# Lyapunov non-typical behavior for linear cocycles
through the lens of semigroup actions

Giovane Ferreira and Paulo Varandas  Giovane Ferreira, Departamento de
Matemática, Universidade Federal do Maranhão
Av. dos Portugueses, 1966, Vila Bacanga, 65065-545 São Luís, MA
<EMAIL_ADDRESS>Paulo Varandas, Departamento de Matemática,
Universidade Federal da Bahia
Av. Ademar de Barros s/n, 40170-110 Salvador, Brazil & CMUP - Universidade do
Porto, Portugal<EMAIL_ADDRESS>
(Date: June 2021)

###### Abstract.

The celebrated Oseledets theorem [58], building over seminal works of
Furstenberg and Kesten on random products of matrices and random variables
taking values on non-compact semisimple Lie groups [31, 32], ensures that the
Lyapunov exponents of $\mathrm{SL}(d,\mathbb{R})$-cocycles $(d\geqslant 2)$
over the shift are well defined for all points in a total probability set, ie,
a full measure subset for all invariant probabilities. Given a locally
constant $\mathrm{SL}(d,\mathbb{R})$-valued cocycle we are interested both in
the set of points on the shift space for which some Lyapunov exponent is not
well defined, and in the set of directions on the projective space
$\mathbf{P}\mathbb{R}^{d}$ along which there exists no well defined
exponential growth rate of vectors for a certain product of matrices. We prove
that if the semigroup generated by finitely many matrices in
$\mathrm{SL}(d,\mathbb{R})$ is not compact and is strongly projectively
accessible then there exists a dense set of directions in
$\mathbf{P}\mathbb{R}^{d}$ along which the Lyapunov exponent of a typical
product of such matrices is not well defined. As a consequence, we deduce that
the presence of Lyapunov non-typical behavior is prevalent among
$SL(3,\mathbb{R})$-valued hyperbolic cocycles. These results arise as a
consequence of the more general description of the set of non-typical points
for Cesàro averages of continuous observables and infinite paths in finitely
generated semigroup actions by homeomorphisms on a compact metric space $X$
under a mild hitting times condition.

###### Key words and phrases:

Random products of matrices, projective linear maps, strong projective
accessibility, Lyapunov irregular behavior, irregular sets, semigroup actions,
frequent hitting times, entropy

###### Contents

  1. 1 Introduction
    1. 1.1 Linear cocycles
    2. 1.2 Semigroup actions
  2. 2 Statement of the main results
    1. 2.1 Lyapunov non-typical behavior
    2. 2.2 Non-typical points for semigroup actions
    3. 2.3 Comparison with previous methods and approaches
    4. 2.4 Organization of the text
  3. 3 Uniform hyperbolicity and specification type properties
    1. 3.1 Hyperbolicity, partial hyperbolicity and dominated splittings
    2. 3.2 Specification and gluing orbit properties
    3. 3.3 Coboundaries and Livšic-like characterization of coboundaries
  4. 4 Entropy and upper capacities
    1. 4.1 Topological entropy of continuous maps and non-autonomous dynamical systems
    2. 4.2 Topological entropies of finitely generated semigroup actions
      1. 4.2.1 Geometric entropy
      2. 4.2.2 Free semigroup actions
    3. 4.3 Fibered topological entropies
    4. 4.4 Some examples
  5. 5 The frequent hitting times property
    1. 5.1 Sufficient condition for the frequent hitting times property
    2. 5.2 Projective linear maps
    3. 5.3 Relation with other gluing orbit properties and hyperbolicity
  6. 6 The set of irregular points for semigroup actions is Baire generic
    1. 6.1 The set of irregular points is non-empty
    2. 6.2 The set of irregular points is Baire generic
    3. 6.3 Proof of Theorem C
  7. 7 Topological complexity on the skew-product
    1. 7.1 Entropy estimates obtained from ergodic measures
    2. 7.2 A Moran set of irregular points in $\\{1,2,...,\kappa\\}^{\mathbb{Z}}\times X$ with large entropy
  8. 8 Topological entropy of irregular points for the semigroup action
    1. 8.1 Ghys-Langevin-Walczak entropy of the set of irregular points
    2. 8.2 Bufetov entropy of the set of irregular points
  9. 9 Topological entropy of fibered irregular sets
    1. 9.1 Fibered complexity of the set of irregular points
      1. 9.1.1 Fibered orbit growth estimates
      2. 9.1.2 Construction of fibered Moran sets of irregular points
      3. 9.1.3 The relative entropy of $\mathfrak{F}_{\underline{\omega}}$
    2. 9.2 Moran sets of irregular points with large fibered entropy
  10. 10 Linear cocycles
    1. 10.1 Projective cocycles and complexities
    2. 10.2 Lyapunov irregular behavior for linear cocycles
    3. 10.3 A dichotomy leading to Lyapunov irregular behavior
    4. 10.4 The absence of Lyapunov irregular behavior is rare
    5. 10.5 Projective accessibility and strong irreducibility

##  1\. Introduction

###  1.1. Linear cocycles

Important results of Furstenberg, Kesten, Oseledets, Kingman, Ledrappier,
Guivarc’h, Raugi, Margulis and other mathematicians, initiated in the sixties
built the study of Lyapunov characteristic exponents into a very active
research field in its own right, with a vast array of interactions with other
areas of Mathematics and Physics. We refer the reader to the monographs by
Duarte, Klein and Viana [28, 80] and references therein for a some historical
remarks and description of the state of the art.

Lyapunov exponents have been studied first in the context of random products
of matrices [31, 32]. Here consider linear cocycles where the iteration of the
matrices is determined by typical orbits of an invariant probability measure
for a deterministic map. Let us recall these notions. Let $(M,\mu)$ be a
probability space, $f\colon M\to M$ a measure-preserving automorphism and
$A\colon M\to\mathrm{SL}(d,\mathbb{R})$ a measurable matrix-valued map. The
pair $(A,f)$ is called a _linear cocycle_ and, for each $n\geqslant 1$ set

$A^{(n)}(x):=A(f^{n-1}(x))\cdots A(f(x))A(x)\quad\text{and}\quad
A^{(-n)}(x)=(A^{(n)}(f^{-n}(x)))^{-1}.$ (1.1)

The dynamics of the linear cocycle can be described by the skew-product

$\begin{array}[]{cccc}F_{A}\colon&M\;\times\;\mathbb{R}^{d}&\longrightarrow&M\;\times\;\mathbb{R}^{d}\\\
&(x,v)&\mapsto&(f(x),A(x)\cdot v),\end{array}$ (1.2)

where the joint base and fiber dynamics is given by
$F^{n}_{A}(x,v)=(f^{n}(x),A^{(n)}(x)v)$ for all $(x,v)\in
M\times\mathbb{R}^{d}$ and every $n\geqslant 1$. Under the integrability
assumption $\log\|A^{\pm 1}\|\in L^{1}(\mu)$, the Oseledets’ theorem [58]
ensures that for $\mu$-a.e. $x\in M$ there exist $1\leqslant k(x)\leqslant d$,
a splitting $\\{x\\}\times\mathbb{R}^{d}=E^{1}_{x}\oplus
E^{2}_{x}\oplus\dots\oplus E_{x}^{k(x)}\;$ (called the _Oseledets splitting_)
varying measurably with $x$ and so that $A(x)E^{i}_{x}=E^{i}_{f(x)}$ for every
$1\leqslant i\leqslant k(x)$, and real numbers
$\lambda_{1}(A,f,x)>\lambda_{2}(A,f,x)>\dots>\lambda_{k(x)}(A,f,x)$ (called
the _Lyapunov exponents_ associated to $f$, $A$ and $x$) such that

$\lambda_{i}(A,f,x)=\lim_{n\to\pm\infty}\frac{1}{n}\log\|A^{(n)}(x)v_{i}\|,\quad\forall
v_{i}\in E^{i}_{x}\setminus\\{\vec{0}\\}.$ (1.3)

If, in addition, $\mu$ is ergodic then both the number and the value of the
Lyapunov exponents are almost everywhere constant, and the Lyapunov exponents
are denoted simply by $\lambda_{i}(A,f,\mu)$.

Moreover, the largest and smallest Lyapunov exponents of the cocycle with
respect to an ergodic probability $\mu$ can be expressed, using Kingman’s
subadditive ergodic theorem, by

$\lambda_{+}(A,\mu):=\lim_{n\to\infty}\frac{1}{n}\log\big{\|}A^{(n)}(x)\big{\|}\quad\text{and}\quad\lambda_{-}(A,\mu):=\lim_{n\to\infty}\frac{1}{n}\log\big{\|}A^{(n)}(x)^{-1}\big{\|}^{-1},$

respectively (see also the previous work by Furstenberg and Kesten [31] on the
products of random matrices).

The projectivization $P_{A}$ of the skew-product $F_{A}$ is the map

$\begin{array}[]{cccc}P_{A}\colon&M\;\times\;\mathbf{P}\mathbb{R}^{d}&\longrightarrow&M\;\times\;\mathbf{P}\mathbb{R}^{d}\\\
&(x,[v])&\mapsto&\Big{(}f(x),\big{[}\frac{A(x)\cdot v}{\|A(x)\cdot
v\|}\big{]}\Big{)}\end{array}$ (1.4)

acting on the compact metric space $M\;\times\;\mathbf{P}\mathbb{R}^{d}$. We
denote by $\pi:M\times\mathbf{P}\mathbb{R}^{d}\to M$ the projection on the
first coordinate. Clearly the fibers
$\pi^{-1}(\\{x\\})=\mathbf{P}\mathbb{R}^{d}$ are compact for every $x\in M$.
The random products of projectivized matrices can be traced to the work of
Furstenberg [32] on random products of matrices. In other words, Furstenberg
considered the case when $f$ coincides with the full shift
$\sigma:\Sigma_{\kappa}\to\Sigma_{\kappa}$ on $\kappa$ symbols
($\kappa\geqslant 2$) and $\mu=\nu^{\mathbb{Z}}$ is a Bernoulli probability
measure on $\Sigma_{\kappa}$.

In view of Oseledets theorem, the set of points for which the Lyapunov
exponents are well defined, often referred as the set of _Lyapunov typical
points_ , form a total probability set in the ambient space. Moreover, if
$x\in M$ is a Lyapunov typical point for the linear cocycle $A$ it is clear
from the invariance of the Oseledets splitting together with the existence of
the limits in (1.3), that for every $v\in\mathbb{R}^{d}\setminus\\{\vec{0}\\}$
the limit

$\lim_{n\to+\infty}\frac{1}{n}\log\|A^{(n)}(x)v\|$

is well defined: it is equal to $\lambda_{k(x)}(A,f,x)$ whenever $v\in
E^{k(x)}_{x}$ and, otherwise, it coincides with $\lambda_{i-1}(A,f,x)$ where
$2\leqslant i\leqslant k(x)$ is the largest integer so that
$v\in\mathbb{R}^{d}\setminus(E^{k(x)}_{x}\oplus
E^{k(x)-1}_{x}\oplus\dots\oplus E^{i}_{x})$. In the late eighties Mañé
suggested that the set of points for which the Lyapunov exponents are not well
defined should to be prevalent in the Baire topological sense. This idea
should be tested in the following two distinct contexts, eventually with
completely different techniques:

  * $\circ$

$d=1$: the cocycle $A:M\to\mathbb{R}$ takes values on the abelian group
$(\mathbb{R},+)$;

  * $\circ$

$d\geqslant 2$: the cocycle $A:M\to\mathrm{SL}(d,\mathbb{R})$ takes values on
the non-abelian group $(\mathrm{SL}(d,\mathbb{R}),\circ)$.

