BHR19]. Due to 2.37, for large $m$, we can obtain an
open set $E(m)$ with $\mu^{-}(E(m))>1-\epsilon/2$ such that for $V\in E(m)$,
we have $|F_{m}(\pi_{V^{\perp}}\mu)-\alpha|<\epsilon/2$. Applying 2.22 to
$E(m)$, for large $n>N_{2}$, we have Eq. 2.80 with
$\frac{1}{m}H(\cdot,\mathcal{Q}_{m})$ replaced $F_{m}(\cdot)$. Then replace
$F_{m}$ back to $\frac{1}{m}H(\cdot,\mathcal{Q}_{m})$. The resulting $O(1/m)$
error can be absorbed in $\epsilon$, which proves the claim. ∎

We recall the notion of Furstenberg entropy, which appears in the statement of
our main results.

######  Definition 2.39.

Let $\mu$ be a $\nu$-stationary measure on ${\mathbb{P}}({\mathbb{R}}^{3})$ or
$\mathcal{F}({\mathbb{R}}^{3}),$ (the flag variety whose precise definition is
given in 3.3) the Furstenberg entropy is given by

$h_{\mathrm{F}}(\mu,\nu)=\int\log\frac{\mathrm{d}g\mu}{\mathrm{d}\mu}(\xi)\left(\frac{\mathrm{d}g\mu}{\mathrm{d}\mu}(\xi)\right)\mathrm{d}\nu(g)\mathrm{d}\mu(\xi).$
(2.81)

The Furstenberg entropy is mysterious and might be difficult to compute. To
obtain a more calculable dimension formula, it is expected the Furstenberg
entropy is equal to the random walk entropy, which holds in some concrete
examples; see our second paper with Jiao [JLPX].

##  3 Non-concentration on arithmetic sequences

One of the main ingredients to show Theorem 1.10 is to prove the porosity
property of projections of the stationary measure, see Section 4. To obtain
the porosity of measures, a key step which we show below is to get the non-
concentration at arithmetic sequences property. We identify
$\mathbb{P}(\mathbb{R}^{2})$ with $\mathbb{R}/\mathbb{Z}$ (see Section 2.1).

######  Definition 3.1.

A probability measure $\tau$ on $\mathbb{P}(\mathbb{R}^{2})$ is called non-
concentrated on arithmetic sequences (across scales) if for any $\delta>0$,
there exist $l,k_{0}\in\mathbb{N}$ such that for all $k>k_{0}$, we have

$\tau\left(\bigcup_{0\leqslant
n<q^{k}}B\left(\frac{n}{q^{k}},\frac{1}{q^{k+l}}\right)\right)\leqslant\delta.$

In [BHR19], a property called uniformly continuous across scales (UCAS) was
defined for measures on metric spaces, which plays an important role in the
proof of porosity in their setting (Lemma 3.5 in [BHR19]). This property is
quite strong and can imply the uniform Hölder regularity of measures. However,
it is not clear whether it holds for a (projection of a) general stationary
measure. In our paper, we apply the knowledge of Fourier decay of Furstenberg
measures in [Li22] to obtain the weaker substitution of UCAS, i.e. non-
concentration at arithmetic sequences, which implies Lemma 4.3 that plays a
role as that of Lemma 3.5 in [BHR19].

Recall that the Fourier coefficients of
$\tau\in\mathbf{P}(\mathbb{P}(\mathbb{R}^{2}))$ are defined as follows: for
any $m\in\mathbb{Z}$,

$\hat{\tau}(m):=\int_{x\in\mathbb{R}/\mathbb{Z}}e^{2\pi
imx}\mathrm{d}\tau(x).$

We say $\tau$ is a Rajchman measure (or Fourier decay) if
$\hat{\tau}(m)\rightarrow 0$ as $|m|\rightarrow\infty$.

