Remark 5.5, for every $\xi>0$ there exists $K(\xi)\geqslant 1$ so that for any
balls $B_{1},B_{2}$ in $X$ such that $|B_{1}|=\xi$ there exists $0\leqslant
p\leqslant K(\xi)$, ${\underline{\omega}}\in\Sigma_{\kappa}$ and a ball
$B_{2}^{\prime}\subset B_{2}$ of radius
$r\geqslant\min\\{|B_{1}|/4,|B_{2}|/2\\}$ so that
$g_{\underline{\omega}}^{p}(B_{1})\supset B_{2}^{\prime}$.

The idea is to use the frequent hitting time property to create points with
oscillating time averages by making them shadow pieces of orbits associated
with points with distinct time averages. The following auxiliary lemma
provides a control on the size of dynamic balls and images of dynamic balls
that will be instrumental in the argument.

######  Lemma 6.1.

If $f$ is a bi-Lipschitz homeomorphism and $L>1$ is a Lipschitz constant for
both $f$ and $f^{-1}$ then

$B_{f}(x,n,\varepsilon)\supset B(x,L^{-n}\varepsilon)\quad\text{and}\quad
f^{n}(B_{f}(x,n,\varepsilon))\supset B(f^{n}(x),L^{-2n}\varepsilon)$ (6.3)

for any $\varepsilon>0$ and $n\geqslant 1$.

###### Proof.

If $x,y\in X$ and $d(x,y)<L^{-n}\varepsilon$ then
$d(f^{j}(x),f^{j}(y))\leqslant L^{j}d(x,y)\leqslant
L^{j-n}\varepsilon<\varepsilon$ for every $0\leqslant j\leqslant n-1$. This
proves that $B_{f}(x,n,\varepsilon)\supset B(x,L^{-n}\varepsilon)$.
Analogously, if $y\in B(f^{n}(x),L^{-2n}\varepsilon)$ then
$d(f^{-j}(y),f^{n-j}(x))\leqslant L^{j}d(y,f^{n}(x))\leqslant
L^{j-2n}\varepsilon<L^{-n}\varepsilon$ for every $0\leqslant j\leqslant n-1$.
This ensures that

$B(f^{n}(x),L^{-2n}\varepsilon)\subset f^{n}(B(x,L^{-n}\varepsilon))\subset
f^{n}(B_{f}(x,n,\varepsilon)),$

which finishes the proof of the lemma. ∎

######  Remark 6.2.

The previous lemma holds more generally of $X$ is a compact and connected
metric space and $f$ is a Lipschitz continuous local homeomorphism whose
inverse branches are Lipschitz continuous. Indeed, in this context there
exists $\delta>0$ so that for every $x\in X$ there are well defined inverse
branches $f_{i}^{-1}:B(x,\delta)\to f_{i}^{-1}(B(x,\delta))$, and the
conclusion of Lemma 6.1 holds for every small $0<\varepsilon<\delta$.

In view of the estimates (6.3), one can see the frequent hitting times
property will be enough to guarantee that the transition times may grow
exponentially fast with the size of the finite pieces of orbits to be
shadowed. This can be observed in the following schematic Figure 6.1 below.

Figure 6.1. On the left: the image by $f_{\kappa}^{n_{1}}$ of the ball
$B(x_{1},L^{-n_{1}}\varepsilon)$ contains a ball of radius
$L^{-2n_{1}}\varepsilon$ in green color. On the center: the frequent hitting
times property ensures that the image of the green region by a suitable
iterate of $f_{0}$ covers a ball of radius $L^{-n_{2}}\varepsilon/2$

The difficulty now reduces to guarantee that the previous property is still
enough to ensure the divergence of the time averages. For that purpose we will
make the following choice of constants. Let us consider a strictly increasing
sequence of positive integers $(n_{k})_{k\geqslant 3}$ such that

$n_{j+1}>2n_{j}+\frac{2\log 2}{\log L}\;\text{for all $j\geqslant
2$}\quad\text{and}\quad\lim_{k\to\infty}\frac{\sum_{j=0}^{k}\big{[}n_{j}+K(\frac{L^{-2j}\varepsilon}{2})\big{]}}{n_{k+1}}=0,$
(6.4)

and for each $k\geqslant 1$ set

$z_{k}=\begin{cases}x_{1},\text{\; if $k$ odd}\\\ x_{2},\text{\; if $k$
even.}\end{cases}$

The following proposition guarantees that there exist points which shadows the
finite size orbits of the prescribed points $x_{1}$ and $x_{2}$, alternately,
and that these have divergent time-averages. More precisely:

######  Proposition 6.3.

There exists $x\in X$ and ${\underline{\omega}}\in\Sigma_{\kappa}$ so that:

  * (a)

$x\in B_{f_{\kappa}}(x_{1},n_{1},\varepsilon)$

  * (b)

for every $k\geqslant 1$ there exists $0\leqslant p_{k}\leqslant
K(\frac{L^{-2k}\varepsilon}{2})$ and a finite word
${\underline{\omega}}^{(k)}:=\omega^{(k)}_{i_{0}}\omega^{(k)}_{i_{1}}\dots\omega^{(k)}_{i_{p_{k}}}\in\\{1,2,\dots,\kappa\\}^{p_{k}}$
such that

$f_{\underline{\omega}}^{\sum_{i=1}^{k}(n_{i}+p_{i})}(x):=f_{{\underline{\omega}}^{(k)}}^{p_{k}}f_{\kappa}^{n_{k}}f_{{\underline{\omega}}^{(k-1)}}^{p_{k-1}}\dots
f_{\kappa}^{n_{2}}f_{{\underline{\omega}}^{(1)}}^{p_{1}}f_{\kappa}^{n_{1}}(x)\in
B_{f_{\kappa}}(z_{k},n_{k},\varepsilon)\quad\forall k\geqslant 2,$ (6.5)

where

${\underline{\omega}}=(\underbrace{\kappa,\kappa,\dots\kappa}_{n_{1}},\omega^{(1)}_{i_{0}}\omega^{(1)}_{i_{1}}\dots\omega^{(1)}_{i_{p_{1}}},\underbrace{\kappa,\kappa,\dots\kappa}_{n_{2}},\omega^{(2)}_{i_{0}}\omega^{(2)}_{i_{1}}\dots\omega^{(2)}_{i_{p_{2}}},\underbrace{\kappa,\kappa,\dots\kappa}_{n_{3}},\ldots).$
(6.6)

Moreover,

$\liminf_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\psi(f_{\underline{\omega}}^{j}(x)<\limsup_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\psi(f_{\underline{\omega}}^{j}(x).$
(6.7)

###### Proof.

Set $B_{1}:=B_{f_{\kappa}}(x_{1},n_{1},\varepsilon)$. By Lemma 6.1 one has
that $f_{\kappa}^{n_{1}}(B_{1})$ contains the ball of radius
$L^{-2n_{1}}\varepsilon$ around $f_{\kappa}^{n_{1}}(x_{1})$. Now, since the
semigroup action has frequent hitting times and
$L^{-n_{2}}\varepsilon<\frac{L^{-2n_{1}}\varepsilon}{4}$ (recall the first
inequality in (6.4)) there are integers
$K_{1}=K(\frac{L^{-2n_{1}}\varepsilon}{2})\geqslant 1$ and $0\leqslant
p_{1}\leqslant K_{1}$, a finite word
${\underline{\omega}}^{(1)}:=\omega^{(1)}_{i_{0}}\omega^{(1)}_{i_{1}}\dots\omega^{(1)}_{i_{p_{1}}}\in\\{1,2,\dots,\kappa\\}^{p_{1}}$
and a ball $B_{2}$ of radius $\frac{L^{-n_{2}}\varepsilon}{2}$ so that

$B_{2}\subset
f_{{\underline{\omega}}^{(1)}}^{p_{1}}\big{(}B(f_{\kappa}^{n_{1}}(x_{1}),L^{-2n_{1}}\varepsilon)\big{)}\cap
B(x_{2},L^{-n_{2}}\varepsilon)\subset
f_{{\underline{\omega}}^{(1)}}^{p_{1}}(B(f_{\kappa}^{n_{1}}(x_{1}),L^{-2n_{1}}\varepsilon))\cap
B_{f_{\kappa}}(x_{2},n_{2},\varepsilon)$

(recall Figure 6.1). By Lemma 6.1, $f_{\kappa}^{n_{2}}(B_{2})$ contains a ball
of radius $\frac{L^{-2n_{2}}\varepsilon}{2}$. Using that
$L^{-n_{3}}\varepsilon<\frac{L^{-2n_{2}}\varepsilon}{4}$, there exist
$K_{2}=K(\frac{L^{-2n_{2}}\varepsilon}{2})\geqslant 1$ and $0\leqslant
p_{2}\leqslant K_{2}$, a finite word
${\underline{\omega}}^{(2)}:=\omega^{(2)}_{i_{0}}\omega^{(2)}_{i_{1}}\dots\omega^{(2)}_{i_{p_{2}}}\in\\{1,2,\dots,\kappa\\}^{p_{2}}$
and a ball $B_{3}$ of radius $\frac{L^{-n_{3}}\varepsilon}{2}$ so that

$B_{3}\subset
f_{{\underline{\omega}}^{(2)}}^{p_{2}}\big{(}f_{\kappa}^{n_{2}}(B_{2})\big{)}\,\cap\,B(x_{3},L^{-n_{3}}\varepsilon)\subset
f_{{\underline{\omega}}^{(2)}}^{p_{2}}(f_{\kappa}^{n_{2}}(B_{2}))\cap
B_{f_{\kappa}}(x_{3},n_{3},\varepsilon).$

Proceeding recursively, we conclude that for every $s\geqslant 1$ there exists
$0\leqslant p_{s-1}\leqslant K(\frac{L^{-2n_{s-1}}\varepsilon}{2})$, a finite
word
${\underline{\omega}}^{(s)}:=\omega^{(s)}_{i_{0}}\omega^{(s)}_{i_{1}}\dots\omega^{(s)}_{i_{p_{s}}}\in\\{1,2,\dots,\kappa\\}^{p_{s}}$
and a ball $B_{s}$ of radius $\frac{L^{-n_{s}}\varepsilon}{2}$ contained in
$f_{{\underline{\omega}}^{(s)}}^{p_{s-1}}(f_{\kappa}^{n_{s-1}}(B_{s-1}))\cap
B_{f_{\kappa}}(x_{s},n_{s},\varepsilon)$. Moreover, if
${\underline{\omega}}\in\Sigma_{\kappa}$ is defined by (6.6), we claim that
any point

$x\in
B_{1}\cap\bigcap_{k>1}\big{(}f_{\underline{\omega}}^{\sum_{j=1}^{k}(n_{j}+p_{j})}\big{)}^{-1}(B_{k}).$

is irregular in the sense of (6.7) (the intersection is non-empty because the
right hand-side above consists of an intersection of compact and nested sets).
Indeed, according to (6.5), for each $k\geqslant 1$ one has that
$y_{k}:=f_{\underline{\omega}}^{\sum_{i=1}^{k}(n_{i}+p_{i})}(x)\in
B_{f_{\kappa}}(z_{k},n_{k},\varepsilon)$. In consequence, recalling (6.1),
(6.2) and (6.4), if $k\geqslant 1$ is even then

$\displaystyle\Big{|}\sum_{j=0}^{n_{k+1}+\sum_{i=1}^{k}(n_{i}+p_{i})}\psi(f_{\underline{\omega}}^{j}(x))-n_{k+1}\,\int\psi\,d\mu_{1}\Big{|}$
$\displaystyle\leqslant\Big{|}\sum_{j=0}^{n_{k+1}+\sum_{i=1}^{k}(n_{i}+p_{i})}\psi(f_{\underline{\omega}}^{j}(x))-\sum_{j=0}^{n_{k+1}}\psi(f_{\underline{\omega}}^{j}(y_{k}))\Big{|}$
$\displaystyle+\Big{|}\sum_{j=0}^{n_{k+1}}\psi(f_{\underline{\omega}}^{j}(y_{k}))-n_{k+1}\,\int\psi\,d\mu_{1}\Big{|}$
$\displaystyle\leqslant\|\psi\|_{\infty}\sum_{i=1}^{k}(n_{i}+p_{i})+n_{k+1}\frac{\int\psi\,d\mu_{2}-\int\psi\,d\mu_{1}}{4}.$

In consequence, if $k\geqslant 1$ is even and large then

$\displaystyle\frac{1}{n_{k+1}+\sum_{i=1}^{k}(n_{i}+p_{i})}\sum_{j=0}^{n_{k+1}+\sum_{i=1}^{k}(n_{i}+p_{i})}\psi(f_{\underline{\omega}}^{j}(x))$
$\displaystyle\leqslant\frac{n_{k+1}}{n_{k+1}+\sum_{i=1}^{k}(n_{i}+p_{i})}\int\psi\,d\mu_{1}$
$\displaystyle+\|\psi\|_{\infty}\,\frac{\sum_{j=1}^{k}\big{[}n_{j}+K(\frac{L^{-2j}\varepsilon}{2})\big{]}}{n_{k+1}}+\frac{\int\psi\,d\mu_{2}-\int\psi\,d\mu_{1}}{4}$
$\displaystyle<\int\psi\,d\mu_{1}+\frac{\int\psi\,d\mu_{2}-\int\psi\,d\mu_{1}}{3}.$

Analogously, it is not hard to check that

$\displaystyle\frac{1}{n_{k+1}+\sum_{i=1}^{k}(n_{i}+p_{i})}\sum_{j=1}^{n_{k+1}+\sum_{i=1}^{k}(n_{i}+p_{i})}\psi(f_{\underline{\omega}}^{j}(x))>\int\psi\,d\mu_{2}-\frac{\int\psi\,d\mu_{2}-\int\psi\,d\mu_{1}}{3}$

for every $k\geqslant 1$ odd and large. This implies that the Birkhoff
averages
$\big{(}\frac{1}{n}\sum_{j=0}^{n-1}\psi(f_{\underline{\omega}}^{j}(x)\big{)}_{n\geqslant
1}$ do not converge and completes the proof of the proposition. ∎

######  Remark 6.4.

The crucial point in (6.4) is that the constants $K(\cdot)$ involved in the
proof may grow arbitrarily fast but are independent of the shadowing points.
For instance, in the special case that $f$ is an irrational rotation on the
circle, Remark 5.3 ensures that $K(L^{-2n}\varepsilon/2)\simeq
3L^{2n}[\frac{1}{\varepsilon}]\gg n$ grows exponentially fast with $n$.

###  6.2. The set of irregular points is Baire generic

The strategy used in the proof of Theorem C is slightly different from the
classical strategies (see e.g [5, 48]), where the use of the specification or
the gluing orbit property is used to make successive approximations of points
with some preestablished behavior. The difference does not rely only on the
fact that in the process of shadowing a finite piece of orbit, time lags
between the shadowing times to become larger as these depend on the proximity
scale. In fact, while specification allows to make successive approximations
$(z_{k})_{k}$ of a certain initial point $x_{0}$ by points $z_{k}$ whose time
averages diverge and so that $\sum_{k\geqslant 1}d(z_{k},x_{0})$ is small, the
strategy here is to construct nested sequences of sets with a irregular
behavior by the analysis of the images of dynamic balls. Such a finer control,
provided by the frequent hitting times property, is not imediate even if the
dynamical system satisfies the specification property. This will become
evident in the proof of the following proposition:

######  Proposition 6.5.

Under the assumptions of Theorem C, the set $I_{F}(\varphi_{\psi})$ is a Baire
residual subset of $\Sigma_{\kappa}\times X$.

The proof of Proposition 6.5, which occupies the remainder of this subsection,
is divided in a three-step construction. The first two steps, whose content is
closer to the arguments presented in Subsection 6.1, consider non-typical
behavior along paths in the semigroup $G$. In the third step, one constructs
nested sets with oscilating behavior and prescribed initial condition for the
skew-product.

Consider a strictly decreasing sequence of positive numbers
$(\delta_{k})_{k\geqslant 1}$ tending to zero. Associated to these, pick a
strictly increasing sequence of large positive integers $(n_{k})_{k\geqslant
1}$ satisfying (6.4) and in such a way that for each $\ell\geqslant 1$ the
sets

$\Gamma_{2\ell-1}:=\Big{\\{}x\in
X:\big{|}\frac{1}{n_{2\ell-1}}\sum_{j=0}^{n_{2\ell-1}-1}\psi(f_{\kappa}^{j}(x))-\int\psi\,d\mu_{1}\big{|}<\delta_{2\ell-1}\Big{\\}}$

and

$\Gamma_{2\ell}:=\Big{\\{}x\in
X:\big{|}\frac{1}{n_{2\ell}}\sum_{i=0}^{n_{2\ell}-1}\psi(f_{\kappa}^{j}(x))-\int\psi\,d\mu_{2}\big{|}<\delta_{2\ell}\Big{\\}}$

are non-empty. This is possible by ergodicity of the measures $\mu_{i}$ with
respect to $f_{\kappa}$ ($i=1,2$), and the ergodic theorem.

Let $D\subset\Sigma_{\kappa}\times X$ be a countable and dense set. Fix
$({\underline{\omega}}_{0},x_{0})\in D$ and set
$S_{0}=\\{({\underline{\omega}}_{0},x_{0})\\}$. Now, for each $\ell\geqslant
1$ let $S_{2\ell-1}\subset\Gamma_{2\ell-1}$ be a
$(f_{\kappa},n_{2\ell-1},8\varepsilon)$-separated set in $X$, i.e. a set so
that

$\max_{0\leqslant j\leqslant
n_{2\ell-1}}d(f_{\kappa}^{j}(x),f_{\kappa}^{j}(y))>8\varepsilon$

for every distinct $x,y\in S_{2\ell-1}$. Analogously, let
$S_{2\ell}\subset\Gamma_{2\ell}$ be a
$(f_{\kappa},n_{2\ell},8\varepsilon)$-separated set. Now consider a sequence
of positive integers $(N_{k})_{k\geqslant 1}$ (independent of
${\underline{\omega}}_{0}$ and $x_{0}$) so that

$L^{-n_{k+1}}\varepsilon<\min\Big{\\{}\frac{L^{-2^{N_{k}-1}n_{k}}\varepsilon}{2^{2N_{k}-1}},L^{\lfloor\log(\varepsilon^{-1})\rfloor}\varepsilon/4\Big{\\}}\\\
\;\;\;\mbox{and}\;\;\;\\\
\lim_{k\to\infty}\frac{\sum_{i=0}^{k}\left[N_{i}n_{i}+\sum_{j=1}^{N_{i}-1}K(\frac{L^{-2^{j}\cdot
n_{j}}\varepsilon}{2^{j}})+K(\frac{L^{-2^{N_{i}-1}\cdot
n_{i}}\varepsilon}{2^{N_{i}-1}})\right]}{n_{k+1}}=0.$ (6.8)

We explain the construction dividing it in steps, which involve the frequent
hitting times property. First we explain the construction of sets which
replicate the behavior of the points in $S_{k}$ for a certain number $N_{k}$
of times. Then we repeat such argument to construct a nested sequence of sets
points which combine the different behaviors.

$1^{st}$ Step: _Replication of the behavior of points in each set $S_{k}$_

Fix $k\geqslant 1$. For each collection
$\underline{x}_{k}=(x_{1}^{k},\cdots,x_{N_{k}}^{k})\in S_{k}^{N_{k}}$ we will
explore ideas similar the ones used in Proposition 6.3 to construct a set
$C(\underline{x}_{k})\subset X$ of points which shadow pieces of orbits of
points in $S_{k}$. Let us be more precise. First recall that
$B_{1}^{k}:=B(x_{1}^{k},L^{-n_{k}}\varepsilon)\subset
B_{f_{\kappa}}(x_{1}^{k},n_{k},\varepsilon)$ and that Lemma 6.1 ensures
$f_{\kappa}^{n_{k}}(B(x_{1}^{k},L^{-n_{k}}\varepsilon))$ contains the ball of
radius $L^{-2n_{k}}\varepsilon$ around $f_{\kappa}^{n_{k}}(x_{1}^{k})$. As the
semigroup action has frequent hitting times there are constants
$K_{1}=K(L^{-2n_{k}}\varepsilon)\geqslant 1$ and $0\leqslant
p^{k}_{1}\leqslant K_{1}$, a finite word
${\underline{\omega}}^{(1)}:=\omega^{(1)}_{i_{0}}\omega^{(1)}_{i_{1}}\dots\omega^{(1)}_{i_{p^{k}_{1}}}\in\\{1,2,\dots,\kappa\\}^{p^{k}_{1}}$
and a ball $B_{2}^{k}$ of radius $\frac{L^{-2n_{k}}\varepsilon}{2^{2}}$ so
that

$\displaystyle B_{2}^{k}$ $\displaystyle\subset
f_{{\underline{\omega}}^{(1)}}^{p^{k}_{1}}\Big{(}B(f_{\kappa}^{n_{k}}(x_{1}^{k}),L^{-2n_{k}}\varepsilon)\Big{)}\cap
B\Big{(}x_{2}^{k},\frac{L^{-2n_{k}}\varepsilon}{2}\Big{)}$
$\displaystyle\subset
f_{{\underline{\omega}}^{(1)}}^{p^{k}_{1}}\Big{(}B(f_{\kappa}^{n_{k}}(x_{1}),L^{-2n_{k}}\varepsilon)\Big{)}\cap
B_{f_{\kappa}}(x_{2}^{k},n_{k},\varepsilon)$ $\displaystyle\subset
f_{{\underline{\omega}}^{(1)}}^{p^{k}_{1}}\Big{(}\,f_{\kappa}^{n_{k}}\big{(}B_{f_{\kappa}}(x_{1}^{k},n_{k},\varepsilon)\big{)}\,\Big{)}\cap
B_{f_{\kappa}}(x_{2}^{k},n_{k},\varepsilon).$ (6.9)

In particular

$\displaystyle
B_{f_{\kappa}}(x_{1}^{k},n_{k},\varepsilon)\;\cap\;\big{(}\,f_{{\underline{\omega}}^{(1)}}^{p^{k}_{1}}\circ
f_{\kappa}^{n_{k}}\,\big{)}^{-1}\Big{(}\,B_{f_{\kappa}}(x_{2}^{k},n_{k},\varepsilon)\,\Big{)}\neq\emptyset.$
(6.10)

Using Lemma 6.1 once more, we conclude that

$f_{\kappa}^{n_{k}}(B_{2}^{k})\;\text{contains a ball of
radius}\;L^{-2n_{k}}\times\frac{L^{-2n_{k}}\varepsilon}{2^{2}}=\frac{L^{-2^{2}n_{k}}\varepsilon}{2^{2}}.$

Again, using the frequent hitting times property and (6.9), there exist
$K_{2}=K(\frac{L^{-2^{2}n_{k}}\varepsilon}{2^{2}})\geqslant 1$, $0\leqslant
p_{2}^{k}\leqslant K_{2}$, a finite word
${\underline{\omega}}^{(2)}:=\omega^{(2)}_{i_{0}}\omega^{(2)}_{i_{1}}\dots\omega^{(2)}_{i_{p_{2}^{k}}}\in\\{0,1,\dots,\kappa\\}^{p^{k}_{2}}$
and a ball $B_{3}^{k}$ of radius $\frac{L^{-2^{2}n_{k}}\varepsilon}{2^{4}}$ so
that

$\displaystyle B_{3}^{k}$ $\displaystyle\subset
f_{{\underline{\omega}}^{(2)}}^{p_{2}^{k}}(f_{\kappa}^{n_{k}}(B_{2}^{k}))\cap
B(x_{3}^{k},\frac{L^{-2^{2}n_{k}}\varepsilon}{2^{3}})$ $\displaystyle\subset
f_{{\underline{\omega}}^{(2)}}^{p_{2}^{k}}(f_{\kappa}^{n_{k}}\Big{(}f_{{\underline{\omega}}^{(1)}}^{p^{k}_{1}}\Big{(}\,f_{\kappa}^{n_{k}}\big{(}B_{f_{\kappa}}(x_{1}^{k},n_{k},\varepsilon)\big{)}\,\Big{)}\cap
B_{f_{\kappa}}(x_{2}^{k},n_{k},\varepsilon)\Big{)}\cap
B_{f_{\kappa}}(x_{3}^{k},n_{k},\varepsilon).$

and its image $f_{\kappa}^{n_{k}}(B_{3}^{k})$ contains a ball of radius
$\frac{L^{-2^{3}n_{k}}\varepsilon}{2^{4}}$ (recall the first choice of
constants in (6.8)).

