$H$ of $G$, we say that ${\cal C}$ is $H$-subordinated modulo $\mu$ if for
$\mu$-almost every $y\in\mathcal{Y}$, there exists $r=r_{y}\in\;]0,1]$ such
that we have

$B_{r}^{H}y\subset[y]_{{\cal C}}\subset B_{1/r}^{H}y\;.$

If ${\cal C}$ is $U^{+}$-subordinated modulo $\mu$ and if furthermore $\mu$ is
${\mathfrak{a}}$-invariant, since ${\mathfrak{a}}$ normalises $U^{+}$ and by
Lemma 5.1, for every $j\in\mathbb{Z}$, the $\sigma$-algebra
${\mathfrak{a}}^{j}{\cal C}$ is also $U^{+}$-subordinated modulo $\mu$.

For every $\sigma$-algebra ${\cal A}$ of subsets of $\mathcal{Y}$, for all
$a,b$ in $\mathbb{Z}\cup\\{\pm\infty\\}$ with $a<b$, we define a
$\sigma$-algebra ${\cal A}_{a}^{b}$ of subsets of $\mathcal{Y}$ by

${\cal A}_{a}^{b}=\bigvee_{i=a}^{b}\;{\mathfrak{a}}^{i}{\cal
A}=\sigma\Big{(}\;\bigcup_{a\leq i\leq b}{\mathfrak{a}}^{i}{\cal
A}\;\Big{)}\;.$

Note that if ${\cal A}$ is countably generated, then so is ${\cal A}_{a}^{b}$.

###### Proposition 5.7

For every $r\in\;]0,1[\,$, there exists a countably generated
sub-$\sigma$-algebra ${\cal A}^{U^{+}}$ of the Borel $\sigma$-algebra of
$\mathcal{Y}$ such that

  1. (1)

the countably generated $\sigma$-algebra ${\cal A}^{U^{+}}$ is
${\mathfrak{a}}^{-1}$-descending,

  2. (2)

for every $y\in\mathcal{Y}(r)$, we have $[y]_{{\cal A}^{U^{+}}}\subset
B_{r}^{U^{+}}y$,

  3. (3)

for every $y\in\mathcal{Y}$, we have $B_{\delta_{r}}^{U^{+}}y\subset[y]_{{\cal
A}^{U^{+}}}$, where $\delta_{r}\in\;]0,r]$ is as in Lemma 5.6.

Let $\mu$ be a Borel ${\mathfrak{a}}$-invariant ergodic probability measure on
$\mathcal{Y}$ with $\mu(\mathcal{Y}(r))>0$. Then ${\cal A}^{U^{+}}$ is
$U^{+}$-subordinated modulo $\mu$.

Proof. Fix $r\in\;]0,1[\,$. Let ${\cal P}=\\{P_{1},\dots,P_{N},P_{\infty}\\}$
be a partition given by Lemma 5.6 for this $r$. We prove a preliminary result
on the countably generated sub-$\sigma$-algebra $\sigma({\cal
P})_{0}^{\infty}$.

###### Lemma 5.8

For every $y\in\mathcal{Y}$, we have
$B_{\delta_{r}}^{U^{+}}y\subset[y]_{\sigma({\cal P})_{0}^{\infty}}$.

Proof. Let $h\in B_{\delta_{r}}^{U^{+}}$. Assume for a contradiction that
$hy\notin[y]_{\sigma({\cal P})_{0}^{\infty}}$. Then there exists
$k\in\mathbb{Z}_{\geq 0}$ such that ${\mathfrak{a}}^{-k}hy$ and
${\mathfrak{a}}^{-k}y$ belong to different atoms of the partition ${\cal P}$.
Let $\alpha=\min\mathbf{r}>0$. By Lemma 5.1, we have

$d_{\mathcal{Y}}({\mathfrak{a}}^{-k}hy,{\mathfrak{a}}^{-k}y)\leq
d_{G}({\mathfrak{a}}^{-k}h{\mathfrak{a}}^{k},\operatorname{id})=d_{U^{+}}({\mathfrak{a}}^{-k}h{\mathfrak{a}}^{k},\operatorname{id})<q_{v}^{-k\alpha}\delta_{r}\leq\delta_{r}\leq
r\;.$ (41)

It follows that both ${\mathfrak{a}}^{-k}hy$ and ${\mathfrak{a}}^{-k}y$ belong
to the $\delta_{r}$-boundary $\partial_{\delta_{r}}{\cal P}$ of ${\cal P}$.
But the set $\partial_{\delta_{r}}{\cal P}$ is empty by Lemma 5.6 (3), which
gives a contradiction. $\Box$

By Lemma 5.6, for every $i\in\\{1,\dots,N\\}$, there exist
$y_{i}\in\mathcal{Y}(r)$ and a Borel subset $V_{i}$ of $\mathcal{Y}$ contained
in $B_{r}^{G}$ such that $P_{i}=V_{i}y_{i}$. Let ${\cal P}^{U^{+}}$ be the
sub-$\sigma$-algebra of the Borel $\sigma$-algebra of $\mathcal{Y}$ generated
by the subsets $P_{\infty}\cap\pi^{-1}(W)$, where $W$ is a Borel subset of
$\mathcal{X}$, and the subsets $((U^{+}B)\cap V_{i})y_{i}$, where
$i\in\\{1,\dots,N\\}$ and $B$ is a Borel subset of $G$. Then ${\cal
P}^{U^{+}}$ is countably generated, since the Borel $\sigma$-algebra of
$\mathcal{X}$ is countably generated and $U^{+}$ is a closed subgroup of $G$.
For every $y\in\mathcal{Y}$, the atom of $y$ for ${\cal P}^{U^{+}}$ is equal
to

$[y]_{{\cal P}^{U^{+}}}=\left\\{\begin{array}[]{ll}Uy&{\rm if}\;\;y\in
P_{\infty}\\\ P_{i}\cap(B^{U^{+}}_{r}y)&{\rm
if}\;\;\exists\;i\in\\{1,\dots,N\\},\;y\in P_{i}\;.\end{array}\right.$ (42)

Let us now define ${\cal A}^{U^{+}}=({\cal P}^{U^{+}})_{0}^{\infty}$, which is
a countably generated sub-$\sigma$-algebra of the Borel $\sigma$-algebra of
$\mathcal{Y}$, since so is ${\cal P}^{U^{+}}$. Note that
${\mathfrak{a}}\,{\cal A}^{U^{+}}=({\cal P}^{U^{+}})_{1}^{\infty}\subset{\cal
A}^{U^{+}}$, which proves Assertion (1).

For every $y\in\mathcal{Y}(r)$, since
$P_{\infty}\subset\mathcal{Y}-\mathcal{Y}(r)$ by Lemma 5.6 (1) and by Equation
(42), we have $[y]_{{\cal A}^{U^{+}}}\subset[y]_{{\cal P}^{U^{+}}}\subset
B_{r}^{U^{+}}y$, which proves Assertion (2).

In order to prove the last Assertion (3), let us take $y\in\mathcal{Y}$ and
$h\in B_{\delta_{r}}^{U^{+}}$ and let us prove that $hy\in[y]_{{\cal
A}^{U^{+}}}$. Since we have $hy\in[y]_{\sigma({\cal P})_{0}^{\infty}}$ by
Lemma 5.8, for every $k\geq 0$, there exists $i\in\\{1,\dots,N,\infty\\}$ such
that the points ${\mathfrak{a}}^{-k}y$ and
${\mathfrak{a}}^{-k}hy={\mathfrak{a}}^{-k}h{\mathfrak{a}}^{k}({\mathfrak{a}}^{-k}y)$
both belong to $P_{i}\in{\cal P}$. If $i=\infty$, then by Equation (42), the
points ${\mathfrak{a}}^{-k}y$ and ${\mathfrak{a}}^{-k}hy$ lie in the same atom
$[{\mathfrak{a}}^{-k}y]_{{\cal P}^{U^{+}}}=U{\mathfrak{a}}^{-k}y$ since
${\mathfrak{a}}^{-k}h{\mathfrak{a}}^{k}\in U^{+}$. Assume that $1\leq i\leq
N$. Since $h\in B_{\delta_{r}}^{U^{+}}$, it follows from Equation (41) that
${\mathfrak{a}}^{-k}h{\mathfrak{a}}^{k}\in B_{r}^{U^{+}}$. Hence by Equation
(42), the points ${\mathfrak{a}}^{-k}y$ and ${\mathfrak{a}}^{-k}hy$ lie in the
same atom $[{\mathfrak{a}}^{-k}y]_{{\cal
P}^{U^{+}}}=P_{i}\cap(B_{r}^{U^{+}}{\mathfrak{a}}^{-k}y)$ of ${\cal
P}^{U^{+}}$. This proves Assertion (3).

Now let $\mu$ be an ${\mathfrak{a}}$-invariant ergodic probability measure on
$\mathcal{Y}$ with $\mu(\mathcal{Y}(r))>0$. By ergodicity, for $\mu$-almost
every $y\in\mathcal{Y}$, there exists $k\in\mathbb{Z}_{\geq 1}$ such that
${\mathfrak{a}}^{-k}y\in\mathcal{Y}(r)$. Since ${\mathfrak{a}}^{k}{\cal
A}^{U^{+}}\subset{\cal A}^{U^{+}}$, by Assertion (1) and by Lemma 5.1, we have

$[y]_{{\cal A}^{U^{+}}}\subset[y]_{{\mathfrak{a}}^{k}{\cal
A}^{U^{+}}}={\mathfrak{a}}^{k}[{\mathfrak{a}}^{-k}y]_{{\cal
A}^{U^{+}}}\subset{\mathfrak{a}}^{k}B_{r}^{U^{+}}{\mathfrak{a}}^{-k}y\subset
B_{q_{v}^{k\max\mathbf{r}}}^{U^{+}}y\;.$

With Assertion (3), this proves that ${\cal A}^{U^{+}}$ is
$U^{+}$-subordinated modulo $\mu$. $\Box$

Let us introduce some material before stating and proving our next Lemma 5.9.
The map $d_{K_{v}^{\,m},\mathbf{r}}:K_{v}^{\,m}\times
K_{v}^{\,m}\rightarrow[0,+\infty[$ defined by

$\forall\;\bm{\xi},\bm{\xi}^{\prime}\in
K_{v}^{\,m},\;\;\;\;d_{K_{v}^{\,m},\mathbf{r}}(\bm{\xi},\bm{\xi}^{\prime})=\|\,\bm{\xi}-\bm{\xi}^{\prime}\,\|_{\mathbf{r}}$
(43)

is an ultrametric distance on $K_{v}^{\,m}$, since the $\mathbf{r}$-pseudonorm
$\|\;\|_{\mathbf{r}}$ satisfies the ultrametric inequality : for all
$\bm{\xi},\bm{\xi}^{\prime}\in K_{v}^{\,m}$, we have

$\|\,\bm{\xi}+\bm{\xi}^{\prime}\,\|_{\mathbf{r}}\leq\max\\{\|\,\bm{\xi}\,\|_{\mathbf{r}},\;\|\,\bm{\xi}^{\prime}\,\|_{\mathbf{r}}\\}\;,$
(44)

with equality if
$\|\,\bm{\xi}\,\|_{\mathbf{r}}\neq\|\,\bm{\xi}^{\prime}\,\|_{\mathbf{r}}$.
Note that the map similar to $d_{K_{v}^{\,m},\mathbf{r}}$ in the real case of
[KKL] is not a distance if $m\geq 2$ for general $\mathbf{r}$. For every
$\epsilon>0$, we denote by $B^{K_{v}^{\,m},\mathbf{r}}_{\epsilon}$ the open
ball of center $0$ and radius $\epsilon$ in $K_{v}^{\,m}$ for
$d_{K_{v}^{\,m},\mathbf{r}}$. Note that the distance
$d_{K_{v}^{\,m},\mathbf{r}}$ is bihölder equivalent to the standard one: For
all $\bm{\xi},\bm{\xi}^{\prime}\in K_{v}^{\,m}$ such that
$\|\,\bm{\xi}-\bm{\xi}^{\prime}\,\|\leq 1$, we have

$\|\,\bm{\xi}-\bm{\xi}^{\prime}\,\|^{\frac{1}{\min\mathbf{r}}}\leq
d_{K_{v}^{\,m},\mathbf{r}}(\bm{\xi},\bm{\xi}^{\prime})\leq\|\,\bm{\xi}-\bm{\xi}^{\prime}\,\|^{\frac{1}{\max\mathbf{r}}}\;.$
(45)

We also endow the quotient space $\mathbb{T}^{m}=K_{v}^{\,m}/R_{v}^{\,m}$ with
the quotient distance $d_{\mathbb{T}^{m},\mathbf{r}}$ of the distance
$d_{K_{v}^{\,m},\mathbf{r}}$ on $K_{v}^{\,m}$ defined by Equation (43). For
every $A\in\mathcal{M}_{m,n}(K_{v})$, we denote by
$d_{U^{+}y_{A,0},\mathbf{r}}$ the distance on the orbit
$U^{+}y_{A,0}=\phi_{A}(\mathbb{T}^{m})$ such that the homeomorphism $\phi_{A}$
defined in Lemma 5.3 is also an isometry for the distances
$d_{\mathbb{T}^{m},\mathbf{r}}$ and $d_{U^{+}y_{A,0},\mathbf{r}}$.

Using the identification $w\mapsto{\widehat{w}}$ between $K_{v}^{\,m}$ and
$U^{+}$ (see Subsection 5.1), for every $\epsilon>0$, we denote by
$B_{\epsilon}^{U^{+},\mathbf{r}}$ the open ball of radius $\epsilon$ in
$U^{+}$ centered at the identity element for the distance
$d_{U^{+},\mathbf{r}}$ on $U^{+}$ isometric to $d_{K_{v}^{\,m},\mathbf{r}}$.
The map $u\mapsto u\,y_{A,0}$ from $U^{+}$ onto $U^{+}y_{A,0}$ is
$1$-Lipschitz and locally isometric for the distances $d_{U^{+},\mathbf{r}}$
and $d_{U^{+}y_{A,0},\mathbf{r}}$. Improving Lemma 5.1, for all $\epsilon>0$
and $k\in\mathbb{Z}$, we have

${\mathfrak{a}}^{-k}B_{\epsilon}^{U^{+},\mathbf{r}}{\mathfrak{a}}^{k}=B_{\epsilon\,q_{v}^{-k}}^{U^{+},\mathbf{r}}\;.$
(46)

Again using the (locally compact) topological group identification
$w\mapsto{\widehat{w}}$ between $(K_{v}^{\,m},+)$ and $U^{+}$, we endow
$U^{+}$ with the Haar measure $m_{U^{+}}$ which corresponds to the normalized
Haar measure $\operatorname{vol}_{v}^{m}$ of $K_{v}^{\,m}$ (see Section 2.1).
For every $j\in\mathbb{Z}$, the Jacobian $\operatorname{Jac}_{j}$ with respect
to the measure $m_{U^{+}}$ of the homeomorphism
$\varphi_{j}:u\mapsto{\mathfrak{a}}^{j}\;u\;{\mathfrak{a}}^{-j}$ from $U^{+}$
to $U^{+}$ (which is constant since $\varphi_{j}$ is a group automorphism and
$m_{U^{+}}$ is bi-invariant) is easy to compute: we have

$\operatorname{Jac}_{j}=q_{v}^{\;j\;|\mathbf{r}|}\;.$ (47)

###### Lemma 5.9

For every $r\in\;]0,1[\,$, let ${\cal A}^{U^{+}}$ be as in Proposition 5.7.
Let $\mu$ be an ${\mathfrak{a}}$-invariant ergodic probability measure on
$\mathcal{Y}$. Then

$h_{\mu}({\mathfrak{a}}^{-1}|\,{\cal A}^{U^{+}})\leq|\mathbf{r}|\;.$

Furthermore, if $\mu(\mathcal{Y}(r))>0$, then

$h_{\mu}({\mathfrak{a}}^{-1}|\,{\cal A}^{U^{+}})\leq H_{\mu}({\cal
A}^{U^{+}}|\,{\mathfrak{a}}\,{\cal A}^{U^{+}})\;.$

Proof. The proof of the latter assertion is formally the same one as for the
real case in [KKL, Lemma 2.8] by replacing $a$, $L$, $\mu_{y_{0}}^{{\cal E}}$
therein by ${\mathfrak{a}}$, $U^{+}$, $\mu$ herein.