In the first context, of real valued observables, the Lyapunov non-typical
points (also known as irregular points) of a smooth map
$f:\mathbf{S}^{1}\to\mathbf{S}^{1}$ are exactly those points which are non-
typical for the Birkhoff averages of $A(x)=\log|f^{\prime}(x)|$ ($x\in M$)
provided that $f$ is smooth. Mañé’s perception could be formalized in rather
strong ways. First, in the context of $C^{1+\alpha}$-uniformly expanding
circle maps, Barreira and Schmelling [4] proved that although irregular points
have zero measure for every invariant measure, these may have full topological
entropy and even full Hausdorff dimension. Here, the key mechanism to
construct points with non-typical behavior is the notion of specification
introduced by Bowen in [16]. This turned out to be a useful tool to develop a
multifractal analysis of Lyapunov exponents and to prove that irregular sets
are either empty or Baire generic, have full Hausdorff dimension, topological
entropy or mean dimension (cf. [3, 5, 12, 48, 82] and references therein).
Actually, while part of these results can be extended for topological
dynamical systems having the specification property shadowing or the gluing
orbit properties (see e.g. [27, 48] and references therein), these properties
fail for large classes of partially hyperbolic diffeomorphisms [13, 66].
Nevertheless, the set points with non-convergent Birkhoff averages form a
Baire residual subset of the ambient space under the mild assumption that
there are dense subsets of points with distinct largest possible Birkhoff
averages (see [23] for the precise statement). This is a very mild assumption
which, in particular, seems not enough to determine whether the set of non-
typical points carry large entropy.

In the second context, of linear cocycles taking values on non-abelian groups,
the situation is subtler and fewe results are known both on the level sets of
Lyapunov exponents and the set of Lyapunov non-typical points. Inspired by
Herman [37], Furman [30] proved that, even though irrational rotations are
minimal and uniquely ergodic, some cocycles over irrational rotations have a
residual subset of Lyapunov irregular points. More recently, Tian [73, 74]
proved that Hölder cocycles over homeomorphisms with an exponential
specification property either have the same top Lyapunov exponent or the set
of Lyapunov irregular points is Baire generic and carries full topological
entropy. In the aforementioned results, the use of the exponential
specification property - which holds for uniformly expanding dynamics - is
used to obtain bounded distortion properties as in the work of Kalinin [41],
and the author asks whether the specification property could be enough to
prove the theorem. In the complementary direction, the level sets of Lyapunov
exponents of certain $\mathrm{SL}(2,\mathbb{R})$-valued cocycles have been
described in terms of the Legendre-Fenchel transform of a certain pressure
function (cf [26]).

Here we consider cocycles satisfying a strong projective accessibility
condition (see Definition 2.2), a condition which can be expressed more
generally in terms of the semigroup generated by the projective linear maps
(cf. Definition 2.7 and Remark 5.5). We will not require any fiber bunching
condition on the cocycle (see e.g. [79] for the definition), hence the
projectivized cocycle $P_{A}$ need not be partially hyperbolic. Moreover, even
if this was the case, a typical fiber-bunched cocycle $A$ generates a
partially hyperbolic skew-product $P_{A}$ which fails to satisfy the
specification, the gluing orbit property and their nonuniform versions (see
[14, 66] and Example 5.10). Inspired by [32], in Theorem A we prove that if
the group generated by a finite number of invertible matrices is non-compact
and is strongly projectively accessible then there exist a Baire generic set
of points in the shift space for which Lyapunov exponents are not well defined
on a dense set of directions in $\mathbf{P}\mathbb{R}^{d}$. We use this fact
to conclude that, in the special context of projective actions for
$SL(3,\mathbb{R})$-valued hyperbolic cocycles, the presence of non-typical
behavior is prevalent (see Theorem B for the precise statement).

###  1.2. Semigroup actions

Group, semigroup and pseudo-group actions have been of increasing interest in
dynamical systems. Apart from its genuine interest, in which some aspects of
the group action are determined by the algebraic structure of the underlying
group (e.g. existence of invariant measures), such objects appear as well in
the description of classical dynamical systems. For instance, these are used
to provide a notion of entropy for invariant foliations and to model the
central dynamics for certain partially hyperbolic diffeomorphisms (see e.g.
[25, 33, 39] and references therein).

In the last three decades, there has been an intense study about pointwise
ergodic theorems for actions of amenable, Lie or free groups, and we refer the
reader to [17, 19, 35, 49, 54, 72], just to mention a few contributions.
Again, the situation is much subtler than the case of $\mathbb{Z}$-actions. In
order to illustrate that, we note that, in the case of free groups, the
spherical averages of $L^{p}$-observables $(p>1)$ converge almost everywhere
and in $L^{1}$ [19, 55] but this is no longer true for general
$L^{1}$-observable [70]. In the case of measure preserving Markov semigroups
an extra Cesàro averaging is enough to guarantee the convergence of spherical
averages (see [20]). It seem to exist almost no results concerning the set of
points presenting irregular behavior for such pointwise ergodic theorems, an
exception made to the case of amenable group actions satisfying a suitable
specification property (cf. [62]). However, as the specification property
turns is rare even among $\mathbb{Z}$ or $\mathbb{R}$-actions (cf. [66, 67])
it is natural to look for other mechanisms which can be used as a tool to
construct points with non-typical behavior.

Here we focus on the case of semigroup actions acting by bi-Lipschitz
homeomorphisms on a compact metric space $X$. Let
$G_{1}=\\{id,f_{1},f_{2},\dots,f_{\kappa}\\}$ denote the generators of the
semigroup $G$. For that reason, inspired by the context of random products of
matrices, for each continuous map $\psi:X\to\mathbb{R}$ we consider
alternatively the problem of describing both the set of infinite paths on the
semigroup $G$ (determined by an infinite word
${\underline{\omega}}\in\\{1,2,\dots,\kappa\\}^{\mathbb{N}}$ and the
homeomorphisms $g_{{\underline{\omega}}_{n}}\circ\dots\circ
g_{{\underline{\omega}}_{2}}\circ g_{{\underline{\omega}}_{1}}\in G$) and
points $x\in X$ for which the Cesàro averages

$\frac{1}{n}\sum_{j=0}^{n-1}\psi(\,g_{{\underline{\omega}}_{j}}\circ\dots\circ
g_{{\underline{\omega}}_{2}}\circ g_{{\underline{\omega}}_{1}}(x)\,)$ (1.5)

diverge as $n\to\infty$. Under a mild assumption on the semigroup action,
concerning a quantitative estimate on the hitting times to balls of fixed size
- which we refer to as frequent hitting times (cf. Definition 2.7) - we have
the following goals: given a continuous observable $\psi:X\to\mathbb{R}$

  * $\circ$

describe the set of points $x\in X$ for which there exist
${\underline{\omega}}\in\\{1,2,\dots,\kappa\\}^{\mathbb{N}}$ which determines
a concatenation of elements in the semigroup along which (1.5) diverges as
$n\to\infty$;

  * $\circ$

construct a ’relevant’ set of infinite paths on the semigroup $G$
(parameterized by infinite words
${\underline{\omega}}\in\\{1,2,\dots,\kappa\\}^{\mathbb{N}}$) so that the set
$I_{\underline{\omega}}(\psi)\subset X$ of points so that the Cesàro averages
(1.5) diverge is a Baire generic subset of $X$ and carries large entropy.

We refer the reader to Theorems C and E for positive results on this
direction, using the notions of entropy for the semigroup introduced by Ghys,
Langevin and Walczak [33] and by Bufetov [18], and the notion of entropy for
non-autonomous dynamical systems introduced by Kolyada and Snoha [45]. As the
notion of entropy of a free semigroup is related to the the notion of entropy
for a certain skew-product [18], part of the argument in Theorem E is a
consequence of a result for non-typical points on skew-products (Theorem D).
It is worth mentioning that, in opposition to the action by projective linear
maps on projective spaces, the generators in $G_{1}$ may have positive
topological entropy, hence one can specifically address the set of paths which
are not only non-typical but which have large entropy (see Theorem F).

##  2\. Statement of the main results

This section is devoted to the statement of the main results. The main results
concerning matrix valued cocycles are stated in Subsection 2.1, while the ones
concerning real valued cocycles associated to semigroup actions are stated in
Subsection 2.2.

In Subsection 2.3 we discuss the methods in the proofs and previous works. In
Subsection 2.4 we describe the organization of the arguments.

###  2.1. Lyapunov non-typical behavior

Let $\sigma~{}:~{}\Sigma_{\kappa}\to\Sigma_{\kappa}$ be a the full shift on
$\kappa\geqslant 1$ symbols, where
$\Sigma_{\kappa}\subset\\{1,2,\dots,\kappa\\}^{\mathbb{Z}}$ and consider the
distance

$d_{\Sigma_{\kappa}}({\underline{\omega}},{\underline{\omega}}^{\prime})=e^{-N({\underline{\omega}},{\underline{\omega}}^{\prime})},\quad\text{where}\;N({\underline{\omega}},{\underline{\omega}}^{\prime})=\inf\\{k\geqslant
1\colon\omega_{k}\neq\omega_{k}^{\prime}\\}.$

We denote the 1-cylinder sets by
$[j]:=\\{(\omega_{i})_{i\in\mathbb{Z}}\in\Sigma_{\kappa}\colon\omega_{0}=j\\}$,
for $1\leqslant j\leqslant\kappa$ and, for any integers $m,n\in\mathbb{Z}$
with $m\leqslant n$ we will use the following notation

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

for the cylinder set in the shift determined by the entries in the interval
$[m,n]$.

Denote by $C^{0}(\Sigma_{\kappa},\mathscr{G})$ the Banach space of continuous
linear cocycles $A:\Sigma_{\kappa}\to\mathscr{G}$ taking values on a closed
subgroup $\mathscr{G}\subset\mathrm{SL}(d,\mathbb{R})$, endowed with the
$\|A\|_{\infty}$ norm. Let
$C_{\text{loc}}^{0}(\Sigma_{\kappa},\mathscr{G})\subset
C^{0}(\Sigma_{\kappa},\mathscr{G})$ denote the closed set of locally constant
cocycles on cylinders of size 1, which is isomorphic to
${\mathscr{G}}^{\kappa}$. As mentioned before, the joint base and fiber
dynamics is given by $F^{n}_{A}(x,v)=(\sigma^{n}(x),A^{(n)}(x)v)$, where the
matrix $A^{(n)}(x)\in\mathscr{G}$ is defined by (1.1). We say that a linear
cocycle $A$ is locally constant if is is constant on cylinder sets of
$\Sigma_{\kappa}$. Up to a slight change of the alphabet these can be assumed
to be step skew-products, meaning that the cocycle $A$ is constant on the

We start by stating the following consequence of the pioneering work of
Furstenberg on the products of random variables taking values on semisimple
Lie groups, which serves as a motivation for Theorem A below.