######  Lemma 3.2.

Let $\tau\in\mathbf{P}(\mathbb{P}(\mathbb{R}^{2}))$ be a Rajchman measure.
Then $\tau$ satisfies Definition 3.1. In particular, we can choose
$l,k_{0}\in\mathbb{N}$ in 3.1 such that

$\hat{\tau}(jq^{k_{0}})\leqslant\delta^{3}/(10^{4}q^{2})\,\,\,\text{ for
any}\,\,\,j\neq 0\,\,\,\text{and}\,\,\,10<q^{l}\delta\leqslant 10q.$

###### Proof.

Let $f$ be a smooth function supported on
$B(0,2/q^{l})\subset\mathbb{R}/\mathbb{Z}$ such that it equals to $1$ on
$B(0,1/q^{l})$ and $\|f^{\prime\prime}\|_{L^{1}}\leqslant 4q^{2l}$. Define
$F:\mathbb{R}/\mathbb{Z}\to\mathbb{R}$ by $F(x)=\sum_{0\leqslant
n<q^{k}}f(xq^{k}-n)$. Then

$\displaystyle\tau\left(\bigcup_{0\leqslant
n<q^{k}}B\left(nq^{-k},q^{-(k+l)}\right)\right)\leqslant\int
F\mathrm{d}\tau=\sum_{m\in\mathbb{Z}}\hat{\tau}(m)\hat{F}(-m),$

where $\hat{F}(m)=\int_{\mathbb{R}/\mathbb{Z}}F(x)e^{i2\pi mx}dx$. We have

$\displaystyle\hat{F}(m)$
$\displaystyle=\int_{\mathbb{R}/\mathbb{Z}}\sum_{0\leqslant
n<q^{k}}f(q^{k}x-n)e^{i2\pi
mx}\mathrm{d}x=\frac{1}{q^{k}}\int_{\mathbb{R}/\mathbb{Z}}\sum_{1\leqslant
n\leqslant q^{k}}f(y)e^{i2\pi m(y+n)/q^{k}}\mathrm{d}y$
$\displaystyle=\frac{\sum_{1\leqslant n\leqslant q^{k}}e^{i2\pi
mn/q^{k}}}{q^{k}}\int_{\mathbb{R}/\mathbb{Z}}f(y)e^{i2\pi
my/q^{k}}\mathrm{d}y.$

It is non-zero only if $m=jq^{k}$. Therefore, we have
$\hat{F}(jq^{k})=\hat{f}(j)$ and

$\tau\left(\cup_{0\leqslant
n<q^{k}}B\left(nq^{-k},q^{-(k+l)}\right)\right)\leqslant\sum_{j\in\mathbb{Z}}\hat{\tau}(q^{k}j)\hat{f}(-j).$

When $j=0$, we have $\hat{f}(0)=\int f\leqslant 4q^{-l}$. For the sum of the
remaining terms, we have

$\left|\sum_{j\neq
0}\hat{\tau}(q^{k}j)\hat{f}(j)\right|\leqslant\left(\sup_{j\neq
0}|\hat{\tau}(q^{k}j)|\right)\sum_{j\neq 0}|\hat{f}(j)|.$

Since $f$ is smooth, we have

$\sum_{j\neq 0}|\hat{f}(j)|\leqslant\sum_{j\neq
0}\frac{1}{j^{2}}\|f^{\prime\prime}\|_{L^{1}}\leqslant
4\|f^{\prime\prime}\|_{L^{1}}\leqslant 16q^{2l}<\infty.$

Since $\tau$ is a Rajchman measure, we have for any $k$ large enough,
$\sup_{j\neq 0}|\hat{\tau}(q^{k}j)|\leqslant\delta^{3}/(10^{4}q^{2})$. Choose
$l\in\mathbb{N}$ such that $10<q^{l}\delta\leqslant 10q$. We have

$\sum_{j\in\mathbb{Z}}\hat{\tau}(q^{k}j)\hat{f}(j)\leqslant\int
f+\left(\sup_{j\neq 0}|\hat{\tau}(q^{k}j)|\right)\sum_{j\neq
0}|\hat{f}(j)|\leqslant 4/q^{l}+(\delta^{3}/10^{4}q^{2})\times
16q^{2l}<\delta.$

Therefore $\tau$ satisfies Definition 3.1. ∎

For higher dimensional cases, the Fourier decay of Furstenberg measures on
flag varieties and projective spaces is known in [Li22], cf. Appendix E for
$3$-dimensional case. For our purpose, we only consider the flag variety
$\mathcal{F}(\mathbb{R}^{3})$ here.

######  Definition 3.3.

  1. 1.