Let us prove the first assertion. By [EL, Prop. 7.44], there exists a
countable Borel-measurable partition ${\cal G}$ with finite entropy which is a
generator for ${\mathfrak{a}}$ modulo $\mu$, such that $\sigma({\cal
G})_{0}^{\infty}$ is ${\mathfrak{a}}^{-1}$-descending and $G^{+}$-subordinated
modulo $\mu$. We first claim that

$h_{\mu}({\mathfrak{a}}^{-1}|\,{\cal A}^{U^{+}})\leq H_{\mu}(\sigma({\cal
G})_{0}^{\infty}\vee{\cal A}^{U^{+}}|\,{\mathfrak{a}}\,(\sigma({\cal
G})_{0}^{\infty}\vee{\cal A}^{U^{+}})).$ (48)

Indeed, for every countable Borel-measurable partition $\xi$ with
$H_{\mu}(\xi)<\infty$, using Assertions (3), (4), and (6) in [KKL, Prop. A.2],
the fact that ${\cal A}^{U^{+}}$ is ${\mathfrak{a}}^{-1}$-descending and the
continuity of entropy [ELW, Prop. 2.14], we have

$\begin{split}&h_{\mu}({\mathfrak{a}}^{-1},\xi\,|\,{\cal A}^{U^{+}})\\\
\underset{\mathrm{(3)}}{\leq}\;&h_{\mu}({\mathfrak{a}}^{-1},\bigvee_{i=-k}^{k}{\mathfrak{a}}^{i}{\cal
G}\,|\,{\cal A}^{U^{+}})+H_{\mu}(\sigma(\xi)\,|\,\sigma({\cal
G})_{-k}^{k}\vee{\cal A}^{U^{+}})\\\
\underset{\mathrm{(4)}}{=}\;&h_{\mu}({\mathfrak{a}}^{-1},{\cal
G}\,|\,{\mathfrak{a}}^{k}\mathcal{A}^{U^{+}})+H_{\mu}(\sigma(\xi)\,|\,\sigma({\cal
G})_{-k}^{k}\vee{\cal A}^{U^{+}})\\\
\underset{\mathrm{(6)}}{=}\;&H_{\mu}(\sigma({\cal G})\,|\,\sigma({\cal
G})_{1}^{\infty}\vee({\mathfrak{a}}^{k}{\cal
A}^{U^{+}})_{0}^{\infty})+H_{\mu}(\sigma(\xi)\,|\,\sigma({\cal
G})_{-k}^{k}\vee{\cal A}^{U^{+}})\\\ =\;&H_{\mu}(\sigma({\cal
G})\,|\,\sigma({\cal G})_{1}^{\infty}\vee({\cal
A}^{U^{+}})_{0}^{\infty})+H_{\mu}(\sigma(\xi)\,|\,\sigma({\cal
G})_{-k}^{k}\vee{\cal A}^{U^{+}})\\\
\xrightarrow[k\to\infty]{}\;&H_{\mu}(\sigma({\cal G})\,|\,\sigma({\cal
G})_{1}^{\infty}\vee({\cal
A}^{U^{+}})_{0}^{\infty})+H_{\mu}(\sigma(\xi)\,|\,\sigma({\cal
G})_{-\infty}^{\infty}\vee{\cal A}^{U^{+}})\\\ \underset{{\cal
G}\text{generator}}{=}\;&H_{\mu}(\sigma({\cal G})\,|\,\sigma({\cal
G})_{1}^{\infty}\vee({\cal A}^{U^{+}})_{0}^{\infty})\leq H_{\mu}(\sigma({\cal
G})\,|\,\sigma({\cal G})_{1}^{\infty}\vee{\cal A}^{U^{+}})\\\
=\;&H_{\mu}(\sigma({\cal G})_{0}^{\infty}\vee{\cal A}^{U^{+}}|\,\sigma({\cal
G})_{1}^{\infty}\vee{\cal A}^{U^{+}})\leq H_{\mu}(\sigma({\cal
G})_{0}^{\infty}\vee{\cal A}^{U^{+}}|\,{\mathfrak{a}}(\sigma({\cal
G})_{0}^{\infty}\vee{\cal A}^{U^{+}})).\end{split}$

This proves the claim (48).

As in the proof of [LSS, Prop. 3.1], the $\sigma$-algebra $\sigma({\cal
G})_{0}^{\infty}\vee{\cal A}^{U^{+}}$ is countably generated,
${\mathfrak{a}}^{-1}$-descending, and $U^{+}$-subordinated since $[y]_{{\cal
A}^{U^{+}}}\subset Uy$ for all $y\in\mathcal{Y}$ and since $\sigma({\cal
G})_{0}^{\infty}$ is $G^{+}$-subordinated. Thus by [EL, Prop. 7.34] (recalling
that we are using logarithms with base $q_{v}$), we have

$H_{\mu}(\sigma({\cal G})_{0}^{\infty}\vee{\cal
A}^{U^{+}}|\,{\mathfrak{a}}\,(\sigma({\cal G})_{0}^{\infty}\vee{\cal
A}^{U^{+}}))=\lim_{k\to\infty}\frac{\log_{q_{v}}\mu_{x}^{U^{+}}({\mathfrak{a}}^{k}B_{1}^{U^{+}}{\mathfrak{a}}^{-k})}{k}\;,$

where $\mu_{x}^{U^{+}}$ is the leaf-wise measure of $\mu$ at $x\in\mathcal{Y}$
with respect to $U^{+}$ as defined in [EL, Theo. 6.3]. By [EL, Theo. 6.30]
(which applies since $U^{+}$ is abelian, hence unimodular) and by Equation
(47) (see also [EL, §7.42]), we have

$\limsup_{k\to\infty}\frac{\mu_{x}^{U^{+}}({\mathfrak{a}}^{k}B_{1}^{U^{+}}{\mathfrak{a}}^{-k})}{k^{2}\,q_{v}^{k|\mathbf{r}|}}=0\;,$

hence we have

$\lim_{k\to\infty}\frac{\log_{q_{v}}\mu_{x}^{U^{+}}({\mathfrak{a}}^{k}B_{1}^{U^{+}}{\mathfrak{a}}^{-k})}{k}\leq|\mathbf{r}|\;.$

This proves the first assertion of the lemma. $\Box$

Let us introduce some more material before stating and proving our final
Proposition 5.10 of Subsection 5.3. Let $\mathcal{A}$ be a countably generated
sub-$\sigma$-algebra of the Borel $\sigma$-algebra of $\mathcal{Y}$. For all
$j\in\mathbb{Z}_{\geq 0}$ and $y\in\mathcal{Y}$, let

$V_{y}^{{\mathfrak{a}}^{j}\mathcal{A}}=\\{u\in
U^{+}:u\,y\in[y]_{{\mathfrak{a}}^{j}\mathcal{A}}\\}\;,$ (49)

which is a Borel subset of $U^{+}$, called the $U^{+}$-shape of the atom
$[y]_{{\mathfrak{a}}^{j}\mathcal{A}}$. Note that for every
$j\in\mathbb{Z}_{\geq 0}$, we have

$V_{y}^{{\mathfrak{a}}^{j}\mathcal{A}}={\mathfrak{a}}^{j}\;V_{{\mathfrak{a}}^{-j}y}^{\mathcal{A}}\;{\mathfrak{a}}^{-j}\;.$

Let us define a Borel-measurable family
$\big{(}\tau_{y}^{{\mathfrak{a}}^{j}\mathcal{A}}\big{)}_{y\in\mathcal{Y}}$ of
Borel measures on $\mathcal{Y}$, that we call the $U^{+}$-subordinated Haar
measure of ${\mathfrak{a}}^{j}\mathcal{A}$, as follows:

$\bullet$ if $m_{U^{+}}(V_{y}^{{\mathfrak{a}}^{j}\mathcal{A}})$ is equal to
$0$ or $\infty$, we set $\tau_{y}^{{\mathfrak{a}}^{j}\mathcal{A}}=0$,

$\bullet$ otherwise, $\tau_{y}^{{\mathfrak{a}}^{j}\mathcal{A}}$ is the push-
forward of the normalized measure
$\frac{1}{m_{U^{+}}(V_{y}^{{\mathfrak{a}}^{j}\mathcal{A}})}\;m_{U^{+}}\\!\\!\mid_{V_{y}^{{\mathfrak{a}}^{j}\mathcal{A}}}$
by the map $u\mapsto u\,y$.

Now let $\mu$ be a Borel ${\mathfrak{a}}$-invariant probability measure on
$\mathcal{Y}$, such that $\mathcal{A}$ is $U^{+}$-subordinated modulo $\mu$.
In particular, for $\mu$-almost every $y\in\mathcal{Y}$, the atom
$V_{y}^{{\mathfrak{a}}^{j}\mathcal{A}}$ has positive and finite
$m_{U^{+}}$-measure, hence the measure
$\tau_{y}^{{\mathfrak{a}}^{j}\mathcal{A}}$ is a probability measure with
support in $[y]_{{\mathfrak{a}}^{j}\mathcal{A}}$. Furthermore, if
$z\in[y]_{{\mathfrak{a}}^{j}\mathcal{A}}$ then there exists $u\in U^{+}$ such
that $z=u\,y$,
$V_{z}^{{\mathfrak{a}}^{j}\mathcal{A}}=V_{y}^{{\mathfrak{a}}^{j}\mathcal{A}}\,u^{-1}$,
and
$\tau_{z}^{{\mathfrak{a}}^{j}\mathcal{A}}=\tau_{y}^{{\mathfrak{a}}^{j}\mathcal{A}}$,
by the right-invariance of $m_{U^{+}}$.

The following proposition is a function field analog of the effective real
case version [KKL, Prop. 2.10, §2.4] of [EL, §7.55].

###### Proposition 5.10

Let $\mu$ be a Borel ${\mathfrak{a}}$-invariant ergodic probability measure on
$\mathcal{Y}$ and let ${\cal A}$ be a countably generated sub-$\sigma$-algebra
of the Borel $\sigma$-algebra of $\mathcal{Y}$ which is
${\mathfrak{a}}^{-1}$-descending and $U^{+}$-subordinated modulo $\mu$. Fix
$j\in\mathbb{Z}_{\geq 1}$ and a $U^{+}$-saturated Borel subset $K^{\prime}$ of
$\mathcal{Y}$. Suppose that there exists $\epsilon>0$ such that $[z]_{{\cal
A}}\subset B_{\epsilon}^{U^{+},\mathbf{r}}z$ for every $z\in K^{\prime}$. Then
we have

$H_{\mu}({\cal A}|{\mathfrak{a}}^{j}{\cal A})\leq
j\,|\mathbf{r}|+\int_{\mathcal{Y}}\log\tau_{y}^{{\mathfrak{a}}^{j}{\cal
A}}((\mathcal{Y}-K^{\prime})\cup
B^{U^{+},\mathbf{r}}_{\epsilon}\operatorname{Supp}\mu)\;d\mu(y).$

Proof. We fix $\mu$, $\mathcal{A}$, $j$, $K^{\prime}$ and $\epsilon$ as in the
statement. By for instance [EL, Theo. 5.9], let
$\big{(}\mu_{y}^{{\mathfrak{a}}^{j}\mathcal{A}}\big{)}_{y\in\mathcal{Y}}$ be a
measurable family of conditional measures of $\mu$ with respect to
${\mathfrak{a}}^{j}\mathcal{A}$, so that for $\mu$-almost every
$y\in\mathcal{Y}$, the measure $\mu_{y}^{{\mathfrak{a}}^{j}\mathcal{A}}$ is a
probability measure on $\mathcal{Y}$ giving full measure to the atom
$[y]_{{\mathfrak{a}}^{j}\mathcal{A}}$, with
$\mu_{z}^{{\mathfrak{a}}^{j}\mathcal{A}}=\mu_{y}^{{\mathfrak{a}}^{j}\mathcal{A}}$
if $z\in[y]_{{\mathfrak{a}}^{j}\mathcal{A}}$, and such that the following
disintegration formula holds true:

$\mu=\int_{y\in\mathcal{Y}}\mu_{y}^{{\mathfrak{a}}^{j}\mathcal{A}}\;d\mu(y)\;.$
(50)

Let
$p_{\mu}:y\mapsto\mu_{y}^{{\mathfrak{a}}^{j}\mathcal{A}}([y]_{\mathcal{A}})$
and
$p_{\tau}:y\mapsto\tau_{y}^{{\mathfrak{a}}^{j}\mathcal{A}}([y]_{\mathcal{A}})$,
which are nonnegative and measurable functions on $\mathcal{Y}$. Since
$\mathcal{A}$ is ${\mathfrak{a}}^{-1}$-descending and $U^{+}$-subordinated
modulo $\mu$, the atom $[y]_{\mathcal{A}}$ contains an open neighborhood of
$y$ in the atom $[y]_{{\mathfrak{a}}^{j}\mathcal{A}}$ for $\mu$-almost every
$y\in\mathcal{Y}$. In particular, the function $p_{\tau}$ is $\mu$-almost
everywhere positive.

Since $\mathcal{A}$ is countably generated and
${\mathfrak{a}}^{-1}$-descending, for every $y\in\mathcal{Y}$, the atom of $y$
for ${\mathfrak{a}}^{j}\mathcal{A}$ is countably partitioned into atoms for
$\mathcal{A}$ up to measure $0$, that is, there exist a finite or countable
subset $I_{y}$ of $[y]_{{\mathfrak{a}}^{j}\mathcal{A}}$ and a
$\mu_{y}^{{\mathfrak{a}}^{j}\mathcal{A}}$-measure zero subset $N_{y}$ of
$[y]_{{\mathfrak{a}}^{j}\mathcal{A}}$ such that

$[y]_{{\mathfrak{a}}^{j}\mathcal{A}}=N_{y}\sqcup\bigsqcup_{x\in
I_{y}}\;[x]_{\mathcal{A}}\;.$ (51)

Let $I^{\prime}_{y}=\\{x\in
I_{x}:[x]_{\mathcal{A}}\cap\operatorname{Supp}\mu\neq\emptyset\\}$.