Proceeding recursively, for each $3\leqslant i\leqslant N_{k}-1$ there exists
a ball $B_{i}^{k}$ of radius $\frac{L^{-2^{i-1}n_{k}}\varepsilon}{2^{2i-3}}$
so that the set $f_{\kappa}^{n_{k}}(B_{i}^{k})$ contains a ball of radius
$\frac{L^{-2^{i}n_{k}}\varepsilon}{2^{2(i-1)}}$ and there are $0\leqslant
p^{k}_{i}\leqslant K_{i}=K(\frac{L^{-2^{i}n_{k}}\varepsilon}{2^{2(i-1)}})$ and
a finite word
${\underline{\omega}}^{(i)}\in\\{1,2,\dots,\kappa\\}^{p_{i}^{k}}$ so that
$f_{{\underline{\omega}}^{(i)}}^{p_{i}^{k}}(f_{\kappa}^{n_{k}}(B_{i}^{k}))\cap
B_{f_{\kappa}}(x_{i}^{k},n_{k},\varepsilon)$ contains a ball $B_{i+1}^{k}$ of
radius $\frac{L^{-2^{i}n_{k}}\varepsilon}{2^{2i-1}}$. For each
$\underline{x}_{k}=(x_{1}^{k}\cdots x_{N_{k}}^{k})\in S_{k}^{N_{k}}$ we
collect the data

$\displaystyle\big{(}\,n_{k}\,;\,({\underline{\omega}}^{(1)},p^{k}_{1}),\dots,({\underline{\omega}}^{(N_{k}-1)},p^{k}_{N_{k}-1})\,;\,(B_{1}^{k},\dots,B_{N_{k}}^{k})\,\big{)}$
(6.11)

given by the previous algorithm, where the times $p^{k}_{i}$ (depending both
on the size of the orbits and the points to be shadowed) are bounded by
$K_{i}$. Moreover, by construction, the set

$\displaystyle
C(\underline{x}_{k}):=B_{1}^{k}(x^{k}_{1})\cap\bigcap_{i=2}^{N_{k}-1}\left(f_{{\underline{\omega}}}^{\sum_{j=1}^{i-1}(n_{k}+\,p_{j}^{k})}\right)^{-1}\big{(}B^{k}_{i}(x_{i}^{k})\big{)}$
(6.12)

is non-empty, where
${\underline{\omega}}={\underline{\omega}}^{(k)}(\underline{x}_{k})$ is
defined by

$\displaystyle{\underline{\omega}}^{(k)}(\underline{x}_{k}):=(\,\underbrace{\kappa,\kappa,\dots\kappa}_{n_{k}\,\text{times}},\underbrace{\omega^{(1)}_{i_{0}}\omega^{(1)}_{i_{1}}\dots\omega^{(1)}_{i_{p_{1}}}}_{=\,\underline{\omega}^{(1)}},\underbrace{\kappa,\kappa,\dots\kappa}_{n_{k}\,\text{times}},\ldots,\underbrace{\omega^{(k-1)}_{i_{0}}\omega^{(k-1)}_{i_{1}}\,\dots\,\omega^{(k-1)}_{i_{p_{k-1}}}}_{{}_{=\,\underline{\omega}^{(N_{k}-1)}}},\underbrace{\kappa,\kappa,\dots\kappa}_{n_{k}\,\text{times}}\,).$
(6.13)

For future reference we denote by $B^{k}_{i}(x_{i}^{k})$ the balls $B^{k}_{i}$
as before, to emphasize that the latter are contained in a neighborhood of the
point $x_{i}^{k}$. Moreover, any point $x\in C(\underline{x}_{k})$ satisfies

$\displaystyle\begin{cases}x\in
B_{f_{\kappa}}(x_{1}^{k},n_{k},\varepsilon)\vskip 6.0pt plus 2.0pt minus
2.0pt\\\ f_{{\underline{\omega}}}^{\sum_{j=1}^{i-1}(n_{k}+\,p_{j}^{k})}(x)\in
B_{f_{\kappa}}(x_{i}^{k},n_{k},\varepsilon),\quad\forall\,1\leqslant
i\leqslant N_{k}.\end{cases}$

$2^{nd}$ Step: _Construction of nested sets with oscilating behavior_

In this step we explain the construction of nested sets which shadow points in
successive collections $S_{k}^{N_{k}}$. We first make explicit such
construction involving collections of points in $S_{1}^{N_{1}}$ and
$S_{2}^{N_{2}}$.

Given $\underline{x}_{1}=(x_{1}^{1},\cdots,x_{N_{1}}^{1})\in S_{1}^{N_{1}}$,
let $C(\underline{x}_{1})$ be the set defined by (6.12) and consider the ball
$B_{N_{1}}^{1}(x_{N_{1}}^{1})\subset
B_{f_{\kappa}}(x_{N_{1}}^{1},n_{1},\varepsilon)$ be the ball of radius
$\frac{L^{-2^{N_{1}-1}n_{1}}\varepsilon}{2^{2N_{1}-3}}$ whose image
$f_{\kappa}^{n_{1}}(B_{N_{1}}^{1}(x_{N_{1}}^{1}))$ contains a ball of radius
$\frac{L^{-2^{N_{1}-1}n_{1}}\varepsilon}{2^{2(N_{1}-1)}}$. By choice of the
constants (recall (6.8)) we have
$L^{-n_{2}}\varepsilon<\frac{1}{2}\frac{L^{-2^{N_{1}-1}n_{1}}\varepsilon}{2^{2(N_{1}-1)}}$.
Hence, the frequent hitting times property implies that there exists
$0\leqslant P_{1}\leqslant
K(\frac{L^{-2^{N_{1}-1}n_{1}}\varepsilon}{2^{2(N_{1}-1)}})$, a finite word
${\underline{\omega}}^{(12)}:=\omega^{(12)}_{i_{0}}\omega^{(12)}_{i_{1}}\dots\omega^{(12)}_{i_{P_{1}}}\in\\{1,2,\dots,\kappa\\}^{P_{1}}$
and ball $B_{1}^{2}$ of radius $\frac{1}{2}L^{-n_{2}}\varepsilon$ so that

$B_{1}^{2}\subset
f_{\underline{\omega}^{(12)}}^{P_{1}}(f_{\kappa}^{n_{1}}(B_{N_{1}}^{1}))\cap
B(x_{2}^{1},L^{-n_{2}}\varepsilon)\subset
f_{\underline{\omega}^{(12)}}^{P_{1}}(f_{\kappa}^{n_{1}}(B_{N_{1}}^{1}))\cap
B_{f_{\kappa}}(x_{2}^{1},n_{2},\varepsilon).$

Then, we can apply the argument used in the $1^{st}$ Step recursively (noting
that when $k=2$ the only difference is that the ball $B_{1}^{2}$ is half the
radius of the corresponding object in the $1^{st}$ Step) and deduce that, in
addition to the data in (6.11), for each $2\leqslant\ell\leqslant k$ there
exists an integer $0\leqslant P_{\ell}\leqslant
K(\frac{L^{-2^{N_{i}-1}n_{i}}\varepsilon}{2^{2(N_{i}-1)}})$, a finite word
${\underline{\omega}}^{(\ell(\ell+1))}:=\omega^{(\ell(\ell+1))}_{i_{0}}\omega^{(\ell(\ell+1))}_{i_{1}}\dots\omega^{(\ell(\ell+1))}_{i_{P_{\ell}}}\in\\{1,2,\dots,\kappa\\}^{P_{\ell}}$
so that the set

$\displaystyle
C(\underline{x}_{1},\underline{x}_{2},\dots,\underline{x}_{k}):=$
$\displaystyle
B_{1}^{1}(x^{1}_{1})\;\cap\;\bigcap_{i=2}^{N_{1}-1}\left(f_{{\underline{\omega}}}^{\sum_{j=1}^{i-1}(n_{j}+\,p_{j}^{k})}\right)^{-1}\big{(}B^{1}_{i}(x_{i}^{1})\big{)}$
$\displaystyle\;\cap\;\bigcap_{\ell=2}^{k}\bigcap_{i=2}^{N_{\ell}-1}\left(f_{{\underline{\omega}}}^{\sum_{s=1}^{\ell-1}P_{s}+\sum_{j=1}^{i-1}(n_{j}+\,p_{j}^{\ell})}\right)^{-1}\big{(}B^{\ell}_{i}(x_{i}^{\ell})\big{)}$

is non-empty, where

${\underline{\omega}}=(\,{\underline{\omega}}^{(1)}(\underline{x}_{1}),\,{\underline{\omega}}^{(12)},\,{\underline{\omega}}^{(2)}(\underline{x}_{2}),\,{\underline{\omega}}^{(23)},\,\dots,\,{\underline{\omega}}^{(k)}(\underline{x}_{k})\,)$

It is immediate from the construction that
$C(\underline{x}_{1},\underline{x}_{2},\dots,\underline{x}_{k},\underline{x}_{k+1})\subset
C(\underline{x}_{1},\underline{x}_{2},\dots,\underline{x}_{k})$ for every
$k\geqslant 1$ and any collections $\underline{x}_{i}\in S_{i}^{N_{i}}$.

$3^{rd}$ Step: _Construction of nested sets in $\Sigma_{\kappa}\times X$ with
oscilating behavior and prescribed initial condition for the skew-product map_

We first explain how to construct points
$({\underline{\omega}},x)\in\Sigma_{\kappa}\times X$ whose orbit by $F$ starts
close to the point $({\underline{\omega}}_{0},x_{0})\in S_{0}$ and,
subsequently, its projection $\pi_{X}(F^{j}({\underline{\omega}},x))$ over $X$
shadows the finite pieces of orbits of any chosen collection of points
$\underline{x}_{1}=(x_{1}^{1},\cdots,x_{N_{1}}^{1})\in S_{1}^{N_{1}}$. Let us
be more precise.

Considering the product metric on $\Sigma_{\kappa}\times X$, the ball
$B(({\underline{\omega}}_{0},x_{0}),\varepsilon)$ of radius $\varepsilon$ in
$\Sigma_{\kappa}\times X$ coincides with the product of the cylinder
$[{\underline{\omega}}_{0}]_{[\lfloor\log\varepsilon\rfloor,\lfloor\log(\varepsilon^{-1})\rfloor]}$
by the ball $B(x_{0},\varepsilon)$. For that reason, in order to consider non-
typical points for the skew-product it is natural to proceed in the
construction of consider the set
$f_{{\underline{\omega}}_{0}}^{\lfloor\log(\varepsilon^{-1})\rfloor}(B(x_{0},\varepsilon))\subset
X$, which contains the ball of radius
$L^{\lfloor\log(\varepsilon^{-1})\rfloor}\varepsilon$ around
$f_{{\underline{\omega}}_{0}}^{\lfloor\log(\varepsilon^{-1})\rfloor}(x_{0})$.
Fix a collection $\underline{x}_{1}=(x_{1}^{1},\cdots,x_{N_{1}}^{1})\in
S_{1}^{N_{1}}$ and observe that
$L^{-n_{1}}\varepsilon<L^{\lfloor\log(\varepsilon^{-1})\rfloor}\varepsilon/4$
(recall (6.8)). Then, the frequent hitting times property implies that there
exists $0\leqslant P_{0}\leqslant K(\varepsilon)$ and a finite word
${\underline{\omega}}^{(01)}\in\\{1,2,\dots,\kappa\\}^{P_{0}}$ so that the
image
$f_{{\underline{\omega}}^{(01)}}^{P_{0}}(B(f_{{\underline{\omega}}_{0}}^{\lfloor\log(\varepsilon^{-1})\rfloor}(x_{0}),L^{\lfloor\log(\varepsilon^{-1})\rfloor}\varepsilon))$
contains a ball $B_{1}^{1}\subset B_{f_{\kappa}}(x_{1}^{1},n_{1},\varepsilon)$
of radius $L^{-n_{1}}\varepsilon/2$. Proceeding _ipsis literis_ as in the
$2^{nd}$ Step (with $B_{1}^{1}$ as above) we conclude that the modified sets

$\displaystyle
C(x_{0},\,\underline{x}_{1},\dots,\underline{x}_{k})=B(x_{0},\varepsilon)\cap\big{(}f_{{\underline{\omega}}^{01}}^{P_{0}}f_{{\underline{\omega}}_{0}}^{\lfloor\log(\varepsilon^{-1})\rfloor}\big{)}^{-1}\Big{(}C(\underline{x}_{1},\underline{x}_{2},\dots,\underline{x}_{k})\Big{)}$
(6.14)

are non-empty and nested, as desired. Even though we omit the dependence of
the sets $C(x_{0},\,\underline{x}_{1},\dots,\underline{x}_{k})$ on the finite
word

$\displaystyle{\underline{\omega}}=(\,[{\underline{\omega}}_{0}]_{[\lfloor\log\varepsilon\rfloor,\lfloor\log(\varepsilon^{-1})\rfloor]},\,{\underline{\omega}}^{(01)},{\underline{\omega}}^{(1)}(\underline{x}_{1}),\,{\underline{\omega}}^{(12)},\,{\underline{\omega}}^{(2)}(\underline{x}_{2}),\,{\underline{\omega}}^{(23)},\,\dots,\,{\underline{\omega}}^{(k)}(\underline{x}_{k})\,)$
(6.15)

depending on $(x_{0},\,\underline{x}_{1},\dots,\underline{x}_{k})$, used to
determine it, we need to bookkeep it. Here, by some abuse of notation we are
denoting by
$[{\underline{\omega}}_{0}]_{[\lfloor\log\varepsilon\rfloor,\lfloor\log(\varepsilon^{-1})\rfloor]}$
the finite word that defines the cylinder set.

Now we are in a position to resume the proof of Proposition 6.5 and to
construct a $G_{\delta}$-dense subset of points with non-typical behavior for
the skew-product. Take the non-empty compact set

$\mathfrak{F}^{\prime}_{k}(\varepsilon,({\underline{\omega}}_{0},x_{0}))=\bigcup\;\;[{\underline{\omega}}]\times\overline{C(x_{0},\,\underline{x}_{1},\dots,\underline{x}_{k})}$

where the union is taken among all possibilities
$(\underline{x}_{1}\cdots\underline{x}_{k})\in S_{1}^{N_{1}}\times...\times
S_{k}^{N_{k}}$ and
${\underline{\omega}}={\underline{\omega}}(x_{0},\underline{x}_{1}\cdots\underline{x}_{k})$
is the finite word determined by (6.15). Define

$\mathfrak{F}^{\prime}(\varepsilon,({\underline{\omega}}_{0},x_{0})):=\bigcap_{k=1}^{\infty}\mathfrak{F}^{\prime}_{k}(\varepsilon,({\underline{\omega}}_{0},x_{0}))\quad\text{and}\quad\mathfrak{F}^{\prime}:=\bigcup_{j=\lfloor\frac{1}{\varepsilon_{0}}\rfloor}^{\infty}\bigcup_{({\underline{\omega}}_{0},x_{0})\in
D}\mathfrak{F}^{\prime}\big{(}\frac{1}{j},({\underline{\omega}}_{0},x_{0})\big{)},$

where $\varepsilon_{0}$ was chosen in (6.2). By construction the sets in
(6.14) are nested as well as the cylinders of increasing size in
$\Sigma_{\kappa}$, and so
$\mathfrak{F}^{\prime}_{k+1}(\varepsilon,({\underline{\omega}}_{0},x_{0}))\subseteq\mathfrak{F}^{\prime}_{k}(\varepsilon,({\underline{\omega}}_{0},x_{0}))$
for every point $({\underline{\omega}}_{0},x_{0})\in D$, $\varepsilon>0$ and
$k\geqslant 1$. We consider, alternatively, the family

$\mathfrak{F}_{k}(\varepsilon)=\bigcup\;\;[{\underline{\omega}}]\times\overline{C(x_{0},\,\underline{x}_{1},\dots,\underline{x}_{k})}$

where the union is taken among all possible elements
$({\underline{\omega}}_{0},x_{0})\in D$ and
$(\underline{x}_{1},\dots,\underline{x}_{k})\in S_{1}^{N_{1}}\times...\times
S_{k}^{N_{k}}$, and the finite words ${\underline{\omega}}$ are once more
determined by (6.15). In this case, define

$\mathfrak{F}_{k}=\bigcup_{j=\lfloor\frac{1}{\varepsilon_{0}}\rfloor}^{\infty}\mathfrak{F}_{k}\big{(}\frac{1}{j}\big{)}\quad\text{and}\quad\mathfrak{F}=\bigcap_{k=1}^{\infty}\mathfrak{F}_{k}$

By construction, $\mathfrak{F}_{k+1}\subset\mathfrak{F}_{k}$ for every
$k\geqslant 1$ and $\mathfrak{F}^{\prime}\subset\mathfrak{F}$. The proof of
the proposition is now reduced to the proof of the following two lemmas.

######  Lemma 6.6.

$\mathfrak{F}$ is a $G_{\delta}$-dense subset of $\Sigma_{\kappa}\times X$.

###### Proof.

We claim that $\mathfrak{F}^{\prime}$ is dense in $\Sigma_{\kappa}\times X$.
Indeed, for $r>0$ and $({\underline{\omega}},x)\in\Sigma_{\kappa}\times X$
there exist $j\geqslant 1$ and $({\underline{\omega}}_{0},x_{0})\in D$ such
that $d_{\Sigma_{\kappa}\times
X}(({\underline{\omega}},x),({\underline{\omega}}_{0},x_{0}))<1/j<r/2$. Any
point
$(\tilde{\underline{\omega}},y)\in\mathfrak{F}^{\prime}(1/j,({\underline{\omega}},x_{0}))$
satisfies $d_{\Sigma_{\kappa}\times
X}((\tilde{\underline{\omega}},y),({\underline{\omega}}_{0},x_{0}))<1/j$ and,
consequently, $d_{\Sigma_{\kappa}\times
X}((\tilde{\underline{\omega}},y),({\underline{\omega}},x))<2/j<r$. As
$\mathfrak{F}^{\prime}\subset\mathfrak{F}$, this shows that $\mathfrak{F}$ is
dense in $\Sigma_{\kappa}\times X$ as well.

Now, in order to prove that $\mathfrak{F}$ is a $G_{\delta}$-set, we define
the sets

$\mathfrak{G}_{k}(\varepsilon,({\underline{\omega}}_{0},x_{0})):=\bigcup\big{\\{}\;[{\underline{\omega}}]\times
C(x_{0},\,\underline{x}_{1},\dots,\underline{x}_{k}):({\underline{\omega}}_{0},x_{0})\in
D,\;(\,\underline{x}_{1},\dots,\underline{x}_{k})\in
S_{1}^{N_{1}}\times...\times S_{k}^{N_{k}}\big{\\}},$

with the finite words ${\underline{\omega}}$ given as in (6.15). Note that
$\mathfrak{G}_{k+1}(\varepsilon,({\underline{\omega}}_{0},x_{0}))\subset\mathfrak{G}_{k}(\varepsilon,({\underline{\omega}}_{0},x_{0}))$
and
$\mathfrak{G}_{k}(\varepsilon,x_{0})\subset\mathfrak{F}^{\prime}_{k}(\varepsilon,({\underline{\omega}}_{0},x_{0}))$
for every $k\geqslant 1$, $\varepsilon>0$ and
$({\underline{\omega}}_{0},x_{0})\in D$. We also observe that
$\mathfrak{F}^{\prime}_{k+1}(\varepsilon,({\underline{\omega}}_{0},x_{0}))\subset\mathfrak{G}_{k}(\varepsilon,({\underline{\omega}}_{0},x_{0}))$
for every $k\geqslant 1$. In fact, given
$(\tilde{\underline{\omega}},z)\in\mathfrak{F}^{\prime}_{k+1}(\varepsilon,({\underline{\omega}}_{0},x_{0}))$,
there exists a $k$-uple $(\,\underline{x}_{1},\dots,\underline{x}_{k+1})\in
S_{1}^{N_{1}}\times...\times S_{k+1}^{N_{k+1}}$ and a finite word
${\underline{\omega}}={\underline{\omega}}(x_{0},\,\underline{x}_{1},\dots,\underline{x}_{k+1})\in\\{1,2,\dots,\kappa\\}^{\mathbb{N}}$
determined by (6.15) in such a way that

$(\tilde{\underline{\omega}},z)\in[{\underline{\omega}}]\times\overline{C(x_{0}\,\underline{x}_{1}\cdots\underline{x}_{k+1})}\quad\text{and}\quad
f_{{\underline{\omega}}}^{\sum_{i=1}^{j}(N_{i}n_{i}+\sum_{j=1}^{N_{i}-1}p_{j}^{i}+P_{i-1})}(z)\in\overline{B_{N_{j}}^{j}},\;\forall
1\leqslant j\leqslant k+1,$

where $B_{N_{j}}^{j}$ is a ball contained in
$B_{f_{\kappa}}(x^{j}_{N_{j}},n_{j},\varepsilon)$ as described above (one can
replace ${\underline{\omega}}$ by $\tilde{\underline{\omega}}$ in the second
expression above because the cylinder $[{\underline{\omega}}]$ determines
${\sum_{i=1}^{j}(N_{i}n_{i}+\sum_{j=1}^{N_{i}-1}p_{j}^{i}+P_{i-1})}$ positive
coordinates of an element in $\Sigma_{\kappa}$). Furthermore, it is immediate
from the construction that

$f_{\tilde{\underline{\omega}}}^{-(N_{k+1}n_{k+1}+\sum_{j=1}^{N_{k+1}-1}p^{k}_{j}+P_{k})}(\overline{B}_{N_{k+1}}^{k+1})\subset
B_{N_{k}}^{k}.$