######  Theorem 2.1.

([32, Theorem 8.6]) Let $\mathscr{G}\subset\mathrm{SL}(d,\mathbb{R})$ be a
closed and non-compact subgroup, $A\in
C_{\text{loc}}^{0}(\Sigma_{\kappa},\mathscr{G})$ and
$\nu=\nu_{0}^{\mathbb{Z}}$ be a Bernoulli probability measure on
$\Sigma_{\kappa}$. Assume that:

  1. (1)

the semigroup generated by the matrices in the support of $\nu_{0}$ is not
contained in a compact subgroup of $\mathscr{G}$,

  2. (2)

the cocycle is strongly irreducible on $\operatorname{supp}\nu$ (ie, there is
no finite union of proper subspaces of $\mathbb{R}^{d}$ invariant by
$A({\underline{\omega}})$ for every
${\underline{\omega}}\in\operatorname{supp}\nu$).

Then $\lambda_{+}(A,\nu)>0$.

The previous theorem provides geometric conditions under which the top
Lyapunov exponent, almost everywhere defined, is positive. Other refinements,
still in the i.i.d. setting, have been obtained by Guivarc’h and Raugi [36],
Gol’dsheid and Margulis [34], among others.

Our first goal here is to understand the complementary problem of determining
which directions, in the projective space $\mathbf{P}\mathbb{R}^{d}$, one can
observe Lyapunov non-typical behavior (meaning that there exists some point in
the shift such that the Lyapunov exponent is not well defined along that
direction, under forward iteration). While the set of Lyapunov non-typical
directions may be meager in the projective space (see e.g. Example 10.10), we
will introduce a geometric property on a finite family of matrices under which
the generated cocycle has plenty Lyapunov non-typical directions.

######  Definition 2.2.

_Let $\mathscr{G}\subset\mathrm{SL}(d,\mathbb{R})$ be a closed and non-compact
subgroup. A cocycle $A\in C_{\text{loc}}^{0}(\Sigma_{\kappa},\mathscr{G})$ is
_strongly projectively accessible_ if for any $\varepsilon>0$ there exists an
integer $K(\varepsilon)\geqslant 1$ so that for any ball
$B_{1}\subset\mathbf{P}\mathbb{R}^{d}$ of radius $\varepsilon$ and a ball
$B_{2}\subset\mathbf{P}\mathbb{R}^{d}$ of radius
$0<\delta\leqslant\varepsilon$ there exists $0\leqslant p\leqslant
K(\varepsilon)$ and ${\underline{\omega}}\in\Sigma_{\kappa}$ so that the image
of $B_{1}$ by the projectivization of $A^{(p)}({\underline{\omega}})$ contains
a ball $B_{2}^{\prime}\subset B_{2}$ of radius
$r\geqslant\min\\{\varepsilon/4,\delta/2\\}$. _

The previous condition is natural among cocycles which have elements in the
orthogonal Lie group $SO(d,\mathbb{R})$ (also known as rotation group) but can
also be used to depict the projective action inside two-dimensional hyperbolic
subbundles (see Section 10 for more details). Our first main result says that,
under the previous condition, the set of Lyapunov non-typical directions is
Baire generic and has large entropy.

######  Theorem A.

Assume that $\mathscr{G}\subset\mathrm{SL}(d,\mathbb{R})$ is a closed subgroup
and that $A\in C^{0}_{loc}(\Sigma_{\kappa},\mathscr{G})$. is a locally
constant cocycle, and write $A_{i}=A\mid_{[i]}$ for each $1\leqslant
i\leqslant\kappa$. Assume that

  1. (1)

the subgroup generated by the matrices $\\{A_{1},A_{2},\dots,A_{\kappa}\\}$ is
non-compact,

  2. (2)

$A$ is strongly projectively accessible.

Then for each $v\in\mathbf{P}\mathbb{R}^{d}$ there exists a Baire residual
subset $\mathscr{R}_{v}\subset\Sigma_{\kappa}$ of entropy at least
$h_{*}(\varphi_{A})$ (defined by (10.2)) so that, for every
${\underline{\omega}}\in\mathscr{R}_{v}$,

$\liminf_{n\to\infty}\frac{1}{n}\log\|A^{n}({\underline{\omega}})v\|<\limsup_{n\to\infty}\frac{1}{n}\log\|A^{n}({\underline{\omega}})v\|$
($\star$)

In particular, there exists a Baire residual subset
$\mathscr{R}\subset\Sigma_{\kappa}$ and a dense subset
$\mathscr{D}\subset\mathbf{P}\mathbb{R}^{d}$ so that $(\star)$ holds for every
${\underline{\omega}}\in\mathscr{R}$ and every $v\in\mathscr{D}$.

Note that the assumptions of Theorem A are related to the ones of Theorem 2.1,
the difference being that the strong irreducibility condition is replaced by
the condition that the cocycle is strongly projectively accessible which,
nonetheless, have the same flavour of indecomposability of the the action on
the projective space $\mathbf{P}\mathbb{R}^{d}$. However, their conclusions
are not, as there exist cocycles so that all invariant measures have a
positive Lyapunov exponent while the conclusion of Theorem A fails (cf.
Example 10.7). The proof of Theorem A is given in Section 10.

######  Remark 2.3.

Projective linear maps on the projective space have a simple Morse-Smale type
of structure. In particular, these maps (and their non-autonomous
compositions) have zero topological entropy (see Subsection 10.1 for more
details).

######  Remark 2.4.

First results on irregular points for the top Lyapunov exponent appeared first
in [30, 37, 73, 74]. The results in [73, 74] can be summarized as follows: if
$A:\Sigma_{\kappa}\to\mathscr{G}$ is a Hölder continuous cocycle then either
$\lambda_{+}(A,\nu_{1})=\lambda_{+}(A,\nu_{2})$ for every $\sigma$-invariant
probability measures $\nu_{1},\nu_{2}$, or there exists a Baire residual and
large topological entropy subset of the shift $\Sigma_{\kappa}$ formed by
points for which the top Lyapunov exponent is not well defined.

The projective accessibility can be also useful to describe the behavior of a
linear cocycle restricted to some invariant subbundle. We will be particularly
interested in cocycles displaying dominated splittings, whose notion we now
recall. Given a cocycle $A\in C^{0}(\Sigma_{\kappa},\mathscr{G})$, we say that
a splitting $\Sigma_{\kappa}\times\mathbb{R}^{d}=E^{1}\oplus E^{2}$ is
$A$-invariant if
$A({\underline{\omega}})E^{i}_{\underline{\omega}}=E^{i}_{\sigma({\underline{\omega}})}$
for every ${\underline{\omega}}\in\Sigma_{\kappa}$. An $A$-invariant splitting
$\Sigma_{\kappa}\times\mathbb{R}^{d}=E^{1}\oplus E^{2}$ is _dominated_ if
there exists $C>0$ and $\lambda\in(0,1)$ so that

$\|A^{(n)}({\underline{\omega}})\mid_{E_{\underline{\omega}}^{2}}\|.\|(A^{(n)}({\underline{\omega}})\mid_{E^{1}_{\underline{\omega}}})^{-1}\|\leqslant
C\lambda^{n}\text{ for every $n\geqslant 1$ and
${\underline{\omega}}\in\Sigma_{\kappa}$.}$

It is well known that the existence of a dominated splitting is $C^{0}$-open
on the space of continuous linear cocycles over a compact space and that the
subbundles vary continuously with the point ${\underline{\omega}}$. In
particular, as the shift $\sigma$ is transitive, the dimensions of the
subbundles is constant. A simple example of a dominated splitting occurs
whenever
$\|A^{(n)}({\underline{\omega}})\mid_{E^{2}_{\underline{\omega}}}\|\leqslant
C\lambda^{n}$ and
$\|(A^{(n)}({\underline{\omega}})\mid_{E^{1}_{\underline{\omega}}})^{-1}\|\leqslant
C\lambda^{n}$ for every ${\underline{\omega}}\in\Sigma_{\kappa}$ and
$n\geqslant 1$, in which case we say that the cocycle is _uniformly
hyperbolic_ , and we refer to $\Sigma_{\kappa}\times\mathbb{R}^{d}=E^{1}\oplus
E^{2}$ as its hyperbolic splitting. We refer the reader to [81] for other
characterizations of uniform hyperbolicity in the context of
$SL(2,\mathbb{R})$-cocycles. We say that $A$ has a _one-dimensional finest
dominated splitting_ if it admits a continuous $A$-invariant splitting
$\Sigma_{\kappa}\times\mathbb{R}^{d}=E^{1}_{\underline{\omega}}\oplus
E^{2}_{\underline{\omega}}\oplus\dots\oplus E_{{\underline{\omega}}}^{d}$ in
one-dimensional subspaces at every ${\underline{\omega}}\in\Sigma_{\kappa}$
such that the splitting $E^{i}\oplus E^{i+1}$ is dominated for each
$1\leqslant i\leqslant d-1$. The next result, and the discussion following it,
ensures that the irregular behavior prevails.