###### Lemma 5.11

Let $x\in I_{y}$

  1. (1)

If $x\notin I^{\prime}_{y}$, then
$\mu_{y}^{{\mathfrak{a}}^{j}\mathcal{A}}([x]_{\mathcal{A}})=0$.

  2. (2)

If $x\in I^{\prime}_{y}$, then $[x]_{\mathcal{A}}$ is contained in
$(\mathcal{Y}-K^{\prime})\cup
B_{\epsilon}^{U^{+},\mathbf{r}}\operatorname{Supp}\mu$.

Proof. (1) This follows since
$\operatorname{Supp}\mu_{y}^{{\mathfrak{a}}^{j}\mathcal{A}}$ is contained in
$\operatorname{Supp}\mu$.

(2) If $x\in I^{\prime}_{y}$, there exists
$z\in[x]_{\mathcal{A}}\cap\operatorname{Supp}\mu$. For every
$z^{\prime}\in[x]_{\mathcal{A}}$, we have either
$z^{\prime}\in\mathcal{Y}-K^{\prime}$ or $z^{\prime}\in K^{\prime}$. In the
second case, since $\mathcal{A}$ is $U^{+}$-subordinated and $K^{\prime}$ is
$U^{+}$-saturated, we have
$z\in[x]_{\mathcal{A}}=[z^{\prime}]_{\mathcal{A}}\subset
U^{+}z^{\prime}\subset K^{\prime}$. Hence by the assumption of Proposition
5.10, we have $z^{\prime}\in[x]_{\mathcal{A}}=[z]_{\mathcal{A}}\subset
B_{\epsilon}^{U^{+},\mathbf{r}}z\subset
B_{\epsilon}^{U^{+},\mathbf{r}}\operatorname{Supp}\mu$, which proves the
result. $\Box$

By the definition of the $U^{+}$-subordinated Haar measure of
${\mathfrak{a}}^{j}\mathcal{A}$, for $\mu$-almost every $y\in\mathcal{Y}$, we
have

$p_{\tau}(y)=\frac{m_{U^{+}}(V_{y}^{\mathcal{A}})}{m_{U^{+}}(V_{y}^{{\mathfrak{a}}^{j}\mathcal{A}})}=\frac{m_{U^{+}}(V_{y}^{\mathcal{A}})}{m_{U^{+}}({\mathfrak{a}}^{j}\;V_{{\mathfrak{a}}^{-j}y}^{\mathcal{A}}\;{\mathfrak{a}}^{-j})}=\frac{m_{U^{+}}(V_{y}^{\mathcal{A}})}{\operatorname{Jac}_{j}\;m_{U^{+}}(V_{{\mathfrak{a}}^{-j}y}^{\mathcal{A}})}\;.$

Hence, by the ${\mathfrak{a}}$-invariance of $\mu$ and by Equation (47), we
have

$\int_{z\in\mathcal{Y}}\log_{q_{v}}p_{\tau}(z)\;d\mu(z)=-\log_{q_{v}}\operatorname{Jac}_{j}=-\,j\,|\mathbf{r}|\;.$

Respectively

$\bullet$ by the definition of the conditional entropy in Equation (38),

$\bullet$ by the disintegration formula (50),

$\bullet$ since $\mu_{y}^{{\mathfrak{a}}^{j}\mathcal{A}}$ gives full measure
to $[y]_{{\mathfrak{a}}^{j}\mathcal{A}}$ which is partitionned as in Equation
(51), and by Lemma 5.11 (1),

$\bullet$ since when $z$ varies in
$[x]_{\mathcal{A}}\subset[y]_{{\mathfrak{a}}^{j}\mathcal{A}}$, the values
$p_{\mu}(z)=\mu_{z}^{{\mathfrak{a}}^{j}\mathcal{A}}([z]_{\mathcal{A}})=\mu_{y}^{{\mathfrak{a}}^{j}\mathcal{A}}([x]_{\mathcal{A}})$
and
$p_{\tau}(z)=\tau_{z}^{{\mathfrak{a}}^{j}\mathcal{A}}([z]_{\mathcal{A}})=\tau_{y}^{{\mathfrak{a}}^{j}\mathcal{A}}([x]_{\mathcal{A}})$
are constant,

$\bullet$ by the concavity property of the logarithm,

$\bullet$ by Lemma 5.11 (2),

we hence have

$\displaystyle
H_{\mu}(\mathcal{A}\,|\,{\mathfrak{a}}^{j}\mathcal{A})\;-j\;|\mathbf{r}|$
$\displaystyle=\;$
$\displaystyle-\int_{z\in\mathcal{Y}}\big{(}\log_{q_{v}}p_{\mu}(z)-\log_{q_{v}}p_{\tau}(z)\big{)}\;d\mu(z)$
$\displaystyle=\;$
$\displaystyle\int_{y\in\mathcal{Y}}\int_{z\in\mathcal{Y}}\big{(}\log_{q_{v}}p_{\tau}(z)-\log_{q_{v}}p_{\mu}(z)\big{)}\;d\mu_{y}^{{\mathfrak{a}}^{j}\mathcal{A}}(z)\;d\mu(y)$
$\displaystyle=\;$ $\displaystyle\int_{y\in\mathcal{Y}}\sum_{x\in
I^{\prime}_{y}}\int_{z\in[x]_{\mathcal{A}}}\big{(}\log_{q_{v}}p_{\tau}(z)-\log_{q_{v}}p_{\mu}(z)\big{)}\;d\mu_{y}^{{\mathfrak{a}}^{j}\mathcal{A}}(z)\;d\mu(y)$
$\displaystyle=\;$ $\displaystyle\int_{y\in\mathcal{Y}}\sum_{x\in
I^{\prime}_{y}}\log_{q_{v}}\frac{\tau_{y}^{{\mathfrak{a}}^{j}\mathcal{A}}([x]_{\mathcal{A}})}{\mu_{y}^{{\mathfrak{a}}^{j}\mathcal{A}}([x]_{\mathcal{A}})}\;\mu_{y}^{{\mathfrak{a}}^{j}\mathcal{A}}([x]_{\mathcal{A}})\;d\mu(y)$
$\displaystyle\leq\;$
$\displaystyle\int_{y\in\mathcal{Y}}\log_{q_{v}}\Big{(}\sum_{x\in
I^{\prime}_{y}}\tau_{y}^{{\mathfrak{a}}^{j}\mathcal{A}}([x]_{\mathcal{A}})\Big{)}\;d\mu(y)$
$\displaystyle\leq\;$
$\displaystyle\int_{y\in\mathcal{Y}}\log_{q_{v}}\big{(}\tau_{y}^{{\mathfrak{a}}^{j}\mathcal{A}}((\mathcal{Y}-K^{\prime})\cup
B_{\epsilon}^{U^{+},\mathbf{r}}\operatorname{Supp}\mu)\big{)}\;d\mu(y)\;.$

This proves the result. $\Box$

##  6 Upper bound on the Hausdorff dimension of $\mathbf{Bad}_{A}(\epsilon)$

###  6.1 Constructing measures with large entropy

In this subsection, we construct, as in [KKL, Prop. 4.1] in the real case, an
${\mathfrak{a}}$-invariant probability measure on $\mathcal{Y}$ giving an
appropriate lower bound on the conditional entropy of ${\mathfrak{a}}$
relative to the $\sigma$-algebra ${\cal A}^{U^{+}}$ constructed in Proposition
5.7.

For any point $x$ in a measurable space, we denote by $\Delta_{x}$ the unit
Dirac measure at $x$. We denote by
$\stackrel{{\scriptstyle*}}{{\rightharpoonup}}$ the weak-star convergence of
Borel measures on any locally compact space.

Let us denote by
${\overline{\mathcal{X}}}=\mathcal{X}\cup\\{\infty_{\mathcal{X}}\\}$ and
${\overline{\mathcal{Y}}}=\mathcal{Y}\cup\\{\infty_{\mathcal{Y}}\\}$ the one-
point compactifications of $\mathcal{X}$ and $\mathcal{Y}$, respectively. We
denote by
${\overline{\pi}}:{\overline{\mathcal{Y}}}\rightarrow{\overline{\mathcal{X}}}$
the unique continuous extension of the natural projection
$\pi:\mathcal{Y}\rightarrow\mathcal{X}$, mapping $\infty_{\mathcal{Y}}$ to
$\infty_{\mathcal{X}}$. The left actions of ${\mathfrak{a}}$ on $\mathcal{X}$
and $\mathcal{Y}$ continuously extend to actions on ${\overline{\mathcal{X}}}$
and ${\overline{\mathcal{Y}}}$ fixing the points at infinity
$\infty_{\mathcal{X}}$ and $\infty_{\mathcal{Y}}$. For every countably
generated $\sigma$-algebra ${\cal A}$ of subsets of $\mathcal{X}$ or
$\mathcal{Y}$, we denote by ${\overline{\cal A}}$ the countably generated
$\sigma$-algebra of subsets of ${\overline{\mathcal{X}}}$ or
${\overline{\mathcal{Y}}}$ generated by $\mathcal{A}$ and its point at
infinity. For a finite partition ${\cal Q}=\\{Q_{1},\dots,Q_{N},Q_{\infty}\\}$
of $\mathcal{Y}$ with only one unbounded atom $Q_{\infty}$, we denote by
${\overline{\cal Q}}$ the finite partition
$\\{Q_{1},\dots,Q_{N},{\overline{Q}}_{\infty}=Q_{\infty}\cup\\{\infty_{\mathcal{Y}}\\}\\}$
of ${\overline{\mathcal{Y}}}$. Note that
$\overline{\bigvee_{i=a}^{b}{\mathfrak{a}}^{-i}{\cal
Q}}=\bigvee_{i=a}^{b}{\mathfrak{a}}^{-i}\,{\overline{\cal Q}}$ for all $a,b$
in $\mathbb{Z}$ with $a<b$.

For every $\eta\in[0,1]$, we say that an element $x\in\mathcal{X}$ has
$\eta$-escape of mass on average under the action of ${\mathfrak{a}}$ if for
every compact subset $Q$ of $\mathcal{X}$,

$\liminf_{N\to\infty}\frac{1}{N}\;{\operatorname{Card}}\,\big{\\{}\ell\in\\{1,\cdots,N\\}:{\mathfrak{a}}^{\ell}x\notin
Q\big{\\}}\geq\eta\;.$

When $\eta=1$, as defined in the Introduction and in Proposition 5.2, we say
that $x$ diverges on average in $\mathcal{X}$ under the action of
${\mathfrak{a}}$. For every $A\in\mathcal{M}_{m,n}(K_{v})$, we denote by
$x_{A}=u_{A}R^{\,m}_{v}\in\mathcal{X}$ its associated unimodular lattice (see
Section 5.2), and by $\eta_{A}\in[0,1]$ the upper bound of the elements
$\eta\in[0,1]$ such that $x_{A}$ has $\eta$-escape of mass on average. Note
that this upper bound is actually a maximum.

###### Proposition 6.1

For every $A\in\mathcal{M}_{m,n}(K_{v})$, there exists a Borel probability
measure $\mu_{A}$ on ${\overline{\mathcal{X}}}$ with
$\mu_{A}(\mathcal{X})=1-\eta_{A}$ such that for every $\epsilon>0$, there
exists an ${\mathfrak{a}}$-invariant Borel probability measure
${\overline{\mu}}$ on ${\overline{\mathcal{Y}}}$ satisfying the following
properties.

  1. (1)

The support of ${\overline{\mu}}$ is contained in
$\mathcal{L}_{\epsilon}\cup\\{\infty_{\mathcal{Y}}\\}$, where
$\mathcal{L}_{\epsilon}$ is defined in Equation (35).

  2. (2)

We have ${\overline{\pi}}_{*}{\overline{\mu}}=\mu_{A}$. In particular, there
exists an ${\mathfrak{a}}$-invariant Borel probability measure $\mu$ on
$\mathcal{Y}$ such that

${\overline{\mu}}=(1-\eta_{A})\mu+\eta_{A}\Delta_{\infty_{\mathcal{Y}}}.$

  3. (3)

For every $r\in\;]0,1[\,$, let ${\cal A}^{U^{+}}$ be the $\sigma$-algebra of
subsets of $\mathcal{Y}$ constructed in the proof of Proposition 5.7. Then

$h_{{\overline{\mu}}}\big{(}{\mathfrak{a}}^{-1}|\,{\overline{{\cal
A}^{U^{+}}}}\,\big{)}\geq|\mathbf{r}|(1-\eta_{A})-\max\mathbf{r}\;(m-\dim_{\rm
Haus}\mathbf{Bad}_{A}(\epsilon))\;.$

Proof. Since $x_{A}$ has $\eta_{A}$-escape of mass on average but does not
have $(\eta_{A}+\delta)$-escape of mass on average for any $\delta>0$, there
exists an increasing sequence of positive integers
$(k_{i})_{i\in\mathbb{Z}_{\geq 1}}$ such that, for the weak-star convergence
of Borel probability measures on the compact space ${\overline{\mathcal{X}}}$,
as $i\rightarrow+\infty$, we have

$\frac{1}{k_{i}}\sum_{k=0}^{k_{i}-1}\Delta_{\,{\mathfrak{a}}^{k}x_{A}}\;\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\;\mu_{A}\;,$
(52)

and $\mu_{A}$ is a Borel probability measure on ${\overline{\mathcal{X}}}$
with $\mu_{A}(\mathcal{X})=1-\eta_{A}$. This is equivalent to
$\mu_{A}(\\{\infty_{\mathcal{X}}\\})=\eta_{A}$.