Thus
$f_{\tilde{\underline{\omega}}}^{\sum_{i=1}^{k}(N_{i}n_{i}+\sum_{j=1}^{N_{i}-1}p_{j}^{i}+P_{i-1})}(z)\in
B_{N_{k}}^{k},$ which proves that
$(\tilde{\underline{\omega}},z)\in\mathfrak{G}_{k}$. Altogether, we conclude
that

$\mathfrak{F}=\bigcap_{k=1}^{\infty}\;\Big{[}\bigcup_{j=\lfloor\frac{1}{\varepsilon_{0}}\rfloor}^{\infty}\bigcup_{({\underline{\omega}}_{0},x_{0})\in
D}\mathfrak{G}_{k}(\frac{1}{j},({\underline{\omega}}_{0},x_{0}))\Big{]}$

is obtainable as countable intersection of open sets, hence it is a
$G_{\delta}$-set. This finishes the proof of the lemma. ∎

Finally, in order to complete the proof of Proposition 6.5 it remains to prove
that all points in $\mathfrak{F}$ are Birkhoff irregular for the skew-product
$F$ and the continuous observable $\varphi_{\psi}:\Sigma_{\kappa}\times
X\to\mathbb{R}$. More precisely:

######  Lemma 6.7.

For any $({\underline{\omega}},x)\in\mathfrak{F}$ it holds that

$\liminf_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\psi(f^{j}_{{\underline{\omega}}}(x))<\limsup_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\psi(f^{j}_{{\underline{\omega}}}(x)).$

###### Proof.

By construction, for each $({\underline{\omega}},x)\in\mathfrak{F}$ and
$k\geqslant 1$ there are $j\geqslant 1$, $({\underline{\omega}}_{0},x_{0})\in
D$ and $(\,\underline{x}_{1},\dots,\underline{x}_{k})\in
S_{1}^{N_{1}}\times...\times S_{k}^{N_{k}}$ so that the first entries of the
word ${\underline{\omega}}$ are of the form
$(\,{\underline{\omega}}^{(1)}(\underline{x}_{1}),\,{\underline{\omega}}^{(12)},\,{\underline{\omega}}^{(2)}(\underline{x}_{2}),\,{\underline{\omega}}^{(23)},\,\dots,\,{\underline{\omega}}^{(kk+1)}\,)$
and

$f_{{\underline{\omega}}}^{(\ell
n_{i}+\sum_{s=1}^{\ell-1}p_{s}^{j}+P_{i-1})+\sum_{j=1}^{i-1}(N_{j}n_{j}+\sum_{s=1}^{N_{j}-1}p_{s}^{j}+P_{j-1})}(x)\in\overline{B^{i}_{\ell}}\qquad\;\forall
1\leqslant\ell\leqslant N_{i}\quad\forall 1\leqslant i\leqslant k,$

where $B^{i}_{\ell}$ is a ball contained in the dynamic ball
$B_{f_{\kappa}}(x^{i}_{\ell},n_{i},\frac{1}{j})$ as described in the previous
subsection. In particular, for each $k\geqslant 1$ the iterate
$y_{k}:=f_{\underline{\omega}}^{\sum_{i=1}^{k}(N_{i}n_{i}+\sum_{j=1}^{N_{i}-1}p_{j}^{i}+P_{i-1})}(x)$
belongs to $B_{f_{\kappa}}(x_{1}^{k+1},n_{k+1},\frac{1}{j})$. By triangular
inequality and the choice of constants in (6.2) and (6.8), if $k\geqslant 1$
is even and large then

$\displaystyle\Big{|}\sum_{j=0}^{n_{k+1}+\sum_{i=1}^{k}(N_{i}n_{i}+\sum_{j=1}^{N_{i}-1}p_{j}^{i}+P_{i-1})}$
$\displaystyle\psi(f_{\underline{\omega}}^{j}(x))-n_{k+1}\,\int\psi\,d\mu_{1}\Big{|}$
$\displaystyle\leqslant\Big{|}\sum_{j=0}^{n_{k+1}+\sum_{i=1}^{k}(N_{i}n_{i}+\sum_{j=1}^{N_{i}-1}p_{j}^{i}+P_{i-1})}\psi(f_{\underline{\omega}}^{j}(x))-\sum_{j=0}^{n_{k+1}}\psi(f_{\kappa}^{j}(y_{k}))\Big{|}$
$\displaystyle+\Big{|}\sum_{j=0}^{n_{k+1}}\psi(f_{\kappa}^{j}(y_{k}))-n_{k+1}\,\int\psi\,d\mu_{1}\Big{|}$
$\displaystyle\leqslant\|\psi\|_{\infty}\sum_{i=1}^{k}(N_{i}n_{i}+\sum_{j=1}^{N_{i}-1}p_{j}^{i}+P_{i-1})$
$\displaystyle{+n_{k+1}\frac{\int\psi\,d\mu_{2}-\int\psi\,d\mu_{1}}{4}}$
$\displaystyle<\Big{[}\int\psi\,d\mu_{1}+\frac{\int\psi\,d\mu_{2}-\int\psi\,d\mu_{1}}{3}\Big{]}\;\Big{(}n_{k+1}+\sum_{i=1}^{k}(N_{i}n_{i}+\sum_{j=1}^{N_{i}-1}p_{j}^{i}+P_{i-1})\Big{)}.$

Analogously, we get that

$\displaystyle\frac{1}{n_{k}+\sum_{i=1}^{k}(N_{i}n_{i}+\sum_{j=1}^{N_{i}-1}p_{j}^{i}+P_{i-1})}\sum_{j=0}^{n_{k}+\sum_{i=1}^{k}(N_{i}n_{i}+\sum_{j=1}^{N_{i}-1}p_{j}^{i}+P_{i-1})}\psi(f_{\underline{\omega}}^{j}(x))>\int\psi\,d\mu_{2}-\frac{\int\psi\,d\mu_{2}-\int\psi\,d\mu_{1}}{3}$

for every large and odd integer $k\geqslant 1$. This implies that the Birkhoff
averages
$\big{(}\frac{1}{n}\sum_{j=0}^{n-1}\psi(f_{\underline{\omega}}^{j}(x)\big{)}_{n\geqslant
1}$ do not converge, proving the lemma. ∎

###  6.3. Proof of Theorem C

The proof of the theorem is now a direct consequence of the results presented
in the previous sections. Indeed, one knows that $I_{F}(\varphi_{\psi})$ is a
Baire generic subset of $\\{1,2,\dots,\kappa\\}^{\mathbb{Z}}\times X$.
Recalling that $\pi:\\{1,2,\dots,\kappa\\}^{\mathbb{Z}}\times
X\to\\{1,2,\dots,\kappa\\}^{\mathbb{Z}}$ denotes the projection on the first
coordinate, it follows from the Kuratowski-Ulam theorem (cf. [46]) that the
set

$\Big{\\{}{\underline{\omega}}\in\Sigma_{\kappa}\colon
I_{F}(\varphi_{\psi})\cap\pi^{-1}({\underline{\omega}})\;\text{is a Baire
residual subset of}\,\\{{\underline{\omega}}\\}\times X\Big{\\}}$

is a Baire residual subset of $\Sigma_{\kappa}$. As
$I_{F}(\varphi_{\psi})\cap\pi^{-1}({\underline{\omega}})=\\{{\underline{\omega}}\\}\times
I_{\mathcal{F}_{\underline{\omega}}}(\psi)$, this proves the first statement
in the theorem. The second one is a direct consequence of the inclusion (2.4)
and the first statement. This completes the proof of the theorem.

##  7\. Topological complexity on the skew-product

In this section we prove Theorem D.

###  7.1. Entropy estimates obtained from ergodic measures

Let $X$ be a compact metric space,
$G_{1}=\\{id,f_{1},f_{2},\dots,f_{\kappa}\\}$ be a collection of bi-Lipschitz
homeomorphisms on $X$,
$\sigma:\\{1,2,\dots,\kappa\\}^{\mathbb{Z}}\to\\{1,2,\dots,\kappa\\}^{\mathbb{Z}}$
be the full shift and $F:\\{1,2,\dots,\kappa\\}^{\mathbb{Z}}\times
X\to\\{1,2,\dots,\kappa\\}^{\mathbb{Z}}\times X$ be the skew product

$F({\underline{\omega}},x)=(\sigma({\underline{\omega}}),f_{\omega_{0}}(x)),\quad\text{for
every}\;({\underline{\omega}},x)\in\\{1,2,\dots,\kappa\\}^{\mathbb{Z}}\times
X.$

Assume that the semigroup action generated by $G_{1}$ has frequent hitting
times. In rough terms, this ensures that $F$ has frequent hitting times along
the fibers. Let $\varphi:\\{1,2,\dots,\kappa\\}^{\mathbb{Z}}\times
X\to\mathbb{R}$ be a continuous observable. Assume that $h_{*}(\varphi)$,
given by (2.5), is strictly positive (otherwise there is nothing to prove).
Given an arbitrary and fixed $\gamma\in(0,1)$, let $\mu_{1},\mu_{2}$ be
$F$-invariant and ergodic probabilities such that
$\int\varphi\,d\mu_{1}<\int\varphi\,d\mu_{2}$ and
$h_{\mu_{i}}(F)>h_{*}(\varphi)-\gamma$. Now, consider a strictly decreasing
sequence of positive numbers $(\delta_{k})_{k\geqslant 1}$ tending to zero so
that $\delta_{1}<\frac{1}{4}(\int\varphi\,d\mu_{2}-\int\varphi\,d\mu_{1})$
and, associated to these, pick a strictly increasing sequence of large
positive integers $(\mathfrak{n}_{k})_{k\geqslant 1}$ so that, for each
$\ell\geqslant 1$, the set

$\Gamma_{2\ell-1}:=\Big{\\{}({\underline{\omega}},x)\in\\{1,2,...,\kappa\\}^{\mathbb{Z}}\times
X:\Big{|}\frac{1}{n}\sum_{j=0}^{n-1}\varphi(F^{j}({\underline{\omega}},x))-\int\varphi\,d\mu_{1}\Big{|}<\delta_{2\ell-1},\;\;\forall
n\geqslant\mathfrak{n}_{2\ell-1}\Big{\\}}$

has $\mu_{1}$-measure larger than $\frac{1}{2}$ and

$\Gamma_{2\ell}:=\Big{\\{}({\underline{\omega}},x)\in\\{1,2,...,\kappa\\}^{\mathbb{Z}}\times
X:\Big{|}\frac{1}{n}\sum_{j=0}^{n-1}\varphi(F^{j}({\underline{\omega}},x))-\int\varphi\,d\mu_{2}\Big{|}<\delta_{2\ell},\;\;\forall
n\geqslant\mathfrak{n}_{2\ell}\Big{\\}}$

has $\mu_{2}$-measure larger than $\frac{1}{2}$. This is possible by
ergodicity of the measures $\mu_{1}$ and $\mu_{2}$ together with the ergodic
theorem and Lusin’s theorem. By uniform continuity of $\varphi$ and Katok’s
entropy formula (recall Subsection 4), there exists $\varepsilon>0$ such that

$\displaystyle|\varphi(\omega,x)-\varphi(\omega^{\prime},x^{\prime})|<\frac{\int\varphi\,d\mu_{2}-\int\varphi\,d\mu_{1}}{4}$
(7.1)

for every
$(\omega,x),(\omega^{\prime},x^{\prime})\in\\{1,2,...,\kappa\\}^{\mathbb{Z}}\times
X$ so that $d((\omega,x),(\omega^{\prime},x^{\prime}))<4\varepsilon$ and, for
each $i=1,2$,

$\displaystyle\underline{h}_{\mu_{i}}(F,8\varepsilon,\frac{1}{2})=\liminf_{n\rightarrow+\infty}\frac{1}{n}\log
N^{\mu_{i}}_{n}(8\varepsilon,\frac{1}{2})>h_{*}(\varphi)-2\gamma.$ (7.2)

This constant $\varepsilon>0$ will be fixed throughout the remainder of the
argument.

The next lemma allows one to select a special class of dynamically separated
points inside each of the sets $\Gamma_{\ell}$. More precisely:

######  Lemma 7.1.

There exists a sequence $(n_{\ell})_{\ell\geqslant 1}$ of positive integers
such that:

  1. (1)

$n_{\ell}\geqslant\max\\{\;\mathfrak{n}_{\ell},\;(\ell+2)\cdot
n_{\ell-1}+\Big{[}\frac{2\log 2}{\log
L}{+\log\varepsilon^{-1}}\Big{]},\;\ell\cdot
K(L^{-2n_{\ell-1}}\,L^{\log(\varepsilon^{-1})^{-1}}\,2\varepsilon)\;\\}$ for
every $\ell\geqslant 2$

  2. (2)

there exists an $(n_{\ell},4\varepsilon)$-separated set
$\mathcal{S}_{\ell}\subset\Gamma_{\ell}$ (with respect to $F$) of cardinality

$\\#\mathcal{S}_{\ell}\geqslant\exp(\,(h_{*}(\varphi)-3\gamma)n_{\ell}\,)\quad\text{for
every $\ell\geqslant 1$.}$

###### Proof.

The proof is by induction on $\ell$. Set $\ell=1$. Since
$\mu_{1}(\Gamma_{1})\geqslant\frac{1}{2}$, inequality (7.2) implies that the
minimum cardinality of a $(n,8\varepsilon)$-generating set for $\Gamma_{1}$ is
bounded below by $N^{\mu_{i}}_{n}(8\varepsilon,\frac{1}{2})$. Pick
$n_{1}\geqslant\mathfrak{n}_{1}$ so that

$N^{\mu_{1}}_{n}(8\varepsilon,\frac{1}{2})\geqslant\exp(\,(h_{*}(\varphi)-3\gamma)n\,),\quad\forall
n\geqslant n_{1}.$

Recall that any $(n,8\varepsilon)$-generating set is a
$(n,4\varepsilon)$-separated set. Hence, we deduce that there exists an
$(n_{1},4\varepsilon)$-separated set $\mathcal{S}_{1}\subset\Gamma_{1}$ with
cardinality $\\#\mathcal{S}_{1}\geqslant
e^{\,(h_{*}(\varphi)-3\gamma)n_{1}\,}$.

Now, assume that the lemma is proven for every $1\leqslant\ell<k$. If $\ell=k$
then pick a large integer

$n_{\ell}\geqslant\max\\{\;\mathfrak{n}_{\ell},\;(\ell+2)\cdot
n_{\ell-1}+\Big{[}\frac{2\log 2}{\log
L}{+\log\varepsilon^{-1}}\Big{]},\;\ell\cdot
K(L^{-2n_{\ell-1}}\,L^{\log(\varepsilon^{-1})^{-1}}\,2\varepsilon)\;\\}$

so that

$N^{\mu_{\ell({\tiny\mbox{mod}}1)}}_{n}(8\varepsilon,\frac{1}{2})\geqslant\exp(\,(h_{*}(\varphi)-3\gamma)n\,),\quad\forall
n\geqslant n_{\ell}.$

Then, taking $n=n_{\ell}$ above we conclude that that there exists an
$(n_{\ell},4\varepsilon)$-separated set
$\mathcal{S}_{\ell}\subset\Gamma_{\ell}$ with cardinality
$\\#\mathcal{S}_{\ell}\geqslant e^{\,(h_{*}(\varphi)-3\gamma)n_{\ell}\,}$, as
claimed. This proves the induction assumption and completes the proof of the
lemma. ∎

By construction, the sequence $(n_{\ell})_{\ell\geqslant 1}$ is strictly
increasing and it satisfies

$\lim_{\ell\to+\infty}\frac{n_{\ell-1}}{n_{\ell}}=0\quad\text{and}\quad\lim_{\ell\to+\infty}\frac{K(L^{-2n_{\ell-1}}\,L^{\log(\varepsilon^{-1})^{-1}}\,2\varepsilon)}{n_{\ell}}=0.$

In particular this ensures that The second expression reflects that the size
of the orbits in $\Gamma_{\ell}$ are much larger than the largest possible
transition time necessary for shadowing pieces of orbits in $\Gamma_{\ell-1}$
and $\Gamma_{\ell}$ consecutively.

###  7.2. A Moran set of irregular points in
$\\{1,2,...,\kappa\\}^{\mathbb{Z}}\times X$ with large entropy

The main goal of this subsection is the construction of a Moran set, contained
in the set $I_{F}(\varphi)$ of irregular in $\Sigma_{\kappa}\times X$, having
large upper topological entropy.

$1^{st}$ Step: _Understanding dynamic balls for points in
$\mathcal{S}_{\ell}$_

Fix $\ell\geqslant 1$, and let
$B_{F}(({\underline{\omega}},x),n_{\ell},\varepsilon)\subset\Sigma_{\kappa}\times
X$ be the dynamic ball with respect to $F$, centered at
$({\underline{\omega}},x)$, of radius $\varepsilon$ and length $n_{\ell}$.
Given ${\underline{\omega}}\in\Sigma_{\kappa}$ and integers $m\leqslant n$ we
denote by

${\underline{\omega}}_{[m,n]}:=\big{\\{}\,\tilde{\underline{\omega}}\in\Sigma_{\kappa}\colon{\underline{\omega}}_{j}=\widetilde{{\underline{\omega}}}_{j},\,\forall
j\in[m,n]\,\big{\\}}$

the cylinder set determined by the infinite word ${\underline{\omega}}$ at the
positions in the interval $[m,n]$. We need the following auxiliary lemma.

######  Lemma 7.2.

For any $\varepsilon>0$, $({\underline{\omega}},x)\in\Sigma_{\kappa}\times X$
and $\ell\geqslant 1$ one has that

$B_{F}(({\underline{\omega}},x),n_{\ell},\varepsilon)=B_{\sigma}({\underline{\omega}},n_{\ell},\varepsilon)\times
B_{{\underline{\omega}}}(x,n_{\ell},\varepsilon)$

where $B_{\sigma}({\underline{\omega}},n_{\ell},\varepsilon)$ is the cylinder
set
${\underline{\omega}}_{\,[-\lfloor\log(\varepsilon^{-1})\rfloor\,,\,n_{\ell}+\lfloor\log(\varepsilon^{-1})\rfloor]}$
and

$B_{{\underline{\omega}}}(x,n_{\ell},\varepsilon):=\big{\\{}y\in X\colon
d_{X}(g_{\underline{\omega}}^{j}(x),g_{\underline{\omega}}^{j}(y))<\varepsilon\,\text{
for every }\,0\leqslant j\leqslant n_{\ell}\big{\\}}.$

In particular,

  1. (1)

$F^{n_{\ell}}\big{(}\,B_{F}(({\underline{\omega}},x),n_{\ell},\varepsilon)\,\big{)}\supseteq\sigma^{n_{\ell}}\big{(}{\underline{\omega}}_{\,[-n_{\ell}+\lfloor\log(\varepsilon^{-1})\rfloor,\,\lfloor-\log(\varepsilon^{-1})\rfloor]}\big{)}\times
B(f^{n_{\ell}}_{\underline{\omega}}(x),L^{-2n_{\ell}}\varepsilon);$

  2. (2)

$F^{n_{\ell}+\lfloor\log(\varepsilon^{-1})\rfloor}\big{(}\,B_{F}(({\underline{\omega}},x),n_{\ell},\varepsilon)\,\big{)}$

  3. $\supseteq\sigma^{n_{\ell}+\lfloor\log(\varepsilon^{-1})\rfloor}\big{(}{\underline{\omega}}_{\,[-n_{\ell}+\lfloor\log(\varepsilon^{-1})\rfloor,\,\lfloor-\log(\varepsilon^{-1})\rfloor]}\big{)}\times B(f^{n_{\ell}+\lfloor\log(\varepsilon^{-1})\rfloor}_{\underline{\omega}}(x),L^{-(2{n_{\ell}}+\lfloor\log(\varepsilon^{-1})\rfloor)}\varepsilon)$

###### Proof.

The first claim is an immediate consequence of the fact that the step skew-
products are constant on cylinders of size one in the shift, and that we are
taking the maximum metric. Then, item (1) follows from the definition and
Lemma 6.1, which ensures that
$f_{\omega}^{n}(B_{{\underline{\omega}}}(x,n,\varepsilon))$ contains the ball
of radius $L^{-2n}\varepsilon$ around $f_{\omega}^{n}(x)$. Finally, the
inclusion in item (2) is obtained from item (1) using the same argument as
above and iteration by $F^{\lfloor\log(\varepsilon^{-1})\rfloor}$. ∎

Comparing the expressions appearing in items (1) and (2) in the lemma, we
conclude that any point $(\tilde{\underline{\omega}},\tilde{x})$ which
$\varepsilon$-shadows the piece of orbit of a point $({\underline{\omega}},x)$
during $n_{\ell}$ iterates is such $\sigma^{n}(\tilde{\underline{\omega}})$
belongs to a cylinder of deepth $\lfloor\log(\varepsilon^{-1})\rfloor$. As a
consequence, one needs to make $\lfloor\log(\varepsilon^{-1})\rfloor$ further
iterations to drop this dependence (cf. item (2) in Lemma 7.2 above).