######  Theorem B.

Assume that $A\in C^{0}_{loc}(\Sigma_{\kappa},SL(3,\mathbb{R}))$ is uniformly
hyperbolic and its hyperbolic spliting
$\Sigma_{\kappa}\times\mathbb{R}^{3}=E^{s}\oplus E^{u}$ satisfies $\dim
E^{s}=2$. There exists a $C^{0}$-open neighborhood $\mathscr{U}\subset
C^{0}_{\text{loc}}(\Sigma_{\kappa},SL(3,\mathbb{R}))$ and a $C^{0}$-Baire
residual and full Haar measure subset $\mathscr{R}\subset\mathscr{U}$ so that
for every $B\in\mathscr{R}$ either:

  1. (1)

$B$ admits a finest one-dimensional dominated splitting
$\mathbb{R}^{3}=E_{B,{\underline{\omega}}}^{u}\oplus
E_{B,{\underline{\omega}}}^{s,1}\oplus E_{B,{\underline{\omega}}}^{s,2}$, or

  2. (2)

there exists a Baire residual subset $\mathscr{R}\subset\Sigma_{\kappa}$ and
for each ${\underline{\omega}}\in\mathscr{R}$ there exists a dense subset
$\mathscr{D}_{\underline{\omega}}\subset\mathbf{P}E^{s}_{\underline{\omega}}$
so that

$\liminf_{n\to\infty}\frac{1}{n}\log\|B^{n}({\underline{\omega}})v\|<\limsup_{n\to\infty}\frac{1}{n}\log\|B^{n}({\underline{\omega}})v\|\quad\text{for
every $v\in\mathscr{D}_{\underline{\omega}}$.}$

######  Remark 2.5.

The property of having a one-dimensional finest dominated splitting is a
$C^{0}$-open condition in
$C^{0}_{\text{loc}}(\Sigma_{\kappa},SL(3,\mathbb{R}))$. Hence, in the context
of Theorem B, if the $C^{0}$-open set $\mathscr{U}_{1}\subset\mathscr{U}$ of
cocycles having a one-dimensional finest dominated splitting is non-empty and
$B\in\mathscr{U}_{1}$ then it follows from the description of irregular sets
for Birkhoff averages for the shift and the real valued observables
$\varphi_{i}$ that either these functions are cohomologous to a constant or
the space of points in $\Sigma_{\kappa}$ displaying irregular behavior is
Baire generic and carries full topological entropy [5, 24]. Actually we prove
that, in the context of Theorem B, a typical cocycle whose Lyapunov irregular
set is empty is cohomologous to a constant matrix cocycle (see Proposition
10.6 for the precise statement). As the class of cocycles in $\mathscr{U}_{1}$
that are cohomologous to a constant cocycle form a closed set with empty
interior, one may say that the irregular behavior prevails.

######  Remark 2.6.

The dichotomy in Theorem B is related to the proof of [6, Theorem 2.5 and
Corollary 2.6] which, in this context of locally constant linear cocycles
acting on $\mathbb{R}^{3}$, ensures that if the Lyapunov exponents at periodic
points are all distinct by small $C^{0}$-perturbations of the cocycle then
there exists a finest one-dimensional dominated splitting, which coincides
with the Oseledets splitting in a total probability set.

###  2.2. Non-typical points for semigroup actions

Given a semigroup action $\mathbb{S}$ generated by a collection
$G_{1}=\\{id,f_{1},f_{2},\dots,f_{\kappa}\\}$ of continuous maps on a compact
metric space $X$ and a continuous observable $\psi:X\to\mathbb{R}$, we denote
by

$I_{\mathbb{S}}(\psi)=\Big{\\{}x\in
X\colon\exists\,{\underline{\omega}}\in\Sigma_{\kappa}\;\text{s.t.}\;\lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\psi(f_{\underline{\omega}}^{j}(x))\;\text{does
not exist}\Big{\\}}$ (2.1)

the set on _Birkhoff non-typical points_ for the semigroup action $\mathbb{S}$
(with respect to $\psi$) where, for each
${\underline{\omega}}\in\Sigma_{\kappa}:=\\{1,2,\dots,\kappa\\}^{\mathbb{N}}$
and $n\geqslant 1$, we write
$f^{n}_{\underline{\omega}}:=f_{{\underline{\omega}}_{n-1}}\circ\dots\circ
f_{{\underline{\omega}}_{1}}\circ f_{{\underline{\omega}}_{0}}$ for the
composition of maps in $G_{1}$. It will be a standing assumption that the
generators semigroup action are bi-Lipschitz homeomorphisms on $X$ and we set

$L:=\max\big{\\{}1\,,\;\max_{1\leqslant i\leqslant\kappa}\text{Lip}(f_{i}^{\pm
1})\big{\\}}.$

The dynamics of the semigroup action generated by $G_{1}$, endowed with a
random walk, is intimately related to the dynamics of the skew-product

$\begin{array}[]{cccc}F\colon&\\{1,2,\dots,\kappa\\}^{\mathbb{Z}}\times
X&\longrightarrow&\\{1,2,\dots,\kappa\\}^{\mathbb{Z}}\times X\\\
&(\underline{\omega},x)&\mapsto&(\sigma(\underline{\omega}),f_{\omega_{0}}(x)),\end{array}$
(2.2)

through the observation that
$F^{n}({\underline{\omega}},x)=(\sigma^{n}({\underline{\omega}}),g_{\underline{\omega}}^{n}(x))$
for every $(\underline{\omega},x)\in\\{1,2,\dots,\kappa\\}^{\mathbb{Z}}\times
X$ and every $n\geqslant 1$. Moreover, if
$\varphi=\\{1,2,\dots,\kappa\\}^{\mathbb{N}}\times X\to\mathbb{R}$ is
continuous and one defines the set of Birkhoff irregular points for the skew-
product $F$ as

$I_{F}(\varphi):=\Big{\\{}({\underline{\omega}},x)\in\\{1,2,\dots,\kappa\\}^{\mathbb{Z}}\times
X:\lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\varphi(F^{j}({\underline{\omega}},x))\;\text{does
not exist}\Big{\\}},$ (2.3)

then one has that

$I_{F}(\varphi_{\psi})\subset\\{1,2,\dots,\kappa\\}^{\mathbb{N}}\times
I_{\mathbb{S}}(\psi)\subset\\{1,2,\dots,\kappa\\}^{\mathbb{N}}\times X,$ (2.4)

where $\varphi_{\psi}({\underline{\omega}},x):=\psi(x)$ for every
$({\underline{\omega}},x)\in\\{1,2,\dots,\kappa\\}^{\mathbb{N}}\times X$.

We aim to describe the topological complexity (Baire genericity and
topological entropy) of these sets of Birkhoff irregular points observing them
as follows:

  * $\circ$

$I_{\mathbb{S}}(\psi)\subset X$ measured through the geometric entropy
introduced in Ghys, Langevin and Walczak for a finitely generated semigroup
action;

  * $\circ$

$I_{\mathbb{S}}(\psi)\subset X$ measured through the entropy introduced by
Bufetov for a free semigroup action;

  * $\circ$

$I_{\mathcal{F}_{\underline{\omega}}}(\psi)\subset X$ as a set of points which
are Birkhoff irregular for the non-autonomous dynamical system
$\mathcal{F}_{\underline{\omega}}=(f_{\omega_{i_{j}}})_{j\geqslant 1}$
determined by a realization ${\underline{\omega}}\in\Sigma_{\kappa}$;

  * $\circ$

$I_{F}(\varphi_{\psi})$ as an $F$-invariant subset of
$\\{1,2,\dots,\kappa\\}^{\mathbb{N}}\times X$.

The entropy in the second item above corresponds to the annealed topological
entropy of the skew-product $F$ and, in some special cases, the third one
corresponds to the notion of Pinsker relative entropy (cf. Subsection 4.3).
The Lipschitz assumption on the generators implies that both notions of
entropy on the semigroup $\mathbb{S}$ and the non-autonomous dynamical system
are finite. These sets have been successively described for both single
dynamics and a semigroup actions satisfying a specification property (see e.g.
[4, 5, 24, 83] and references therein). However, most smooth dynamical systems
beyond hyperbolicity fail to satisfy the specification property and this is
also the case for the skew-products which arise in the study of linear
cocycles [14]. For that reason we were led us to define the following mild
topological concept which measures recurrence between balls of controlled
radius. We denote by $|J|$ the radius of a ball $J\subset X$.

######  Definition 2.7.

_We say that the semigroup action $\mathbb{S}$ generated by the collection
$G_{1}=\\{id,f_{1},f_{2},\dots,f_{\kappa}\\}$ of continuous maps has _frequent
hitting times_ if for any $\varepsilon>0$ there exists
$K(\varepsilon)\geqslant 1$ so that for any ball $B_{1}$ of radius
$|B_{1}|=\varepsilon$ and any ball $B_{2}$ of radius
$0<|B_{2}|\leqslant\frac{\varepsilon}{2}$ there exists $0\leqslant p\leqslant
K(\varepsilon)$ and an element
${\underline{\omega}}\in\Sigma_{\kappa}:=\\{1,2,\dots,\kappa\\}^{\mathbb{N}}$
so that $f^{p}_{\underline{\omega}}(B_{1})$ contains a ball
$B_{2}^{\prime}\subset B_{2}$ of radius $|B_{2}|/2$. _

As defined above, the frequent hitting times property for a single map $f$
differs substantially from the specification or gluing orbit property. It
combines not only the control of the hitting time between balls of comparable
radius and the size of their intersection. In particular, the latter implies
that the set of hitting times $N(A,B)=\\{n\in\mathbb{N}\colon f^{n}(A)\cap
B\neq\emptyset\\}$ between two balls of the same radius is syndetic (cf. [2]
for the definition) and that the frequency of visits to balls of a definite
radius is bounded away from zero by some constant which depends only on the
radius of the balls (cf. [13]). We refer the reader to Section 5 for an ample
discussion on this property and examples.

######  Theorem C.

Let $X$ be a compact metric space and consider the semigroup action
$\mathbb{S}$ generated by the collection of bi-Lipschitz homeomorphisms
$G_{1}=\\{id,f_{1},f_{2},\dots,f_{\kappa}\\}$. Assume that
$\psi:X\to\mathbb{R}$ is a continuous observable and that:

  * (a)

the semigroup action $\mathbb{S}$ has frequent hitting times,

  * (b)

there exists $1\leqslant i\leqslant\kappa$ so that $\psi$ is not a coboundary
with respect to $f_{i}$.

The set

$\mathscr{R}_{\Sigma}=\Big{\\{}{\underline{\omega}}\in\Sigma_{\kappa}\colon
I_{\mathcal{F}_{\underline{\omega}}}(\psi)\;\text{is a Baire residual subset
of}\,X\Big{\\}}$

is a Baire residual subset of $\Sigma_{\kappa}$. In particular,
$I_{\mathbb{S}}(\psi)$ is a Baire residual subset of $X$.

The previous result can be compared with the historic behavior obtained by
Nakano [53] in the context of random compositions of expanding maps in the
circle. Indeed, exploring the uniformly expanding behavior of the maps, it is
proven that for almost all $\omega\in\Sigma_{\kappa}$ (with respect to some
reference probability $\mathbb{P}$) there exists a residual subset
$\mathcal{R}_{\omega}\subset\mathbf{S}^{1}$ such that for each
$x\in\mathcal{R}_{\omega}$ the random orbit $(f_{\omega}^{n}(x))_{n\geqslant
1}$ has historic behavior. Here we notice that no hyperbolic behavior is
assumed on the fibers. Theorem C will be proven in Section 6.

The remaining results concern the entropy of the set of irregular points using
two important concepts of entropy in the context of semigroup actions, namely
the entropy of free semigroup actions endowed with a random walk introduced by
Bufetov [18] (see [21, 22, 40, 83] and references therein for further studies)
and the geometric entropy introduced by Ghys, Langevin and Walczak [33] (see
also [8] and references therein), which we recall in Subsection 4.2.