Let $\epsilon>0$. For every $T\in\mathbb{Z}_{\geq 0}$, with the notation of
Subsection 5.2 (see in particular Equations (35) and (36)), let

$R_{T}=\\{\bm{\theta}\in\mathbb{T}^{m}:\forall k\geq
T,{\mathfrak{a}}^{k}\phi_{A}(\bm{\theta})\in\mathcal{L}_{\epsilon}\\}\cap\mathbf{Bad}_{A}(\epsilon)\;.$

By Proposition 5.4, since a countable subset of $K_{\nu}^{\,m}$ has Hausdorff
dimension $0$, we have $\dim_{\rm
Haus}\big{(}\bigcup_{T=1}^{\infty}R_{T}\big{)}=\dim_{\rm
Haus}\mathbf{Bad}_{A}(\epsilon)$. Thus, for every $j\in\mathbb{Z}_{\geq 1}$,
there exists $T_{j}\in\mathbb{Z}_{\geq 0}$ satisfying

$\dim_{\rm Haus}R_{T_{j}}\geq\dim_{\rm
Haus}\mathbf{Bad}_{A}(\epsilon)-\frac{1}{j}\;.$

For all $i,j\in\mathbb{Z}_{\geq 1}$ such that $k_{i}\geq T_{j}$, let $S_{i,j}$
be a maximal $q_{v}^{-k_{i}}$-separated subset of $R_{T_{j}}$ for the distance
$d_{\mathbb{T}^{m},\mathbf{r}}$ defined after Equation (45). Then $R_{T_{j}}$
can be covered by ${\operatorname{Card}}\;S_{i,j}$ open balls of radius
$q_{v}^{-k_{i}}$ for $d_{\mathbb{T}^{m},\mathbf{r}}$. Each open ball of radius
$q_{v}^{-k_{i}}$ for $d_{\mathbb{T}^{m},\mathbf{r}}$ can be covered by
$\prod_{j=1}^{m}q_{v}^{-k_{i}r_{j}}/q_{v}^{-k_{i}\max\mathbf{r}}=q_{v}^{k_{i}(m\max\mathbf{r}-|\mathbf{r}|)}$
open balls of radius $q_{v}^{-k_{i}\max\mathbf{r}}$ with respect to the
standard distance $d_{\mathbb{T}^{m}}$ (defining the Hausdorff dimension of
subsets of $\mathbb{T}^{m}$). Since the lower Minskowski dimension is at least
equal to the Hausdorff dimension, we have

$\liminf_{i\to\infty}\frac{\log_{q_{v}}\big{(}q_{v}^{k_{i}(m\max\mathbf{r}-|\mathbf{r}|)}{\operatorname{Card}}\;S_{i,j}\big{)}}{-\log_{q_{v}}\big{(}q_{v}^{-k_{i}\max\mathbf{r}}\big{)}}\geq\dim_{\rm
Haus}R_{T_{j}}\geq\dim_{\rm Haus}\mathbf{Bad}_{A}(\epsilon)-\frac{1}{j}\;,$

which implies that

$\liminf_{i\to\infty}\frac{\log_{q_{v}}{\operatorname{Card}}\;S_{i,j}}{k_{i}}\geq|\mathbf{r}|-\max\mathbf{r}\;\big{(}m+\frac{1}{j}-\dim_{\rm
Haus}\mathbf{Bad}_{A}(\epsilon)\big{)}\;.$ (53)

Let us define the Borel probability measures

$\nu_{i,j}=\frac{1}{{\operatorname{Card}}\;S_{i,j}}\sum_{\bm{\theta}\in
S_{i,j}}\Delta_{\phi_{A}(\bm{\theta})}\;,$

which is the normalized counting measure on the finite subset
$\phi_{A}(S_{i,j})$ of the $U^{+}$-orbit
$\phi_{A}({\mathbb{T}}^{m})=U^{+}y_{A,0}\subset\pi^{-1}(x_{A})$, and

${\widetilde{\nu}}_{i,j}=\frac{1}{k_{i}}\sum_{0\leq k\leq
k_{i}-1}{\mathfrak{a}}_{*}^{k}\nu_{i,j}\;,$

which is the average of the previous one on the first $k_{i}$ points of the
${\mathfrak{a}}$-orbit. Since ${\overline{\mathcal{Y}}}$ is compact,
extracting diagonally a subsequence if necessary, we may assume that
${\widetilde{\nu}}_{i,j}$ weak-star converges as $i\rightarrow+\infty$ towards
an ${\mathfrak{a}}$-invariant Borel probability measure
${\widetilde{\mu}}_{j}$, and that ${\widetilde{\mu}}_{j}$ weak-star converges
as $j\rightarrow+\infty$ towards an ${\mathfrak{a}}$-invariant Borel
probability measure ${\overline{\mu}}$. Let us prove that ${\overline{\mu}}$
satisfies the three assertions of Proposition 6.1.

(1) For all $k\geq T_{j}$ and $\bm{\theta}\in S_{i,j}\subset R_{T_{j}}$, we
have ${\mathfrak{a}}^{k}\phi_{A}(\bm{\theta})\in\mathcal{L}_{\epsilon}$ by the
definition of $R_{T_{j}}$. Since ${\mathfrak{a}}^{k}_{*}\nu_{i,j}$ is a
probability measure, we hence have

${\widetilde{\nu}}_{i,j}(\mathcal{Y}-\mathcal{L}_{\epsilon})=\frac{1}{k_{i}}\sum_{k=0}^{k_{i}-1}{\mathfrak{a}}^{k}_{*}\nu_{i,j}(\mathcal{Y}-\mathcal{L}_{\epsilon})=\frac{1}{k_{i}}\sum_{k=0}^{T_{j}}{\mathfrak{a}}^{k}_{*}\nu_{i,j}(\mathcal{Y}-\mathcal{L}_{\epsilon})\leq\frac{T_{j}}{k_{i}}\;.$

Since $\mathcal{L}_{\epsilon}\cup\\{\infty_{\mathcal{Y}}\\}$ is closed in
${\overline{\mathcal{Y}}}$ and by taking limits first as $i\rightarrow+\infty$
then as $j\rightarrow+\infty$, we therefore have
${\overline{\mu}}(\mathcal{Y}-\mathcal{L}_{\epsilon})=0$. This proves
Assertion (1).

(2) Since $\phi_{A}(S_{i,j})$ is contained in the fiber above $x_{A}$ of
${\overline{\pi}}$ and since $\nu_{i,j}$ is a probability measure, we have
${\overline{\pi}}_{*}\nu_{i,j}=\Delta_{x_{A}}$. By the linearity and
equivariance of ${\overline{\pi}}_{*}$, we hence have

${\overline{\pi}}_{*}{\widetilde{\nu}}_{i,j}=\frac{1}{k_{i}}\sum_{0\leq k\leq
k_{i}-1}{\mathfrak{a}}_{*}^{k}\;{\overline{\pi}}_{*}\,\nu_{i,j}=\frac{1}{k_{i}}\sum_{0\leq
k\leq k_{i}-1}\Delta_{{\mathfrak{a}}^{k}x_{A}}\;.$

By the weak-star continuity of ${\overline{\pi}}_{*}$ and Equation (52), we
thus have

${\overline{\pi}}_{*}{\overline{\mu}}=\lim_{j\rightarrow+\infty}\lim_{i\rightarrow+\infty}{\overline{\pi}}_{*}{\widetilde{\nu}}_{i,j}=\lim_{j\rightarrow+\infty}\mu_{A}=\mu_{A}\;.$

Note that the point at infinity $\infty_{\mathcal{Y}}$ is an isolated point in
the support of ${\overline{\mu}}$ by Assertion (1), since
$\mathcal{L}_{\epsilon}$ is compact. We hence have

${\overline{\mu}}(\\{\infty_{\mathcal{Y}}\\})={\overline{\mu}}(\,{\overline{\pi}}^{-1}(\\{\infty_{\mathcal{X}}\\}))=\mu_{A}(\\{\infty_{\mathcal{X}}\\})=\eta_{A}\;.$
(54)

(3) Suppose that ${\cal Q}$ is any finite Borel-measurable partition of
$\mathcal{Y}$ satisfying

  1. (i)

the partition ${\cal Q}$ contains an atom $Q_{\infty}$ of the form
$\pi^{-1}(Q_{\infty}^{*})$, where $\mathcal{X}-Q_{\infty}^{*}$ has compact
closure,

  2. (ii)

there exists $\ell_{0}\in\mathbb{Z}_{\geq 1}$ such that for every atom
$Q\in{\cal Q}$ different from $Q_{\infty}$, we have
${\operatorname{diam}}\;Q<q_{v}^{-\ell_{0}\max\mathbf{r}}$ for the distance
$d_{\mathcal{Y}}$.

  3. (iii)

for all $Q\in{\cal Q}$ and $j\in\mathbb{Z}_{\geq 1}$, we have
${\widetilde{\mu}}_{j}(\partial Q)=0$ and ${\overline{\mu}}(\partial Q)=0$.

We first prove the following entropy bound: For every $M\in\mathbb{Z}_{\geq
1}$,

$\frac{1}{M}H_{{\overline{\mu}}}\big{(}\,\overline{\sigma({\cal
Q})}\;_{0}^{M-1}|\,\overline{{\cal
A}^{U^{+}}}\,\big{)}\geq|\mathbf{r}|(1-{\overline{\mu}}(\,{\overline{Q}}_{\infty}))-\max\mathbf{r}\;(m-\dim_{\rm
Haus}\mathbf{Bad}_{A}(\epsilon))\;.$ (55)

Since Equation (55) is clear if
${\overline{\mu}}(\,{\overline{Q}}_{\infty})=1$, we may assume that
${\overline{\mu}}(\,\overline{Q}_{\infty})<1$, hence that
${\widetilde{\mu}}_{j}(\,\overline{Q}_{\infty})<1$ for all large enough $j\geq
1$. Now, we fix such a $j\geq 1$.

Take $\rho>0$ small enough so that
${\widetilde{\mu}}_{j}(\,\overline{Q}_{\infty})+\rho<1$ and let

$\beta={\widetilde{\mu}}_{j}(\,\overline{Q}_{\infty})+\rho\;.$ (56)

Then for all large enough $i\in\mathbb{Z}_{\geq 1}$, since
$\phi_{A}(S_{i,j})\subset\pi^{-1}(x_{A})$ and
$Q_{\infty}=\pi^{-1}(Q_{\infty}^{*})$ by Property (i) of ${\cal Q}$, we have

$\begin{split}\beta={\widetilde{\mu}}_{j}(\,\overline{Q}_{\infty})+\rho>{\widetilde{\nu}}_{i,j}(Q_{\infty})&=\frac{1}{k_{i}{\operatorname{Card}}\;S_{i,j}}\sum_{k=0}^{k_{i}-1}\sum_{\bm{\theta}\in
S_{i,j}}\Delta_{{\mathfrak{a}}^{k}\phi_{A}(\bm{\theta})}(Q_{\infty})\\\
&=\frac{1}{k_{i}}\sum_{k=0}^{k_{i}-1}\Delta_{{\mathfrak{a}}^{k}x_{A}}(Q_{\infty}^{*})\;.\end{split}$

Thus, for every $\bm{\theta}\in{\mathbb{T}}^{m}$, since
${\mathfrak{a}}^{k}\phi_{A}(\bm{\theta})\in Q_{\infty}$ implies that
${\mathfrak{a}}^{k}x_{A}\in Q^{*}_{\infty}$ by Property (i) of ${\cal Q}$, we
have

${\operatorname{Card}}\\{k\in\\{0,\dots,k_{i}-1\\}:{\mathfrak{a}}^{k}\phi_{A}(\bm{\theta})\in
Q_{\infty}\\}<\beta\,k_{i}\;.$ (57)

Let us prove the following counting lemma inspired by [ELMW, Lem. 4.5] and
[LSS, Lem. 2.4], where $\ell_{0}$ is given by Property (ii) of ${\cal Q}$.

###### Lemma 6.2

There exists a constant $C>0$ depending only on $\mathbf{r}$ and $\ell_{0}$
such that for all $A\in\mathcal{M}_{m,n}(K_{v})$,
$\bm{\theta}\in\mathbb{T}^{m}$ and $T\in\mathbb{Z}_{\geq 0}$, defining
$y=\phi_{A}(\bm{\theta})$, $I=\\{k\in\mathbb{Z}_{\geq
0}:{\mathfrak{a}}^{k}y\in Q_{\infty}\\}$, and

$E_{y,T}=\\{z\in
U^{+}y:\forall\;k\in\\{0,\dots,T\\}-I,\;\;d_{\mathcal{Y}}({\mathfrak{a}}^{k}y,{\mathfrak{a}}^{k}z)<q_{v}^{-\ell_{0}\max\mathbf{r}}\;\\}\;,$

the set $E_{y,T}$ can be covered by
$C\,q_{v}^{\,|\mathbf{r}|{\operatorname{Card}}(I\cap\\{0,\dots,T\\})}$ closed
balls of radius $q_{v}^{-(\ell_{0}+T)}$ for the distance
$d_{U^{+}y,\,\mathbf{r}}$.

Proof. As in the proof of [LSS, Lemma 2.4], we proceed by induction on $T$.

By the compactness of ${\mathbb{T}}^{m}$, there exists a constant
$C\in\mathbb{Z}_{\geq 1}$ depending only on $\mathbf{r}$ and $\ell_{0}$ such
that the metric space $({\mathbb{T}}^{m},d_{{\mathbb{T}}^{m},\mathbf{r}})$ can
be covered by $C$ closed balls of radius $q_{v}^{-\ell_{0}}$. Since
$\phi_{A}:{\mathbb{T}}^{m}\rightarrow U^{+}y$ is an isometry for the distances
$d_{{\mathbb{T}}^{m},\mathbf{r}}$ and $d_{U^{+}y,\,\mathbf{r}}$, the orbit
$U^{+}y$ can be covered by $C$ closed balls for $d_{U^{+}y,\,\mathbf{r}}$ of
radius $q_{v}^{-\ell_{0}}$. Thus the lemma holds for $T=0$. Let
$N_{T}=C\,q_{v}^{\,|\mathbf{r}|{\operatorname{Card}}(I\cap\\{0,\dots,T\\})}$.

Assume by induction that $E_{y,T-1}$ can be covered by $N_{T-1}$ balls for
$d_{U^{+}y,\,\mathbf{r}}$ of radius $q_{v}^{-(\ell_{0}+T-1)}$. Note that for
every $k\in\mathbb{Z}$, since
$\pi_{v}^{k}\mathcal{O}_{v}/(\pi_{v}^{k+1}\mathcal{O}_{v})$ has order $q_{v}$,
every closed ball in $K_{v}$ of radius $q_{v}^{-k}$ is the disjoint union of
$q_{v}$ closed ball of radius $q_{v}^{-k-1}$. Hence every closed ball for
$d_{U^{+}y,\,\mathbf{r}}$ of radius $q_{v}^{-(\ell_{0}+T-1)}$ in $U^{+}y$ can
be covered by $q_{v}^{\,|\mathbf{r}|}$ closed balls for
$d_{U^{+}y,\,\mathbf{r}}$ of radius $q_{v}^{-(\ell_{0}+T)}$. Therefore, if
$T\in I$, then $E_{y,T}=E_{y,T-1}$ can be covered by
$N_{T}=q_{v}^{|\mathbf{r}|}N_{T-1}$ closed balls for $d_{U^{+}y,\,\mathbf{r}}$
of radius $q_{v}^{-(\ell_{0}+T)}$.

Suppose conversely that $T\notin I$, so that in particular $N_{T}=N_{T-1}$.
Denote the above covering of $E_{y,T-1}$ by $\\{B_{i}:i=1,\dots,N_{T-1}\\}$.
Since we have $E_{y,T}\subset E_{y,T-1}$, the set $\\{E_{y,T}\cap
B_{i}:i=1,\dots,N_{T-1}\\}$ is a covering of $E_{y,T}$.

Claim.  For all $i=1,\dots,N_{T-1}$ and $z_{1},z_{2}\in E_{y,T}\cap B_{i}$, we
have $d_{U^{+}y,\,\mathbf{r}}(z_{1},z_{2})\leq q_{v}^{-(\ell_{0}+T)}$.