$2^{nd}$ Step: _Construction of a Moran set of points with oscilating
behavior_

Given $({\underline{\omega}}_{1},x_{1})\in\mathcal{S}_{1}$ and
$({\underline{\omega}}_{2},x_{2})\in\mathcal{S}_{2}$, Lemma 7.2 (2) implies
that
$F^{n_{1}+\lfloor\log(\varepsilon^{-1})\rfloor}\big{(}\,B_{F}(({\underline{\omega}}_{1},x),n_{1},\varepsilon)\,\big{)}$
contains the set

$\sigma^{n_{1}+\lfloor\log(\varepsilon^{-1})\rfloor}\big{(}{\underline{\omega}}_{\,[-\lfloor\log(\varepsilon^{-1})\rfloor\,,\,n_{1}+\lfloor\log(\varepsilon^{-1})\rfloor]}\big{)}\times
B(f^{n_{1}+\lfloor\log(\varepsilon^{-1})\rfloor}_{\underline{\omega}}(x),L^{-(2{n_{1}}+\lfloor\log(\varepsilon^{-1})\rfloor)}\varepsilon).$

On the other hand, $B_{{\underline{\omega}}_{2}}(x_{2},n_{2},\varepsilon)$
contains the ball centered at $x_{2}$ with radius $L^{-n_{2}}\varepsilon$.
Then, the frequent hitting times property along the fiber of $F$ ensures that
there exists $0\leqslant p_{1}\leqslant
K(L^{-(2{n_{1}}+\lfloor\log(\varepsilon^{-1})\rfloor)}\varepsilon)$ and a
finite word ${\underline{\omega}}^{(12)}\in\\{1,2,\dots,\kappa\\}^{p}_{1}$ so
that

$f^{p_{1}}_{{\underline{\omega}}^{(12)}}(B(f^{n_{1}+\lfloor\log(\varepsilon^{-1})\rfloor}_{\underline{\omega}}(x),L^{-(2{n_{1}}+\lfloor\log(\varepsilon^{-1})\rfloor)}\varepsilon))\cap
B_{{\underline{\omega}}_{2}}(x_{2},n_{2},\varepsilon)$

contains a ball $B_{1}^{2}$ of radius $\frac{1}{2}L^{-n_{2}}\varepsilon$. Thus

$\displaystyle
C(\,({\underline{\omega}}_{1},x_{1}),({\underline{\omega}}_{2},x_{2})\,)$
$\displaystyle:={{\underline{\omega}}_{1}}_{\,[-\log(\varepsilon^{-1})\,,\,n_{1}+\log(\varepsilon^{-1})]}\times
B_{{\underline{\omega}}_{1}}(x_{1},n_{1},\varepsilon)$
$\displaystyle\;\cap\;F^{-(p_{1}+\lfloor\log(\varepsilon^{-1})\rfloor+n_{1})}\,\Big{(}{{\underline{\omega}}_{2}}_{\,[-\log(\varepsilon^{-1})\,,\,n_{2}+\log(\varepsilon^{-1})]}\times
B_{1}^{2}\Big{)}$

is a compact and non-empty subset of $\Sigma_{\kappa}\times X$. Observe now
that
$f_{{\underline{\omega}}_{2}}^{n_{2}+\lfloor\log(\varepsilon^{-1})\rfloor}(B_{1}^{2})$
contains a ball of radius
$\frac{1}{2}L^{-(2n_{2}+\lfloor-\log(\varepsilon^{-1}\rfloor))}\varepsilon\geqslant
L^{-n_{3}}\varepsilon$. Thus, if
$({\underline{\omega}}_{3},x_{3})\in\mathcal{S}_{3}$, repeating the argument,
there exists $0\leqslant p_{2}\leqslant K(L^{-n_{3}}\varepsilon)$, a finite
word ${\underline{\omega}}^{(23)}\in\\{1,2,\dots,\kappa\\}^{p_{2}}$ and a ball
$B_{2}^{3}\subset B_{{\underline{\omega}}_{3}}(x_{3},n_{3},\varepsilon)$ of
radius $\frac{1}{2}L^{-n_{3}}\varepsilon$ such that

$\displaystyle
C(\,({\underline{\omega}}_{1},x_{1}),({\underline{\omega}}_{2},x_{2}),({\underline{\omega}}_{3},x_{3})\,)$
$\displaystyle:={{\underline{\omega}}_{1}}_{\,[-\log(\varepsilon^{-1})\,,\,n_{1}+\log(\varepsilon^{-1})]}\times
B_{{\underline{\omega}}_{1}}(x_{1},n_{1},\varepsilon)$
$\displaystyle\;\cap\;F^{-(p_{1}+\lfloor\log(\varepsilon^{-1})\rfloor+n_{1})}\,\Big{(}{{\underline{\omega}}_{2}}_{\,[-\log(\varepsilon^{-1})\,,\,n_{2}+\log(\varepsilon^{-1})]}\times
B_{1}^{2}\Big{)}$
$\displaystyle\;\cap\;F^{-\sum_{i=1}^{2}(p_{i}+\lfloor\log(\varepsilon^{-1})\rfloor+n_{i})}\,\Big{(}{{\underline{\omega}}_{3}}_{\,[-\log(\varepsilon^{-1})\,,\,n_{3}+\log(\varepsilon^{-1})]}\times
B_{2}^{3}\Big{)}$

is non-empty subset of
$C(\,({\underline{\omega}}_{1},x_{1}),({\underline{\omega}}_{2},x_{2})\,)$.
More generally, given $k\geqslant 1$ and a $k$-uple
$(({\underline{\omega}}_{1},x_{1}),\dots,({\underline{\omega}}_{k},x_{k}))\in\mathcal{S}_{1}\times\dots\times\mathcal{S}_{k}$,
for each $1\leqslant i\leqslant k$ there exist $0\leqslant p_{i}\leqslant
K(L^{-n_{i}}\varepsilon)$, a finite word
${\underline{\omega}}^{(i(i+1))}\in\\{1,2,\dots,\kappa\\}^{p_{i}}$ and a ball
$B_{i}^{i+1}\subset
B_{{\underline{\omega}}_{i+1}}(x_{i+1},n_{i+1},\varepsilon)$ of radius
$\frac{1}{2}L^{-n_{i+1}}\varepsilon$ so that

$\displaystyle
C(\,({\underline{\omega}}_{1},x_{1}),\dots,({\underline{\omega}}_{k},x_{k})\,)$
$\displaystyle:={{\underline{\omega}}_{1}}_{\,[-\log(\varepsilon^{-1})\,,\,n_{1}+\log(\varepsilon^{-1})]}\times
B_{{\underline{\omega}}_{1}}(x_{1},n_{1},\varepsilon)$
$\displaystyle\;\cap\;F^{-(p_{1}+\lfloor\log(\varepsilon^{-1})\rfloor+n_{1})}\,\Big{(}{{\underline{\omega}}_{2}}_{\,[-\log(\varepsilon^{-1})\,,\,n_{2}+\log(\varepsilon^{-1})]}\times
B_{1}^{2}\Big{)}$ $\displaystyle\quad\dots$
$\displaystyle\;\cap\;F^{-\sum_{i=1}^{k-1}(p_{i}+\lfloor\log(\varepsilon^{-1})\rfloor+n_{i})}\,\Big{(}{{\underline{\omega}}_{k}}_{\,[-\log(\varepsilon^{-1})\,,\,n_{k}+\log(\varepsilon^{-1})]}\times
B_{k-1}^{k}\Big{)}$

is non-empty. By construction, the previous family of cylinder sets satisfy

$C(\,({\underline{\omega}}_{1},x_{1}),\dots,({\underline{\omega}}_{k},x_{k}),({\underline{\omega}}_{k+1},x_{k+1})\,)\subset
C(\,({\underline{\omega}}_{1},x_{1}),\dots,({\underline{\omega}}_{k},x_{k})\,)$

for every
$(({\underline{\omega}}_{1},x_{1}),\dots,({\underline{\omega}}_{k},x_{k}),({\underline{\omega}}_{k+1},x_{k+1}))\in\mathcal{S}_{1}\times\dots\times\mathcal{S}_{k}\times\mathcal{S}_{k+1}$
and $k\geqslant 1$.

For the sake of completeness, let us emphasize that the transition times
$p_{i}$ are functions of the collections
$(\,({\underline{\omega}}_{1},x_{1}),\dots,({\underline{\omega}}_{k},x_{k}))$.
Moreover, since transition times are not constant, one cannot ensure that
points in different cylinder sets of

$\mathcal{\tilde{C}}_{k}:=\big{\\{}C(\,({\underline{\omega}}_{1},x_{1}),\dots,({\underline{\omega}}_{k},x_{k})\,)\colon(({\underline{\omega}}_{1},x_{1}),\dots,({\underline{\omega}}_{k},x_{k}))\in\mathcal{S}_{1}\times\dots\times\mathcal{S}_{k}\big{\\}}$

are separated by some suitable iterate of $F$. We proceed to show this is the
case, at least for a relevant proportion of cylinder sets. Let us be more
precise. For each $1\leqslant i\leqslant k-1$ consider the
$i^{\text{th}}$-transition time function

$\begin{array}[]{rccc}p_{i}:&\mathcal{\tilde{C}}_{k}&\to&\\{0,1,\dots,K(L^{-(2{n_{i}}+\lfloor\log(\varepsilon^{-1})\rfloor)}\varepsilon)\\}.\end{array}$

By the pigeonhole principle, for each $1\leqslant i\leqslant k-1$ there exists
$0\leqslant t_{i}\leqslant
K(L^{-(2{n_{i}}+\lfloor\log(\varepsilon^{-1})\rfloor)}\varepsilon)$ and a
collection $\mathcal{C}_{k}\subset\mathcal{\tilde{C}}_{k}$ such that

$\\#{\mathcal{C}_{k}}\geqslant\frac{\\#\mathcal{S}_{k}\,\dots\,\\#\mathcal{S}_{2}\,\\#\mathcal{S}_{1}}{\prod_{i=1}^{k-1}\,K(L^{-(2{n_{i}}+\lfloor\log(\varepsilon^{-1})\rfloor)}\varepsilon)}$
(7.3)

and $p_{i}(C)=t_{i}$ for every $C\in{\mathcal{C}_{k}}$.

######  Lemma 7.3.

For each $k\geqslant 1$ let the subcollection
${\mathcal{C}}_{k}\subset\tilde{\mathcal{C}}_{k}$ and the integers
$(t_{i})_{1\leqslant i\leqslant k-1}$ be given as above. There exists
$K_{0}>0$ such that:

  1. (1)

for any distinct $C_{1},C_{2}\in{\mathcal{C}}_{k}$,
$({\underline{\omega}},x)\in C_{1}$ and
$(\tilde{\underline{\omega}},\tilde{x})\in C_{2}$ it holds that
$d_{\alpha_{k}}(({\underline{\omega}},x),(\widetilde{{\underline{\omega}}},\tilde{x}))>\varepsilon$,

  2. (2)

$\\#{\mathcal{C}}_{k}\geqslant
K_{0}\,e^{(h_{*}(\varphi)-4\gamma)\,\alpha_{k}}$

where
$\alpha_{k}=n_{k}+\sum_{i=1}^{k-1}(n_{i}+\lfloor\log(\varepsilon^{-1})\rfloor+t_{i})$.

###### Proof.

Fix $k\geqslant 1$. For any distinct $k$-uples
$C_{1}=\tilde{C}(\,({\underline{\omega}}_{1},x_{1}),\dots,({\underline{\omega}}_{k},x_{k})\,)$
and
$C_{2}=\tilde{C}(\,({\underline{\omega}}_{1},\tilde{x}_{1}),\dots,(\tilde{\underline{\omega}}_{k},\tilde{x}_{k})\,)$
belonging to $\tilde{\mathcal{C}}_{k}$ there exists $1\leqslant j\leqslant k$
so that
$({\underline{\omega}}_{j},x_{j})\neq(\tilde{\underline{\omega}}_{j},\tilde{x}_{j})$.
Thus, if $({\underline{\omega}},x)\in C_{1}$ and
$(\widetilde{{\underline{\omega}}},\tilde{x})\in C_{2}$ then

$\displaystyle
d(F^{\sum_{i=1}^{j}(n_{i}+\lfloor\log(\varepsilon^{-1})^{-1}\rfloor+t_{i})}({\underline{\omega}},x),$
$\displaystyle
F^{\sum_{i=1}^{j}(n_{i}+\lfloor\log(\varepsilon^{-1})^{-1}\rfloor+t_{i})}(\tilde{\underline{\omega}},\tilde{x}))$
$\displaystyle\geqslant
d(({\underline{\omega}}_{j},x_{j}),(\tilde{\underline{\omega}}_{j},\tilde{x}_{j}))-d(F^{\sum_{i=1}^{j}(n_{i}+\lfloor\log(\varepsilon^{-1})^{-1}\rfloor+t_{i})}({\underline{\omega}},x),({\underline{\omega}}_{j},x_{j}))$
$\displaystyle-d(F^{\sum_{i=1}^{j}(n_{i}+\lfloor\log(\varepsilon^{-1})^{-1}\rfloor+t_{i})}(\tilde{\underline{\omega}},\tilde{x})),(\tilde{\underline{\omega}}_{j},\tilde{x}_{j}))>\varepsilon,$

thus proving the first item in the lemma. The proof of the second item follows
from quantitative information concerning (7.3). Indeed, using Lemma 7.1(1) one
has that
${n_{i+1}}\geqslant{K(L^{-(2{n_{i}}+\lfloor\log(\varepsilon^{-1})\rfloor)}\varepsilon)}$
for every $1\leqslant i\leqslant k-1$ and

$\displaystyle\frac{1}{n_{i+1}}\;\;e^{-(h_{*}(\varphi)-3\gamma)\,[\lfloor\log(\varepsilon^{-1})\rfloor+K(L^{-(2{n_{i}}+\lfloor\log(\varepsilon^{-1})\rfloor)}\varepsilon)]}\geqslant
e^{-\gamma[n_{i+1}+\lfloor\log(\varepsilon^{-1})\rfloor+t_{i}]}$

for every large $i\geqslant 1$. Together with Lemma 7.1(2), this ensures that
there exists $K_{0}>0$ so that

$\displaystyle\frac{\\#\mathcal{S}_{k}\,\dots\,\\#\mathcal{S}_{2}\,\\#\mathcal{S}_{1}}{\prod_{i=1}^{k-1}\,K(L^{-(2{n_{i}}+\lfloor\log(\varepsilon^{-1})\rfloor)}\varepsilon)}$
$\displaystyle\geqslant
e^{(h_{*}(\varphi)-3\gamma)\,[n_{k}+\dots+n_{2}+n_{1}]}\prod_{i=1}^{k-1}\frac{1}{n_{i+1}}\,\frac{n_{i+1}}{K(L^{-(2{n_{i}}+\lfloor\log(\varepsilon^{-1})\rfloor)}\varepsilon)}$
$\displaystyle\geqslant\prod_{i=0}^{k-1}\frac{1}{n_{i+1}}\;e^{(h_{*}(\varphi)-3\gamma)\,[n_{i+1}+\lfloor\log(\varepsilon^{-1})\rfloor+t_{i}]}\;e^{-(h_{*}(\varphi)-3\gamma)\,[\lfloor\log(\varepsilon^{-1})\rfloor+K(L^{-(2{n_{i}}+\lfloor\log(\varepsilon^{-1})\rfloor)}\varepsilon)]}$
$\displaystyle\geqslant K_{0}\,e^{(h_{*}(\varphi)-4\gamma)\,\alpha_{k}}$

for every $k\geqslant 1$, as claimed. ∎

Now, consider the Moran set $\mathfrak{F}$, obtained as a countable
intersection of compact and nested subsets, defined by
$\mathfrak{F}:=\bigcap_{k\geqslant 1}\mathfrak{F}_{k}$ where

$\mathfrak{F}_{k}:=\bigcup\Big{\\{}\overline{C(\,({\underline{\omega}}_{1},x_{1}),\dots,({\underline{\omega}}_{k},x_{k})\,)}:C(\,({\underline{\omega}}_{1},x_{1}),\dots,({\underline{\omega}}_{k},x_{k})\,)\in{\mathcal{C}}_{k}\Big{\\}}.$

We now observe that the compact and non-empty set $\mathfrak{F}$ is contained
in $I_{F}(\varphi)$.

######  Lemma 7.4.

For any $({\underline{\omega}},x)\in\mathfrak{F}$ we have

$\liminf_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\varphi(F^{j}({\underline{\omega}},x))<\limsup_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\varphi(F^{j}({\underline{\omega}},x)).$

###### Proof.

This proof is identical to the proof of Lemma 6.7, and for that reason we
shall omit it. ∎

$3^{rd}$ Step: _Estimating the topological entropy of $\mathfrak{F}$_

In order to estimate the topological entropy of the Moran set $\mathfrak{F}$
for the skew-product $F$ we will make use of the generalized entropy
distribution principle stated as Theorem 4.1. For that, we define a special
sequence of probability measures $\mu_{k}$ supported in points selected among
the elements of sets in the family $\mathcal{C}_{k}$ as follows: for each
$C\in\mathcal{C}_{k}$ choose a point $z_{C}\in C$ and consider the probability
measure

$\mu_{k}=\frac{1}{\\#\mathcal{C}_{k}}\sum_{C\in\mathcal{C}_{k}}\delta_{z_{C}}.$

######  Lemma 7.5.

If $\lim_{k\to\infty}\mu_{k}=\mu$ (in the weak∗-topology) then
$\mu(\mathfrak{F})=1$.

###### Proof.

Suppose $\lim_{k\to\infty}\mu_{k}=\mu$ in the weak∗-topology. For each
$\ell>0$ and $i>0$ we have $\mu_{\ell+i}(\mathfrak{F}_{\ell+i})=1$ and as
$\mathfrak{F}_{\ell+i}\subset\mathfrak{F}_{\ell}$ then
$\mu_{\ell+i}(\mathfrak{F}_{\ell})=1$. Since $\mathfrak{F}_{\ell}$ is a closed
set, we have
$\mu(\mathfrak{F}_{\ell})\geqslant\limsup_{k\rightarrow\infty}\mu_{k}(\mathfrak{F}_{\ell})=1$.
Therefore, $\mu(\mathfrak{F})=\lim_{\ell\to\infty}\mu(\mathfrak{F}_{\ell})=1$.
∎

In order to apply the generalized entropy distribution principle, one needs to
estimate the measure of dynamic balls intersecting the Moran set. Let
$\varepsilon>0$ be small and $\mathbf{n}\geqslant 1$ be fixed and large,
satisfying

$n_{j}\geqslant j\cdot
K(L^{-2n_{j-1}+\lfloor\log(\varepsilon^{-1})\rfloor}\,2\varepsilon)>\frac{1}{\gamma}\log
K(L^{-2n_{j-1}+\lfloor\log(\varepsilon^{-1})\rfloor}\,2\varepsilon)$ (7.4)

for every $j\geqslant\mathbf{n}$ (recall Lemma 7.1). Pick also the unique
$\ell=\ell(\mathbf{n})\geqslant 1$ so that
$\alpha_{\ell-1}\leqslant\mathbf{n}<\alpha_{\ell}$, where the sequence
$\alpha_{\ell}$ was defined in Lemma 7.3. Consider an arbitrary dynamic ball
$B_{F}(({\underline{\omega}},x),\mathbf{n},\varepsilon)\subset\Sigma_{\kappa}\times
X$ which intersects the set $\mathfrak{F}$.

######  Lemma 7.6.

$\mu_{k+\ell}(B_{F}(({\underline{\omega}},x),\mathbf{n},\varepsilon))\leqslant
e^{-\mathbf{n}(h_{*}(\varphi)-5\gamma)}$ for every large $k\geqslant 1$.

###### Proof.

If $\mu_{k+\ell}(B_{F}(({\underline{\omega}},x),\mathbf{n},\varepsilon))=0$
there is nothing to prove. Otherwise
$\mu_{k+\ell}(B_{F}(({\underline{\omega}},x),\mathbf{n},\varepsilon))>0$ and,
using that the probability $\mu_{k+\ell}$ is an atomic probability supported
on points of $\mathfrak{F}_{k+\ell}$, we conclude that
$\mathfrak{F}_{k+\ell}\cap
B_{F}(({\underline{\omega}},x),\mathbf{n},\varepsilon)\neq\emptyset$. We claim
that all points in such intersection belong to a single fixed cylinder
$C_{*}=C(\,({\underline{\omega}}_{1},x_{1}),\dots,({\underline{\omega}}_{\ell},x_{\ell})\,)\in\mathcal{C}_{\ell}$.
In fact, if $({\underline{\omega}}^{\prime},x^{\prime})\in
B_{F}(({\underline{\omega}},x),\mathbf{n},\varepsilon)\cap C^{\prime}$,
$(\widetilde{{\underline{\omega}}},\tilde{x})\in
B_{F}(({\underline{\omega}},x),\mathbf{n},\varepsilon)\cap\tilde{C}$ with
$C^{\prime},\tilde{C}\in\mathcal{C}_{\ell}$ and $C^{\prime}\neq C$ then item
$(1)$ in Lemma 7.3 guarantees that
$d_{\alpha_{\ell}}(({\underline{\omega}},x),(\widetilde{{\underline{\omega}}},\tilde{x}))>\varepsilon$.
As $\alpha_{\ell}>\mathbf{n}$ this leads to a contradiction with the fact that
both points belong to the same dynamic ball of radius $\varepsilon$ and length
$\mathbf{n}$.

These estimates show that the number of cylinder sets constructed at each step
(which involve the selection of those having the same transition times) are
close to the exponential rate provided by entropy, a fact which imposes
constraints of size on the transition times in comparison to the finite orbit
size. Let us be more precise. By construction and inequality (7.4), for each
fixed $C_{\ell}\in\mathcal{C}_{\ell}$ one has

$\\#\\{C\in\mathcal{C}_{\ell+1}\colon C\cap
C_{\ell}\neq\emptyset\\}\geqslant\frac{\\#\mathcal{S}_{\ell+1}}{K(L^{-2n_{\ell}+\lfloor\log(\varepsilon^{-1})\rfloor}\,2\varepsilon)}>e^{(h_{*}(\varphi)-3\gamma)\,n_{\ell+1}}\,e^{-\gamma\,n_{\ell+1}}$

provided that $\ell$ is large (which means large $\mathbf{n}$). Using the same
argument recursively we conclude that $\\#\\{C\in\mathcal{C}_{k+\ell}\colon
C\cap
C_{\ell}\neq\emptyset\\}>e^{(h_{*}(\varphi)-4\gamma)\,[n_{k+\ell}+\dots+n_{k+1}]}$
for every large $k$. Thus

$\displaystyle\\#\\{C\in\mathcal{C}_{k+\ell}\colon C\cap
C_{*}\neq\emptyset\\}$
$\displaystyle\leqslant\frac{\\#\mathcal{C}_{k+\ell}}{\min\limits_{C_{\ell}\in\mathcal{C}_{\ell}}\;\\#\\{C\in\mathcal{C}_{k+\ell}\colon
C\cap C_{\ell}\neq\emptyset\\}}$
$\displaystyle<e^{-(h_{*}(\varphi)-4\gamma)\,[n_{k+\ell}+\dots+n_{k+1}]}\;\\#\mathcal{C}_{k+\ell}.$
(7.5)

Therefore,

$\displaystyle\mu_{k+\ell}(B_{F}(({\underline{\omega}},x),\mathbf{n},\varepsilon))$
$\displaystyle=\displaystyle\frac{1}{\\#\mathcal{C}_{k+\ell}}\sum_{\begin{subarray}{c}C\in\mathcal{C}_{k+\ell}\\\
C\cap
C_{*}\neq\emptyset\end{subarray}}\delta_{z_{C}}\left(B_{F}(({\underline{\omega}},x),\mathbf{n},\varepsilon)\right)$
$\displaystyle\leqslant\displaystyle\frac{\\#\\{C\in\mathcal{C}_{k+\ell}\colon
C\cap
C_{*}\neq\emptyset\\}}{\\#\mathcal{C}_{k+\ell}}<e^{-(h_{*}(\varphi)-4\gamma)\,[n_{k+\ell}+\dots+n_{k+1}]}.$

Notice that

$\displaystyle(h_{*}(\varphi)-4\gamma)\cdot\sum_{i=k+1}^{k+\ell}n_{i}$
$\displaystyle\geqslant(h_{*}(\varphi)-4\gamma)\Big{[}n_{k+\ell}+\sum_{i=1}^{k+\ell-1}(n_{i}+\lfloor\log(\varepsilon^{-1})^{-1}\rfloor+t_{i})-\sum_{i=1}^{k+\ell-1}(\lfloor\log(\varepsilon^{-1})^{-1}\rfloor+t_{i})\Big{]}$
$\displaystyle=(h_{*}(\varphi)-4\gamma)\Big{[}\alpha_{k+\ell}-\frac{\sum_{i=1}^{k+\ell-1}(\lfloor\log(\varepsilon^{-1})^{-1}\rfloor+t_{i})}{\alpha_{k+\ell}}\Big{]}$
$\displaystyle\geqslant(h_{*}(\varphi)-5\gamma)\,\alpha_{k+\ell}\geqslant(h_{*}(\varphi)-5\gamma)\,\mathbf{n}$

for every large $k\geqslant 1$. Altogether this ultimately implies that
$\mu_{k+\ell}(B_{F}(({\underline{\omega}},x),\mathbf{n},\varepsilon))\leqslant
e^{-\mathbf{n}\,(h_{*}(\varphi)-5\gamma)\,}$, which completes the proof of the
lemma. ∎

We are now in a position to prove the theorem. Indeed, on the one hand any
accumulation point of $(\mu_{k})_{k}$ is supported on $\mathfrak{F}$, by Lemma
7.5. On the other hand, Lemma 7.6 (taking the $\limsup$ as $\ell$ tends to
infinity) ensures that
$\limsup_{k\rightarrow\infty}\mu_{k}(B_{F}(({\underline{\omega}},x),\mathbf{n},\varepsilon))\leqslant
e^{-\mathbf{n}(h_{*}(\varphi)-5\gamma)}$. Then, using the generalized entropy
distribution principle we conclude that $h_{top}(F,I_{F}(\varphi))\geqslant
h_{top}(F,\mathfrak{F})\geqslant h_{*}(\varphi)-5\gamma$. As $\gamma>0$ was
chosen arbitrary, this shows that $h_{top}(F,I_{F}(\varphi))\geqslant
h_{*}(\varphi)$, and completes the proof of Theorem D.