(Definitions and notations) Let

$\mathcal{F}({\mathbb{R}}^{3}):=\left\\{(V_{1},V_{2}):V_{1}\subset
V_{2},V_{i}\text{ is a linear subspace of }{\mathbb{R}}^{3}\text{ of dimension
}i\right\\}$

be the space of flags in ${\mathbb{R}}^{3}$, which admits a natural
$\mathrm{SL}_{3}({\mathbb{R}})$-action. We abbreviate
$\mathcal{F}({\mathbb{R}}^{3})$ to $\mathcal{F}$ in later discussion.

  2. 2.

(Metrics on $\mathcal{F}$) For
$\eta=(V_{1}(\eta),V_{2}(\eta)),\eta^{\prime}=(V_{1}(\eta^{\prime}),V_{2}(\eta^{\prime}))\in\mathcal{F}$,
let

$d_{1}(\eta,\eta^{\prime})=d_{\mathbb{P}(\mathbb{R}^{3})}(V_{1}(\eta),V_{1}(\eta^{\prime}))\,\,\,\text{and}\,\,\,d_{2}(\eta,\eta^{\prime})=d_{\mathbb{P}(\mathbb{R}^{3})}(V_{2}(\eta),V_{2}(\eta^{\prime})).$

Here $d_{\mathbb{P}(\mathbb{R}^{3})}(V_{2}(\eta),V_{2}(\eta^{\prime}))$ is the
Hausdorff distance between two projective lines in
$(\mathbb{P}(\mathbb{R}^{3}),d)$. Then the metric $d$ on $\mathcal{F}$ can be
defined as the following:

$d(\eta,\eta^{\prime}):=\max\\{d_{1}(\eta,\eta^{\prime}),d_{2}(\eta,\eta^{\prime})\\}.$
(3.1)

  3. 3.

(Lifting functions from $\mathbb{P}(\mathbb{R}^{3})$)A function $\varphi$ on
$\mathcal{F}$ is called _lifted from $\mathbb{P}(\mathbb{R}^{3})$_ if
$\varphi(\eta)$ only depends on the first coordinate $V_{1}(\eta)$.

Recall that $\mu$ is the $\nu$-stationary measure on
$\mathbb{P}(\mathbb{R}^{3})$. In the following, for any
$V\in\mathbb{P}(\mathbb{R}^{3})$, by identifying $\mathbb{P}(V^{\perp})$
isometrically with $\mathbb{P}(\mathbb{R}^{2})$, we view $\pi_{V^{\perp}}\mu$
as a measure on $\mathbb{P}(\mathbb{R}^{2})$.

######  Proposition 3.4.

For any $\delta>0$, there exist $l,k_{0}\in\mathbb{N}$ such that for any
$V\in\mathbb{P}(\mathbb{R}^{3})$ and any $k\geqslant k_{0}$

$\pi_{V^{\perp}}\mu(\bigcup_{0\leqslant
n<q^{k}}B(nq^{-k},q^{-(k+l)}))\leqslant\delta.$ (3.2)

###### Proof.

By the proof of 3.2, it is sufficient to prove
$\widehat{\pi_{V^{\perp}}\mu}(m)\rightarrow 0$ uniformly for all $V$ as
$|m|\rightarrow\infty$. To obtain it our strategy is to apply E.2. Recall the
constants $\epsilon_{0},\epsilon_{1}>0$ from E.2. Let
$\epsilon_{2}=\epsilon_{0}/8$. Fix an arbitrary
$V\in\mathbb{P}(\mathbb{R}^{3})$ and an integer $m\in\mathbb{Z}$ such that
$|m|^{\epsilon_{2}}>10$.

##### Step 1: Define, lift and localize $\varphi$.

Consider the map

$\displaystyle\varphi:\mathbb{P}(\mathbb{R}^{3})-\\{V\\}$
$\displaystyle\to\mathbb{P}(V^{\perp}),$ (3.3) $\displaystyle x$
$\displaystyle\mapsto\pi_{V^{\perp}}x.$

Then $\varphi$ can be lifted as a map from
$\mathcal{F}-\\{\eta=(V_{1}(\eta),V_{2}(\eta)):V_{1}(\eta)\neq V\\}$. From now
to the end of the proof of Proposition 3.4 we only consider the lifting of
$\varphi$ and we still denote by $\varphi$. By the definition, $\varphi$ is
neither a well-defined map on whole $\mathcal{F}$ nor a well-defined real-
valued function, to which we cannot apply Fourier decay (i.e. E.2) directly.
To overcome this difficulty we localize $\varphi$ through partition of unity.