The first notion is related, but does not coincide, with the entropy of the
skew-product $F$ defined in (2.2). Hence, given a continuous observable
$\psi:X\to\mathbb{R}$ it is natural to consider the quantity

$\displaystyle h_{*}(\varphi_{\psi}):=\sup\Big{\\{}c\geqslant 0\colon$
$\displaystyle\text{there exist}\,\mu_{1},\mu_{2}\in\mathcal{M}_{erg}(F)$
$\displaystyle\,\text{so that}\,h_{\mu_{i}}(F)\geqslant
c\;\text{and}\,\int\varphi_{\psi}\,d\mu_{1}<\int\varphi_{\psi}\,d\mu_{2}\Big{\\}}$
(2.5)

(where $\varphi_{\psi}$ is defined as before), which is the largest value of
entropy almost attainable by ergodic probabilities which evaluate the
observable $\varphi_{\psi}$ differently. The following result estimates the
topological entropy of the set of Birkhoff irregular points for the skew-
product $F$ with respect to the potential $\varphi_{\psi}$.

######  Theorem D.

Let $X$ be a compact metric space and consider the semigroup action generated
by the collection $G_{1}=\\{id,f_{1},f_{2},\dots,f_{\kappa}\\}$ of bi-
Lipschitz homeomorphisms, and denote by
$F:\\{1,2,\dots,\kappa\\}^{\mathbb{Z}}\times
X\to\\{1,2,\dots,\kappa\\}^{\mathbb{Z}}\times X$ the skew-product
$F({\underline{\omega}},x)=(\sigma({\underline{\omega}}),f_{\omega_{0}}(x))$
where $\sigma$ is the shift. Assume that
$\varphi:\\{1,2,\dots,\kappa\\}^{\mathbb{Z}}\times X\to\mathbb{R}$ is a
continuous observable. If

  * (a)

the semigroup $\mathbb{S}$ generated by $G_{1}$ has frequent hitting times,
and

  * (b)

$\varphi$ is not a coboundary with respect to $F$

then $h_{\operatorname{top}}(F,I_{F}(\varphi))\geqslant h_{*}(\varphi)$.

Some comments are in order. First, we observe that if the skew-product $F$
satisfies the specification property and the entropy map $\mu\mapsto
h_{\mu}(F)$ is upper-semicontinuous then
$h_{*}(\varphi)=h_{\operatorname{top}}(F)$ for every continuous map $\varphi$
which is not cohomologous to a constant (see [29, Theorem B]). However, the
skew-product in (2.2) may even fail to satisfy a non-uniform specification
property (cf. Example 5.10) making it unclear if ergodic probabilities are
dense or entropy dense in the simplex of $F$-invariant probabilities. Second,
the entropy $h_{\operatorname{top}}(F,I_{F}(\varphi))$ appearing in the
theorem is the entropy of the $F$-invariant subset $I_{F}(\varphi)$ of
$\\{1,2,\dots,\kappa\\}^{\mathbb{Z}}\times X$, as defined in Subsection 4.1.
The proof of the previous theorem will occupy Section 7.

The notion of geometric entropy $h^{GLW}(\mathbb{S})$ for the semigroup
$\mathbb{S}$ depends on the generating set $G_{1}$ but does not depend on any
intrinsic structure of the semigroup. Indeed, its definition encloses the
exponential behavior of all possible concatenations of maps with the same
number of generators (cf. Subsection 4.2 for the definition) and,
consequently, it is larger or equal than the entropy
$h_{\operatorname{top}}(\mathcal{F}_{\underline{\omega}})$ of any sequential
(or non-autonomous) dynamical systems $\mathcal{F}_{\underline{\omega}}$
obtained with generating set $G_{1}$ (cf. Subsection 4.1 and equation (4.3)).
For that reason it is natural to consider the following quantity:

$\displaystyle
H^{KS}(\psi):=\sup\big{\\{}h_{\operatorname{top}}(\mathcal{F}_{\underline{\omega}})\colon{\underline{\omega}}\in\Sigma_{\kappa}\big{\\}}.$
(2.6)

It could occur that the elements ${\underline{\omega}}\in\Sigma_{\kappa}$
leading to a large entropy
$h_{\operatorname{top}}(\mathcal{F}_{\underline{\omega}})$ above do not belong
to the ergodic basin of attraction of some $\sigma$-invariant probability, in
which case no information is known about the time averages of $\psi$ along the
non-autonomous dynamical system $\mathcal{F}_{\underline{\omega}}$. For that
reason we introduce the following alternative quantities:

$\displaystyle H^{\text{Pinsker}}(\psi):=\sup\Big{\\{}c\geqslant 0\colon$
$\displaystyle\text{for every}\,\varepsilon>0\,\text{there
exist}\,\mu_{1},\mu_{2}\in\mathcal{M}_{erg}(F)\,\text{so that}$ $\displaystyle
h_{\mu_{i}}(F\mid\sigma)>c-\varepsilon\;\text{and}\,\int\varphi_{\psi}\,d\mu_{1}<\int\varphi_{\psi}\,d\mu_{2}\Big{\\}}.$
(2.7)

and

$\displaystyle H^{\sigma}(\psi):=\sup\Big{\\{}c\geqslant 0\colon$
$\displaystyle\text{for every}\,\varepsilon>0\,\text{there
are}\,\mu_{1},\mu_{2}\in\mathcal{M}_{erg}(F)\,\text{so that}$ $\displaystyle
h_{\pi_{*}\mu_{i}}(\sigma)>c-\varepsilon\;\text{and}\,\int\varphi_{\psi}\,d\mu_{1}<\int\varphi_{\psi}\,d\mu_{2}\Big{\\}},$
(2.8)

where we keep denoting by $\pi:\\{1,2,\dots,\kappa\\}^{\mathbb{Z}}\times
X\to\\{1,2,\dots,\kappa\\}^{\mathbb{Z}}$ the projection on the first
coordinate. The latter quantities (2.7) and (2.8) measure the largest Pinsker
entropy and entropy of the marginal on the shift, respectively, among the
space of probability measures invariant by the skew-product which distinguish
the observable $\varphi_{\psi}$. Once more, it can occur that the measures
with larger entropy on the shift can have small fibered entropy. For that
reason we consider take $H^{\text{Pinsker}}_{\sigma}(\psi):=\lim_{\zeta\to
0}H^{\text{Pinsker}}_{\sigma}(\psi,\zeta)$, where

$\displaystyle
H^{\text{Pinsker}}_{\sigma}(\psi,\zeta):=\sup\Big{\\{}c\geqslant 0\colon$
$\displaystyle\text{there
are}\,\mu_{1},\mu_{2}\in\mathcal{M}_{erg}(F)\,\text{so
that}\;h_{\mu_{i}}(F\mid\sigma)\geqslant H^{\text{Pinsker}}(\psi)-\zeta,$
$\displaystyle h_{\pi_{*}\mu_{i}}(\sigma)\geqslant
c\;\text{and}\,\int\varphi_{\psi}\,d\mu_{1}<\int\varphi_{\psi}\,d\mu_{2}\Big{\\}}.$
(2.9)

We work these quantities and compare them in concrete examples of skew
products associated to semigroup actions by continuous interval maps (see
Examples 4.6 and 4.7). We denote by $h^{B}(\mathbb{S},I_{\mathbb{S}}(\psi))$
the geometric topological entropy of the set $I_{\mathbb{S}}(\psi)\subset X$
under the action of the semigroup $\mathbb{S}$, and by
$h^{GLW}(\mathbb{S},I_{\mathbb{S}}(\psi))$ when we observe it as a free
semigroup action (cf. Subsections 4.2 and 4.3 for the definitions of these
entropies and relations between them). Our next result shows that the
geometric entropy of irregular sets for the semigroup action is bounded below
by these entropies.

######  Theorem E.

Let $X$ be a compact metric space and consider the semigroup action generated
by a collection of bi-Lipschitz homeomorphisms
$G_{1}=\\{id,f_{1},f_{2},\dots,f_{\kappa}\\}$. Assume that
$\psi:X\to\mathbb{R}$ is a continuous observable and that

  * (a)

the semigroup generated by $G_{1}$ has frequent hitting times, and

  * (b)

$\varphi_{\psi}$ is not a coboundary with respect to $F$.

Then:

  1. (1)

$h^{GLW}(\mathbb{S},I_{\mathbb{S}}(\psi))\geqslant H^{\text{Pinsker}}(\psi)$;

  2. (2)

$h^{B}(\mathbb{S},I_{\mathbb{S}}(\psi))\geqslant h_{*}(\varphi_{\psi})$.

This theorem will be proven in Section 8, where item (2) above will be a
consequence of Theorem D and a relation between the entropy of a free
semigroup action with the entropy of the skew-product $F$. Finally, it is
worth noticing that one can replace assumption (a) above by the weaker
assumption that the semigroup $\mathbb{S}$ is _strongly transitive_ : for
every open set $U\subset X$ there exists $N=N(U)\geqslant 1$ so that
$\bigcup_{g\in G_{N}}g(U)=X$. We refer the reader to the proof of Theorem F
where this is made precise in the special case that a single generator in
$G_{1}$ is strongly transitive.

######  Remark 2.8.

The previous semigroup actions $\mathbb{S}$ and skew-product $F$ satisfy a
very mild condition, and in general fail to satisfy any kind of specification.
In the context of free semigroup actions satisfying the specification
property, it is known that either $I_{\mathbb{S}}(\psi)=\emptyset$ or
$h^{B}(\mathbb{S},I_{\mathbb{S}}(\psi))=h^{B}(\mathbb{S})$ (cf. [83, Theorem
1.2]).

The information carried by Theorem E is of global nature, enclosing
information of the whole semigroup action $\mathbb{S}$. In particular it does
not address any kind of information about the sequential dynamics that we now
describe. For each
${\underline{\omega}}\in\\{1,2,\dots,\kappa\\}^{\mathbb{Z}}$ consider the set

$I_{\underline{\omega}}(\psi):=X_{\underline{\omega}}\cap
I_{F}(\varphi_{\psi}),$ (2.10)

which consists of the set of points in $X_{\underline{\omega}}$ which have
irregular behavior along the orbits of
$\mathcal{F}_{\underline{\omega}}=(f_{\omega}^{j})_{j\geqslant 1}$, and set

$h^{path}_{I}(\mathbb{S},\psi):=\sup_{{\underline{\omega}}\in\\{1,2,\dots,\kappa\\}^{\mathbb{Z}}}\,h_{I_{\underline{\omega}}(\psi)}(\mathcal{F}_{\underline{\omega}}),$
(2.11)

where $h_{I_{\underline{\omega}}(\psi)}(\mathcal{F}_{\underline{\omega}})$ is
the relative entropy defined by the end of Subsection 4.3. In other words,
$h_{I}^{fiber}(\mathbb{S},\psi)$ represents the largest possible complexity of
irregular sets obtained along an infinite path on the semigroup $G$. The
assumptions in Theorem E allow one to produce Moran sets with large entropy,
but whose intermediate transition times functions (cf. Subsection 8.1) depend
on the points involved in the construction of such Cantor set and,
consequently, these are not enough to estimate the entropy of irregular sets
along infinite paths in the semigroup. This is possible if one assumes instead
a strong transitivity condition on one of the generators of the semigroup.
More precisely:

######  Theorem F.

Let $X$ be a compact metric space and consider the semigroup action generated
by a collection of bi-Lipschitz homeomorphisms
$G_{1}=\\{id,f_{1},f_{2},\dots,f_{\kappa}\\}$. Assume that
$\psi:X\to\mathbb{R}$ is a continuous observable and that the following hold:

  * (a)

there exists $1\leqslant i\leqslant\kappa$ so that $f_{i}$ is strongly
transitive (i.e. for every open set $U\subset X$ there exists $N=N(U)\geqslant
1$ so that $\bigcup_{n=1}^{N}f_{i}^{n}(U)=X$)

  * (b)

$\varphi_{\psi}$ is not a coboundary with respect to $F$.