##  8\. Topological entropy of irregular points for the semigroup action

This subsection is devoted to the proof of Theorem E. Let $X$ be a compact
metric space, assume that the semigroup action generated by the collection
$G_{1}=\\{id,f_{1},f_{2},\dots,f_{\kappa}\\}$ of bi-Lipschitz homeomorphisms
has frequent hitting times, and let $\psi:X\to\mathbb{R}$ be a continuous
observable which is not a coboundary with respect to $G_{1}$.

###  8.1. Ghys-Langevin-Walczak entropy of the set of irregular points

Here we prove item (1) in Theorem E. It is immediate that

$s((G,G_{1}),n,\varepsilon)\geqslant
s({\underline{\omega}},n,\varepsilon,\delta)$

for every $n\geqslant 1$, $\varepsilon>0$,
${\underline{\omega}}\in\Sigma_{\kappa}$ and $0<\delta<1$. Our mild
assumptions seem not to allow to construct a single sequence
$\tilde{\underline{\omega}}\in\Sigma_{\kappa}$ whose number of irregular
points for the observable $\psi$ and the sequential dynamical system
$\mathcal{F}_{\tilde{\underline{\omega}}}$ grows at least at the exponential
rate of $H^{\text{Pinsker}}(\psi)-2\gamma$. Alternatively, we will produce
sets $E_{n}\subset X$ so that any two points are $(n,\varepsilon)$-separated
by some element $\underline{g}\in G_{n}$ and whose Birkhoff averages of $\psi$
determined by such finite paths in the semigroup $G$ do not converge. Let us
formalize this strategy.

Assume that $H^{\text{Pinsker}}(\psi)>0$ (otherwise there is nothing to prove)
and note that $H^{Katok}(\psi)=H^{\text{Pinsker}}(\psi)>0$ (recall (4.12)).
Hence, for any fixed $0<\gamma,\delta<1$ there exist
$\mu_{1},\mu_{2}\in\mathcal{M}_{erg}(F)$ so that
$\int\varphi_{\psi}\,d\mu_{1}<\int\varphi_{\psi}\,d\mu_{2}$ and
$h_{\mu_{i}}^{\delta}(F)\geqslant H^{\text{Pinsker}}(\psi)-2\gamma$, for
$i=1,2$. Using (4.10), one knows that

$\lim_{\varepsilon\rightarrow 0}\liminf_{n\rightarrow\infty}\frac{1}{n}\log
s({\underline{\omega}},n,\varepsilon,\delta)\geqslant
H^{\text{Pinsker}}(\psi)-2\gamma$ (8.1)

for $\mu_{i}$-almost every ${\underline{\omega}}$, $i=1,2$. One can choose for
each $i=1,2$ a sequence ${\underline{\omega}}_{i}\in\Sigma_{\kappa}$ in the
ergodic basin of attraction of the $\sigma$-invariant probability measure
$\pi_{*}\mu_{i}$ satisfying (8.1). The latter, together with the uniform
continuity of $\psi$, allow us to select $\varepsilon>0$ so that
$|\psi(x)-\psi(x^{\prime})|<\frac{\int\varphi_{\psi}\,d\mu_{2}-\int\varphi_{\psi}\,d\mu_{1}}{4}$
for every $x,x^{\prime}\in X$ so that $d(x,x^{\prime})<4\varepsilon$, and

$\liminf_{n\rightarrow\infty}\frac{1}{n}\log
s({\underline{\omega}}_{i},n,4\varepsilon,\delta)\geqslant
H^{\text{Pinsker}}(\psi)-3\gamma$

for every $i=1,2$. Write
${\underline{\omega}}_{\ell}={\underline{\omega}}_{1}$ when $\ell\geqslant 1$
is odd and ${\underline{\omega}}_{\ell}={\underline{\omega}}_{2}$ for an even
$\ell\geqslant 1$.

######  Lemma 8.1.

There exists a sequence $(n_{\ell})_{\ell\geqslant 1}$ of positive integers
such that:

  1. (1)

$n_{\ell}\geqslant\max\\{\;\mathfrak{n}_{\ell},\;(\ell+2)\cdot
n_{\ell-1}+\Big{[}\frac{2\log 2}{\log
L}\Big{]},\;\ell\cdot\prod_{i=1}^{\ell-1}K(L^{-2{n_{i}}}2\varepsilon)\,\exp(K(L^{-2{n_{i}}}2\varepsilon)\log\kappa)\;\\}$
for every $\ell\geqslant 2$

  2. (2)

for each $\ell\geqslant 1$ there exists a
$(g_{{\underline{\omega}}_{\ell}},n_{\ell},4\varepsilon)$-separated set
$\mathcal{S}_{\ell}\subset X$ with cardinality

$\\#\mathcal{S}_{\ell}\geqslant\exp(\,(H^{\text{Pinsker}}(\psi)-3\gamma)n_{\ell}\,).$

###### Proof.

The proof is identical to the one of Lemma 7.1, the differences being that the
sequence $(n_{\ell})_{\ell}$ is required to grow exponentially with respect to
the terms $K(L^{-2n_{i-1}}\,\,2\varepsilon)$, and the separated sets are
subsets of $X$ instead of subsets of some fibers $X_{\underline{\omega}}$
(which have a natural identification). Moreover, since we need not consider
iterations by the skew-product, we need not take the extra
$\lfloor\log(\varepsilon^{-1})\rfloor$ iterates necessary to control the
dynamical distance with respect to the shift. We shall omit the details. ∎

Now, note that the frequent hitting times property for the semigroup action
ensures that for each pair
$(x_{1},x_{2})\in\mathcal{S}_{1}\times\mathcal{S}_{2}$ there exists
$0\leqslant p_{1}\leqslant K(L^{-2{n_{1}}})$ and a finite word
$\hat{\underline{\omega}}^{(12)}=\hat{\underline{\omega}}^{(12)}(x_{1},x_{2})\in\\{1,2,\dots,\kappa\\}^{p_{1}}$
(depending on $x_{1}$ and $x_{2}$) so that

$f^{p_{1}}_{{\underline{\omega}}^{(12)}}(B(f^{n_{1}}_{{\underline{\omega}}_{1}}(x),L^{-2{n_{1}}}\varepsilon))\cap
B_{{\underline{\omega}}_{2}}(x_{2},n_{2},\varepsilon)$

is a subset of $X$ which contains a ball $B_{1}^{2}$ of radius
$\frac{1}{2}L^{-n_{2}}\varepsilon$. Thus

$\displaystyle
C_{({\underline{\omega}}_{1},\,\hat{\underline{\omega}}^{(12)}(x_{1},x_{2}),\,{\underline{\omega}}_{2})}(\,x_{1},x_{2}\,)$
$\displaystyle:=B_{{\underline{\omega}}_{1}}(x_{1},n_{1},\varepsilon)\;\cap\;(\,f^{p_{1}}_{{\underline{\omega}}^{(12)}}\circ
f_{{\underline{\omega}}_{1}}^{n_{1}}\,)^{-1}\big{(}B_{1}^{2}\big{)}$ (8.2)

is a compact and non-empty subset of $X$. As the number of finite words of
length at most $K(L^{-2{n_{1}}})$ is bounded above by
$K(L^{-2{n_{1}}})\,\kappa^{K(L^{-2{n_{1}}})}$, by the pigeonhole principle,
for each $x_{1}\in\mathcal{S}_{1}$, there exists $0\leqslant t_{1}\leqslant
K(L^{-2{n_{1}}}\varepsilon)$, a finite word
${\underline{\omega}}^{(12)}\in\\{1,2,\dots,\kappa\\}^{t_{1}}$ (depending only
on $x_{1}$), and a collection $\mathcal{\tilde{C}}_{2}$ of elements of the
form (8.2) with
$\hat{\underline{\omega}}^{(12)}(x_{1},x_{2})={\underline{\omega}}^{(12)}$ and

$\\#{\mathcal{C}_{2}}\geqslant\\#\mathcal{S}_{1}\cdot\frac{\\#\mathcal{S}_{2}}{K(L^{-2{n_{1}}})\,\exp(K(L^{-2{n_{1}}})\log\kappa)}.$

In other words, the family $\mathcal{C}_{2}$ is so that map making the
transition between the finite piece of orbit of $x_{1}$ and $x_{2}$ is the
same $f^{t_{1}}_{{\underline{\omega}}^{(12)}}$, where
${\underline{\omega}}^{(12)}$ depends exclusively on $x_{1}$. Let

$\mathcal{I}_{2}=\big{\\{}(x_{1},x_{2})\in\mathcal{S}_{1}\times\mathcal{S}_{2}\colon
C_{({\underline{\omega}}_{1},\,{\underline{\omega}}^{(12)},\,{\underline{\omega}}_{2})}(\,x_{1},x_{2}\,)\in\mathcal{C}_{2}\big{\\}}.$

Proceeding inductively we obtain, for each $k\geqslant 2$, a collection
$\mathcal{I}_{k}\subset\mathcal{S}_{1}\times\dots\times\mathcal{S}_{k}$ so
that

  1. (1)

for each $(x_{1},\dots,x_{k})\in\mathcal{I}_{k}$ and $1\leqslant i\leqslant
k-1$ there are $0\leqslant t_{i}\leqslant K(L^{-2n_{i}}\varepsilon)$ and a
finite word ${\underline{\omega}}^{(i(i+1))}\in\\{1,2,\dots,\kappa\\}^{t_{i}}$
(depending only on $x_{1},x_{2},\dots,x_{i-1}$), and a ball
$B_{i}^{i+1}\subset
B_{{\underline{\omega}}_{i+1}}(x_{i+1},n_{i+1},\varepsilon)$ of radius
$\frac{1}{2}L^{-n_{i+1}}\varepsilon$ so that

$\displaystyle
C_{({\underline{\omega}}_{1},{\underline{\omega}}^{(12)},{\underline{\omega}}_{2},{\underline{\omega}}^{(23)},...,{\underline{\omega}}_{k})}(\,x_{1},\dots,x_{k}\,)$
$\displaystyle\qquad:=B_{{\underline{\omega}}_{1}}(x_{1},n_{1},\varepsilon)\;\cap\;\bigcap_{i=1}^{k-1}(\,f^{p_{i}}_{{\underline{\omega}}^{(i(i+1)}}\circ
f_{{\underline{\omega}}_{i}}^{n_{i}}\circ\dots\circ
f^{p_{1}}_{{\underline{\omega}}^{(12)}}\circ
f_{{\underline{\omega}}_{1}}^{n_{1}}\,)^{-1}(B_{i}^{i+1})$ (8.3)

is a non-empty subset of $X$;

  2. (2)

the collection $\mathcal{C}_{k}$ of elements of the form (8.3), parameterized
by $\mathcal{I}_{k}$, satisfies

$\\#{\mathcal{C}_{k}}\geqslant\frac{\\#\mathcal{S}_{k}\,\dots\,\\#\mathcal{S}_{2}\,\\#\mathcal{S}_{1}}{\prod_{i=1}^{k-1}K(L^{-2{n_{i}}}2\varepsilon)\,\exp(K(L^{-2{n_{i}}}2\varepsilon)\log\kappa)}.$

For each $k\geqslant 1$ let

$m_{k}:=n_{k}+\sum_{i=1}^{k-1}(n_{i}+t_{i})$

and notice that although the transition length $t_{i}$ $(1\leqslant i\leqslant
k-1)$ may vary with the elements in $\mathcal{I}_{k}$, by construction this is
constant for any to $k$-uples
$(\,x_{1},\dots,x_{k}\,),(\,y_{1},\dots,y_{k}\,)\in\mathcal{I}_{k}$ so that
$x_{j}=y_{j}$ for all $1\leqslant j\leqslant i-1$.

######  Lemma 8.2.

Given $k\geqslant 1$, if $E_{k}\subset X$ is a collection of points, each one
selected among a single element of $\mathcal{C}_{k}$ then $E_{k}$ is
$(m_{k},\varepsilon)$-separated by elements of $G$.

###### Proof.

Fix $k\geqslant 1$ and let $E_{k}\subset X$ be an arbitrary collection of
points, each one selected among a single element of $\mathcal{C}_{k}$. Recall
that, for each $\ell\geqslant 1$, the set $\mathcal{S}_{\ell}\subset X$ is
$(g_{{\underline{\omega}}_{\ell}},n_{\ell},4\varepsilon)$-separated. In
consequence, the dynamic balls
$\\{B_{{\underline{\omega}}_{i}}(x,n_{i},\varepsilon)\colon
x\in\mathcal{S}_{i}\\}$ are pairwise disjoint, for every $1\leqslant
i\leqslant k$, and the collection ${\mathcal{C}_{k}}$ is formed by a disjoint
union of compact sets of the form (8.3). Let $x,y$ belong to distinct elements
in ${\mathcal{C}_{k}}$ and let
$(\,x_{1},\dots,x_{k}\,),(\,y_{1},\dots,y_{k}\,)\in\mathcal{I}_{k}$ be so that

$x\in
C_{({\underline{\omega}}_{1},{\underline{\omega}}^{(12)},{\underline{\omega}}_{2},{\underline{\omega}}^{(23)},...,{\underline{\omega}}_{k})}(\,x_{1},\dots,x_{k}\,)\;\text{and}\;y\in
C_{({\underline{\omega}}_{1},\hat{\underline{\omega}}^{(12)},{\underline{\omega}}_{2},\hat{\underline{\omega}}^{(23)},...,{\underline{\omega}}_{k})}(\,y_{1},\dots,y_{k}\,).$

The previous construction ensures that if $1\leqslant i\leqslant k$ is the
smallest integer so that $x_{i}\neq y_{i}$ then
${\underline{\omega}}^{j(j+1)}=\hat{\underline{\omega}}^{j(j+1)}$ for every
$1\leqslant j\leqslant j-1$. In particular,
$g=f^{n_{i}}_{{\underline{\omega}}_{i}}\circ\dots\circ
f_{{\underline{\omega}}_{2}}^{n_{2}}\circ
f_{{\underline{\omega}}^{(12)}}^{t_{1}}\circ
f_{{\underline{\omega}}_{1}}^{n_{1}}\in G_{m_{i}}$ satisfies
$d(g(x),g(y))>\varepsilon$. As $G_{m_{i}}\subset G_{m_{k}}$ we conclude that
$E_{k}$ is $(m_{k},\varepsilon)$-separated by elements of $G$, as claimed. ∎

As $(\mathcal{C}_{k})_{k\geqslant 1}$ is a nested collection of disjoint and
compact sets, the Moran set

$\mathfrak{F}=\bigcap_{k\geqslant
1}\;\Big{[}\bigcup_{C\in\mathcal{C}_{k}}\,C\Big{]}$

is a non-empty compact subset of $X$. Furthermore, as
$\lim_{k\to\infty}\frac{m_{k}}{n_{k}}=1$ (recall item (1) in Lemma 8.1), using
the information of Lemma 8.2 in (4.4) one concludes that

$\displaystyle h_{\operatorname{top}}^{GLW}(\mathbb{S},\mathfrak{F})$
$\displaystyle\geqslant\limsup_{k\to\infty}\frac{1}{m_{k}}\log
s(G,G_{1},\mathfrak{F},m_{k},\varepsilon)\geqslant\limsup_{k\to\infty}\frac{1}{m_{k}}\log\\#{\mathcal{C}_{k}}$
$\displaystyle\geqslant\limsup_{k\to\infty}\frac{1}{m_{k}}\log\frac{\\#\mathcal{S}_{k}\,\dots\,\\#\mathcal{S}_{2}\,\\#\mathcal{S}_{1}}{\prod_{i=1}^{k-1}K(L^{-2{n_{i}}}2\varepsilon)\,\exp(K(L^{-2{n_{i}}}2\varepsilon)\log\kappa)}$
$\displaystyle\geqslant H^{\text{Pinsker}}(\psi)-3\gamma.$

Thus, in order to complete the proof of item (1) in Theorem E it remains to
prove the following:

######  Lemma 8.3.

$\mathfrak{F}$ is contained in the irregular set $I_{\mathbb{S}}(\psi)$,
defined by (2.1).

###### Proof.

For every $x\in\mathfrak{F}$ there exists a nested sequence
$(C_{k})_{k\geqslant 1}$ of compact sets

$C_{k}=C_{({\underline{\omega}}_{1},{\underline{\omega}}^{(12)},{\underline{\omega}}_{2},{\underline{\omega}}^{(23)},...,{\underline{\omega}}_{k})}(\,x_{1},\dots,x_{k}\,)\in\mathcal{C}_{k}$

so that $x\in\bigcap_{k\geqslant 1}C_{k}$. The choice of constants in Lemma
8.1 guarantees, along an argument identical to the one in the proof of Lemma
6.7, that if ${\underline{\omega}}\in\bigcap_{k\geqslant
1}[{\underline{\omega}}_{1}\mid_{[0,n_{1}]}{\underline{\omega}}^{12}{\underline{\omega}}_{2}\mid_{[0,n_{2}]}{\underline{\omega}}^{23}{\underline{\omega}}_{3}\mid_{[0,n_{3}]}\dots\underline{\omega}_{k}\mid_{[0,n_{k}]}]$
then
$\lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\varphi(f^{j}_{{\underline{\omega}}}(x))$
does not exist and, consequently, $x\in I_{\mathbb{S}}(\psi)$. ∎

###  8.2. Bufetov entropy of the set of irregular points

Here we prove item (2) in Theorem E, namely that
$h^{B}(\mathbb{S},I_{\mathbb{S}}(\psi))\geqslant h_{*}(\varphi_{\psi})$.
Indeed, using that $X$ is compact and [40, Theorem 5.1] (see also Remark 2.4
in [83]), if $\Sigma_{\kappa}^{+}=\\{1,2,\dots,\kappa\\}^{\mathbb{N}}$ then
$\overline{Ch}_{\Sigma_{\kappa}^{+}\times
Z}(F)=\log\kappa+\overline{Ch_{\text{top}}}(\mathbb{S},Z,\mathbb{P}_{sym})$
for every $Z\subset X$ (the term in the left-hand side is the topological
capacity of the skew-product from the aforementioned references and the right-
hand side innvolves the topological capacity of the semigroup as in Subsection
4.2). In particular

$\overline{Ch}_{\Sigma_{\kappa}^{+}\times
I_{\mathbb{S}}(\psi)}(F)=\log\kappa+\overline{Ch}_{I_{\mathbb{S}}(\varphi)}(\mathbb{S}).$

By compactness of $X$,

$h_{\text{top}}(F)=\overline{Ch_{\operatorname{top}}}(F,\Sigma_{\kappa}^{+}\times
X)=\log\kappa+\overline{Ch_{\text{top}}}(\mathbb{S},X,\mathbb{P}_{sym})=\log\kappa+h^{B}(\mathbb{S},\mathbb{P}_{sym})$

Combining these two expressions with Theorem D we conclude that

$h_{\operatorname{top}}(F)\geqslant\overline{Ch}_{X}(F)=\overline{Ch}_{I_{\mathbb{S}}(\varphi)}(F)\geqslant
h_{*}(\varphi),$

as desired. Since the two-sided shift $\Sigma_{\kappa}$ is the natural
extension of $\Sigma_{\kappa}^{+}$, hence preserves the same topological
entropy, the previous results remain unaltered when $\Sigma_{\kappa}^{+}$ is
replaced by $\Sigma_{\kappa}$. This finishes the proof of Theorem E.

##  9\.  Topological entropy of fibered irregular sets

This subsection is devoted to the proof of Theorem F. Let $X$ be a compact
metric space, assume that the semigroup action $\mathbb{S}$ is generated by
the collection $G_{1}=\\{id,f_{1},f_{2},\dots,f_{\kappa}\\}$ of bi-Lipschitz
homeomorphisms and let $\psi:X\to\mathbb{R}$ be a continuous observable so
that $\varphi_{\psi}$ is not a coboundary with respect to $F$. Assume also
that there exists a strongly transitive homeomorphism $f_{i}\in G_{1}$, for
some $1\leqslant i\leqslant\kappa$. We will assume without loss of generality
that $i=\kappa$. In particular, by compactness of $X$, for each $\delta>0$
there exists $N_{\delta}\geqslant 1$ so that

$\bigcup_{j=0}^{N_{\delta}}f_{\kappa}^{j}(B(x,\delta))=X\qquad\text{for every
$x\in X$}.$ (9.1)

Let ${\mathfrak{l}}_{x}(\delta)>0$ denote the Lebesgue covering number of
$\\{f_{\kappa}^{j}(B(x,\delta))\\}_{j=0}^{N_{\delta}}$. By compactness of $x$,
there exists ${\mathfrak{l}}_{\delta}:=\min_{x\in
X}{\mathfrak{l}}_{x}(\delta)>0$ so that:

$\forall x,y\in X,\,\forall
0<\varepsilon<{\mathfrak{l}}_{\delta},\exists\,0\leqslant p\leqslant
N_{\delta}$ s.t. $f_{\kappa}^{p}(B(x,\delta))\supset B(y,\varepsilon)$. (9.2)

######  Remark 9.1.

The latter implies that the semigroup $\mathbb{S}$ has the frequent hitting
times property where the transition maps can be obtained by a certain number
of iterates of the map $f_{\kappa}$. The expression (9.1) can be compared to
the quantitative estimates obtained in Remark 6.4 in the special case that
$f_{\kappa}$ is an isometry.