Proof. Since $T\notin I$, we have
$d_{\mathcal{Y}}({\mathfrak{a}}^{T}y,{\mathfrak{a}}^{T}z_{j})<q_{v}^{-\ell_{0}\max\mathbf{r}}$
for each $j=1,2$. Thus we have
$d_{\mathcal{Y}}({\mathfrak{a}}^{T}z_{1},{\mathfrak{a}}^{T}z_{2})<q_{v}^{-\ell_{0}\max\mathbf{r}}$
by the ultrametric inequality property of $\|\cdot\|$. Note that since
$z_{1},z_{2}\in U^{+}y=U^{+}y_{A,\bm{\theta}}$, there exist
$\bm{\theta}_{1}=(\theta_{1,1},\dots,\theta_{1,m})$ and
$\bm{\theta}_{2}=(\theta_{2,1},\dots,\theta_{2,m})$ in ${\mathbb{T}}^{m}$ such
that (denoting in the same way lifts of $\bm{\theta}_{1}$ and
$\bm{\theta}_{2}$ to $K_{v}^{\,m}$) we have $z_{1}=y_{A,\bm{\theta}_{1}}$ and
$z_{2}=y_{A,\bm{\theta}_{2}}$. With $|\langle\,\;\,\rangle|$ the map defined
after Equation (3), it follows that we have

$\displaystyle\max_{1\leq i\leq
m}q_{v}^{r_{i}T}\,|\langle\,\theta_{1,i}-\theta_{2,i}\,\rangle|=d_{{\mathbb{T}}^{m}}({\mathfrak{a}}_{-}^{T}\bm{\theta}_{1},{\mathfrak{a}}_{-}^{T}\bm{\theta}_{2})$
$\displaystyle=d_{\mathcal{Y}}({\mathfrak{a}}^{T}y_{A,\bm{\theta}_{1}},{\mathfrak{a}}^{T}y_{A,\bm{\theta}_{2}})$
$\displaystyle=d_{\mathcal{Y}}({\mathfrak{a}}^{T}z_{1},{\mathfrak{a}}^{T}z_{2})<q_{v}^{-\ell_{0}\max\mathbf{r}}\;.$

Hence, we have

$d_{U^{+}y,\,\mathbf{r}}(z_{1},z_{2})=d_{{\mathbb{T}}^{m},\mathbf{r}}(\bm{\theta}_{1},\bm{\theta}_{2})=\max_{1\leq
i\leq
m}|\langle\,\theta_{1,i}-\theta_{2,i}\,\rangle|^{\frac{1}{r_{i}}}<q_{v}^{-(\ell_{0}+T)}\;,$

which concludes the claim. $\Box$

By the above claim, the intersection $E_{y,T}\cap B_{i}$ is contained in a
single ball for $d_{U^{+}y,\,\mathbf{r}}$ of radius $q_{v}^{-(\ell_{0}+T)}$
for each $i=1,\dots,N_{T-1}$. Thus $E_{y,T}$ can be covered by $N_{T}=N_{T-1}$
balls for $d_{U^{+}y,\,\mathbf{r}}$ of radius $q_{v}^{-(\ell_{0}+T)}$. $\Box$

Recall that as constructed in the proof of Proposition 5.7, there exist a
Borel-measurable partition ${\cal P}=\\{P_{1},\dots,P_{N},P_{\infty}\\}$ of
$\mathcal{Y}$ with $N+1$ elements, and a countably generated Borel-measurable
$\sigma$-algebra ${\cal P}^{U^{+}}$ of subsets of $\mathcal{Y}$, with
$[y]_{{\cal P}^{U^{+}}}=[y]_{\cal P}\cap B^{U^{+}}_{r}y$ for every
$y\in\mathcal{Y}(r)$ by Equation (42), such that we have ${\cal
A}^{U^{+}}=({\cal P}^{U^{+}})_{0}^{\infty}$.

For all $\ell\in\mathbb{Z}_{\geq 1}$ and $y\in\mathcal{Y}(r)$, we have
$[y]_{({\cal P}^{U^{+}})_{0}^{\ell}}\subset[y]_{\sigma({\cal
P})_{0}^{\ell}}\cap B_{r}^{U^{+}}y$. Since the support of $\nu_{i,j}$ is a
finite set of points on a single $U^{+}$-orbit
$\phi_{A}(\mathbb{T}^{m})=U^{+}y_{A,0}$, the measure
${\mathfrak{a}}^{-k}_{*}\nu_{i,j}$ is also supported on a single $U^{+}$-orbit
${\mathfrak{a}}^{k}U^{+}y_{A,0}=U^{+}{\mathfrak{a}}^{-k}y_{A,0}$ for every
$k\in\mathbb{Z}_{\geq 0}$. Recall (see for instance [EL, Def. 5.7]) that two
sub-$\sigma$-algebras ${\cal A}$ and ${\cal A}^{\prime}$ of a measured space
$(X,{\cal B})$ are equivalent modulo a probability measure $\nu$ if for every
$A\in{\cal A}$ there exists $A^{\prime}\in{\cal A}^{\prime}$ such that
$\nu(A\Delta A^{\prime})=0$ (where $\Delta$ is the symmetric difference) and
vice versa.

Claim. For every $k\in\mathbb{Z}_{\geq 0}$, the sub-$\sigma$-algebras ${\cal
P}^{U^{+}}$ and $\sigma({\cal P})$ are equivalent modulo
${\mathfrak{a}}^{-k}_{*}\nu_{i,j}$.

Proof. By the construction of ${\cal P}^{U^{+}}$ above Equation (42), we have
$[y]_{{\cal P}^{U^{+}}}\subset[y]_{{\cal P}}$ for every $y\in\mathcal{Y}$ and
for every $y\notin P_{\infty}$, we have

$U^{+}y\cap[y]_{{\cal P}}=[y]_{{\cal P}^{U^{+}}}.$ (58)

We first consider any point $y\in\mathcal{Y}$ such that $[y]_{{\cal
P}^{U^{+}}}$ intersects $\operatorname{Supp}{\mathfrak{a}}_{*}^{-k}\nu_{i,j}$.
If $y\notin P_{\infty}$, by Equation (58), the symmetric difference
$[y]_{{\cal P}^{U^{+}}}\Delta\,[y]_{{\cal P}}=[y]_{{\cal P}}-[y]_{{\cal
P}^{U^{+}}}$ does not intersect $U^{+}y$. But
$\operatorname{Supp}{\mathfrak{a}}_{*}^{-k}\nu_{i,j}\subset U^{+}y$ by the
single orbit support property, hence we have
${\mathfrak{a}}^{-k}_{*}\nu_{i,j}([y]_{{\cal P}^{U^{+}}}\Delta\,[y]_{{\cal
P}})=0$. If $y\in P_{\infty}$, since $[y]_{{\cal P}^{U^{+}}}=Uy$ by Equation
(42) and $\operatorname{Supp}{\mathfrak{a}}_{*}^{-k}\nu_{i,j}\subset Uy$, we
have ${\mathfrak{a}}^{-k}_{*}\nu_{i,j}([y]_{{\cal
P}^{U^{+}}}\Delta\,[y]_{{\cal
P}})={\mathfrak{a}}^{-k}_{*}\nu_{i,j}(P_{\infty}-Uy)=0$.

For every point $y\in\mathcal{Y}$ such that $[y]_{{\cal P}^{U^{+}}}$ does not
intersect $\operatorname{Supp}{\mathfrak{a}}_{*}^{-k}\nu_{i,j}$, we can take
$\varnothing\in\sigma({\cal P})$ so that
${\mathfrak{a}}^{-k}_{*}\nu_{i,j}([y]_{{\cal
P}^{U^{+}}}\Delta\,\varnothing)={\mathfrak{a}}^{-k}_{*}\nu_{i,j}([y]_{{\cal
P}^{U^{+}}})=0$.

Conversely, consider any point $y\in\mathcal{Y}$ such that $[y]_{{\cal P}}$
intersects $\operatorname{Supp}{\mathfrak{a}}_{*}^{-k}\nu_{i,j}$. Let us fix
$y^{\prime}\in[y]_{{\cal
P}}\cap\operatorname{Supp}{\mathfrak{a}}_{*}^{-k}\nu_{i,j}$, so that in
particular $\operatorname{Supp}{\mathfrak{a}}_{*}^{-k}\nu_{i,j}\subset
U^{+}y^{\prime}\subset Uy^{\prime}$. If $y\in P_{\infty}$, then $y^{\prime}\in
P_{\infty}=[y]_{{\cal P}}$ and $[y^{\prime}]_{{\cal P}^{U^{+}}}=Uy^{\prime}$
by Equation (42), hence ${\mathfrak{a}}^{-k}_{*}\nu_{i,j}([y]_{{\cal
P}}\,\Delta\,[y^{\prime}]_{{\cal
P}^{U^{+}}})={\mathfrak{a}}^{-k}_{*}\nu_{i,j}(P_{\infty}-Uy^{\prime})=0$. If
$y\notin P_{\infty}$, then $y^{\prime}\notin P_{\infty}$ and by Equation (58),
the difference $U^{+}y^{\prime}-[y^{\prime}]_{{\cal P}^{U^{+}}}$ does not
intersect $[y^{\prime}]_{{\cal P}}=[y]_{{\cal P}}$, hence
${\mathfrak{a}}^{-k}_{*}\nu_{i,j}([y]_{{\cal P}}\Delta\,[y^{\prime}]_{{\cal
P}^{U^{+}}})=0$.

For every point $y\in\mathcal{Y}$ such that $[y]_{{\cal P}}$ does not
intersect $\operatorname{Supp}{\mathfrak{a}}_{*}^{-k}\nu_{i,j}$, we take
$\varnothing\in{\cal P}^{U^{+}}$ so that
${\mathfrak{a}}^{-k}_{*}\nu_{i,j}([y]_{{\cal
P}}\Delta\,\varnothing)={\mathfrak{a}}^{-k}_{*}\nu_{i,j}([y]_{{\cal P}})=0$.
This proves the claim. $\Box$

Using the above claim, we have that $({\cal P}^{U^{+}})_{0}^{\ell}$ is
equivalent to $\sigma({\cal P})_{0}^{\ell}$ modulo $\nu_{i,j}$. Hence, it
follows from Equation (37) that

$H_{\nu_{i,j}}\big{(}({\cal P}^{U^{+}})_{0}^{\ell}\big{)}\leq
H_{\nu_{i,j}}\big{(}\bigvee_{k=0}^{\ell}{\mathfrak{a}}^{k}{\cal
P}\big{)}\leq(\ell+1)\log_{q_{v}}(N+1)\;.$ (59)

If $Q$ is any atom of the finite partition ${\cal
Q}_{i,\ell}=\bigvee_{k=0}^{k_{i}-1}{\mathfrak{a}}^{k}{\cal
Q}\vee\bigvee_{k=0}^{\ell}{\mathfrak{a}}^{k}{\cal P}$ of $\mathcal{Y}$, then
fixing any $y\in Q$, by Property (ii) of ${\cal Q}$, the intersection
$\phi_{A}(S_{i,j})\cap Q$ is contained in $E_{y,k_{i}-1}$ with the notation of
Lemma 6.2. It follows from Lemma 6.2 and Equation (57) that
$\phi_{A}(S_{i,j})\cap Q$ can be covered by $C\,q_{v}^{\,|\mathbf{r}|\beta
k_{i}}$ closed balls for $d_{U^{+}y_{A,0},\,\mathbf{r}}$ of radius
$q_{v}^{-(\ell_{0}+k_{i}-1)}=q_{v}^{-\ell_{0}+1}q_{v}^{-k_{i}}$, where $C$
depends only on $\mathbf{r}$ and $\ell_{0}$. Since $S_{i,j}$ is
$q_{v}^{-k_{i}}$-separated (hence
$q_{v}^{-\ell_{0}+1}q_{v}^{-k_{i}}$-separated since $\ell_{0}\geq 1$) with
respect to $d_{{\mathbb{T}}^{m},\mathbf{r}}$, and since
$\phi_{A}:({\mathbb{T}}^{n},d_{{\mathbb{T}}^{m},\mathbf{r}})\rightarrow(U^{+}y_{A,0},d_{U^{+}y_{A,0},\,\mathbf{r}})$
is an isometry, we have

${\operatorname{Card}}(\phi_{A}(S_{i,j})\cap Q)\leq
C\,q_{v}^{\,|\mathbf{r}|\beta k_{i}}\;.$

Since we have $({\cal P}^{U^{+}})_{0}^{\ell}=\sigma({\cal P})_{0}^{\ell}$
modulo $\nu_{i,j}$, since $\nu_{i,j}$ is the normalised counting measure on
$\phi_{A}(S_{i,j})$, and since the map $\Psi=-\log_{q_{v}}$ is nonincreasing,
it hence follows that

$\displaystyle H_{\nu_{i,j}}(\sigma({\cal Q})_{0}^{k_{i}-1}\vee({\cal
P}^{U^{+}})_{0}^{\ell})=H_{\nu_{i,j}}({\cal Q}_{i,\ell})=\sum_{Q\in{\cal
Q}_{i,\ell}}\nu_{i,j}(Q)\Psi\big{(}\nu_{i,j}(Q)\big{)}$ $\displaystyle=\;$
$\displaystyle\sum_{Q\in{\cal
Q}_{i,\ell}}\nu_{i,j}(Q)\Psi\Big{(}\frac{{\operatorname{Card}}(\phi_{A}(S_{i,j})\cap
Q)}{{\operatorname{Card}}\;S_{i,j}}\Big{)}\geq\Psi\Big{(}\frac{C\,q_{v}^{\,|\mathbf{r}|\beta
k_{i}}}{{\operatorname{Card}}\;S_{i,j}}\Big{)}\sum_{Q\in{\cal
Q}_{i,\ell}}\nu_{i,j}(Q)$ $\displaystyle=\;$
$\displaystyle\log_{q_{v}}({\operatorname{Card}}\;S_{i,j})-|\mathbf{r}|\,\beta\,k_{i}-\log_{q_{v}}C\;.$
(60)

Combining Equations (59) and (60), we have

$\begin{split}H_{\nu_{i,j}}(\sigma({\cal Q})_{0}^{k_{i}-1}|({\cal
P}^{U^{+}})_{0}^{\ell})&=H_{\nu_{i,j}}(\sigma({\cal
Q})_{0}^{k_{i}-1}\vee({\cal P}^{U^{+}})_{0}^{\ell})-H_{\nu_{i,j}}(({\cal
P}^{U^{+}})_{0}^{\ell})\\\
&\geq\log_{q_{v}}({\operatorname{Card}}\;S_{i,j})-|\mathbf{r}|\beta
k_{i}-\log_{q_{v}}C-(\ell+1)\log_{q_{v}}(N+1).\end{split}$ (61)

By the subadditivity and concavity properties of the entropy as in the proof
of [LSS, Eq. (2.9)], for every $M\in\mathbb{Z}_{\geq 1}$, we have

$\frac{1}{M}H_{{\widetilde{\nu}}_{i,j}}(\sigma({\cal Q})_{0}^{M-1}|({\cal
P}^{U^{+}})_{0}^{\ell})\geq\frac{1}{k_{i}}H_{\nu_{i,j}}(\sigma({\cal
Q})_{0}^{k_{i}-1}|({\cal
P}^{U^{+}})_{0}^{\ell})-\frac{2\,M\,\log_{q_{v}}({\operatorname{Card}}\;{\cal
Q})}{k_{i}}.$ (62)

Therefore, since $\nu_{i,j}(\infty_{\mathcal{Y}})=0$, it follows from
Equations (62) and (61) that