Then

  1. (1)

$h^{path}_{I}(\mathbb{S},\psi)\geqslant H^{\text{Pinsker}}(\psi)$;

  2. (2)

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
$H^{\text{Pinsker}}_{\sigma}(\psi)$.

Theorem F is proven in Section 9. Notice that assumption (a) above is
equivalent to the minimality of the homeomorphism $f_{i}$ in the generating
set $G_{1}$. We opted to state the condition as a strong transitivity
condition as this equivalent form may turn out to be useful for an extension
of this result to semigroups generated by non-invertible maps (see Subsection
2.3 for more details). Since $g_{i}$ is not assumed to be an isometry, the
semigroup action is not required to have the frequent hitting times property.
Item (2) in the previous theorem ensures that the set of points in
$\Sigma_{\kappa}$ which generate sequential dynamical systems
$\mathcal{F}_{\omega}$ with large irregular sets has large entropy, bounded
below given by the largest possible entropy (on the shift) of the
$F$-invariant probabilities whose fibered entropies can be used to compute
$H^{\text{Pinsker}}(\psi)$. Let us derive the following trivial consequence of
Theorem F whenever all generators coincide.

######  Corollary 1.

Let $X$ be a compact metric space and $f:X\ \to X$ be a minimal bi-Lipschitz
homeomorphism. If $f$ is not uniquely ergodic and $\psi:X\to\mathbb{R}$ is a
continuous observable which is not a coboundary then
$h_{I_{f}(\psi)}(f)\geqslant H^{\text{Pinsker}}(\psi)$.

###  2.3. Comparison with previous methods and approaches

Most well known methods to describe irregular sets make use of topological
methods, as some weak forms of specification or shadowing. In rough terms, in
the additive context these are used in the following two step argument: for
each continuous map $f:X\to X$ on the compact metric space $X$ satisfying the
specification property and every continuous observable
$\varphi:X\to\mathbb{R}$ one has

  1. (I)

if $\mu_{1},\mu_{2}$ are ergodic probabilities so that
$\int\varphi\,d\mu_{1}<\int\varphi\,d\mu_{2}$ there exist points which shadow
arbitrarily long pieces of orbits whose time averages are close to
$\int\varphi\,d\mu_{1}$ and $\int\varphi\,d\mu_{2}$, alternatively, and whose
time between each shadowing process is a constant independing on the size of
the shadowing orbits;

  2. (II)

for each convex combination $\mu$ of ergodic probabilities
$\mu_{1},\mu_{2},\dots,\mu_{s}$ there exist points which shadow finite pieces
of orbits determined by typical points for $\mu_{i}$, alternatively, and whose
time averages converge to $\int\varphi\,d\mu$.

The first step guarantees that the irregular set is non-empty and can be used
in a crucial way to show that this set is actually Baire generic (see e.g. [5]
and references therein). The second step, combined with the fact that the
times between shadowing processes are constant, allows to construct a Moran
set formed by points with oscillatory behavior and carrying full entropy (cf.
[4, 24]). Specification can actually be used to study the saturated sets as
obtained by Pfister and Sullivan [61] and the same holds when time lags
between shadowing processes are bounded by some constant or have sub-linear
growth in comparison to the size of orbits (cf. [48, 75]). The skew-products

$\begin{array}[]{cccc}F\colon&\\{1,2,\dots,\kappa\\}^{\mathbb{Z}}\times
X&\longrightarrow&\\{1,2,\dots,\kappa\\}^{\mathbb{Z}}\times X\\\
&(\underline{\omega},x)&\mapsto&(\sigma(\underline{\omega}),f_{\omega_{0}}(x)),\end{array}$

associated to the semigroup action $\mathbb{S}$ with generating set
$G_{1}=\\{id,f_{1},f_{2},\dots,f_{\kappa}\\}$ does not satisfy any of these
properties as a consequence of [14] and Example 5.10. In [73, 74], this is
overcome by requiring an extra Hölder continuous regularity on the cocycle and
making use of the exponential specification property for the base dynamical
system.

Here, inspired by the projective dynamics of linear cocycles, we introduce an
alternative sufficient condition (satisfied whenever there exists a minimal
sub-action by isometries, see Subsection 5.1) to study irregular points. Such
condition is a quantitative estimate for hitting times between balls of
controlled size and their intersection. The frequent hitting times property is
unrelated to specification and its weaker versions (see Subsection 5.3 for a
discussion), and following diagram presents some relations between the latter
and related concepts:

$\begin{array}[]{cccc}\text{minimal action by
isometries}&\Rightarrow&\text{frequent hitting times}&(\text{cf.
Subsection~{}\ref{sec:vs-gluing}})\\\ &&\not\Uparrow\;\not\Downarrow\\\ \text{
$C^{0}$-generic homeomorphisms}&\Rightarrow&\text{gluing orbit property
}&\text{\cite[cite]{[\@@bibref{}{BeTV}{}{}]}}\\\ &&\Uparrow\;\not\Downarrow\\\
\text{ uniform hyperbolicity }&\Rightarrow&\text{specification property
}&\text{\cite[cite]{[\@@bibref{}{Bo71}{}{}]}}\\\ &&\Downarrow\;\not\Uparrow\\\
\text{ non-uniform hyperbolicity }&\Rightarrow&\text{non-uniform specification
property }&\text{\cite[cite]{[\@@bibref{}{OliTian,Paulo}{}{}]}}\\\ &&\\\
\text{partial hyperbolicity }&\Rightarrow&\text{'typically' {\bf no}
specification }&\text{\cite[cite]{[\@@bibref{}{SVY}{}{}]}}\\\ &&\text{and {\bf
no} gluing orbit property }&\text{\cite[cite]{[\@@bibref{}{BTV2}{}{}]}}\\\
\end{array}$

The assumption that the homeomorphisms $f_{i}$ are bi-Lipschitz is used to
estimate the inner diameter of the images of dynamic balls (cf. Lemma 6.1).
This information, combined with the frequent hitting times assumption,
provides quantitative estimates on the amount of time needed to perform the
shadowing of the finite pieces of orbits. Notably, albeit transition times are
not constant and can be much larger than the pieces of orbits that have been
shadowed (these actually grow exponentially fast), these satisfy a Markov
property, meaning that depend exclusively on the size of the last piece of
orbit. This provides a new argument for step (I) above and, under much weaker
assumptions, still to prove that the irregular sets are non-empty and Baire
generic (we refer the reader to Figure 5.3 for a representation of the control
over the oscillations under the different previous circumstances).

The situation differs substantially in step (II), which is the most relevant
to the construction of Moran sets with arbitrarily large entropy. First, while
in the case of maps with specification one can make successive approximations
of points which shadow very large pieces of orbits, this strategy falls
shortly in our context due to the much larger transitions times associated. We
introduce a new argument, which consists of controlling the forward images of
dynamic balls for all iterates, guaranteeing that a proportion of points
remains possible to iterate at each level of the construction (see
$2^{nd}$Step at Subsection 7.2). The construction is made by choosing points
which, alternatively, shadow pieces of orbits which are typical for ergodic
probabilities $\mu_{1},\mu_{2}$ having different space averages with respect
to $\varphi$. A subtle but key difference is that since one cannot replicate
small orbits by its repetition, the size of $n_{k}$ the finite pieces of
orbits to be shadowed at level $k$ are chosen depending on the maximal
transition time $K$ involving the pieces of orbits of size $n_{k-1}$ used at
the level $k-1$. Selecting points which have the same transition times at each
level of approximation guarantees that one can construct distinguishable
orbits. These choices are performed in Lemma 7.3. Second, the failure of step
(II), imposed by the very long transition times, impart that one can only use
typical points associated to ergodic probabilities as in step (I) for the
construction of a Moran set of irregular points. This explains the lower
bounds on the entropy of irregular sets provided in Theorems E and D, and the
fact that every point is irregular for some path in the semigroup (for
instance, recall the first statement in Theorem A).

Finally, notice that although the main results have been stated for semigroups
generated by homeomorphisms, the argument just makes use of the semigroup
structure. The bi-Lipschitz continuity of the generators guarantees not only
that the semigroup action (and the skew-product) have finite topological
entropy but is also is crucial to control the inner diameter of dynamic balls
and their images (cf. Lemma 6.1). Nevertheless, the Cesàro averages and the
proofs of the main results take into consideration just the forward iterates
of the mappings. A simple modification of the argument is likely to show that
the main results hold as well for local homeomorphisms which are Lipschitz
continuous and whose inverse branches are Lipschitz continuous as well, but we
do not pursue this here.

###  2.4. Organization of the text

The remainder of the paper is organized as follows.

  * $\diamond$

In Section 3 we recall basic definitions concerning uniform and partial
hyperbolicity, dominated splittings and coboundaries.

  * $\diamond$

In Section 4 we include a broad discussion about the different notions of
topological and measure theoretical entropies for a single map, non-autonomous
dynamical systems and semigroup actions that will be used here. We first
recall the notion of fibered entropy for extensions and the entropy of non-
autonomous dynamical systems, and prove a fibered entropy distribution
principle. Then we describe two notions of entropy of a finitely generated
semigroup action due to Ghys, Langevin and Walczak (sometimes known as
geometric entropy) and another due to Bufetov, which exploits a coding by
means of a free semigroup action and makes use of a skew-product map over a
full shift on finitely many symbols.

  * $\diamond$

Section 5 is devoted to the study of the frequent hitting times condition. We
provide sufficient conditions for it to hold, and give some examples on how
this is useful in the context of projective maps arising from linear cocycles
which have some projective accessibility. Furthermore, we observe that this
property is much weaker and unrelated to other gluing orbit properties and
need not occur for uniformly hyperbolic maps.

  * $\diamond$

In Section 6 we start a series of results on the set of irregular points with
respect to a finitely generated semigroup action by continuous maps on a
compact metric space $X$. Here we prove Theorem C, stating that under the mild
frequent hitting times condition, Birkhoff irregular sets for semigroup
actions are either empty or Baire generic on the phase space $X$.