###  9.1. Fibered complexity of the set of irregular points

Here we prove item (1), namely that $h^{path}_{I}(\mathbb{S},\psi)\geqslant
H^{\text{Pinsker}}(\psi)$. Assume that $H^{\text{Pinsker}}(\psi)>0$ (otherwise
there is nothing to prove). By equality (4.12) we know that
$H^{Katok}(\psi)=H^{\text{Pinsker}}(\psi)>0$. Hence, for any fixed
$0<\gamma,\delta<1$ there exist $\mu_{1},\mu_{2}\in\mathcal{M}_{erg}(F)$ so
that $\int\varphi_{\psi}\,d\mu_{1}<\int\varphi_{\psi}\,d\mu_{2}$ and
$h_{\mu_{i}}^{\delta}(F)\geqslant H^{\text{Pinsker}}(\psi)-2\gamma$, for
$i=1,2$. Using the equality (4.11) and the ergodic theorem, one concludes that
there are ${\underline{\omega}}_{i}\in\Sigma_{\kappa}$ so that:

  1. (a)

$\lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\delta_{\sigma^{j}({\underline{\omega}}_{i})}=\pi_{*}\mu_{i}$

  2. (b)

$\lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\psi(f_{{\underline{\omega}}_{i}}^{j}(x))=\int\varphi_{\psi}\,d\mu_{i}$
for $\mu_{{\underline{\omega}}_{i}}$-a.e. $x\in X_{{\underline{\omega}}_{i}}$

  3. (c)

$h_{\mu}^{\delta}({\underline{\omega}}_{i})>H^{\text{Pinsker}}(\psi)-2\gamma$

for $i=1,2$, where for some abuse of notation we denote by
$\mu_{{\underline{\omega}}_{i}}$ denotes the sample measure of $\mu_{i}$ on
$X_{{\underline{\omega}}_{i}}$ obtained by Rokhlin’s disintegration theorem.

Since $\gamma>0$ was chosen arbitrary, in order to prove the claim it is
sufficient to construct ${\underline{\omega}}\in\Sigma_{\kappa}$ and a fibered
Moran subset of the fiber $X_{\underline{\omega}}$ with entropy larger than
$H^{\text{Pinsker}}(\psi)-\gamma$. Such a construction differs from the ones
at Sections 7 and 8 in the sense that, in order to capture information created
by the relative entropies $h_{\mu_{i}}(F\\!\mid\\!\sigma)$, the point
${\underline{\omega}}\in\Sigma_{\kappa}$ is constructed inductively. In fact,
we will provide a sequence of integers $(m_{k})_{k\geqslant 1}$, a nested
sequence of cylinders $[\omega_{1},\omega_{2},\dots,\omega_{m_{k}}]$, and for
each $\tilde{\underline{\omega}}\in\bigcap_{k\geqslant
1}[\omega_{1},\omega_{2},\dots,\omega_{m_{k}}]$ a Moran set
$\mathfrak{F}_{\widetilde{{\underline{\omega}}}}\subset
X_{\widetilde{{\underline{\omega}}}}$ so that

$h_{\mathfrak{F}_{\widetilde{{\underline{\omega}}}}}(\mathcal{F}_{\widetilde{{\underline{\omega}}}})\geqslant
H^{\text{Pinsker}}(\psi)-\gamma$

(here
$h_{\mathfrak{F}_{\widetilde{{\underline{\omega}}}}}(\mathcal{F}_{\widetilde{{\underline{\omega}}}})$
denotes the fibered relative entropy of the set
$\mathfrak{F}_{\widetilde{{\underline{\omega}}}}$, recall Subsection 4.3).
Albeit the construction of the Cantor sets
$\mathfrak{F}_{\widetilde{{\underline{\omega}}}}$ has a different flavor from
the construction of the Cantor set in the previous section, we will use some
of the technical tools developed in the latter sections. We will emphasize the
differences in comparison to the previous construction and skim the details on
the transition times.

####  9.1.1. Fibered orbit growth estimates

Consider a strictly decreasing sequence of positive numbers
$(\delta_{k})_{k\geqslant 1}$ tending to zero so that
$\delta_{1}<\frac{1}{4}[\int\varphi_{\psi}\,d\mu_{2}-\int\varphi_{\psi}\,d\mu_{1}].$
Associated to these, item (b) above together with Lusin’s theorem allows one
to pick a strictly increasing sequence of large positive integers
$(\mathfrak{n}_{k})_{k\geqslant 1}$ so that, for each $\ell\geqslant 1$, the
fibered sets

$\Gamma_{2\ell-1}:=\Big{\\{}x\in
X_{{\underline{\omega}}_{1}}:\Big{|}\frac{1}{n}\sum_{j=0}^{n-1}\psi(g_{\underline{\omega}}^{j}(x))-\int\varphi_{\psi}\,d\mu_{1}\Big{|}<\delta_{2\ell-1},\;\;\forall
n\geqslant\mathfrak{n}_{2\ell-1}\Big{\\}}$

and

$\Gamma_{2\ell}:=\Big{\\{}x\in
X_{{\underline{\omega}}_{2}}:\Big{|}\frac{1}{n}\sum_{j=0}^{n-1}\psi(g_{\underline{\omega}}^{j}(x))-\int\varphi_{\psi}\,d\mu_{2}\Big{|}<\delta_{2\ell},\;\;\forall
n\geqslant\mathfrak{n}_{2\ell}\Big{\\}}$

satisfy $\mu_{{\underline{\omega}}_{1}}(\Gamma_{2\ell-1})>\frac{1}{2}$ and
$\mu_{{\underline{\omega}}_{2}}(\Gamma_{2\ell})>\frac{1}{2}$. Moreover, using
item (c) above and the definition of the fibered Katok $\delta$-entropy
(taking $\delta=\frac{1}{2}$ in (4.10)) one can choose $\varepsilon>0$ small
so that

$\displaystyle\liminf_{n\rightarrow+\infty}\frac{1}{n}\log
s({\underline{\omega}}_{1},n,8\varepsilon,\Gamma_{2\ell-1})>H^{\text{Pinsker}}(\psi)-2\gamma$
(9.3)

and

$\displaystyle\liminf_{n\rightarrow+\infty}\frac{1}{n}\log
s({\underline{\omega}}_{2},n,8\varepsilon,\Gamma_{2\ell})>H^{\text{Pinsker}}(\psi)-2\gamma.$
(9.4)

For each $i\geqslant 3$ we take
${\underline{\omega}}_{i}={\underline{\omega}}_{1}$ if $i$ is odd and set
${\underline{\omega}}_{i}={\underline{\omega}}_{2}$ otherwise. By uniform
continuity of $\psi$, reducing $\varepsilon$ is necessary, we may also assume
that

$\displaystyle|\psi(x)-\psi(x^{\prime})|<\frac{\int\varphi\,d\mu_{2}-\int\varphi\,d\mu_{1}}{4}$
(9.5)

for every $x,x^{\prime}\in X$ so that $d(x,x^{\prime})<4\varepsilon$. The next
lemma provides a lower bound estimate on the cardinality of separated sets
within the family of sets $(\Gamma_{\ell})_{\ell\geqslant 1}$.

######  Lemma 9.2.

There exists a sequence $(n_{\ell})_{\ell\geqslant 1}$ of positive integers
satisfying

$n_{\ell}\geqslant\max\\{\;\mathfrak{n}_{\ell},\;(\ell+2)\cdot
n_{\ell-1}+\Big{[}\frac{2\log 2}{\log
L}\Big{]},\;\ell\cdot(\log\varepsilon^{-1}-\log{\mathfrak{l}}_{L^{-2{n_{\ell-1}}}\varepsilon}),\;\ell\cdot\frac{1}{\gamma}\cdot
N_{L^{-2{n_{\ell-1}}}\varepsilon}\;\\}$ (9.6)

for every $\ell\geqslant 2$, and so that for each $\ell\geqslant 1$ there
exists an $({\underline{\omega}}_{\ell},n_{\ell},4\varepsilon)$-separated set
$\mathcal{S}_{\ell}\subset\Gamma_{\ell}$ with cardinality
$\\#\mathcal{S}_{\ell}\geqslant\exp(\,(H^{\text{Pinsker}}(\psi)-3\gamma)n_{\ell}\,).$

###### Proof.

The proof is similar to the one of Lemma 7.1, and for that reason we shall
omit it. ∎

####  9.1.2. Construction of fibered Moran sets of irregular points

We proceed to construct a point ${\underline{\omega}}\in\Sigma_{\kappa}$ and a
special Moran subset of $X_{\underline{\omega}}$ formed by points with
irregular behavior. By Lemma 9.2, given $k\geqslant 1$ and $1\leqslant
i\leqslant k$ one can choose a
$({\underline{\omega}}_{i},n_{i},4\varepsilon)$-separated set
$\mathcal{S}_{i}\subset\Gamma_{i}$ so that
$\\#\mathcal{S}_{i}\geqslant\exp(\,(H^{\text{Pinsker}}(\psi)-3\gamma)n_{i}\,).$
Given $x_{1}\in\mathcal{S}_{1}$ and $x_{2}\in\mathcal{S}_{2}$, one knows that

$B(f^{n_{1}}_{{\underline{\omega}}_{1}}(x_{1}),L^{-2{n_{1}}}\varepsilon)\subset
f_{{\underline{\omega}}_{1}}^{n_{1}}(\,B_{{\underline{\omega}}_{1}}(x_{1},n_{1},\varepsilon)\,)\quad\text{and}\quad
B(x_{2},L^{-n_{2}}\varepsilon)\subset
B_{{\underline{\omega}}_{2}}(x_{2},n_{2},\varepsilon).$

The choice of constants in (9.6) guarantees that $L^{-n_{2}}\varepsilon$ is
smaller than the Lebesgue covering number
${\mathfrak{l}}_{L^{-2{n_{1}}}\varepsilon}$. Thus, according to (9.2), one can
pick $0\leqslant p_{1}\leqslant N_{L^{-2{n_{1}}}\varepsilon}$ so that

$f_{\kappa}^{p_{1}}(B(f^{n_{1}}_{{\underline{\omega}}_{1}}(x_{1}),L^{-2{n_{1}}}\varepsilon))\supset
B(x_{2},L^{-n_{2}}\varepsilon).$

Thus

$\displaystyle
C_{({\underline{\omega}}_{1},\,\kappa^{p_{1}},\,{\underline{\omega}}_{2})}(\,x_{1},x_{2}\,)$
$\displaystyle:=B_{{\underline{\omega}}_{1}}(x_{1},n_{1},\varepsilon)\;\cap\;(\,f^{p_{1}}_{\kappa}\circ
f_{{\underline{\omega}}_{1}}^{n_{1}}\,)^{-1}\big{(}B(x_{2},L^{-n_{2}}\varepsilon)\big{)}$

is a compact and non-empty subset of $X_{{\underline{\omega}}_{1}}$ and

$\displaystyle(f_{{\underline{\omega}}_{2}}^{n_{2}}\circ
f^{p_{1}}_{\kappa}\circ
f_{{\underline{\omega}}_{1}}^{n_{1}})(\,C_{({\underline{\omega}}_{1},\,\kappa^{p_{1}},\,{\underline{\omega}}_{2})}(\,x_{1},x_{2}\,)\,)\supset
B(f_{{\underline{\omega}}_{2}}^{n_{2}}(x),L^{-2n_{2}}\varepsilon)$

where $\kappa^{p_{1}}$ denotes the finite word
$(\kappa,\kappa,\dots,\kappa)\in\\{1,2,\dots,\kappa\\}^{p_{1}}$. Proceeding
recursively, for each $k\geqslant 1$,
$(x_{1},\dots,x_{k})\in\mathcal{S}_{1}\times\dots\times\mathcal{S}_{k}$ and
$1\leqslant i\leqslant k-1$ we obtain an integer $0\leqslant p_{i}\leqslant
N_{L^{-2{n_{i}}}\varepsilon}$ so that

$\displaystyle
C_{({\underline{\omega}}_{1},\kappa^{p_{1}},{\underline{\omega}}_{2},\kappa^{p_{2}},...,{\underline{\omega}}_{k})}(\,x_{1},\dots,x_{k}\,)$
$\displaystyle\qquad:=B_{{\underline{\omega}}_{1}}(x_{1},n_{1},\varepsilon)\;\cap\;\bigcap_{i=1}^{k-1}(\,f^{p_{i}}_{\kappa}\circ
f_{{\underline{\omega}}_{i}}^{n_{i}}\circ\dots\circ f^{p_{1}}_{\kappa}\circ
f_{{\underline{\omega}}_{1}}^{n_{1}}\,)^{-1}(\,B(x_{i+1},L^{-n_{i+1}}\varepsilon)\,)$

is a non-empty subset of $X_{{\underline{\omega}}_{1}}$. The next result
ensures that one can select a large number of points for which the transition
times function $p_{i}=p_{i}(\,x_{1},\dots,x_{i}\,)$ is constant, and use this
to construct a suitable ${\underline{\omega}}\in\Sigma_{\kappa}$ whose non-
autonomous dynamical system $\mathcal{F}_{\underline{\omega}}$ has a irregular
set with large entropy.

######  Lemma 9.3.

There exists $K_{0}>0$ and for each $k\geqslant 1$ there exists a collection
$\mathcal{I}_{k}\subset\mathcal{S}_{1}\times\mathcal{S}_{2}\times\dots\times\mathcal{S}_{k}$
such that:

  1. (1)

for any $1\leqslant i\leqslant k-1$ there exists $1\leqslant t_{i}\leqslant
N_{L^{-2{n_{i}}}\varepsilon}$ so that $p_{i}(\,x_{1},\dots,x_{k}\,)=t_{k}$ for
every $(\,x_{1},\dots,x_{k}\,)\in\mathcal{I}_{k}$,

  2. (2)

$\\#{\mathcal{I}}_{k}\geqslant
K_{0}\,e^{(H^{\text{Pinsker}}(\psi)-4\gamma)\,m_{k}}$

  3. (3)

$d_{\underline{\omega}}^{m_{k}}(x,y)>2\varepsilon$ for any points $x\in
C_{({\underline{\omega}}_{1},\kappa^{t_{1}},{\underline{\omega}}_{2},\kappa^{t_{2}},...,{\underline{\omega}}_{k})}(\,x_{1},\dots,x_{k}\,)$
and $y\in
C_{({\underline{\omega}}_{1},\kappa^{t_{1}},{\underline{\omega}}_{2},\kappa^{t_{2}},...,{\underline{\omega}}_{k})}(\,y_{1},\dots,y_{k}\,)$
associated to distinct $k$-uples
$(\,x_{1},\dots,x_{k}\,),(\,y_{1},\dots,y_{k}\,)\in\mathcal{I}_{k}$,

where $m_{k}=n_{k}+\sum_{i=1}^{k-1}(n_{i}+t_{i})$ and
${\underline{\omega}}\in\bigcap_{k\geqslant
1}[{\underline{\omega}}_{1},\kappa^{t_{1}},{\underline{\omega}}_{2},\kappa^{t_{2}},...,{\underline{\omega}}_{k}]$.

###### Proof.

By the pigeonhole principle, for each $1\leqslant i\leqslant k-1$ there exists
$0\leqslant t_{i}\leqslant N_{L^{-2{n_{i}}}\varepsilon}$ and a collection
$\mathcal{I}_{k}\subset\mathcal{S}_{1}\times\mathcal{S}_{2}\times\dots\times\mathcal{S}_{k}$
such that item (1) above holds and, recalling that $n_{i}\gg
N_{L^{-2{n_{i}}}\varepsilon}$ and $\lim_{k\to\infty}\frac{m_{k}}{n_{k}}=1$
(recall (9.6)),

$\displaystyle\\#{\mathcal{I}_{k}}$
$\displaystyle\geqslant\frac{\\#\mathcal{S}_{k}\,\dots\,\\#\mathcal{S}_{2}\,\\#\mathcal{S}_{1}}{\prod_{i=1}^{k-1}\,N_{L^{-2{n_{i}}}\varepsilon}}$
$\displaystyle\geqslant
e^{(H^{\text{Pinsker}}(\psi)-3\gamma)\,[n_{k}+\dots+n_{2}+n_{1}]}\prod_{i=1}^{k-1}\frac{1}{n_{i+1}}\,\frac{n_{i+1}}{N_{L^{-2{n_{i}}}\varepsilon}}$
$\displaystyle\geqslant e^{(H^{\text{Pinsker}}(\psi)-4\gamma)\,m_{k}}$

for every large $k\geqslant 1$, where
$m_{k}=n_{k}+\sum_{i=1}^{k-1}(n_{i}+t_{i})$. In particular,
$\\#{\mathcal{I}}_{k}\geqslant Ke^{(H^{\text{Pinsker}}(\psi)-4\gamma)\,m_{k}}$
for every $k\geqslant 1$. Now, if ${\underline{\omega}}\in\bigcap_{k\geqslant
1}[{\underline{\omega}}_{1},\kappa^{t_{1}},{\underline{\omega}}_{2},\kappa^{t_{2}},...,{\underline{\omega}}_{k}]$
and $x\in
C_{({\underline{\omega}}_{1},\kappa^{t_{1}},{\underline{\omega}}_{2},\kappa^{t_{2}},...,{\underline{\omega}}_{k})}(\,x_{1},\dots,x_{k}\,)$
and $y\in
C_{({\underline{\omega}}_{1},\kappa^{t_{1}},{\underline{\omega}}_{2},\kappa^{t_{2}},...,{\underline{\omega}}_{k})}(\,y_{1},\dots,y_{k}\,)$
are points associated to distinct $k$-uples
$(\,x_{1},\dots,x_{k}\,),(\,y_{1},\dots,y_{k}\,)\in\mathcal{I}_{k}$ and
$j=\inf\\{1\leqslant i\leqslant k\colon x_{i}\neq y_{i}\\}$ then
$d_{\underline{\omega}}^{m_{k}}(x,y)\geqslant
d(x_{j},y_{j})-d(f_{{\underline{\omega}}}^{m_{j}}(x),x_{j})-d(f_{{\underline{\omega}}}^{m_{j}}(y),y_{j})>\varepsilon.$
This completes the proof of the lemma. ∎

The sequence ${\underline{\omega}}\in\Sigma_{k}$ above was determined by the
concatenation of the cylinders ${{\underline{\omega}}_{i}}_{[0,n_{i}]}$ and
the cylinders determined by the finite words $\kappa^{p_{i}}$, i.e.,

${\underline{\omega}}\in({{\underline{\omega}}_{1}}_{[0,n_{1}]}\cdot\kappa^{p_{1}})\,\cap\,\bigcap_{i\geqslant
1}\sigma^{-\sum_{j=1}^{i}(p_{j}+n_{j})}\;({{\underline{\omega}}_{i+1}}_{[0,n_{i+1}]}\cdot\kappa^{p_{i+1}}).$
(9.7)

Consider also the sets

$\mathfrak{F}_{m_{k}}({\underline{\omega}})=\bigcup_{(x_{1},\dots,x_{k})\in\mathcal{S}_{1}\times\dots\times\mathcal{S}_{k}}C_{({\underline{\omega}}_{1},\kappa^{p_{1}},{\underline{\omega}}_{2},\kappa^{p_{2}},...,{\underline{\omega}}_{k})}(\,x_{1},\dots,x_{k}\,)$

identified as a subset of $X_{\underline{\omega}}$, where
$m_{k}=n_{k}+\sum_{i=1}^{k-1}(n_{i}+t_{i})$, and the Moran set

$\mathfrak{F}_{{{\underline{\omega}}}}:=\bigcap_{k\geqslant
1}\mathfrak{F}_{m_{k}}({\underline{\omega}})\;\cap\;X_{{{\underline{\omega}}}}.$

######  Lemma 9.4.

$\mathfrak{F}_{{{\underline{\omega}}}}$ is contained in the irregular set
$I_{\underline{\omega}}(\psi)$ defined by (2.10).

###### Proof.

The argument follows the lines of the proof of Lemma 6.7, and for that reason
we shall omit it. ∎

####  9.1.3. The relative entropy of $\mathfrak{F}_{\underline{\omega}}$

In order to provide lower bounds for the fiber topological entropy of the
fibered Moran set $\mathfrak{F_{\underline{\omega}}}$ for the non-autonomous
dynamical system $\mathcal{F}_{\underline{\omega}}$ we will make use of the
fibered version of the entropy distribution principle (cf. Proposition 4.4).
For each $k\geqslant 1$ consider the probability measure
$\mu_{{\underline{\omega}},k}$ supported on points selected among the elements
of sets in the family

$\mathcal{C}_{k}=\\{C_{({\underline{\omega}}_{1},\kappa^{p_{1}},{\underline{\omega}}_{2},\kappa^{p_{2}},...,{\underline{\omega}}_{k})}(\,x_{1},\dots,x_{k}\,)\colon(x_{1},x_{2},\dots,x_{k})\in\mathcal{I}_{k}\\}.$

More precisely, for each $C\in\mathcal{C}_{k}$ choose a point $z_{C}\in C$ and
consider the probability measure

$\mu_{{\underline{\omega}},k}=\frac{1}{\\#\mathcal{C}_{k}}\sum_{C\in\mathcal{C}_{k}}\delta_{z_{C}}.$

######  Lemma 9.5.

Any weak∗ accumulation point $\nu_{\underline{\omega}}$ of the sequence
$(\mu_{{\underline{\omega}},k})_{k\geqslant 1}$ satisfies
$\nu_{\underline{\omega}}(\mathfrak{F}_{\underline{\omega}})=1$.

###### Proof.

Let $\nu_{\underline{\omega}}$ be an weak∗ accumulation point of the sequence
$(\mu_{{\underline{\omega}},k})_{k\geqslant 1}$. Given $k\geq 1$, using
Portmanteau theorem on the characterization of weak∗ convergence, as
$\mu_{{\underline{\omega}},\ell}(\mathfrak{F}_{m_{k}}({\underline{\omega}}))=1$
for every $\ell\geqslant k$ then
$\nu_{{\underline{\omega}}}(\mathfrak{F}_{m_{k}}({\underline{\omega}}))=1$.
Thus $\nu_{{\underline{\omega}}}\big{(}\bigcap_{k\geqslant
1}\mathfrak{F}_{m_{k}}({\underline{\omega}})\big{)}=1$ as claimed. ∎

Now, let $\varepsilon>0$ be small and $\mathbf{n}\geqslant 1$ be fixed and
large. Let $B_{\underline{\omega}}(x,\mathbf{n},\varepsilon)\subset
X_{\underline{\omega}}$ be a dynamic ball which intersects the set
$\mathfrak{F}_{\underline{\omega}}$ and let $\ell=\ell(\mathbf{n})\geqslant 1$
be so that $m_{\ell}\leqslant\mathbf{n}<m_{\ell+1}$, where the sequence
$(m_{\ell})_{\ell\geqslant 1}$ was defined in Lemma 9.3. In order to apply
Proposition 4.4 we need the following:

######  Lemma 9.6.

$\mu_{{\underline{\omega}},k+\ell}(B_{\underline{\omega}}((x,\mathbf{n},\varepsilon))\leqslant
e^{-\mathbf{n}(H^{\text{Pinsker}}(\psi)-5\gamma)}$ for every large $k\geqslant
1$.