Let $\rho_{m}(t)$ be a $C^{1}$-function on $[0,1]$ such that $\rho_{m}(t)=0$
if $t\leqslant 1/|m|^{\epsilon_{2}}$ and $\rho_{m}(x)=1$ if $t\geqslant
2/|m|^{\epsilon_{2}}$. We further assume that the Lipchitz constant of
$\rho_{m}$ satisfies $\operatorname{\mathrm{Lip}}(\rho_{m})\leqslant
2|m|^{\epsilon_{2}}$. Let $r_{m}$ be a cutoff function on $\mathcal{F}$ given
by

$r_{m}(\eta)=\rho_{m}(d_{\mathbb{P}(\mathbb{R}^{3})}(V_{2}(\eta),V)).$ (3.4)

Roughly speaking the support of $r_{m}$ consists of points
$\eta=(V_{1}(\eta),V_{2}(\eta))$ such that $V$ is not too close to
$V_{2}(\eta)$; in particular, $V$ is also not too close to $V_{1}(\eta)$ and
we have

$d(V_{1}(\eta),V)\geqslant 1/|m|^{\epsilon_{2}}\text{ if $r_{m}(\eta)>0$}.$
(3.5)

Our idea to localize $\varphi$ is to locally identify $\varphi$ with smooth
real-valued functions $\varphi_{j}$ on $\mathcal{F}$ such that
$\varphi=\varphi_{j}\ \mathrm{mod}\ \mathbb{Z}$. Then the value of the
oscillatory integral (i.e. Eq. E.3) remains the same. Here is how we proceed:
take smooth functions $p_{1}$ and $p_{2}$ on $\mathbb{R}/\mathbb{Z}$ as a
partition of unity with respect to open intervals $d(z,\mathbb{Z})<1/3$ and
$d(z,\frac{1}{2}+\mathbb{Z})<1/3$. Let
$r_{m,j}(\eta)=r_{m}(\eta)p_{j}(\pi_{V^{\perp}}(V_{1}(\eta)))$ with
$\eta=(V_{1}(\eta),V_{2}(\eta))$, for $j=1,2$. Then on a neighbourhood of the
support of $r_{m,j}$, the map $\varphi$ takes value in a proper subset of
$\mathbb{R}/\mathbb{Z}$ which can be identified with a subset of $\mathbb{R}$
continuously. Therefore on the neighbourhood of the support of $r_{m,j}$,
$\varphi$ can be identified with smooth real-valued lifted functions
$\varphi_{j}$, without changing the integral.

##### Step 2: Verify E.3.

Our plan is to apply E.2 to $\varphi_{j}$. Since both $\varphi_{j}$ are
functions lifted from $\mathbb{P}(\mathbb{R}^{3})$, we only need to verify the
conditions in E.3. A remarkable fact for $\mathcal{F}$ is that for any
$\eta\in\mathcal{F}$, its coordinate provides not only its projection
$V_{1}(\eta)$ to $\mathbb{P}(\mathbb{R}^{3})$, but also an important tangent
direction $V_{\alpha_{2}}\subset T_{\eta}\mathcal{F}$ (see Appendix E).

  1. 1.

(Verify Eq. E.4 and Eq. E.7 with $C=64m^{4\epsilon_{2}}$). Using the spherical
law of sines, we have

$\displaystyle\partial_{1}\varphi_{j}(\eta)=\lim_{t\rightarrow
0}\frac{\varphi_{j}(V_{1})-\varphi_{j}(y(t))}{t}=\lim_{t\rightarrow
0}\frac{\angle_{V}(V_{1},y(t))}{d_{R}(V_{1},y(t))}=\lim_{t\rightarrow
0}\frac{\sin(\angle_{V}(V_{1},y(t)))}{d(V_{1},y(t))}=\lim_{t\rightarrow
0}\frac{\sin(\angle_{V_{1}}(V,y(t)))}{d(y(t),V)},$

where $y(t)$ is a smooth curve on $\mathbb{P}(\mathbb{R}^{3})$ which is
contained in $V_{2}$ with $d_{R}(V_{1},y(t))=t$. The angle
$\angle_{V_{1}}(V,y(t))$ is equal to angle between two hyperplane
$V_{2}=V_{1}\oplus y(t)$ and $V_{1}\oplus V$, so