$\begin{split}&\frac{1}{M}H_{{\widetilde{\nu}}_{i,j}}\big{(}\;\overline{\sigma({\cal
Q})}\;_{0}^{M-1}|(\,\overline{{\cal
P}^{U^{+}}}\,)_{0}^{\ell}\big{)}=\frac{1}{M}H_{{\widetilde{\nu}}_{i,j}}\big{(}\sigma({\cal
Q})_{0}^{M-1}|({\cal P}^{U^{+}})_{0}^{\ell}\big{)}\\\
&\geq\frac{1}{k_{i}}\left(\log_{q_{v}}({\operatorname{Card}}\;S_{i,j})-|\mathbf{r}|\beta
k_{i}-\log_{q_{v}}C-(\ell+1)\log_{q_{v}}(N+1)-2\,M\,\log_{q_{v}}({\operatorname{Card}}\;{\cal
Q})\right).\end{split}$

Now we can take $i\to\infty$ since the atoms $Q$ of the partition
${\overline{\cal Q}}$ and hence of the partition
$\bigvee_{k=0}^{M-1}{\mathfrak{a}}^{k}\,{\overline{\cal Q}}$, satisfy
${\widetilde{\mu}}_{j}(\partial Q)=0$ by the property (iii) of ${\cal Q}$.
Also, the constants $C$, $N$, $\ell$, and ${\operatorname{Card}}\;{\cal Q}$
are independent of $k_{i}$. Thus it follows from Equation (53) that

$\frac{1}{M}H_{{\widetilde{\mu}}_{j}}\big{(}(\,\overline{\sigma({\cal
Q})}\,)_{0}^{M-1}|\,(\,{\overline{{\cal
P}^{U^{+}}}}\,)_{0}^{\ell}\big{)}\geq|\mathbf{r}|(1-\beta)-\max\mathbf{r}\;(m+\frac{1}{j}-\dim_{\rm
Haus}\mathbf{Bad}_{A}(\epsilon))\;.$

By taking $\rho\to 0$ in Equation (56), we have

$\frac{1}{M}H_{{\widetilde{\mu}}_{j}}\big{(}(\,\overline{\sigma({\cal
Q})}\,)_{0}^{M-1}|\,(\,{\overline{{\cal
P}^{U^{+}}}}\,)_{0}^{\ell}\big{)}\geq|\mathbf{r}|(1-{\widetilde{\mu}}_{j}(\,{\overline{Q}}_{\infty}))-\max\mathbf{r}\;(m+\frac{1}{j}-\dim_{\rm
Haus}\mathbf{Bad}_{A}(\epsilon))\;.$

Hence, it follows by taking $j\to\infty$ and by using the property (iii) of
${\cal Q}$ that

$\frac{1}{M}H_{{\overline{\mu}}}\big{(}(\,\overline{\sigma({\cal
Q})}\,)_{0}^{M-1}|\,(\,{\overline{{\cal
P}^{U^{+}}}}\,)_{0}^{\ell}\big{)}\geq|\mathbf{r}|(1-{\overline{\mu}}(\,{\overline{Q}}_{\infty}))-\max\mathbf{r}\;(m-\dim_{\rm
Haus}\mathbf{Bad}_{A}(\epsilon))\;.$

Since $(\,{\overline{{\cal
P}^{U^{+}}}}\,)_{0}^{\ell}\nearrow\;{\overline{{\cal A}^{U^{+}}}}$ as
$\ell\to\infty$, by the continuity of entropy, we finally have

$\frac{1}{M}H_{{\overline{\mu}}}\big{(}(\,\overline{\sigma({\cal
Q})}\,)_{0}^{M-1}|\;{\overline{{\cal
A}^{U^{+}}}}\big{)}\geq|\mathbf{r}|(1-{\overline{\mu}}({\overline{Q}}_{\infty}))-\max\mathbf{r}\;(m-\dim_{\rm
Haus}\mathbf{Bad}_{A}(\epsilon))\;,$

which proves Equation (55).

Hence, by taking $M\to\infty$, we have

$h_{{\overline{\mu}}}({\mathfrak{a}}^{-1}|\;\overline{{\cal
A}^{U^{+}}}\,)\geq|\mathbf{r}|(1-{\overline{\mu}}({\overline{Q}}_{\infty}))-\max\mathbf{r}\;(m-\dim_{\rm
Haus}\mathbf{Bad}_{A}(\epsilon))\;,$

provided that we have a partition ${\cal Q}$ satisfying the above requirements
(i), (ii) and (iii). After taking a sufficiently small neighborhood of
infinity $Q_{\infty}^{*}$ in $\mathcal{X}$, so that if
$Q_{\infty}=\pi^{-1}(Q_{\infty}^{*})$, then
${\overline{\mu}}({\overline{Q}}_{\infty})$ is sufficiently close to
${\overline{\mu}}(\infty_{\mathcal{Y}})=\eta_{A}$, we can indeed construct a
finite Borel-measurable partition $\mathcal{Q}$ of $\mathcal{Y}$ satisfying
Properties (i), (ii) and (iii), by following the procedure in [LSS, Proof of
Theorem 4.2, Claim 2]. This proves Assertion (3). $\Box$

###  6.2 Effective upper bound on $\dim_{\rm Haus}\mathbf{Bad}_{A}(\epsilon)$

For every $\ell\in\mathbb{Z}_{\leq 1}$, with $\lambda_{1}$ the shortest length
function of a nonzero vector of an $R_{v}$-lattice (see Subsection 2.2), we
define

$\mathcal{X}^{\geq q_{v}^{\ell}}=\\{x\in\mathcal{X}:\lambda_{1}(x)\geq
q_{v}^{\ell}\\}\quad\text{and}\quad\mathcal{Y}^{\geq
q_{v}^{\ell}}=\pi^{-1}(\mathcal{X}^{\geq q_{v}^{\ell}})\;.$

Note that by Corollary 2.2, we have $\lambda_{1}(x)\leq q_{v}$ for all
$x\in\mathcal{X}$, thus
$\mathcal{X}=\bigcup_{\ell=-\infty}^{1}\mathcal{X}^{\geq q_{v}^{\ell}}$. By
Mahler’s compactness criterion (see for instance [KlST, Theo. 1.1]), the
subsets $\mathcal{X}^{\geq q_{v}^{\ell}}$ and $\mathcal{Y}^{\geq
q_{v}^{\ell}}$ are compact.

###### Lemma 6.3

Let $\mu^{\prime}$ be an ${\mathfrak{a}}$-invariant Borel probability measure
on $\mathcal{Y}$ and let ${\cal A}$ be a countably generated
sub-$\sigma$-algebra of the Borel $\sigma$-algebra of $\mathcal{Y}$ which is
${\mathfrak{a}}^{-1}$-descending and $U^{+}$-subordinated modulo
$\mu^{\prime}$. For all $r^{\prime}\geq\delta^{\prime}>0$,
$\epsilon\in\;]0,1]$ and $\ell\in\mathbb{Z}_{\leq 0}$, let $j_{1},j_{2}$ be
integers satisfying

$j_{1}>\frac{d-(d-1)\ell}{\min\mathbf{r}}-\log_{q_{v}}\delta^{\prime}\quad\text{and}\quad
j_{2}>\frac{d-(d-1)\ell}{\min\mathbf{s}}-\frac{n}{d}\log_{q_{v}}\epsilon\>.$

If $y\in\mathcal{Y}^{\geq q_{v}^{\ell}}$ satisfies
$B_{\delta^{\prime}}^{U^{+},\mathbf{r}}{\mathfrak{a}}^{-j_{1}}y\subset[{\mathfrak{a}}^{-j_{1}}y]_{\cal
A}\subset B_{r^{\prime}}^{U^{+},\mathbf{r}}{\mathfrak{a}}^{-j_{1}}y$, then we
have

$\tau_{y}^{{\mathfrak{a}}^{j_{1}}{\cal
A}}({\mathfrak{a}}^{-j_{2}}\mathcal{L}_{\epsilon})\leq
1-\left(q_{v}^{-(j_{1}+j_{2})}(r^{\prime})^{-1}\epsilon^{\frac{m}{d}}\right)^{|\mathbf{r}|}\;.$

Proof. Let $x=\pi(y)$, which belongs to $\mathcal{X}^{\geq q_{v}^{\ell}}$.
Since $x$ is a unimodular $R_{v}$-lattice, by Minkowski’s theorem 2.1, we
hence have

$q_{v}^{(d-1)\ell}\lambda_{d}(x)\leq(\lambda_{1}(x))^{d-1}\lambda_{d}(x)\leq\lambda_{1}(x)\lambda_{2}(x)\cdots\lambda_{d}(x)\leq
q_{v}^{d}\;,$

therefore $\lambda_{d}(x)\leq q_{v}^{d-(d-1)\ell}$. There are linearly
independent vectors $v_{1},\dots,v_{d}$ in the $R_{v}$-lattice $x$ such that
$\|v_{i}\|\leq q_{v}^{d-(d-1)\ell}$. Let $\Delta$ be the parallelepiped in
$K_{v}^{\,d}$ generated by $v_{1},\dots,v_{d}$, that is,

$\Delta=\\{t_{1}v_{1}+\cdots+t_{d}v_{d}\in
K_{v}^{\,d}:\;\forall\;i=1,\dots,d,\;\;|\,t_{i}\,|\leq 1\\}\;.$

We identify $K_{v}^{\,d}$ with $K_{v}^{\,m}\times K_{v}^{\,n}$. Then for every
$\mathbf{b}=(\mathbf{b}^{-},\mathbf{b}^{+})\in\Delta$ with $\mathbf{b}^{-}\in
K_{v}^{\,m}$ and $\mathbf{b}^{+}\in K_{v}^{\,n}$, we have
$\|\,\mathbf{b}\,\|\leq q_{v}^{d-(d-1)\ell}$, hence
$\|\,\mathbf{b}^{-}\,\|_{\mathbf{r}}\leq
q_{v}^{\frac{d-(d-1)\ell}{\min\mathbf{r}}}$ and
$\|\,\mathbf{b}^{+}\,\|_{\mathbf{s}}\leq
q_{v}^{\frac{d-(d-1)\ell}{\min\mathbf{s}}}$ since $\ell\leq 0$. Note that the
fiber $\pi^{-1}(x)$ can be parametrized as follows: Fixing $g\in G_{0}$ with
$x=g\Gamma_{0}$, since $\Delta$ is a fondamental domain for the action of
$R_{v}^{\,d}$ on $K_{v}^{\,d}$, we have

$\pi^{-1}(x)=\\{w(\mathbf{b})g\Gamma:\mathbf{b}\in\Delta\\},\text{ where
}w(\mathbf{b})=\begin{pmatrix}I_{d}&\mathbf{b}\\\ 0&1\end{pmatrix}\;.$

In particular, there exists
$\mathbf{b}_{0}=(\mathbf{b}_{0}^{-},\mathbf{b}_{0}^{+})\in\Delta$ such that
$y=w(\mathbf{b}_{0})g\Gamma$.

With a slightly simplified notation, let $V_{y}$ be the $U^{+}$-shape of the
atom $[y]_{{\mathfrak{a}}^{j_{1}}{\cal A}}$ (see Equation (49)), so that we
have $V_{y}y=[y]_{{\mathfrak{a}}^{j_{1}}{\cal A}}$. Let $\Xi=\\{\bm{\theta}\in
K_{v}^{\,m}:w(\bm{\theta},0)\in V_{y}\\}$ be the Borel set corresponding to
$V_{y}$ by the canonical bijection $\bm{\theta}\mapsto w(\bm{\theta},0)$ (see
above Equation (34)) between $K_{v}^{\,m}$ and $U^{+}$. Note that $0\in\Xi$ as
$I_{d+1}\in V_{y}$. Since ${\mathfrak{a}}_{-}^{j_{1}}$ expands the
$\mathbf{r}$-quasinorm on $K_{v}^{\,m}$ with ratio exactly $q_{v}^{j_{1}}$
(see Equation (32)), and by the assumption on $y$ in the statement of Lemma
6.3, we have
$B_{q_{v}^{j_{1}}\delta^{\prime}}^{U^{+},\mathbf{r}}y\subset[y]_{{\mathfrak{a}}^{j_{1}}{\cal
A}}\subset B_{q_{v}^{j_{1}}r^{\prime}}^{U^{+},\mathbf{r}}y$, hence

$B_{q_{v}^{j_{1}}\delta^{\prime}}^{K_{v}^{\,m},\mathbf{r}}\subset\Xi\subset
B_{q_{v}^{j_{1}}r^{\prime}}^{K_{v}^{\,m},\mathbf{r}}\;.$ (63)

The atom $[y]_{{\mathfrak{a}}^{j_{1}}{\cal A}}$ can be parametrized by

$[y]_{{\mathfrak{a}}^{j_{1}}{\cal
A}}=\big{\\{}w(\mathbf{b})g\Gamma:\;\exists\;\mathbf{b}^{-}\in\mathbf{b}_{0}^{-}+\Xi,\;\;\mathbf{b}=(\mathbf{b}^{-},\mathbf{b}_{0}^{+})\big{\\}}\;,$

and $\tau_{y}^{{\mathfrak{a}}^{j_{1}}{\cal A}}$ is the pushforward measure of
the normalized Haar measure on the Borel set (with positive measure)
$\mathbf{b}_{0}^{-}+\Xi$ of $K_{v}^{\,m}$.

Let us consider the sets

$\Theta^{-}=\\{\mathbf{b}^{-}\in
K_{v}^{\,m}:\|\,\mathbf{b}^{-}\,\|_{\mathbf{r}}<q_{v}^{-j_{2}}\epsilon^{\frac{m}{d}}\\}\;\text{
and }\;\Theta^{+}=\\{\mathbf{b}^{+}\in
K_{v}^{\,n}:\|\,\mathbf{b}^{+}\,\|_{\mathbf{s}}<q_{v}^{j_{2}}\epsilon^{\frac{n}{d}}\\}\;.$

If $\mathbf{b}=(\mathbf{b}^{-},\mathbf{b}^{+})\in\Theta^{-}\times\Theta^{+}$,
then
$\|{\mathfrak{a}}_{-}^{j_{2}}\mathbf{b}^{-}\|_{\mathbf{r}}<\epsilon^{\frac{m}{d}}$
and
$\|{\mathfrak{a}}_{+}^{j_{2}}\mathbf{b}^{+}\|_{\mathbf{s}}<\epsilon^{\frac{n}{d}}$
by Equation (32). By the definition of $\mathcal{L}_{\epsilon}$ in Equation
(35), and since the grid
${\mathfrak{a}}^{j_{2}}gR_{v}^{\,m}+({\mathfrak{a}}_{-}^{j_{2}}\mathbf{b}^{-},{\mathfrak{a}}_{+}^{j_{2}}\mathbf{b}^{+})$
contains the vector
$({\mathfrak{a}}_{-}^{j_{2}}\mathbf{b}^{-},{\mathfrak{a}}_{+}^{j_{2}}\mathbf{b}^{+})$,
we have

${\mathfrak{a}}^{j_{2}}w(\mathbf{b})g\Gamma=w({\mathfrak{a}}_{-}^{j_{2}}\mathbf{b}^{-},{\mathfrak{a}}_{+}^{j_{2}}\mathbf{b}^{+}){\mathfrak{a}}^{j_{2}}g\Gamma\notin\mathcal{L}_{\epsilon}\;.$

Hence we have
$w(\mathbf{b})g\Gamma\notin{\mathfrak{a}}^{-j_{2}}\mathcal{L}_{\epsilon}$, so
that

$[y]_{{\mathfrak{a}}^{j_{1}}{\cal
A}}-{\mathfrak{a}}^{-j_{2}}\mathcal{L}_{\epsilon}\supset
w\big{(}\,\big{(}(\mathbf{b}_{0}^{-}+\Xi)\times\\{\mathbf{b}_{0}^{+}\\}\big{)}\cap(\Theta^{-}\times\Theta^{+})\big{)}g\Gamma\;.$
(64)

Claim.  We have the inclusion
$\Theta^{-}\times\\{\mathbf{b}_{0}^{+}\\}\subset\big{(}(\mathbf{b}_{0}^{-}+\Xi)\times\\{\mathbf{b}_{0}^{+}\\}\big{)}\cap(\Theta^{-}\times\Theta^{+})$.