  * $\diamond$

In Section 7, motivated by the study of free semigroup actions, we consider
step skew-products with the frequent hitting times property, and prove that
its set of irregular points in $\Sigma_{\kappa}\times X$ carries at least as
much entropy as the one of any pair of ergodic probabilities with different
space averages (recall Theorem D). This lower bound reflects the fact that
under this mild assumptions one should not expect the entropy denseness of the
space of ergodic probability measures.

  * $\diamond$

In Section 8 we obtain lower bounds for the topological entropy of irregular
points for the semigroup action, considering both the Ghys-Langevin-Walczak
and Bufetov entropies. While Bufetov entropy measures the average number of
points with irregular behavior for the non-autonomous dynamical systems
associated to the different concatenations of the generators of the group and
can be studied in relation to the skew-product map, the lower bound for the
Ghys-Langevin-Walczak entropy demands a careful choice of a set of points
which can be separated by some element in the group.

  * $\diamond$

In Section 9 we consider the topological entropy of fibered irregular sets,
that is, the largest possible entropy of irregular sets associated to all
possible non-autonomous dynamical systems. There are two main differences in
comparison to the results of Sections 8 and 9. First, in order to construct
Moran subsets formed by points with irregular behavior for certain (actually a
large entropy set of) non-autonomous dynamical systems we require some
generator in $G_{1}$ to be strongly transitive. Second, the strong
transitivity is weaker than the frequent hitting times condition, hence this
furnishes an alternative mechanism for homeomorphisms that do not arise as
projective linear maps.

  * $\diamond$

In Section 10 we apply the theory developed to general finitely generated
semigroup actions to the context for projective linear cocycles. In Subsection
10.1 we present some preliminary results on the of projective linear maps and
invariant measures for the projectivized skew-product. In Subsection 10.2 the
non-compactness of the semigroup generated by the $SL(d,\mathbb{R})$ matrices
allows to relate hyperbolic behavior of the concatenations of matrices along
specific vectors with its recurrence to certain regions of the projective
space, and use this fact to prove Theorem A. Actually in the special case of
$SL(3,\mathbb{R})$-cocycles the absence of irregular behavior leads to a
rigidity that the cocycle is cohomologous to a constant matrix cocycle (cf.
Theorem B and Subsection 10.3). We finish this section with few examples.

##  3\. Uniform hyperbolicity and specification type properties

In this brief section, included for making the paper self-contained, we recall
some basic concepts concerning uniform, partial and dominated splittings for
linear and derivative cocycles, the concept of specification and some of its
weakenings, and recall some characterizations of coboundaries due to Livšic.

###  3.1. Hyperbolicity, partial hyperbolicity and dominated splittings

Given a continuous linear cocycle
$A:{\Sigma_{\kappa}}\to\mathrm{SL}(d,\mathbb{R})$ we say that an $A$-invariant
splitting $\Sigma_{\kappa}\times\mathbb{R}^{d}=E\oplus F$ is _dominated_ if
there exists an integer $k\in\mathbb{N}$ such that

$\frac{\|A^{k}({\underline{\omega}})v\|}{\|A^{k}({\underline{\omega}})w\|}<\frac{1}{2},$

for every ${\underline{\omega}}\in{\Sigma_{\kappa}}$ and every pair of unit
vectors $v\in E_{\underline{\omega}}$ and $w\in F_{\underline{\omega}}$. More
generally, an $A$-invariant splitting
$\Sigma_{\kappa}\times\mathbb{R}^{d}=E_{1}\oplus\cdots\oplus E_{k}$ is
_dominated_ if $(E_{1}\oplus\cdots\oplus
E_{l})\oplus(E_{l+1}\oplus\cdots\oplus E_{k})$ is dominated for any $1\leq
l\leq k-1$. The $A$-invariant sub-bundle $E$ is _uniformly contracting_ (resp.
_expanding_) if there are $C>0$ and $0<\lambda<1$ such that for every $n>0$
one has $\|A^{n}({\underline{\omega}})v\|\leqslant C\lambda^{n}\|v\|$ (resp.
$\|(A^{n}({\underline{\omega}}))^{-1}(v)\|\leq C\lambda^{n}\|v\|$) for all
${\underline{\omega}}\in\Sigma_{\kappa}$ and $v\in E_{\underline{\omega}}$.
Finally, we say that an $A$-invariant splitting $TM=E^{s}\oplus E^{c}\oplus
E^{u}$ is: (i) _partially hyperbolic_ (resp. _strongly partially hyperbolic_)
if $E^{s}$ and $E^{u}$ are uniformly contracting and uniformly expanding
respectively and at least one of them is (resp. both of them are) not trivial;
and (ii) _hyperbolic_ if it is strongly partially hyperbolic and
$E^{c}=\Sigma_{\kappa}\times\\{0\\}$. These notions have a natural counterpart
for $C^{1}$-diffeomorphisms acting on a compact Riemannian manifold $M$.
Indeed, such a $C^{1}$-diffeomorphism $f:M\to M$ is _uniformly hyperbolic_ (or
an Anosov diffeomorphism) (resp. _partially hyperbolic_) if the derivative
cocycle $Df:TM\to TM$ on the tangent bundle $TM$ is uniformly hyperbolic
(resp. partially hyperbolic).

###  3.2. Specification and gluing orbit properties

First we recall the concept of specification, introduced by Bowen [16] in the
context of hyperbolic maps, which in rough terms describes the ability to
shadow an arbitrary number of finite pieces of orbits of arbitrary length
having a prefixed number of iterates in between. Let us be more precise.

Given a continuous map $f\colon X\to X$ on a compact metric space $X$, we say
that $f$ satisfies the _gluing orbit property_ if for any $\varepsilon>0$
there exists a positive integer $N(\varepsilon)\geqslant 1$ such that for any
points $x_{1},\dots,x_{k}\in X$ and any integers $n_{1},\dots,n_{k}\geqslant
0$ and $N_{1},N_{2},\dots,N_{k-1}\geqslant N(\varepsilon)$ there exists a
point $z\in X$ so that $d(f^{n}(z),f^{n}(x_{1}))<\varepsilon$ for every
$n=0\ldots n_{1}$ and $d(f^{n+\sum_{1\leq
i<j}(N_{i}+n_{i})}(z),f^{n}(x_{j}))<\varepsilon$ for every $2\leqslant
j\leqslant k$ and $0\leqslant n\leqslant n_{j}$. It is clear that the latter
implies that $f$ is topological mixing.

A somewhat weaker notion has been used ever since, especially in the symbolic
context, which some authors sometimes referred to as the specification
property. In the context of topological dynamical systems this was formalized
in [15] as follows. We say that a continuous map $f\colon X\to X$ on a compact
metric space $X$, satisfies the _gluing orbit property_ if: given
$\varepsilon>0$ there exists a positive integer $N(\varepsilon)\geqslant 1$
such that for any points $x_{1},\dots,x_{k}\in X$ and times
$n_{1},\dots,n_{k}\geqslant 0$ there exist gluing times $0\leqslant
p_{1},\dots,p_{k-1}\leqslant N(\varepsilon)$ and a point $z\in X$ so that
$d(f^{n}(z),f^{n}(x_{1}))<\varepsilon$ for every $n=0\ldots n_{1}$ and
$d(f^{n+\sum_{1\leq i<j}(p_{i}+n_{i})}(z),f^{n}(x_{j}))<\varepsilon$ for every
$2\leqslant j\leqslant k$ and $0\leqslant n\leqslant n_{j}$.

Even though the previous two notions can seem similar at first sight, the
gluing orbit property is substantially weaker than the notion of
specification, and it holds for non-mixing hyperbolic basic sets, minimal
rotations and some partially hyperbolic maps with neutral direction. Hence,
while the specification property implies on topological mixing and positive
topological entropy, the same does not hold for all continuous maps with the
gluing orbit property. A characterization of zero entropy maps with the gluing
orbit property appeared in [69]. We refer the reader to [13] and references
therein for a more embracing discussion on these topological invariants.

Finally, we shall refer to a measure theoretical non-uniform specification
property, that proved to hold for invariant measures of local diffeomorphisms
with no zero Lyapunov exponents (cf. [56, 78]). Indeed, if $f$ is a
$C^{1}$-local diffeomorphism and $\mu$ is an $f$-invariant probability
measure, we say that $(f,\mu)$ satisfies the _non-uniform specification
property_ if there exists $\delta>0$ and for $\mu$-almost every $x$ and
$0<\varepsilon<\delta$ there exists an integer $p(x,n,\varepsilon)\geqslant 1$
satisfying

$\lim_{\varepsilon\to 0}\limsup_{n\to\infty}\frac{1}{n}p(x,n,\varepsilon)=0$
(3.1)

and such that the following property holds: given $x_{1},\dots,x_{k}$ in a
full $\mu$-measure set any integers $n_{1},\dots,n_{k}\geqslant 1$ and
$p_{i}\geq p(x_{i},n_{i},\varepsilon)$ then there exists $z\in
B(x_{1},n_{1},\varepsilon)$ such that $f^{\sum_{j=1}^{i-1}(n_{j}+p_{j})}(z)\in
B(x_{i},n_{i},\varepsilon)$ for every $2\leqslant i\leqslant k.$ In contrast
to their topological counterparts, in this context the time lag between any
two consecutive pieces of orbits may depend on the sizes of the orbits, but it
is required to have sub-linear grow in respect to these (cf. (3.1)).

###  3.3. Coboundaries and Livšic-like characterization of coboundaries

Given a continuous map $f:X\to X$ on a compact metric space $X$, we say that a
continuous observable $\varphi:X\to\mathbb{R}$ is _not a coboundary_ (with
respect to $f$) if

$\inf_{\mu\in\mathcal{M}_{1}(f)}\int\varphi\,d\mu<\sup_{\mu\in\mathcal{M}_{1}(f)}\int\varphi\,d\mu$
(3.2)

where $\mathcal{M}_{1}(f)$ denotes the space of $f$-invariant probability
measures. In the special case that $f$ satisfies the specification property
the latter is equivalent to say that $\varphi$ satisfies (3.2) if and only if
$\varphi$ is not $C^{0}$-approximated by a sequence of functions
$(\varphi_{n})_{n}$ of the form $\varphi_{n}=u_{n}-u_{n}\circ f+c_{n}$ for
some $c_{n}\in\mathbb{R}$ and some continuous $u_{n}:X\to\mathbb{R}$. Any such
function $\varphi_{n}$ is called coholomogous to a constant and, if it is
Hölder continuous then it is characterized through Birkhoff averages at
periodic points by Livšic theorem [51]:

######  Theorem 3.1.

Let $G$ be a metric group, $f:M\to M$ be an Anosov diffeomorphism and let
$A:M\to G$ be Hölder continuous cocycle. If either $G=\mathbb{R}$ or $G$ is an
abelian group and $A,B:M\to G$ are so that $A^{n}(p)=B^{n}(p)$ whenever
$f^{n}(p)=p$ then there exists a continuous cocycle $C:M\to G$ so that
$A(x)=C(f(x))B(x)C(x)^{-1}$

In the context of an Anosov diffeomorphism $f$, as the space of periodic
probability measures is dense in the space of invariant probability measures,
it is not hard to use the previous theorem to ensure that, if the observable
$\varphi$ is Hölder continuous then condition (3.2) fails if and only if there
exists a continuous function $\psi:M\to\mathbb{R}$ and $c\in\mathbb{R}$ so
that $\varphi=\psi\circ f-\psi+c$. The set of irregular points associated to a
continuous observable $\varphi$ satisfying (3.2) and a continuous map with the
specification property is a Baire residual subset of $X$ and it has full
topological entropy (cf. [4, 24, 77]).