$\displaystyle\partial_{1}\varphi_{j}(\eta)=\frac{d(V_{2},(V_{1}\oplus
V))}{d(V_{1},V)}=\frac{d(V_{2},V)}{d(V_{1},V)^{2}},$

where we use again spherical law of sines to obtain $d(V_{2},(V_{1}\oplus
V))=d(V_{2},V)/d(V_{1},V)$666 The Hausdorff distance $d(V_{2},(V_{1}\oplus
V))$ equals $\sin(\angle_{V_{1}}(V,y(t)))$. Let $V^{\prime}$ be the point in
$V_{2}$ such that $d(V,V^{\prime})=d(V,V_{2})$. Then
$\angle_{V^{\prime}}(V,V_{1})=\pi/2$. Hence by spherical law of sines,
$\sin(\angle_{V_{1}}(V,y(t)))/d(V,V_{2})=\sin(\angle_{V_{1}}(V,y(t)))/d(V,V^{\prime})=\sin(\pi/2)/d(V,V_{1})$.
for the last equality. By the choice of $r_{m,j}$, on its support, due to Eq.
3.4 and Eq. 3.5, we have (See Fig. 6)

$\partial_{1}\varphi_{j}(\eta)\in[1/|m|^{\epsilon_{2}},|m|^{2\epsilon_{2}}],$

which implies Eq. E.4 and Eq. E.7 with $C=64m^{4\epsilon_{2}}$.

Figure 6: $\alpha_{1}$-direction

  2. 2.

(Verify Eq. E.5). Eq. E.5 can be obtained directly from B.2.

  3. 3.

(Verify Eq. E.6). We have

$\partial_{1}\varphi_{j}(\eta)-\partial_{1}\varphi_{j}(\eta^{\prime})=\frac{d(V_{2},V)}{d(V_{1},V)^{2}}-\frac{d(V_{2}^{\prime},V)}{d(V_{1}^{\prime},V)^{2}}=\frac{d(V_{2},V)d(V_{1}^{\prime},V)^{2}-d(V_{2}^{\prime},V)d(V_{1},V)^{2}}{d(V_{1},V)^{2}d(V_{1}^{\prime},V)^{2}}.$

where $V_{i}^{\prime}:=V_{i}(\eta^{\prime})$. Due to the triangle inequality,
the numerator is less than

$|(d(V_{2},V)-d(V_{2}^{\prime},V))d(V_{1}^{\prime},V)^{2}+d(V_{2}^{\prime},V)(d(V_{1}^{\prime},V)^{2}-d(V_{1},V)^{2})|\leqslant
2(d(V_{2},V_{2}^{\prime})+d(V_{1},V_{1}^{\prime})).$

Due to Eq. 3.5, for $\eta,\eta^{\prime}$ in the $1/C$-neighborhood of the
support of $r$, the denominator is greater than $1/(2|m|^{\epsilon_{2}})^{4}$.
Hence, we have

$|\partial_{1}\varphi_{j}(\eta)-\partial_{1}\varphi_{j}(\eta^{\prime})|\leqslant
64|m|^{4\epsilon_{2}}d(\eta,\eta^{\prime})$

which is Eq. E.6, where $d(\eta,\eta^{\prime})$ is defined in Eq. 3.1.

  4. 4.