###### Proof.

Suppose that
$\mu_{{\underline{\omega}},k+\ell}(B_{\underline{\omega}}(x,\mathbf{n},\varepsilon))>0$
(otherwise there is nothing to prove), hence
$B_{\underline{\omega}}(x,\mathbf{n},\varepsilon)$ intersects
$\mathfrak{F}_{{\underline{\omega}},m_{k+\ell}}$. We observe that if $y_{1}\in
B_{\underline{\omega}}(x,\mathbf{n},\varepsilon)\cap C_{1}$ and $y_{2}\in
B_{\underline{\omega}}(x,\mathbf{n},\varepsilon)\cap C_{2}$ where
$C_{1},C_{2}\in\mathcal{C}_{\ell}$ then item (3) in Lemma 9.3, guarantees that
$d_{{\underline{\omega}}}^{\mathbf{n}}(y_{1},y_{2})\geqslant
d_{{\underline{\omega}}}^{m_{\ell}}(y_{1},y_{2})>2\varepsilon$ whenever
$C_{1}$ and $C_{2}$ are distinct. Thus, all points in the set
$B_{\underline{\omega}}(x,\mathbf{n},\varepsilon)\cap\mathfrak{F}_{{\underline{\omega}},m_{k+\ell}}$
belong to a single fixed set $C_{\ell}\in\mathcal{C}_{\ell}$. Furthermore, the
construction in the proof of Lemma 9.3 yields that, for every
$D\in\mathcal{C}_{\ell}$,

$\\#\Big{\\{}C\in\mathcal{C}_{k+\ell}\colon C\cap
D\neq\emptyset\Big{\\}}\geqslant\frac{\\#\mathcal{S}_{k+\ell}\,\dots\,\\#\mathcal{S}_{\ell+1}}{\prod_{i=\ell}^{k+\ell-1}\,N_{L^{-2{n_{i}}}\varepsilon}}>e^{(H^{\text{Pinsker}}(\psi)-4\gamma)\,[n_{k+\ell}+\dots+n_{k+1}]}$

for every large $k$. In particular, using that

$\displaystyle\\#\\{C\in\mathcal{C}_{k+\ell}\colon C\cap
C_{\ell}\neq\emptyset\\}$
$\displaystyle\leqslant\frac{\\#\mathcal{C}_{k+\ell}}{\min\limits_{D\in\mathcal{C}_{\ell}}\;\\#\\{C\in\mathcal{C}_{k+\ell}\colon
C\cap D\neq\emptyset\\}}$
$\displaystyle<e^{-(H^{\text{Pinsker}}(\psi)-4\gamma)\,[n_{k+\ell}+\dots+n_{k+1}]}\;\\#\mathcal{C}_{k+\ell},$

that $m_{k+\ell}>\mathbf{n}$, that $\lim_{k\to\infty}\frac{m_{k}}{n_{k}}=1$
and the choices at item (1) in Lemma 9.3, we obtain that

$\displaystyle\mu_{{\underline{\omega}},k+\ell}(B_{\underline{\omega}}(x,\mathbf{n},\varepsilon))$
$\displaystyle=\displaystyle\frac{1}{\\#\mathcal{C}_{k+\ell}}\sum_{\begin{subarray}{c}C\in\mathcal{C}_{k+\ell}\end{subarray}}\delta_{z_{C}}\left(B_{\underline{\omega}}(x,\mathbf{n},\varepsilon)\right)$
$\displaystyle\leqslant\displaystyle\frac{\\#\\{C\in\mathcal{C}_{k+\ell}\colon
C\cap C_{\ell}\neq\emptyset\\}}{\\#\mathcal{C}_{k+\ell}}$
$\displaystyle<e^{-(H^{\text{Pinsker}}(\psi)-4\gamma)\,[n_{k+\ell}+\dots+n_{k+1}]}$
$\displaystyle<e^{-(H^{\text{Pinsker}}(\psi)-5\gamma)\,m_{k+\ell}}<e^{-\mathbf{n}\,(H^{\text{Pinsker}}(\psi)-5\gamma)}$

for every large $k\geqslant 1$, thus concluding the proof of the lemma.

∎

We can now complete the proof of item (1) in Theorem F. Indeed, as the
assumptions of the entropy distribution principle are satisfied (as a
consequence of Lemmas 9.5 and 9.6 above) we conclude that

$h^{path}_{I}(\mathbb{S},\psi)\geqslant
h_{I_{\underline{\omega}}(\psi)}(\mathcal{F}_{\underline{\omega}})\geqslant
H^{\text{Pinsker}}(\psi)-5\gamma.$

As $\gamma>0$ was chosen arbitrary, we get that
$h^{path}_{I}(\mathbb{S},\psi)\geqslant H^{\text{Pinsker}}(\psi)$, as desired.

###  9.2. Moran sets of irregular points with large fibered entropy

In this subsection we explain the necessary modifications in the argument
explored in the previous subsections of in order to prove item (2) in Theorem
F, namely that the topological entropy of the set
$\Sigma:=\\{{\underline{\omega}}\in\Sigma_{\kappa}\colon
h_{I_{\underline{\omega}}(\psi)}(\mathcal{F}_{\underline{\omega}})\geqslant
H^{\text{Pinsker}}(\psi)\\}$ is larger or equal than
$H^{\text{Pinsker}}_{\sigma}(\psi)$.

Fix $\gamma>0$. By definition (cf. (2.9)) there exists $\zeta>0$ so that
$H^{\text{Pinsker}}_{\sigma}(\psi,\zeta)\geqslant
H^{\text{Pinsker}}_{\sigma}(\psi)-\gamma$. In particular there exist
$\mu_{1},\mu_{2}\in\mathcal{M}_{erg}(F)$ so that
$\int\varphi_{\psi}\,d\mu_{1}<\int\varphi_{\psi}\,d\mu_{2}$,

$\displaystyle h_{\mu_{i}}(F\mid\sigma)\geqslant
H^{\text{Pinsker}}(\psi)-\zeta\quad\text{and}\quad
h_{\pi_{*}\mu_{i}}(\sigma)\geqslant H^{\text{Pinsker}}_{\sigma}(\psi)-\gamma$
(9.8)

for $i=1,2$. As the probability measures $\mu_{i}$ are ergodic (taking
$\delta=\frac{1}{4}$ in (4.11)) it follows that
$h_{\mu_{i}}^{1/4}(\underline{\omega})=h_{\mu_{i}}(F\\!\mid\\!\sigma)\geqslant
H^{\text{Pinsker}}(\psi)-\zeta$ for $(\pi)_{*}\mu_{i}$-almost every
${\underline{\omega}}\in\Sigma_{\kappa}$. Fix $0<\delta<\frac{1}{8}$. By the
definition of the fibered Katok entropy (recall (4.10)) and Lusin’s theorem,
for each $i=1,2$ there exists $\tilde{\Omega}_{i}\subset\Sigma_{\kappa}$ and
$\varepsilon>0$ small so that $(\pi_{*}\mu_{i})(\tilde{\Omega}_{i})>1-\delta$
and $\liminf_{n\rightarrow+\infty}\frac{1}{n}\log
s({\underline{\omega}},n,4\varepsilon,\frac{1}{4})>H^{\text{Pinsker}}(\psi)-2\zeta,$
for every ${\underline{\omega}}\in\tilde{\Omega}_{i}$, $i=1,2$. Using Lusin’s
theorem once more, we may take an integer $\mathfrak{n}_{0}\geqslant 1$ and a
subset $\Omega_{i}\subset\tilde{\Omega}_{i}$ so that
$(\pi_{*}\mu_{i})(\Omega_{i})\geqslant(1-2\delta)>\frac{3}{4}$ and

$s({\underline{\omega}},n,4\varepsilon,\frac{1}{4})>e^{(H^{\text{Pinsker}}(\psi)-3\zeta)n}\quad\text{for
every $n\geqslant\mathfrak{n}_{0}$, ${\underline{\omega}}\in\Omega_{i}$ and
$i=1,2$.}$ (9.9)

We diminish $\varepsilon$, if necessary, so that
$|\psi(x)-\psi(x^{\prime})|<\frac{\int\varphi\,d\mu_{2}-\int\varphi\,d\mu_{1}}{4}$
whenever $d(x,x^{\prime})<4\varepsilon$.

Now we turn our attention to the convergence of Birkhoff averages on
$\Sigma_{\kappa}\times X$. Consider a strictly decreasing sequence of positive
numbers $(\delta_{k})_{k\geqslant 1}$ tending to zero so that
$\delta_{1}<\frac{1}{4}[\int\varphi_{\psi}\,d\mu_{2}-\int\varphi_{\psi}\,d\mu_{1}].$
Using Birkhoff ergodic theorem and Lusin theorem, there exists a strictly
increasing sequence of large positive integers $(\mathfrak{n}_{k})_{k\geqslant
1}$ so that $\mathfrak{n}_{1}>\mathfrak{n}_{0}$ and, for each $\ell\geqslant
1$, the sets

$\Gamma_{2\ell-1}:=\Big{\\{}({\underline{\omega}},x)\in\\{1,2,...,\kappa\\}^{\mathbb{Z}}\times
X:\Big{|}\frac{1}{n}\sum_{j=0}^{n-1}\varphi(F^{j}({\underline{\omega}},x))-\int\varphi\,d\mu_{1}\Big{|}<\delta_{2\ell-1},\;\;\forall
n\geqslant\mathfrak{n}_{2\ell-1}\Big{\\}}$

and

$\Gamma_{2\ell}:=\Big{\\{}({\underline{\omega}},x)\in\\{1,2,...,\kappa\\}^{\mathbb{Z}}\times
X:\Big{|}\frac{1}{n}\sum_{j=0}^{n-1}\varphi(F^{j}({\underline{\omega}},x))-\int\varphi\,d\mu_{2}\Big{|}<\delta_{2\ell},\;\;\forall
n\geqslant\mathfrak{n}_{2\ell}\Big{\\}}$

satisfy $\mu_{1}(\Gamma_{2\ell-1})>1-\delta$ and
$\mu_{{\underline{\omega}}_{2}}(\Gamma_{2\ell})>1-\delta$ for every
$\ell\geqslant 1$. These sets are not suitable to our purposes as we need to
decouple information on the shift space $\Sigma_{\kappa}$ and the fibers. This
is done as follows. Let
$\mu_{i}=(\mu_{i,{\underline{\omega}}})_{\underline{\omega}}$ be the
disintegration of $\mu_{i}$ in sample measures given by Rokhlin’s
disintegration theorem. More precisely, there exists a $\mu_{i}$-almost
everywhere defined and measurable family of probability measures
${\underline{\omega}}\mapsto\mu_{i,{\underline{\omega}}}$ so that
$\operatorname{supp}\mu_{i,{\underline{\omega}}}\subset
X_{\underline{\omega}}$ and
$\mu_{i}(A)=\int_{\Sigma_{\kappa}}\,\mu_{i,{\underline{\omega}}}(A\cap
X_{\underline{\omega}})\,d(\pi_{*}\mu_{i})({\underline{\omega}})$ for every
measurable set $A\subset\Sigma_{\kappa}\times X$. We will use the following
quantitative estimate.

######  Lemma 9.7.

For each $\ell\geqslant 1$ there exist subsets
$\Xi_{2\ell-1},\Xi_{2\ell}\subset\Sigma_{\kappa}$ so that

  1. (1)

$(\pi_{*}\mu_{1})(\Xi_{2\ell-1})>\frac{1}{4}$ and
$(\pi_{*}\mu_{2})(\Xi_{2\ell})>\frac{1}{4}$,

  2. (2)

$\mu_{1,{\underline{\omega}}}\big{(}X_{\underline{\omega}}\cap\Gamma_{2\ell-1}\big{)}>\frac{1}{4}$
for every ${\underline{\omega}}\in\Xi_{2\ell-1}$, and

  3. (3)

$\mu_{2,{\underline{\omega}}}\big{(}X_{\underline{\omega}}\cap\Gamma_{2\ell}\big{)}>\frac{1}{4}$
for every ${\underline{\omega}}\in\Xi_{2\ell}$.

###### Proof.

We use the following Fubini type estimate involving to the sample measures
$(\mu_{1,{\underline{\omega}}})_{\underline{\omega}}$:

$\displaystyle 1-\delta<\mu_{1}(\Gamma_{2\ell-1})=$
$\displaystyle\int_{\Sigma_{\kappa}}\,\mu_{i,{\underline{\omega}}}(X_{\underline{\omega}}\cap\Gamma_{2\ell-1})\,d(\pi_{*}\mu_{1})({\underline{\omega}})$
$\displaystyle\leqslant\pi_{*}\mu_{1}\Big{(}{\underline{\omega}}\in\Sigma_{\kappa}\colon\mu_{i,{\underline{\omega}}}\big{(}X_{\underline{\omega}}\cap\Gamma_{2\ell-1}\big{)}>a\Big{)}$
$\displaystyle+a\cdot\pi_{*}\mu_{1}\Big{(}{\underline{\omega}}\in\Sigma_{\kappa}\colon\mu_{i,{\underline{\omega}}}\big{(}X_{\underline{\omega}}\cap\Gamma_{2\ell-1}\big{)}\leqslant
a\Big{)}$

and, consequently,
$\pi_{*}\mu_{1}\Big{(}{\underline{\omega}}\in\Sigma_{\kappa}\colon\mu_{i,{\underline{\omega}}}\big{(}X_{\underline{\omega}}\cap\Gamma_{2\ell-1}\big{)}>a\Big{)}>1-\delta-a$
for every $0<a<1$. The choice $0<\delta<\frac{1}{4}$ guarantees that
$\pi_{*}\mu_{1}\Big{(}{\underline{\omega}}\in\Sigma_{\kappa}\colon\mu_{i,{\underline{\omega}}}\big{(}X_{\underline{\omega}}\cap\Gamma_{2\ell-1}\big{)}>\frac{1}{4}\Big{)}>\frac{1}{2}.$
The claim concerning the sample measures $\mu_{2,{\underline{\omega}}}$
follows similarly. ∎

Observe that
$\pi_{*}\mu_{1}(\Omega_{1}\cap\Xi_{2\ell-1})>1-2\delta-\frac{1}{4}>0$ and
$\pi_{*}\mu_{2}(\Omega_{2}\cap\Xi_{2\ell})>1-2\delta-\frac{1}{4}>0$ for every
$\ell\geqslant 1$. Moreover, by Katok’s entropy formula (recall (4.1)) one can
compute $h_{\pi_{*}\mu_{i}}(\sigma)$ ($i=1,2$) using the sets
$\Omega_{1}\cap\Xi_{2\ell-1}$ and $\Omega_{2}\cap\Xi_{2\ell}$:

$h_{\pi_{*}\mu_{i}}(\sigma)=\lim_{\varepsilon\to
0}\;\Big{[}\liminf_{n\rightarrow+\infty}\frac{1}{n}\log
N^{\pi_{*}\mu_{i}}_{n}(8\varepsilon,1-2\delta-\frac{1}{4})\,\Big{]},$

where $N^{\pi_{*}\mu_{i}}_{n}(8\varepsilon,1-2\delta-\frac{1}{4})$ stands for
the minimal cardinality of an $(n,8\varepsilon)$-generating subset for a
subset of $\Sigma_{\kappa}$ of $(\pi_{*}\mu_{i})$-measure at least
$1-2\delta-\frac{1}{4}$. In particular, for each $\ell\geqslant 1$ and
$\varepsilon>0$ is small then there exist $(n,4\varepsilon)$-separated sets
$\Xi_{2\ell-1,n}\subset\Omega_{1}\cap\Xi_{2\ell-1}$ and
$\Xi_{2\ell,n}\subset\Omega_{2}\cap\Xi_{2\ell}$ such that

$\\#\Xi_{2\ell,n}\geqslant
e^{(H^{\text{Pinsker}}_{\sigma}(\psi)-2\gamma)n}\quad\text{and}\quad\\#\Xi_{2\ell-1,n}\geqslant
e^{(H^{\text{Pinsker}}_{\sigma}(\psi)-2\gamma)n}$

for every large $n\geqslant 1$ (depending on $\ell$). Moreover, combining the
latter with (9.9), and item (2) in Lemma 9.7 one can produce a strictly
increasing sequence $(n_{\ell})_{\ell\geqslant 1}$ of positive integers
satisfying (9.6) and such that for each $\ell\geqslant 1$:

  * (i)

$\\#\Xi_{2\ell,n_{2\ell}}\geqslant
e^{(H^{\text{Pinsker}}_{\sigma}(\psi)-2\gamma)n_{2\ell}}\quad\text{and}\quad\\#\Xi_{2\ell-1,n_{2\ell-1}}\geqslant
e^{(H^{\text{Pinsker}}_{\sigma}(\psi)-2\gamma)n_{2\ell-1}}$

  * (ii)

for each ${\underline{\omega}}\in\Xi_{2\ell-1}$ there exists a
$({\underline{\omega}},n_{2\ell-1},4\varepsilon)$-separated set

$\mathcal{S}_{{\underline{\omega}},2\ell-1}\subset
X_{\underline{\omega}}\cap\Gamma_{2\ell-1}\quad\text{so
that}\quad\\#\mathcal{S}_{{\underline{\omega}},2\ell-1}\geqslant\exp(\,(H^{\text{Pinsker}}(\psi)-3\zeta)n_{2\ell-1}\,).$

  * (iii)

for each ${\underline{\omega}}\in\Xi_{2\ell}$ there exists a
$({\underline{\omega}},n_{2\ell},4\varepsilon)$-separated set

$\mathcal{S}_{{\underline{\omega}},2\ell}\subset
X_{\underline{\omega}}\cap\Gamma_{2\ell}\quad\text{so
that}\quad\\#\mathcal{S}_{{\underline{\omega}},2\ell}\geqslant\exp(\,(H^{\text{Pinsker}}(\psi)-3\zeta)n_{2\ell}\,).$

These informations can now be used to construct a Moran subset of
$\Sigma_{\kappa}\times X$ of skew-product type whose both the projection on
$\Sigma_{\kappa}$ and the fibers have large entropy (see Figure 8.1 below for
an illustration).

Figure 9.1. Construction of a specific Moran subset of $\Sigma_{\kappa}\times
X$, formed from typical points for $\mu_{1}$ (represented in gray) and typical
points for $\mu_{2}$ (represented in brown).

.

More precisely:

######  Proposition 9.8.

There exists $K>0$ and a sequence $(\mathfrak{S}_{k})_{k\geqslant 1}$ of
compact subsets of $\Sigma_{\kappa}$ and a sequence
$(\mathfrak{F}_{k})_{k\geqslant 1}$ of compact subsets of
$\Sigma_{\kappa}\times X$ so that, for each $k\geqslant 1$:

  1. (1)

$\mathfrak{S}_{k}$ is a finite union of cylinders in $\Sigma_{\kappa}$ with
the same length,

  2. (2)

$\mathfrak{F}_{k}\subset\pi^{-1}(\mathfrak{S}_{k})$ and for each
${\underline{\omega}}\in\mathfrak{S}_{k}$ there exist
$m_{k}=m_{k}({\underline{\omega}})\geqslant 1$ so that
$\lim_{k\to\infty}\frac{m_{k}}{n_{k}}=1$ and there exist
$e^{(H^{\text{Pinsker}}_{\sigma}(\psi)-4\gamma)\,[n_{k}+\dots+n_{2}+n_{1}]}$
points in $\mathfrak{F}_{k}\cap X_{\underline{\omega}}$ that are
$({\underline{\omega}},m_{k},\varepsilon)$-separated,

  3. (3)

The compact set

$\mathfrak{F}:=\bigcap_{k\geqslant
1}\mathfrak{F}_{k}\subset\Sigma_{\kappa}\times X$

is such that $\mathfrak{F}\,\cap\,X_{\underline{\omega}}\subset
I_{\underline{\omega}}(\psi)$ for every
${\underline{\omega}}\in\mathfrak{S}:=\bigcap_{k\geqslant 1}\mathfrak{S}_{k}$,

  4. (4)

$h_{I_{\underline{\omega}}(\psi)}(\mathcal{F}_{\underline{\omega}})\geqslant
H^{\text{Pinsker}}(\psi)-5\gamma$ for every $x\in\mathfrak{S}$.