Proof. We only have to prove that $\mathbf{b}_{0}^{+}\in\Theta^{+}$ and that
$\Theta^{-}\subset\mathbf{b}_{0}^{-}+\Xi$. Since
$(\mathbf{b}_{0}^{-},\mathbf{b}_{0}^{+})\in\Delta$, we have
$\|\,\mathbf{b}_{0}^{+}\,\|_{\mathbf{s}}\leq
q_{v}^{\frac{d-(d-1)\ell}{\min\mathbf{s}}}$, hence the former assertion
follows from the assumption that
$j_{2}>\frac{d-(d-1)\ell}{\min\mathbf{s}}-\frac{n}{d}\log_{q_{v}}\epsilon$.

In order to prove the latter assertion, let us fix
$\mathbf{b}^{-}\in\Theta^{-}$. Recall that the $\mathbf{r}$-quasinorm
$\|\cdot\|_{\mathbf{r}}$ satisfies the ultrametric inequality property, see
Equation (44). Hence, it follows from the assumptions
$j_{2}>\frac{d-(d-1)\ell}{\min\mathbf{s}}-\frac{n}{d}\log_{q_{v}}\epsilon$ and
$j_{1}>\frac{d-(d-1)\ell}{\min\mathbf{r}}-\log_{q_{v}}\delta^{\prime}$, since
$\epsilon\leq 1$, that

$\begin{split}\|\,\mathbf{b}^{-}-\mathbf{b}_{0}^{-}\,\|_{\mathbf{r}}&\leq\max\\{\|\,\mathbf{b}^{-}\,\|_{\mathbf{r}},\|\,\mathbf{b}_{0}^{-}\,\|_{\mathbf{r}}\\}\leq\max\big{\\{}q_{v}^{-j_{2}}\epsilon^{\frac{m}{d}},q_{v}^{\frac{d-(d-1)\ell}{\min\mathbf{r}}}\big{\\}}\\\
&\leq\max\big{\\{}q_{v}^{-\frac{d-(d-1)\ell}{\min\mathbf{s}}}\,\epsilon\,,q_{v}^{\frac{d-(d-1)\ell}{\min\mathbf{r}}}\big{\\}}=q_{v}^{\frac{d-(d-1)\ell}{\min\mathbf{r}}}<q_{v}^{\,j_{1}}\delta^{\prime}\;.\end{split}$

Hence by the left inclusion in Equation (63), we have
$\mathbf{b}^{-}\in\mathbf{b}_{0}^{-}+B_{q_{v}^{\,j_{1}}\delta^{\prime}}^{K_{v}^{\,m},\mathbf{r}}\subset\mathbf{b}_{0}^{-}+\Xi$,
which concludes the latter assertion. $\Box$

Now by Equation (64), by the above claim and by the right inclusion in
Equation (63), we have

$\displaystyle 1-\tau_{y}^{{\mathfrak{a}}^{\,j_{1}}{\cal
A}}({\mathfrak{a}}^{-j_{2}}\mathcal{L}_{\epsilon})$
$\displaystyle=\tau_{y}^{{\mathfrak{a}}^{\,j_{1}}{\cal
A}}([y]_{{\mathfrak{a}}^{j_{1}}{\cal
A}}-{\mathfrak{a}}^{-j_{2}}\mathcal{L}_{\epsilon})\geq\frac{m_{K_{v}^{\,m}}(\Theta^{-})}{m_{K_{v}^{\,m}}(\mathbf{b}_{0}^{-}+\Xi)}\geq\frac{m_{K_{v}^{\,m}}\big{(}B_{q_{v}^{-j_{2}}\epsilon^{\frac{m}{d}}}^{K_{v}^{\,m},\mathbf{r}}\big{)}}{m_{K_{v}^{\,m}}\big{(}B_{q_{v}^{\,j_{1}}r^{\prime}}^{K_{v}^{\,m},\mathbf{r}}\big{)}}$
$\displaystyle=\Big{(}\frac{q_{v}^{-j_{2}}\epsilon^{\frac{m}{d}}}{q_{v}^{\,j_{1}}r^{\prime}}\Big{)}^{|\mathbf{r}|}=\big{(}q_{v}^{-(j_{1}+j_{2})}(r^{\prime})^{-1}\epsilon^{\frac{m}{d}}\big{)}^{|\mathbf{r}|}\;.$

This proves the lemma. $\Box$

Proof of Theorem 1.2.  We fix a matrix $A\in\mathcal{M}_{m,n}(K_{v})$ which is
not $(\mathbf{r},\mathbf{s})$-singular on average, or equivalently by
Proposition 5.2 and the definition of $\eta_{A}$ just before Lemma 6.1, that
$\eta_{A}<1$. We also fix $\epsilon\in\;]0,1]$.

By Proposition 6.1, there exist an ${\mathfrak{a}}$-invariant Borel
probability measure ${\overline{\mu}}$ on ${\overline{\mathcal{Y}}}$
(depending on $\epsilon$) and an ${\mathfrak{a}}$-invariant Borel probability
measure $\mu$ on $\mathcal{Y}$ (unique since $\eta_{A}<1$) such that

$\operatorname{Supp}{{\overline{\mu}}}\subset\mathcal{L}_{\epsilon}\cup\\{\infty_{\mathcal{Y}}\\},\;\;{\overline{\pi}}_{*}{\overline{\mu}}=\mu_{A},\;\;\text{and}\;\;\;{\overline{\mu}}=(1-\eta_{A})\mu+\eta_{A}\Delta_{\infty_{\mathcal{Y}}}\;.$

Take a compact subset $K_{0}$ of $\mathcal{X}$ such that
$\mu_{A}(K_{0})>0.99\,\mu_{A}(\mathcal{X})=0.99\,(1-\eta_{A})$. Write
$K=\pi^{-1}(K_{0})$ and choose $r\in\;]0,1[$ such that
$K\subset\mathcal{Y}(r)$. Then $\mu(\mathcal{Y}(r))\geq\mu(K)>0.99$ since
$\eta_{A}<1$. Note that the choices of $K$ and $r$ are independent of
$\epsilon$ since the measure $\mu_{A}$ depends only on $A$ (see Proposition
6.1 and Equation (52)).

For such an $r>0$, let ${\cal A}^{U^{+}}$ be the $\sigma$-algebra of subsets
of $\mathcal{Y}$ constructed in Proposition 5.7. Proposition 6.1 (3) gives the
inequality

$h_{{\overline{\mu}}}({\mathfrak{a}}^{-1}|\;{\overline{{\cal
A}^{U^{+}}}}\,)\geq|\mathbf{r}|(1-\eta_{A})-\max\mathbf{r}\;(m-\dim_{\rm
Haus}\mathbf{Bad}_{A}(\epsilon))\;.$

By the linearity of entropy (and since the entropy of ${\mathfrak{a}}^{-1}$
vanishes on the fixed set $\\{\infty_{\mathcal{Y}}\\}$), we have

$h_{\mu}({\mathfrak{a}}^{-1}|\;{\cal
A}^{U^{+}})=\frac{1}{1-\eta_{A}}h_{{\overline{\mu}}}({\mathfrak{a}}^{-1}|\;{\overline{{\cal
A}^{U^{+}}}}\,)\geq|\mathbf{r}|-\frac{\max\mathbf{r}}{1-\eta_{A}}\;(m-\dim_{\rm
Haus}\mathbf{Bad}_{A}(\epsilon))\;.$ (65)

In order to use Lemma 5.9 and Proposition 5.10, we need an ergodicity
assumption on the measures that appear in these statements. We will choose an
appropriate ergodic component of $\mu$. Let us denote the ergodic
decomposition of $\mu$ by

$\mu=\int_{y\in\mathcal{Y}}\mu_{y}^{\cal E}\;d\mu(y)$

as in the second equality of Proposition 5.5 with $T={\mathfrak{a}}^{-1}$ and
${\cal A}={\cal A}^{U^{+}}$. Let $E=\\{y\in\mathcal{Y}:\mu_{y}^{{\cal
E}}(K)>0.9\\}$. It follows from $\mu(K)>0.99$ that

$0.99<\int_{\mathcal{Y}}\mu_{y}^{\cal
E}(K)\;d\mu(y)\leq\mu(E)+0.9\,\mu(\mathcal{Y}-E)=0.9+0.1\,\mu(E)\;,$

hence $\mu(E)>0.9$. By Proposition 5.5 and Equation (65), we have

$\int_{\mathcal{Y}}h_{\mu_{y}^{{\cal E}}}({\mathfrak{a}}^{-1}|\;{\cal
A}^{U^{+}})\;d\mu(y)=h_{\mu}({\mathfrak{a}}^{-1}|\;{\cal
A}^{U^{+}})\geq|\mathbf{r}|-\frac{\max\mathbf{r}}{1-\eta_{A}}\;(m-\dim_{\rm
Haus}\mathbf{Bad}_{A}(\epsilon))\;.$

Since $h_{\mu_{y}^{{\cal E}}}({\mathfrak{a}}^{-1}|\;{\cal
A}^{U^{+}})\leq|\mathbf{r}|$ for every $y\in\mathcal{Y}$ by Lemma 5.9, we have

$\int_{\mathcal{Y}-E}h_{\mu_{y}^{{\cal E}}}({\mathfrak{a}}^{-1}|\;{\cal
A}^{U^{+}})\;d\mu(y)\leq|\mathbf{r}|\;\mu(\mathcal{Y}-E)\;.$

Hence

$\displaystyle\int_{E}h_{\mu_{y}^{{\cal E}}}({\mathfrak{a}}^{-1}|\;{\cal
A}^{U^{+}})\;d\mu(y)$
$\displaystyle\geq|\mathbf{r}|\,\mu(E)-\frac{\max\mathbf{r}}{1-\eta_{A}}\;(m-\dim_{\rm
Haus}\mathbf{Bad}_{A}(\epsilon))$
$\displaystyle\geq\mu(E)\Big{(}|\mathbf{r}|-\frac{\max\mathbf{r}}{0.9\,(1-\eta_{A})}\;(m-\dim_{\rm
Haus}\mathbf{Bad}_{A}(\epsilon))\Big{)}\;.$

Therefore, there exists $z\in\mathcal{Y}$ such that $\mu_{z}^{{\cal
E}}(K)>0.9$ and

$h_{\mu_{z}^{{\cal E}}}({\mathfrak{a}}^{-1}|\;{\cal
A}^{U^{+}})\geq|\mathbf{r}|-\frac{\max\mathbf{r}}{0.9\,(1-\eta_{A})}\;(m-\dim_{\rm
Haus}\mathbf{Bad}_{A}(\epsilon))\;.$

We denote $\lambda=\mu_{z}^{{\cal E}}$ for such a $z\in\mathcal{Y}$. Then
$\lambda$ is an ${\mathfrak{a}}$-invariant ergodic Borel probability measure
on $\mathcal{Y}$ and
$\operatorname{Supp}{\lambda}\subset\operatorname{Supp}{\mu}\subset\mathcal{L}_{\epsilon}$.
By Lemma 5.9, we have

$H_{\lambda}({\cal A}^{U^{+}}|\;{\mathfrak{a}}\,{\cal
A}^{U^{+}})\geq|\mathbf{r}|-\frac{\max\mathbf{r}}{0.9\,(1-\eta_{A})}\;(m-\dim_{\rm
Haus}\mathbf{Bad}_{A}(\epsilon))\;.$ (66)

We will apply Lemma 6.3 with $\mu^{\prime}=\lambda$ and ${\cal
A}={\mathfrak{a}}^{-k}{\cal A}^{U^{+}}$ for some $k\geq 1$. Take an integer
$\ell\leq 0$ such that $K\subset\mathcal{Y}^{\geq q_{v}^{\ell}}$, which
depends only on $A$. Set

$j_{1}=\Big{\lceil}\frac{d-(d-1)\ell}{\min\mathbf{r}}-\log_{q_{v}}\delta^{\prime}\Big{\rceil}+1\;\text{
and
}\;j_{2}=\Big{\lceil}\frac{d-(d-1)\ell}{\min\mathbf{s}}-\frac{n}{d}\log_{q_{v}}\epsilon\Big{\rceil}+1\;,$

where $\delta^{\prime}$ will be determined later on.