(Estimate the Lipschitz constants of $r_{m,j}$). Recall a basic estimate of
the Lipschitz constant of a product of two Lipchitz continuous functions
$f,g$:

$\operatorname{\mathrm{Lip}}(fg)\leqslant\operatorname{\mathrm{Lip}}(f)|g|_{\infty}+|f|_{\infty}\sup_{\eta\neq\eta^{\prime},\eta,\eta^{\prime}\in\operatorname{\mathrm{supp}}f}\frac{|g(\eta)-g(\eta^{\prime})|}{d(\eta,\eta^{\prime})}.$

By this estimate and (2) in B.2,

$\displaystyle\operatorname{\mathrm{Lip}}(r_{m,j})$
$\displaystyle\leqslant\operatorname{\mathrm{Lip}}(r_{m})+\sup_{\eta\neq\eta^{\prime},\eta,\eta^{\prime}\in\operatorname{\mathrm{supp}}r_{m}}\frac{|p_{j}(\pi_{V^{\perp}}(V_{1}(\eta)))-p_{j}(\pi_{V^{\perp}}(V_{1}(\eta^{\prime})))|}{d(\eta,\eta^{\prime})}$
$\displaystyle\leqslant|m|^{\epsilon_{2}}+\operatorname{\mathrm{Lip}}(p_{j})|m|^{\epsilon_{2}}\leqslant
4|m|^{\epsilon_{2}}.$

##### Step 3: Estimate of $\widehat{\pi_{V^{\perp}}\mu}(m)$.

From Step 2, each $\varphi_{j}$ is $(m^{\epsilon_{0}},r_{m,j})$-good (due to
the choice of $\epsilon_{2}$). Then for each $j$ and all $m$ with $|m|$ large
enough, by Theorem E.2 we have

$|\int e^{2\pi
im\varphi_{j}(\eta)}r_{m,j}(\eta)\mathrm{d}\mu_{\mathcal{F}}(\eta)|\leqslant|m|^{-\epsilon_{1}},$

where $\mu_{\mathcal{F}}$ is the $\nu$-stationary measure on $\mathcal{F}$,
whose projection on $\mathbb{P}(\mathbb{R}^{3})$ is $\mu$. Therefore, combined
with 2.23 and Eq. 3.5, we obtain

$\displaystyle|\widehat{\pi_{V^{\perp}}\mu}(m)|$ $\displaystyle=$
$\displaystyle|\int e^{2\pi
im\varphi(\eta)}\mathrm{d}\mu_{\mathcal{F}}(\eta)|$ $\displaystyle\leqslant$
$\displaystyle\left|\int e^{2\pi
im\varphi(\eta)}(r_{m,1}(\eta)+r_{m,2}(\eta))\mathrm{d}\mu_{\mathcal{F}}(\eta)\right|+\left|\int
e^{2\pi
im\varphi(\eta)}(1-r_{m,1}(\eta)-r_{m,2}(\eta))\mathrm{d}\mu_{\mathcal{F}}(\eta)\right|$
$\displaystyle\leqslant$ $\displaystyle\sum_{j}\left|\int e^{2\pi
im\varphi_{j}(\eta)}r_{m,j}(\eta)\mathrm{d}\mu_{\mathcal{F}}(\eta)\right|+\mu_{\mathcal{F}}(\\{r_{m}\neq
1\\})$ $\displaystyle\leqslant$ $\displaystyle\sum_{j}\left|\int e^{2\pi
im\varphi_{j}(\eta)}r_{m,j}(\eta)\mathrm{d}\mu_{\mathcal{F}}(\eta)\right|+\mu(\\{V_{2}(\eta):d(V_{2}(\eta),V)<1/|m|^{\epsilon_{2}}\\})\ll|m|^{-\beta},$

for some $\beta>0$ and uniform for all $V$. The proof is complete. ∎

######  Proposition 3.5.

For any $\delta>0$, there exist $l,k_{0}\in\mathbb{N}$ such that for any
$V\in\mathbb{P}(\mathbb{R}^{3})$, any $k\geqslant k_{0}$ and
$g\in\mathrm{SL}_{2}(\mathbb{R})$, we have

$g\pi_{V^{\perp}}\mu\left(\bigcup_{0\leqslant
n<q^{k}}B(nq^{-(k+[\chi_{1}(g)])},q^{-(k+[\chi_{1}(g)]+l)})\right)\leqslant\delta.$
(3.6)

###### Proof.

Abbreviate $\pi_{V^{\perp}}\mu$ by $\tau$. It follows from 2.23 that there
exists $s\in(0,1)$ such that $\tau(b(g^{-},s)^{c})<\delta/10$ and $s$ only
depends on $\mu$ and $\delta$. We decompose the measure $\tau$ as follows:

$\tau=\lambda\cdot\tau_{b(g^{-},s)^{c}}+(1-\lambda)\cdot\tau_{b(g^{-},s)},$

where $\lambda=\tau(b(g^{-},s)^{c})$. We denote $\tau_{b(g^{-},s)^{c}}$ by
$\tau_{I}$ to simply to notation. Set

$t=[\chi_{1}(g)-2|\log s|-1]=[\chi_{1}(g)+2\log s-1].$

Let $S_{t}:\mathbb{R}\to\mathbb{R},x\mapsto q^{t}x$ be the scaling map given
as in 2.19. 2.4 gives that $gb(g^{-},s)\subset
B(g^{+},q^{-\chi_{1}(g)}/s^{2})$. If $t$ is a non-negative integer, then
$S_{t}$ induces a map $\mathbb{R}/\mathbb{Z}\to\mathbb{R}/\mathbb{Z}$, which
we still denote by $S_{t}$ by abusing the notation. 2.4 gives that
$gb(g^{-},s)\subset B(g^{+},q^{-\chi_{1}(g)}/s^{2})$. Hence the restriction of
$S_{t}$ on $gb(g^{-},s)$ is a diffeomorphism. If $t$ is negative, then $S_{t}$
is a contracting map. Identifying $\mathbb{R}/\mathbb{Z}$ (hence
$\mathbb{P}(V^{\perp})$) with $[-\frac{1}{2},\frac{1}{2})$ such that
$V^{+}_{g}=0$, we have that $gb(g^{-},s)\subset(-\frac{1}{2},\frac{1}{2})$ is
an interval containing $0$. Hence $S_{t}$ restricted on $gb(g^{-},s)$ is also
a diffeomorphism to its image in $gb(g^{-},s)$.

We have

$\displaystyle
g\tau_{\mathbf{I}}\left(\cup_{n}B\left(\frac{n}{q^{[\chi_{1}(g)]+k}},\frac{1}{q^{[\chi_{1}(g)]+k+\ell}}\right)\right)$
$\displaystyle=S_{t}(g\tau_{\mathbf{I}})\left(S_{t}(\cup_{n}B\left(\frac{n}{q^{[\chi_{1}(g)]+k}},\frac{1}{q^{[\chi_{1}(g)]+k+\ell}})\right)\right)$
$\displaystyle=S_{t}g\tau_{\mathbf{I}}\left(\cup_{n}B\left(\frac{n}{q^{[\chi_{1}(g)]+k-t}},\frac{1}{q^{[\chi_{1}(g)]+k-t+\ell}}\right)\right),$

where we only sum up $n$’s such that the interval
$B\left(\frac{n}{q^{[\chi_{1}(g)]+k}},\frac{1}{q^{[\chi_{1}(g)]+k+\ell}}\right)$
intersects the support of $\tau_{\mathbf{I}}$.

We use 2.8 to obtain distortion estimates. For $y\in b(g^{-},s)$, we have

$|(S_{t}g)^{\prime}y|=q^{t}|g^{\prime}y|\in[s^{2}/q,1/q],|(S_{t}g)^{\prime\prime}(y)|=q^{t}|g^{\prime\prime}(y)|\leqslant
10/(qs).$ (3.7)

Let $p_{1}$ be a cuffoff function which equals $1$ on $b(g^{-},s)$ and $0$
outside $b(g^{-},s/2)$. Let $p_{1}\tau$ be the measure on
$\mathbb{P}(V^{\perp})$ defined by $p_{1}\tau(f)=\int p_{1}\cdot fd\tau$ for
any measurable function $f$ on $\mathbb{P}(V^{\perp})$. Consider the map
$\psi:\mathbb{P}(\mathbb{R}^{3})-\\{V\\}\to\mathbb{P}(V^{\perp})$ defined by

$\psi(x)=(S_{t}g)\pi_{V^{\perp}}(x)=S_{t}g\varphi(x)$

with $\varphi$ given as in Eq. 3.3. Then $\psi$ can be lifted to a function
defined on $\mathcal{F}-\\{\eta=(V_{1}(\eta),V_{2}(\eta)):V_{1}(\eta)\neq
V\\}$.