###### Proof.

This construction combines information from Subsection 9.1 and the sets
described in items (i)-(iii) above. For each $k\geqslant 1$, each $k$-uple
$({\underline{\omega}}_{1},{\underline{\omega}}_{2},\dots,{\underline{\omega}}_{k})\in\Xi_{1,n_{1}}\times\Xi_{2,n_{2}}\times\dots\times\Xi_{k,n_{k}}$
and
$(x_{1},x_{2},\dots,x_{k})\in\mathcal{S}_{{\underline{\omega}}_{1},1}\times\mathcal{S}_{{\underline{\omega}}_{2},2}\times\dots\mathcal{S}_{{\underline{\omega}}_{k},k}$
and each $1\leqslant i\leqslant k-1$ there exists an integer $0\leqslant
p_{i}\leqslant N_{L^{-2{n_{i}}}\varepsilon}$ (depending on
${\underline{\omega}}_{j}$ and $x_{j}$, for $1\leqslant j\leqslant i-1$) so
that

$\displaystyle
C_{({\underline{\omega}}_{1},\kappa^{p_{1}},{\underline{\omega}}_{2},\kappa^{p_{2}},...,{\underline{\omega}}_{k})}(\,x_{1},\dots,x_{k}\,)$
$\displaystyle\qquad:=B_{{\underline{\omega}}_{1}}(x_{1},n_{1},\varepsilon)\;\cap\;\bigcap_{i=1}^{k-1}(\,f^{p_{i}}_{\kappa}\circ
f_{{\underline{\omega}}_{i}}^{n_{i}}\circ\dots\circ f^{p_{1}}_{\kappa}\circ
f_{{\underline{\omega}}_{1}}^{n_{1}}\,)^{-1}(\,B(x_{i+1},L^{-n_{i+1}}\varepsilon)\,)$

is a non-empty subset of $X$. Proceeding as in the proof of Lemma 9.3, one can
extract a large subcollection of these sets for which the transition time
functions depend only on
${\underline{\omega}}_{1},{\underline{\omega}}_{2},\dots,{\underline{\omega}}_{k}$.
Indeed, for each $1\leqslant i\leqslant k-1$ there exists $0\leqslant
t_{i}=t_{i}(({\underline{\omega}}_{1},\dots,{\underline{\omega}}_{i}))\leqslant
N_{L^{-2{n_{i}}}\varepsilon}$ and a collection
$\mathcal{I}_{k}({\underline{\omega}}_{1},{\underline{\omega}}_{2},\dots,{\underline{\omega}}_{k})\subset\mathcal{S}_{{\underline{\omega}}_{1},1}\times\mathcal{S}_{{\underline{\omega}}_{2},2}\times\dots\mathcal{S}_{{\underline{\omega}}_{k},k}$
such that

$\\#\mathcal{I}_{k}({\underline{\omega}}_{1},{\underline{\omega}}_{2},\dots,{\underline{\omega}}_{k})\geqslant
Ke^{(H^{\text{Pinsker}}(\psi)-4\gamma)\,m_{k}}$

and

$p_{i}(\,x_{1},\dots,x_{i},{\underline{\omega}}_{1},\dots,{\underline{\omega}}_{i}\,)=t_{i}\quad\text{for
every}\;(\,x_{1},\dots,x_{k}\,)\in\mathcal{I}_{k}({\underline{\omega}}_{1},{\underline{\omega}}_{2},\dots,{\underline{\omega}}_{k}).$

Taking $m_{k}=n_{k}+\sum_{i=1}^{k-1}(n_{i}+t_{i})$ (which is a function of
$({\underline{\omega}}_{1},{\underline{\omega}}_{2},\dots,{\underline{\omega}}_{k})$),
if
$(\,x_{1},\dots,x_{k}\,),(\,y_{1},\dots,y_{k}\,)\in\mathcal{I}_{k}({\underline{\omega}}_{1},{\underline{\omega}}_{2},\dots,{\underline{\omega}}_{k})$
are distinct $k$-uples and one considers points $x\in
C_{({\underline{\omega}}_{1},\kappa^{t_{1}},{\underline{\omega}}_{2},\kappa^{t_{2}},...,{\underline{\omega}}_{k})}(\,x_{1},\dots,x_{k}\,)$
and $y\in
C_{({\underline{\omega}}_{1},\kappa^{t_{1}},{\underline{\omega}}_{2},\kappa^{t_{2}},...,{\underline{\omega}}_{k})}(\,y_{1},\dots,y_{k}\,)$
then $d_{\underline{\omega}}^{m_{k}}(x,y)>2\varepsilon$ for any

${\underline{\omega}}\in\mathfrak{S}({\underline{\omega}}_{1},{\underline{\omega}}_{2},\dots,{\underline{\omega}}_{k}):=({{\underline{\omega}}_{1}}_{[0,n_{1}]}\cdot\kappa^{t_{1}})\,\cap\,\bigcap_{i\geqslant
1}\sigma^{-\sum_{j=1}^{i}(t_{j}+n_{j})}\;({{\underline{\omega}}_{i+1}}_{[0,n_{i+1}]}\cdot\kappa^{t_{i+1}}).$

Notice that the cylinder set
$\mathfrak{S}({\underline{\omega}}_{1},{\underline{\omega}}_{2},\dots,{\underline{\omega}}_{k})\subset\Sigma_{\kappa}$
has size $m_{k}$, which is a function of the transition times
$(t_{1},t_{2},\dots,t_{k-1})$ given by the pigeonhole principle applied to the
fibered sets. There are

$\\#\Xi_{1,n_{1}}\times\\#\Xi_{2,n_{2}}\times\dots\times\\#\Xi_{k,n_{k}}\geqslant
e^{(H^{\text{Pinsker}}_{\sigma}(\psi)-2\gamma)[n_{k}+\dots+n_{2}+n_{1}]}$

such cylinders in $\Sigma_{\kappa}$ and, as $0\leqslant t_{i}\leqslant
N_{L^{-2{n_{i}}}\varepsilon}$, using the pigeonhole principle once more we get
a subcollection $\mathfrak{S}_{k}$ of such cylinder sets formed by cylinders
in $\Sigma_{\kappa}$ of the same length so that

$\displaystyle\\#{\mathfrak{S}_{k}}\geqslant\frac{e^{(H^{\text{Pinsker}}_{\sigma}(\psi)-2\gamma)[n_{k}+\dots+n_{2}+n_{1}]}}{\prod_{i=1}^{k-1}\,N_{L^{-2{n_{i}}}\varepsilon}}\geqslant
e^{(H^{\text{Pinsker}}_{\sigma}(\psi)-4\gamma)\,[n_{k}+\dots+n_{2}+n_{1}]}.$

This allow us to consider also the nested sequence of compact sets
$\mathfrak{F}_{k}\subset\Sigma_{\kappa}\times X$ defined by

$\mathfrak{F}_{k}=\bigcup_{\mathfrak{S}({\underline{\omega}}_{1},{\underline{\omega}}_{2},\dots,{\underline{\omega}}_{k})\in\mathfrak{S}_{k}}\;\bigcup_{(\,x_{1},\dots,x_{k}\,)\in\mathcal{I}_{k}({\underline{\omega}}_{1},{\underline{\omega}}_{2},\dots,{\underline{\omega}}_{k})}\mathfrak{S}({\underline{\omega}}_{1},{\underline{\omega}}_{2},\dots,{\underline{\omega}}_{k})\times
C_{({\underline{\omega}}_{1},\kappa^{t_{1}},{\underline{\omega}}_{2},\kappa^{t_{2}},...,{\underline{\omega}}_{k})}(\,x_{1},\dots,x_{k}\,),$

as represented by Figure 8.1. We emphasize that the transition times $t_{i}$
are functions of
$({\underline{\omega}}_{1},{\underline{\omega}}_{2},\dots,{\underline{\omega}}_{i})$.
These sets comply with the requirements of items (1) and (2) in the
proposition.

By construction $\mathfrak{F}:=\bigcap_{k\geqslant 1}\mathfrak{F}_{k}$ is a
Moran set. It is not hard to check that the same ideas explored in previous
sections ensure that items (3) and (4) also hold, we shall omit the details. ∎

##  10\. Linear cocycles

The main goal of this section is to prove Theorems A and B on the irregular
behavior and rigidity for linear cocycles. First we shall comment on the
fibered complexity of the projective cocycle.

###  10.1. Projective cocycles and complexities

The projective cocycle
$P_{A}:\Sigma_{\kappa}\times\mathbf{P}\mathbb{R}^{d}\to\Sigma_{\kappa}\times\mathbf{P}\mathbb{R}^{d}$
was defined in (1.4), and is a skew-product with compact fibers. Consider the
continuous potential
$\varphi_{A}:\Sigma_{\kappa}\times\mathbf{P}\mathbb{R}^{d}\to\mathbb{R}$ given
by

$\varphi_{A}({\underline{\omega}},v)=\log\frac{\|A({\underline{\omega}})\cdot
v\|}{\|v\|},\quad\text{for}\;({\underline{\omega}},v)\in\Sigma_{\kappa}\times\mathbf{P}\mathbb{R}^{d}.$
(10.1)

Given $\nu\in\mathcal{M}_{erg}(\sigma)$, if the cocycle $A$ is strongly
irreducible then Furstenberg formula ensures that

$\lambda_{+}(A,\nu)=\int\varphi_{A}({\underline{\omega}},v)\,d\eta(v)d\nu({\underline{\omega}})$

where $\eta$ is any $\nu$-stationary measure (see [32] or [80, Theorem 6.8]).
While the previous expression allows to compute the top Lyapunov exponent of
the linear cocycle $A$ as an average of a continuous potential, it does not
provide any immediate regularity of the Lyapunov exponent as a function of
$\nu$ as this relies ultimately on the analysis of the space of
$\nu$-stationary measures as $\nu$ varies.

######  Remark 10.1.

As a consequence of the Oseledets theorem, if ${\underline{\omega}}$ belongs
to a total probability subset of $\Sigma_{\kappa}$ then for _every_ normalized
vector $v\in\mathbb{R}^{d}$ the limit

$\lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\varphi(F_{A}^{j}({\underline{\omega}},v))=\lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\log\frac{\|A(\sigma^{j}({\underline{\omega}}))\cdot
A^{(j)}({\underline{\omega}})v\|}{\|A^{(j)}({\underline{\omega}})v\|}=\lim_{n\to\infty}\frac{1}{n}\log\frac{\|A^{(n)}({\underline{\omega}})v\|}{\|v\|}$

does exist and coincides with one of the Lyapunov exponents
$(\lambda_{i}(\mu))_{1\leqslant i\leqslant d}$.

We now observe that the non-wandering set of each projective map
$P_{A_{i}}:\mathbf{P}\mathbb{R}^{d}\to\mathbf{P}\mathbb{R}^{d}$ ($1\leqslant
i\leqslant\kappa$) is contained in the projectivization of invariant subspaces
of $A_{i}$. In fact, as a consequence of the Jordan canonical form theorem, we
have the following possibilities for the non-wandering set of the restriction
of each map $P_{A_{i}}$ to a projectivization of an eigenspace $E$ associated
to $A_{i}$: (i) if $A_{i}\mid_{E}$ has no nilpotent term then
$P_{A_{i}}\mid_{\mathbf{P}E}$ is the identify or an isometry in case of a real
or complex eigenvalue, respectively; (ii) if $A_{i}\mid_{E}$ has nilpotent
terms then there exists $F\subsetneq E$ so that ${\mathbf{P}F}$ is an
attractor for $P_{A_{i}}\mid_{E}$ and $P_{A_{i}}\mid_{\mathbf{P}F}$ is the
identify or an isometry in case of a real or complex eigenvalue, respectively.
In either case, the refinement of open coverings of $\mathbf{P}\mathbb{R}^{d}$
under the action of $P_{A_{i}}$ has sublinear growth, and the same holds for
any composition of such maps.Therefore,
$h_{\mu}(P_{A})=h_{\pi_{*}\mu}(\sigma)$ for every
$\mu\in\mathcal{M}_{inv}(P_{A})$ and every non-autonomous dynamical system
generated by these have zero topological entropy, ie,
$h_{\operatorname{top}}(\mathcal{F}_{\underline{\omega}})=0$.

Recall that for each $P_{A}$-invariant probability $\mu$ one has that
$h_{\mu}(P_{A})\leqslant h_{\pi_{*}\mu}(\sigma)+\int
h_{\operatorname{top}}(\mathcal{F}_{\underline{\omega}})d\pi_{*}\mu$ (cf.
[52]). As $H^{\text{Pinsker}}(\varphi_{A})=0$ then

$\displaystyle h_{*}(\varphi_{A})$ $\displaystyle=\sup\Big{\\{}c\geqslant
0\colon\text{there
exist}\,\mu_{1},\mu_{2}\in\mathcal{M}_{erg}(P_{A})\,\text{so that}\hfill$
$\displaystyle\hskip 56.9055pt\,h_{\pi_{*}\mu_{i}}(\sigma)\geqslant
c\;\text{and}\,\int\varphi_{A}\,d\mu_{1}<\int\varphi_{A}\,d\mu_{2}\Big{\\}}$

is bounded above by $h_{\operatorname{top}}(\sigma)=\log\kappa$. The next
lemma relates the top Lyapunov exponents with integrals for the projective
cocycle by means of lifted measures.

######  Lemma 10.2.

For every $\nu\in\mathcal{M}_{erg}(\sigma)$ there exists
$\mu\in\mathcal{M}_{erg}(P_{A})$ so that
$\int\varphi_{A}({\underline{\omega}},v)\,d\mu({\underline{\omega}},v)\lambda_{+}(A,\nu)$
and $\pi_{*}\mu=\nu.$

###### Proof.

First we construct invariant measures for the projective cocycle which can
express the top Lyapunov exponent in an integral form. Given
$\nu\in\mathcal{M}_{erg}(\sigma)$, the Oseledets theorem ensures there exists
$1\leqslant\tilde{\kappa}\leqslant\kappa$ and that for $\nu$-almost every
${\underline{\omega}}\in\Sigma_{\kappa}$ there exists a splitting
$\\{{\underline{\omega}}\\}\times\mathbb{R}^{d}=E^{1}_{\underline{\omega}}\oplus
E^{2}_{\underline{\omega}}\oplus\dots\oplus
E^{\tilde{\kappa}}_{\underline{\omega}}$, varying measurably with
${\underline{\omega}}$, so that
$\lambda_{i}(A,\mu)=\lim_{n\to\infty}\frac{1}{n}\log\|A^{n}({\underline{\omega}})v\|$
for every $v\in E^{i}_{\underline{\omega}}\setminus\\{0\\}$ and $1\leqslant
i\leqslant\tilde{\kappa}$. As $\lambda^{+}(A,\nu)=\lambda_{1}(A,\nu)$ and
$\omega\mapsto\mathbf{P}E^{1}_{\underline{\omega}}$ is a measurable family of
submanifolds of $\mathbf{P}\mathbb{R}^{d}$ of constant dimension, one may
consider the measurable family of probabilities
$(\mu_{\underline{\omega}})_{\underline{\omega}}$ so that
$\operatorname{supp}\mu_{\underline{\omega}}\subset\\{{\underline{\omega}}\\}\times\mathbf{P}\mathbb{R}^{d}$
and given by

$\mu_{\underline{\omega}}(E)=\begin{cases}\begin{array}[]{ll}\text{Leb}\mid_{\\{{\underline{\omega}}\\}\times\mathbf{P}E^{1}_{\underline{\omega}})}(B)&,\text{if}\;\dim
E^{1}_{\underline{\omega}}>1\\\
\delta_{({\underline{\omega}},\mathbf{P}E^{1}_{\underline{\omega}})}(B)&,\text{otherwise}\end{array}\end{cases}$

for every measurable set
$B\in\\{{\underline{\omega}}\\}\times\mathbf{P}\mathbb{R}^{d}$. It is clear
that the probability
$\mu=\int\mu_{\underline{\omega}}\,d\nu({\underline{\omega}})$ is
$P_{A}$-invariant, possibly non-ergodic. By some abuse of notation we still
denote by $\mu$ an ergodic component of the previous probability. By
construction, and Remark 10.1, one can check that
$\int\varphi_{A}({\underline{\omega}},v)\,d\mu({\underline{\omega}},v)\lambda_{+}(A,\nu)$
and $\pi_{*}\mu=\nu$, as claimed. ∎

This lemma ensures that every $\sigma$-invariant and ergodic probability
measure can be lifted to a $P_{A}$-invariant probability so that its Lyapunov
exponent can be computed as the time average of the potential $\varphi_{A}$
according to its lift. For general linear cocycles the upper semicontinuity of
the top Lyapunov exponent can be used (see e.g. the proof of [74, Proposition
2.12] to prove that

$h_{*}(\varphi_{A})\geqslant\sup\Big{\\{}h_{\nu}(\sigma)\colon\nu\in\mathcal{M}_{erg}(\sigma)\;\text{and}\,\lambda_{+}(A,\nu)>\inf_{\eta\in\mathcal{M}_{erg}(\sigma)}\lambda_{+}(A,\eta)\Big{\\}}.$
(10.2)

######  Corollary 10.3.

Let $\nu_{max}$ be the unique measure of maximal entropy for the shift
$\sigma:\Sigma_{\kappa}\to\Sigma_{\kappa}$. The following hold:

  1. (1)

If
$\lambda_{+}(A,\nu_{max})>\inf_{\eta\in\mathcal{M}_{erg}(\sigma)}\\{\lambda_{+}(A,\eta)\\}$
then $h_{*}(\varphi_{A})=\log\kappa$,

  2. (2)

If the support of $\nu_{max}$ is not contained in a compact subgroup of
$SL(d,\mathbb{R})$, the cocycle is strongly irreducible and some of the
matrices has norm one then $h_{*}(\varphi_{A})=\log\kappa$.

###### Proof.

Item (1) is a trivial consequence of (10.2). As for item (2) it is well known
that the probability $\nu_{max}$ is the Bernoulli probability measure with
equidistributed weights ie.
$\nu_{max}=(\frac{1}{\kappa},\frac{1}{\kappa},\dots,\frac{1}{\kappa})^{\mathbb{Z}}$.
Hence, Furstenberg criterion for positive Lyapunov exponents of random walks
on groups of matrices in Theorem 2.1 ensures that, under the assumptions of
item (2) one has that
$\lambda_{+}(A,\nu_{max})>0=\inf_{\eta\in\mathcal{M}_{erg}(\sigma)}\\{\lambda_{+}(A,\eta)\\}$.
Item (2) now follows from item (1). ∎

We observe that while the previous corollary provides sufficient conditions
for $h_{*}(\varphi_{A})$ to be maximal, it is still unclear whether
$h_{*}(\varphi_{A})$ coincides with $\log\kappa$ this holds in full
generality.

###  10.2. Lyapunov irregular behavior for linear cocycles

Here we prove Theorem A. Notice first that as the cocycle $A$ takes values in
the subgroup $\mathscr{G}$ of $\mathrm{SL}(d,\mathbb{R})$ then
$\|A^{n}({\underline{\omega}})^{-1}\|^{-1}\leqslant
1\leqslant\|A^{n}({\underline{\omega}})\|$ for every
${\underline{\omega}}\in\Sigma_{\kappa}$. In consequence,
$\lambda_{-}(A,\mu)\leqslant 0\leqslant\lambda_{+}(A,\mu)$ for every
$\sigma$-invariant and ergodic probability measure $\mu$. By assumption, the
group generated by the matrices $\\{A_{1},A_{2},\dots,A_{\kappa}\\}$ is a non-
compact subgroup of $SL(d,\mathbb{R})$. In particular there exists a matrix
$A_{j}$ so that $\|A_{j}\|>1$. Assume, without loss of generality that
$j=\kappa$.

The argument to prove that the set of Lyapunov non-typicalpoints is a Baire
residual subset of $\Sigma_{\kappa}$ is somewhat similar to the proof of
Theorem C, taking into account the fibered dynamics of the projectivized
cocycle $P_{A}$, defined in (1.4).. Let $\mu$ denote the Dirac measure at
$\underline{\kappa}=(\kappa,\kappa,\kappa,\dots)$. As
$\lambda_{-}(A,\mu)<\lambda_{+}(A,\mu)$ there exists a non-trivial hyperbolic
splitting
$\\{\underline{\kappa}\\}\times\mathbb{R}^{d}=E_{\underline{\kappa}}^{s}\oplus
E_{\underline{\kappa}}^{u}$ for the matrix $A(\kappa)$, hence it admits a
positively invariant unstable cone $C^{+}$ and a negatively invariant stable
cone $C^{-}$. Using adapted norms (see e.g. [85]) it is folklore in the theory
of uniform hyperbolicity that the previous cones can be chosen of arbitrarily
close diameter, in order to guarantee the following:

######  Lemma 10.4.

There exists $\theta\in(0,1)$ and $\zeta>0$ so that the cones

$C^{+}=\big{\\{}w=w_{+}\,+\,w_{-}\in E^{+}\oplus
E^{-}\colon\|w_{-}\|\leqslant\zeta\|w_{+}\|\big{\\}}$

and

$C^{-}=\big{\\{}w=w_{+}\,+\,w_{-}\in E^{+}\oplus
E^{-}\colon\|w_{+}\|\leqslant\zeta\|w_{-}\|\big{\\}}$

satisfy the following properties:

  1. (i)

$A(\underline{\kappa})(C^{+})\subsetneq C^{+}$ and
$A(\underline{\kappa})^{-1}(C^{-})\subsetneq C^{-}$,

  2. (ii)

$\|A(\underline{\kappa})^{n}\cdot w\|\geqslant\theta^{-n}\|w\|$ for every
$w\in C^{+}$ and $n\geqslant 1$,

  3. (iii)

$\|A(\underline{\kappa})^{n}\cdot w\|\leqslant\theta^{n}\|w\|$ whenever
$A(\underline{\kappa})^{j}\cdot w\in C^{-}$ for every $0\leqslant j\leqslant
n$.

In view of the previous lemma, one can construct points with Lyapunov
irregular behavior provided that one can control recurrence along the fiber
dynamics. By construction, the skew-product $P_{A}$ is locally constant and
for each $1\leqslant j\leqslant\kappa$, the map
$f_{j}:\mathbf{P}\mathbb{R}^{d}\to\mathbf{P}\mathbb{R}^{d}$ defined by
$f_{j}=P_{A}(\underline{j},\cdot)$ (where $\underline{j}=(j,j,j,\dots)$) is a
bi-Lipschitz homeomorphism with Lipschitz constant
$L:=\max_{{\underline{\omega}}\in\Sigma_{\kappa}}\frac{\|A({\underline{\omega}})\|\;\;}{\|A({\underline{\omega}})^{-1}\|^{-1}}$.
Additionally, by the uniform hyperbolicity of the matrix
$A(\underline{\kappa})$, there exist open neighborhoods $D_{A}$ and $D_{R}$ of
an attracting set $A$ and a repelling set $R$ for the projective map

$P_{A}(\underline{\kappa},\cdot):\mathbf{P}\mathbb{R}^{d}\to\mathbf{P}\mathbb{R}^{d}\quad\text{where}\quad
P_{A}(\underline{\kappa},v)=\frac{A(\underline{\kappa})\cdot
v}{\|A(\underline{\kappa})\cdot v\|}$

obtained by the projectivization of unstable and stable cones in Lemma 10.4,
respectively. Even though the attracting and repelling sets need not be
transitive, the projective map $f_{\kappa}$ admit invariant and ergodic
probability measures $\nu_{+}$ and $\nu_{-}$ supported on a topological
attractor and repeller, respectively (see Figures 9.1 and 9.2 below).

Figure 10.1. Unstable and stable eigenspaces for the hyperbolic linear map (on
the left) generating attracting $A$ and repelling $R$ on the double covering
${\mathbf{S}}^{d}$ of the projective space (on the right)

.

Figure 10.2. Unstable and stable cones for the original hyperbolic
automorphism (on the left) inducing neighborhoods of $A$ and $R$ with
attracting and repelling behavior for the projective map (on the right).

By Lemma 10.4, the continuous potential
$\varphi_{A}:\Sigma_{\kappa}\times\mathbf{P}\mathbb{R}^{d}\to\mathbb{R}$
defined in (10.1) satisfies
$\varphi_{A}(\underline{\kappa},w_{1})\leqslant\log\theta<0<-\log\theta\leqslant\varphi_{A}(\underline{\kappa},w_{2})$
for every $w_{1}\in D_{R}$ and $w_{2}\in D_{A}$. Thus,

$\int_{\Sigma_{\kappa}\times\mathbf{P}\mathbb{R}^{d}}\varphi_{A}({\underline{\omega}},v)\,d(\mu\times\nu_{-})<\int_{\Sigma_{\kappa}\times\mathbf{P}\mathbb{R}^{d}}\varphi_{A}({\underline{\omega}},v)\,d(\mu\times\nu_{+})$
(10.3)

The assumption that the cocycle is strongly projectively accessible is
equivalent to say that the semigroup action generated by the collection
$G_{1}=\\{id,f_{1},f_{2},\dots,f_{\kappa}\\}$ has frequent hitting times.
While Theorem C cannot be applied directly here (the potential $\varphi_{A}$
depends on both coordinates of the skew-product) the proof is identical, and
consists of combining both accessibility of the base and fibered dynamics to
produce points with erratic behavior. Let us skim the details.

Fix an arbitrary $v\in\mathbf{P}\mathbb{R}^{d}$. We claim that there exists a
Baire residual subset $\mathscr{R}_{v}\subset\Sigma_{\kappa}$ so that the
Cesàro averages
$\frac{1}{n}\sum_{j=0}^{n-1}\varphi_{A}(F_{A}^{j}({\underline{\omega}},v))$
diverge. In contrast to the previous sections, the potential $\varphi_{A}$
does not depend exclusively on the variable in $\mathbf{P}\mathbb{R}^{d}$.
Assumption (10.3) ensures that a source of different asymptotic behavior is
located at the fiber $\\{\underline{\kappa}\\}\times\mathbf{P}\mathbb{R}^{d}$.
Hence, although the argument is similar to the one in previous sections, we
have now to keep a control on the recurrence of the base dynamics as well. We
will detail the construction of a dense set of points in $\Sigma_{\kappa}$
with respect to which $v$ has Lyapunov irregular behavior, as the remaining
modifications in the construction of the Baire residual set will carry