Let
$k=\big{\lceil}\log_{q_{v}}\big{(}r^{\frac{1}{\max\mathbf{r}}}\,\epsilon^{-\frac{m}{d}}\big{)}\big{\rceil}+j_{2}+1$
and ${\cal A}={\mathfrak{a}}^{-k}{\cal A}^{U^{+}}$. By the properties of
${\cal A}^{U^{+}}$ given in Proposition 5.7 and since
$K\subset\mathcal{Y}(r)$, for every $y\in K$, we have

$B_{\delta_{r}}^{U^{+}}y\subset[y]_{{\cal A}^{U^{+}}}\subset
B_{r}^{U^{+}}y\;.$

It follows from Equation (45) that for any $y\in K$,

$B_{\delta_{r}^{\frac{1}{\min{\mathbf{r}}}}}^{U^{+},\mathbf{r}}y\subset[y]_{{\cal
A}^{U^{+}}}\subset B_{r^{\frac{1}{\max{\mathbf{r}}}}}^{U^{+},\mathbf{r}}y\;.$

Hence, by Equation (46), we have

$B_{q_{v}^{-k}\delta_{r}^{\frac{1}{\min{\mathbf{r}}}}}^{U^{+},\mathbf{r}}{\mathfrak{a}}^{-k}y\subset[{\mathfrak{a}}^{-k}y]_{{\mathfrak{a}}^{-k}{\cal
A}^{U^{+}}}=[{\mathfrak{a}}^{-k}y]_{{\cal A}}\subset
B_{q_{v}^{-k}r^{\frac{1}{\max{\mathbf{r}}}}}^{U^{+},\mathbf{r}}{\mathfrak{a}}^{-k}y\;.$

Thus for every $y\in{\mathfrak{a}}^{k}K$, we have

$B_{\delta^{\prime}}^{U^{+},\mathbf{r}}y\subset[y]_{{\cal A}}\subset
B_{r^{\prime}}^{U^{+},\mathbf{r}}y\;,$ (67)

where, by the definition of $k$, we take

$r^{\prime}=q_{v}^{-j_{2}-1}\epsilon^{\frac{m}{d}}\quad\text{and}\quad\delta^{\prime}=q_{v}^{-1}\;r^{-\frac{1}{\max\mathbf{r}}}\;\delta_{r}^{\frac{1}{\min\mathbf{r}}}\;r^{\prime}\;.$

Equation (67) implies that for every $y\in{\mathfrak{a}}^{j_{1}+k}K$, we have

$B_{\delta^{\prime}}^{U^{+},\mathbf{r}}{\mathfrak{a}}^{-j_{1}}y\subset[{\mathfrak{a}}^{-j_{1}}y]_{{\cal
A}}\subset B_{r^{\prime}}^{U^{+},\mathbf{r}}{\mathfrak{a}}^{-j_{1}}y\;.$ (68)

Now, we will use Proposition 5.10 with $j=j_{1}$,
$K^{\prime}={\mathfrak{a}}^{k}K$ (which is $U^{+}$-saturated since so is $K$
and as ${\mathfrak{a}}$ normalizes $U^{+}$), and $\epsilon=r^{\prime}$ (which
satisfies the assumption of Proposition 5.10 by Equation (67)). We claim that

$B_{q_{v}^{-1}\epsilon^{\frac{m}{d}}}^{U^{+},\mathbf{r}}\mathcal{L}_{\epsilon}\subset\mathcal{L}_{\epsilon}\;.$
(69)

Indeed, for all $y\in\mathcal{L}_{\epsilon}$ and $\bm{\theta}\in K_{v}^{\,m}$
such that $\|\,\bm{\theta}\,\|_{\mathbf{r}}\leq
q_{v}^{-1}\epsilon^{\frac{m}{d}}$, for every vector $u=(u^{-},u^{+})$ in the
grid $w(\bm{\theta},\mathbf{0})y$, we can write $u=v+(\bm{\theta},\mathbf{0})$
for some $v=(v^{-},v^{+})$ in the grid ${\widetilde{\Lambda}}_{y}$ associated
with $y$ (see Equation (33)). Since $y\in\mathcal{L}_{\varepsilon}$, we have
(see Equation (35))
$\|\,v\,\|_{\mathbf{r},\mathbf{s}}=\max\\{\|\,v^{-}\,\|_{\mathbf{r}}^{\;\frac{d}{m}},\|\,v^{+}\,\|_{\mathbf{s}}^{\;\frac{d}{n}}\\}\geq\epsilon$.
Since $u^{+}=v^{+}$, if
$\|\,v^{+}\,\|_{\mathbf{s}}^{\;\frac{d}{n}}\geq\epsilon$, then
$w(\bm{\theta},\mathbf{0})y\in\mathcal{L}_{\epsilon}$. Otherwise
$\|\,v^{-}\,\|_{\mathbf{r}}^{\;\frac{d}{m}}\geq\epsilon$. We then have
$\|\,\bm{\theta}\,\|_{\mathbf{r}}\leq
q_{v}^{-1}\epsilon^{\frac{m}{d}}<\epsilon^{\frac{m}{d}}\leq\|\,v^{-}\,\|_{\mathbf{r}}$.
It follows from the equality case of the ultrametric inequality property of
$\|\;\,\|_{\mathbf{r}}$ that

$\|\,u^{-}\,\|_{\mathbf{r}}=\|\,\bm{\theta}+v^{-}\,\|_{\mathbf{r}}=\max\big{\\{}\|\,\bm{\theta}\,\|_{\mathbf{r}},\;\|\,v^{-}\,\|_{\mathbf{r}}\big{\\}}=\|\,v^{-}\,\|_{\mathbf{r}}\geq\epsilon^{\frac{m}{d}}\;.$

Hence $w(\bm{\theta},\mathbf{0})y\in\mathcal{L}_{\epsilon}$, which proves
Equation (69).

By Proposition 5.7, the $\sigma$-algebra ${\cal A}^{U^{+}}$ is
${\mathfrak{a}}^{-1}$-descending and $U^{+}$-subordinated modulo $\lambda$,
and so does ${\cal A}={\mathfrak{a}}^{-k}{\cal A}^{U^{+}}$ since
${\mathfrak{a}}$ normalizes $U^{+}$. Note that
$\operatorname{Supp}{\lambda}\subset{\mathfrak{a}}^{-j_{2}}\mathcal{L}_{\epsilon}$
since $\lambda$ is ${\mathfrak{a}}$-invariant. By Equations (46) and (69), we
have

$B_{r^{\prime}}^{U^{+},\mathbf{r}}{\mathfrak{a}}^{-j_{2}}\mathcal{L}_{\epsilon}={\mathfrak{a}}^{-j_{2}}B_{q_{v}^{\,j_{2}}r^{\prime}}^{U^{+},\mathbf{r}}\mathcal{L}_{\epsilon}={\mathfrak{a}}^{-j_{2}}B_{q_{v}^{-1}\epsilon^{\frac{m}{d}}}^{U^{+},\mathbf{r}}\mathcal{L}_{\epsilon}\subset{\mathfrak{a}}^{-j_{2}}\mathcal{L}_{\epsilon}\;.$

Note that we have

$\tau_{y}^{{\mathfrak{a}}^{j_{1}}{\cal A}}(\mathcal{Y}-{\mathfrak{a}}^{k}K)=0$

for $\lambda$-almost every $y\in{\mathfrak{a}}^{k}K$, since then (see just
above Proposition 5.10) the support
$\operatorname{Supp}\tau_{y}^{{\mathfrak{a}}^{j_{1}}{\cal A}}$ is contained in
$[y]_{{\mathfrak{a}}^{j_{1}}{\cal A}}$, which is contained in $U^{+}y$, hence
in ${\mathfrak{a}}^{k}K$ since ${\mathfrak{a}}$ normalizes $U^{+}$ and
$K=\pi^{-1}(K_{0})$ is $U^{+}$-saturated. Therefore, it follows from
Proposition 5.10 for the first line, from the fact that the integrated
function is nonpositive (hence its integral on a smaller domain is larger) for
the third line, that

$\displaystyle H_{\lambda}({\cal A}|{\mathfrak{a}}^{j_{1}}{\cal A})$
$\displaystyle\leq
j_{1}|\mathbf{r}|+\int_{\mathcal{Y}}\log_{q_{v}}\tau_{y}^{{\mathfrak{a}}^{j_{1}}{\cal
A}}((\mathcal{Y}-{\mathfrak{a}}^{k}K)\cup
B_{r^{\prime}}^{U^{+},\mathbf{r}}\operatorname{Supp}{\lambda})\;d\lambda(y)$
$\displaystyle\leq
j_{1}|\mathbf{r}|+\int_{\mathcal{Y}}\log_{q_{v}}\tau_{y}^{{\mathfrak{a}}^{j_{1}}{\cal
A}}((\mathcal{Y}-{\mathfrak{a}}^{k}K)\cup{\mathfrak{a}}^{-j_{2}}\mathcal{L}_{\epsilon})\;d\lambda(y)$
$\displaystyle\leq
j_{1}|\mathbf{r}|+\int_{{\mathfrak{a}}^{k}K\cap{\mathfrak{a}}^{j_{1}+k}K\cap\mathcal{Y}^{\geq
q_{v}^{\ell}}}\log_{q_{v}}\tau_{y}^{{\mathfrak{a}}^{j_{1}}{\cal
A}}((\mathcal{Y}-{\mathfrak{a}}^{k}K)\cup{\mathfrak{a}}^{-j_{2}}\mathcal{L}_{\epsilon})\;d\lambda(y)$
$\displaystyle=j_{1}|\mathbf{r}|+\int_{{\mathfrak{a}}^{k}K\cap{\mathfrak{a}}^{j_{1}+k}K\cap\mathcal{Y}^{\geq
q_{v}^{\ell}}}\log_{q_{v}}\tau_{y}^{{\mathfrak{a}}^{j_{1}}{\cal
A}}({\mathfrak{a}}^{-j_{2}}\mathcal{L}_{\epsilon})\;d\lambda(y)\;.$

We now apply Lemma 6.3 with as said above $\mu^{\prime}=\lambda$ and ${\cal
A}={\mathfrak{a}}^{-k}{\cal A}^{U^{+}}$, and with
$y\in{\mathfrak{a}}^{j_{1}+k}K\cap\mathcal{Y}^{\geq q_{v}^{\ell}}$ which
satisfies the assumption of Lemma 6.3 by Equation (68). Thus

$\tau_{y}^{{\mathfrak{a}}^{j_{1}}{\cal
A}}({\mathfrak{a}}^{-j_{2}}\mathcal{L}_{\epsilon})\leq
1-\left(q_{v}^{-(j_{1}+j_{2})}{r^{\prime}}^{-1}\epsilon^{\frac{m}{d}}\right)^{|\mathbf{r}|}=1-q_{v}^{-(j_{1}-1)|\mathbf{r}|}\;.$

Hence

$-\log_{q_{v}}\tau_{y}^{{\mathfrak{a}}^{j_{1}}{\cal
A}}({\mathfrak{a}}^{-j_{2}}\mathcal{L}_{\epsilon})\geq-\log_{q_{v}}\big{(}1-q_{v}^{-(j_{1}-1)|\mathbf{r}|}\big{)}\geq\frac{q_{v}^{-(j_{1}-1)|\mathbf{r}|}}{\operatorname{ln}q_{v}}\;.$

Note that
$\lambda({\mathfrak{a}}^{k}K\cap{\mathfrak{a}}^{j_{1}+k}K\cap\mathcal{Y}^{\geq
q_{v}^{\ell}})\geq\frac{1}{2}$ since $\lambda$ is ${\mathfrak{a}}$-invariant,
$K\subset\mathcal{Y}^{\geq q_{v}^{\ell}}$ and $\lambda(K)>0.9$, so that the
three sets ${\mathfrak{a}}^{k}K$, ${\mathfrak{a}}^{j_{1}+k}K$ and
$\mathcal{Y}^{\geq q_{v}^{\ell}}$ have $\lambda$-measure $>0.9$, hence their
pairwise intersections have $\lambda$-measure $>2\times 0.9-1=0.8\,$, and
their triple intersection has $\lambda$-measure $>2\times 0.8-1=0.6\,$. It
follows from Equation (39) and the invariance under ${\mathfrak{a}}$ of
$\lambda$, hence of the conditional entropy, that

$\displaystyle|\mathbf{r}|-H_{\lambda}({\cal
A}^{U^{+}}|\;{\mathfrak{a}}\,{\cal A}^{U^{+}})$
$\displaystyle=|\mathbf{r}|-\frac{1}{j_{1}}H_{\lambda}({\cal
A}^{U^{+}}|\;{\mathfrak{a}}^{j_{1}}{\cal
A}^{U^{+}})=|\mathbf{r}|-\frac{1}{j_{1}}H_{\lambda}({\cal
A}|\;{\mathfrak{a}}^{j_{1}}{\cal A})$
$\displaystyle\geq-\frac{1}{j_{1}}\int_{{\mathfrak{a}}^{k}K\cap{\mathfrak{a}}^{j_{1}+k}K\cap\mathcal{Y}^{\geq
q_{v}^{\ell}}}\log_{q_{v}}\tau_{y}^{{\mathfrak{a}}^{j_{1}}{\cal
A}}({\mathfrak{a}}^{-j_{2}}\mathcal{L}_{\epsilon})\;d\lambda(y)$
$\displaystyle\geq\frac{q_{v}^{|\mathbf{r}|}}{2\operatorname{ln}{q_{v}}}\frac{q_{v}^{-j_{1}|\mathbf{r}|}}{j_{1}}\;.$

Therefore, by Equation (66), we have

$\frac{\max\mathbf{r}}{0.9\,(1-\eta_{A})}\;(m-\dim_{\rm
Haus}\mathbf{Bad}_{A}(\epsilon))\geq|\mathbf{r}|-H_{\lambda}({\cal
A}^{U^{+}}|\;{\mathfrak{a}}\,{\cal
A}^{U+})\geq\frac{q_{v}^{|\mathbf{r}|}}{2\operatorname{ln}{q_{v}}}\frac{q_{v}^{-j_{1}|\mathbf{r}|}}{j_{1}}\;.$

Observe that

$\displaystyle j_{1}$
$\displaystyle=\Big{\lceil}\;\frac{d-(d-1)\ell}{\min\mathbf{r}}-\log_{q_{v}}\delta^{\prime}\;\Big{\rceil}+1=\Big{\lceil}\;\frac{d-(d-1)\ell}{\min\mathbf{r}}-\log_{q_{v}}\Big{(}\;\frac{\delta_{r}^{\frac{1}{\min\mathbf{r}}}}{q_{v}^{2}\;r^{\frac{1}{\max\mathbf{r}}}}q_{v}^{-j_{2}}\epsilon^{\frac{m}{d}}\;\Big{)}\Big{\rceil}+1$
$\displaystyle=\Big{\lceil}\;\frac{d-(d-1)\ell}{\min\mathbf{r}}+\Big{\lceil}\;\frac{d-(d-1)\ell}{\min\mathbf{s}}-\frac{n}{d}\log_{q_{v}}\epsilon\;\Big{\rceil}+1-\frac{m}{d}\log_{q_{v}}\epsilon-\log_{q_{v}}\frac{\delta_{r}^{\frac{1}{\min\mathbf{r}}}}{q_{v}^{2}\;r^{\frac{1}{\max\mathbf{r}}}}\;\Big{\rceil}+1$
$\displaystyle\leq(\,d-(d-1)\ell\,)\Big{(}\frac{1}{\min\mathbf{r}}+\frac{1}{\max\mathbf{s}}\Big{)}-\log_{q_{v}}\frac{\delta_{r}^{\frac{1}{\min\mathbf{r}}}}{q_{v}^{2}r^{\frac{1}{\max\mathbf{r}}}}+4-\log_{q_{v}}\epsilon\;.$

The constants $\eta_{A}$, $\ell$, $\delta_{r}$, and $r$ depend only on the
fixed matrix $A\in\mathcal{M}_{m,n}(K_{v})$. Hence there exists a constant
$c(A)>0$ depending only on $d$, $\mathbf{r}$, $\mathbf{s}$ and $A$ such that

$m-\dim_{\rm Haus}\mathbf{Bad}_{A}(\epsilon)\geq
c(A)\;\frac{\epsilon^{\,|\mathbf{r}|}}{\log_{q_{v}}(1/\epsilon)}\;.$

This proves Theorem 1.2. $\Box$

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Department of Mathematical sciences
---
Seoul National University
San 56-1, Sillim-Dong, Seoul 151-747, SOUTH KOREA.
e-mail<EMAIL_ADDRESS>

Department of Mathematical sciences and Research Institute of Mathematics
---
Seoul National University
San 56-1, Sillim-Dong, Seoul 151-747, SOUTH KOREA.
e-mail<EMAIL_ADDRESS>

Laboratoire de mathématique d’Orsay, UMR 8628 Univ. Paris-Sud, CNRS
---
Université Paris-Saclay, 91405 ORSAY Cedex, FRANCE
e-mail<EMAIL_ADDRESS>