Take any integer $m$ with $|m|^{\epsilon_{2}}>4$, where
$\epsilon_{2}=\epsilon_{0}/8$ with $\epsilon_{0}$ given as in E.2. Let
$r_{m,1}(\eta)=r_{m}(\eta)p_{1}(\pi_{V^{\perp}}(\eta))$ where
$\eta=(V_{1}(\eta),V_{2}(\eta))$ and $r_{m}$ is defined as in Eq. 3.4. Here is
the key observation: since $S_{t}g$ on $b(g^{-},s)$ is a bi-Lipschitz map to
its image with bounded $C^{2}$-norm (Eq. 3.7), if $\varphi$ is
$(C,r_{m,1})$-good as in E.3, then $(S_{t}g)\varphi$ is
$(C^{\prime},r_{m,1})$-good with $C^{\prime}=(10q/s)^{2}C$. As $s$ is fixed,
we can verify as in 3.4 that $\varphi$ is
$(64m^{\epsilon_{0}/2},r_{m,1})$-good and hence $\psi=(S_{t}g)\varphi$ is
$(m^{\epsilon_{0}},r_{m,1})$-good for all integer $m$ with $|m|$ sufficiently
large. Then the same argument as in 3.4 yields that for all integer $m$ with
$|m|$ large,

$|\widehat{S_{t}g(p_{1}\tau)}(m)|\leqslant|\int e^{2\pi
im\psi(\eta)}r_{m,1}(\eta)\mathrm{d}\mu_{\mathcal{F}}(\eta)|+\mu_{\mathcal{F}}(\\{r_{m}=1\\}^{c})\ll
m^{-\beta}.$

3.2 implies that there exist $l,k_{0}\in\mathbb{N}$ (depending on $\delta$)
such that for all $k\in N$ with $[\chi_{1}(g)]+k-t>k_{0}$,

$\displaystyle
S_{t}g\tau_{\mathbf{I}}\left(\cup_{n}B\left(\frac{n}{q^{[\chi_{1}(g)]+k-t}},\frac{1}{q^{[\chi_{1}(g)]+k-t+\ell}}\right)\right)$
$\displaystyle\leqslant$
$\displaystyle\frac{1}{1-\lambda}S_{t}g(p_{1}\tau)\left(\cup_{n}B\left(\frac{n}{q^{[\chi_{1}(g)]+k-t}},\frac{1}{q^{[\chi_{1}(g)]+k-t+\ell}}\right)\right)<\delta/2.$

Note that $g\tau=(1-\lambda)\cdot g\tau_{\mathbf{I}}+\lambda\cdot
g\tau_{b(g^{-},s)^{c}}$ with $\lambda<\delta/10$. The proof is complete. ∎

##  4 Convolution and porosity

######  Definition 4.1.

Let $\alpha,\epsilon>0$ and $m,i_{1},i_{2}\in\mathbb{N}$ with $i_{1}\leqslant
i_{2}$. We call a probability measure $\tau$ on $\mathbb{R}$ or
$\mathbb{P}(\mathbb{R}^{2})$ is $(\alpha,\epsilon,m)$-entropy porous from
scale $i_{1}$ to $i_{2}$ if

$\mathbb{P}_{i_{1}\leqslant j\leqslant
i_{2}}\left\\{\frac{1}{m}H(\tau_{x,j},\mathcal{Q}_{m+j})<\alpha+\epsilon\right\\}>1-\epsilon.$

The main result of this part is the entropy porosity of projections of
stationary measures, which is a key ingredient in the entropy growth argument.

######  Proposition 4.2.

For every $\epsilon>0,\,\,m\geqslant M(\epsilon),\,\,k\geqslant
K(\epsilon,m),\,\,n\geqslant N(\epsilon,m,k)$,

$\inf_{V\in\mathbb{P}(\mathbb{R}^{3})}\mathbb{P}_{1\leqslant i\leqslant
n}\left\\{\pi_{V^{\perp}}g_{{\mathbf{I}}(i)}\mu\text{ is
}(\alpha,\epsilon,m)\text{-entropy porous from scale $i$ to
$i+k$}\right\\}>1-\epsilon.$ (4.1)

###  4.1 General results on entropy porosity

In this section, we list some general lemmas in [BHR19, Section 3.1] which
show that if a measure $\tau$ decomposes as a convex combination of measures
$\tau_{i}$ whose support has a short length, then many properties of the