In the context of finitely generated semigroup actions common invariant
measures seldom exist, hence the concept of coboundary is extremely rigid. For
instance, it is known that a finitely generated group action admits a common
invariant probability if and only if the group is amenable (see e.g. [57]).
For that reason, we will only refer to coboundaries with respect to individual
dynamical systems or the skew-product determined by the semigroup action.
Furthermore, even though the concepts of orbital specification for semigroup
actions and semigroup action introduced in [63] can probably be used as a tool
to characterize irregular points for certain classes of semigroup actions
(e.g. generated by expanding maps), the class of semigroup actions treated
here do not satisfy any of the traditional gluing orbit properties (see
Example 5.10 for more details).

##  4\. Entropy and upper capacities

In the core of this section we relate several notions of entropy for semigroup
actions and non-autonomous dynamical systems, and prove entropy distribution
principle for fibered maps.

###  4.1. Topological entropy of continuous maps and non-autonomous dynamical
systems

Let $(X,d)$ be a compact metric space and $f:X\to X$ be a continuous map.
Given $\varepsilon>0$ and $n\in\mathbb{N}$, we say that $E\subset X$ is
$(n,\varepsilon)$-separated if for every $x\neq y\in E$ it holds that
$d^{f}_{n}(x,y)>\varepsilon$, where
$d^{f}_{n}(x,y)=\max\\{d(f^{j}(x),f^{j}(y))\colon 0\leqslant j\leqslant
n-1\\}$ denotes Bowen’s distance. The sets $B_{f}(x,n,\varepsilon)=\\{y\in
X:d_{n}^{f}(x,y)<\varepsilon\\}$ are called Bowen dynamic balls. If
$s(n,\varepsilon)$ denotes the maximal cardinality of a
$(n,\varepsilon)$-separated subset of $X$ then the _topological entropy_ of
$f$ is

$\displaystyle\displaystyle h_{top}(f)=\lim_{\varepsilon\to
0}\limsup_{n\to\infty}\frac{1}{n}\log s(n,\varepsilon).$

The classical variational principle for entropy ensures that
$h_{\operatorname{top}}(f)=\sup_{\mu\in\mathcal{M}_{inv}(f)}\\{h_{\mu}(f)\\}.$

Similarly, Kolyada and Snoha [45] introduced the concept of topological
entropy of non-autonomous dynamical systems $\mathcal{F}=(f_{n})_{n\geqslant
1}$, described by the iterates $F_{n}:=f_{n}\circ f_{n-1}\circ\dots\circ
f_{1}$ for each $n\geqslant 1$, as

$\displaystyle\displaystyle h_{top}(\mathcal{F})=\lim_{\varepsilon\to
0}\limsup_{n\to\infty}\frac{1}{n}\log s(\mathcal{F},n,\varepsilon),$

where $s(\mathcal{F},n,\varepsilon)$ denotes the maximal cardinality of a
subset $E\subset X$ so that

$\max\\{d(F_{j}(x),F_{j}(y))\colon 1\leqslant j\leqslant
n-1\\}>\varepsilon\quad\text{for any distinct $x,y\in E$.}$

To each ${\underline{\omega}}\in\Sigma_{\kappa}$ we associate the (one-sided)
non-autonomous dynamical system
$\mathcal{F}_{\underline{\omega}}=(f_{\omega_{j}})_{j\geqslant 1}$.

The topological entropy of invariant and not necessarily compact subsets
$Z\subset X$ is defined using a Charathéodory structure. Given
$\varepsilon>0$, $\alpha\in\mathbb{R}$ and $N\geq 1$, define

$m_{\alpha}(f,Z,\varepsilon,N)=\inf_{\mathcal{G}}\Big{\\{}\sum_{B(x,n,\varepsilon)\in\mathcal{G}}e^{-\alpha
n}\Big{\\}}$

where the infimum is taken over all finite or countable families $\mathcal{G}$
of dynamic balls of length $n\geqslant N$ so that
$\bigcup_{B(x,n,\varepsilon)\in\mathcal{G}}B(x,n,\varepsilon)$ covers $Z$. Let
$m_{\alpha}(f,Z,\varepsilon)=\lim_{N\to\infty}m_{\alpha}(f,Z,\varepsilon,N)$
and

$h_{\operatorname{top}}(f,Z,\varepsilon)=\inf{\\{\alpha:m_{\alpha}(f,Z,\varepsilon)=0\\}}=\sup{\\{\alpha:m_{\alpha}(f,Z,\varepsilon)=+\infty\\}}.$

Then, _topological entropy (also known as topological capacity) of $Z\subset
X$_ is given by $h_{\operatorname{top}}(f,Z)=\lim_{\varepsilon\to
0}h_{\operatorname{top}}(f,Z,\varepsilon)$ and satisfies the partial
variational principle
$h_{\operatorname{top}}(f,Z)\geqslant\sup\left\\{h_{\mu}(f)\right\\}$ where
the supremum is over all $f$-invariant probability measures $\mu$ such that
$\mu(Z)=1$. Moreover, in the special case that $Z\subset X$ is $f$-invariant
and compact we have that
$h_{\operatorname{top}}(f,Z)=h_{\operatorname{top}}(f\mid_{Z})$. We refer the
reader to [59] for the proofs and further details in the case of a single map.

In order to estimate the topological entropy of non-compact sets, we will make
use of the following extremely useful entropy distribution principle, due to
Takens and Verbitski (see [71, Theorem 3.6]):

######  Theorem 4.1.

Let $X$ be a compact metric space, $f:X\to X$ be a continuous map and
$Z\subset X$. Suppose that there exists $s\geqslant 0$ so that for any
sufficiently small $\varepsilon>0$ one can find a Borel probability measure
$\mu=\mu_{\varepsilon}$ and $C(\varepsilon)>0$ satisfying: (1)
$\mu_{\varepsilon}(Z)>0$, and (2)
$\mu_{\varepsilon}(B_{f}(x,n,\varepsilon))\leqslant C(\varepsilon)e^{-s\,n}$
whenever $B_{f}(x,n,\varepsilon)\cap Z\neq\emptyset$. Then
$h_{\operatorname{top}}(f,Z)\geqslant s$.

Finally, we recall a characterization of the Kolmogorov-Sinai metric entropy
due to Katok. As a consequence of [42, Theorem I.I], if $\mu$ is an
$f$-invariant and ergodic probability measure then for any $\rho>0$ it holds
that $h_{\mu}(f)=\lim_{\varepsilon\to
0}\bar{h}_{\mu}(f,\varepsilon,\rho)=\lim_{\varepsilon\to
0}\underline{h}_{\mu}(f,\varepsilon,\rho)$ where

$\displaystyle\bar{h}_{\mu}(f,\varepsilon,\rho)=\limsup_{n\rightarrow+\infty}\frac{1}{n}\log
N^{\mu}_{n}(\varepsilon,\rho),\qquad\underline{h}_{\mu}(f,\varepsilon,\rho)=\liminf_{n\rightarrow+\infty}\frac{1}{n}\log
N^{\mu}_{n}(\varepsilon,\rho),$ (4.1)

and $N^{\mu}_{n}(\varepsilon,\rho)$ denotes the minimal cardinality of an
$(n,\varepsilon)$-generating set for some set $Z\subset X$ with measure at
least $1-\rho$. It will be useful to use that a minimal cardinality
$(n,\varepsilon)$-generating set is an $(n,\frac{\varepsilon}{2})$-separated
set, not necessarily of maximal cardinality.

###  4.2. Topological entropies of finitely generated semigroup actions

Given a compact metric space $X$ and the finite collection
$G_{1}=\\{id,f_{1},f_{2},\dots,f_{\kappa}\\}$ of continuous self maps on $X$,
consider the finitely generated semigroup $(G,\,\circ)$ generated by $G_{1}$,
where the semigroup operation is the composition of maps. Setting
$G_{n}:=\big{\\{}g_{n}\circ\dots g_{2}\circ g_{1}\circ g_{0}\colon\;g_{i}\in
G_{1}\big{\\}}$ it is clear that $G_{n}\subset G_{n+1}$ for every $n\geqslant
1$.

####  4.2.1. Geometric entropy

We first recall the _geometric entropy_ of a semigroup, introduced by Ghys,
Langevin and Walczak [33]. Given $n\in\mathbb{N}$ and $\varepsilon>0$, the
$(n,\varepsilon)$-Bowen ball generated by the semigroup action and centered at
$x$ is defined by $B^{G}_{n}(x,\varepsilon)=\big{\\{}y\in
X\colon\,d(g(x),g(y))<\varepsilon,\;\forall\,g\in G_{n}\big{\\}}$. One says
that two points $x,\,y\in X$ are $(n,\varepsilon)$-separated by $G$ if there
exists $g\in G_{n}$ such that $d(g(x),\,g(y))\geqslant\varepsilon$. A subset
$E$ of $X$ is $(n,\varepsilon)$-separated if any two distinct points of $E$
are $(n,\varepsilon)$-separated by $G$. Having fixed $n\geqslant 1$ and
$\varepsilon>0$, consider

$s(G,G_{1},n,\varepsilon)=\max\,\big{\\{}|E|:\,\,E\subset X\text{ is
$(n,\varepsilon)$-separated}\big{\\}}.$

Since $X$ is compact, $s(G,G_{1},n,\varepsilon)$ is finite for every
$n\in\mathbb{N}$ and $\varepsilon>0$. The _geometric entropy_ of the free
semigroup $G$ generated by $G_{1}$ is defined by

$h^{GLW}(\mathbb{S})=\lim_{\varepsilon\,\to\,0}\,\limsup_{n\,\to\,+\infty}\,\frac{1}{n}\,\log\,s(G,G_{1},n,\varepsilon)$
(4.2)

(the limit as $\varepsilon$ tends to zero exists by monotonicity), where we
omit the dependence on $(G,G_{1})$ for notational simplicity. It is clear from
the definition that $s(G,G_{1},n,\varepsilon)\geqslant
s(\mathcal{F}_{\underline{\omega}},n,\varepsilon)$ for every $n\geqslant 1$,
$\varepsilon>0$ and ${\underline{\omega}}\in\Sigma_{\kappa}$ and,
consequently,

$h^{GLW}(\mathbb{S})\geqslant
h_{\operatorname{top}}(\mathcal{F}_{\underline{\omega}}),\qquad\forall{\underline{\omega}}\in\Sigma_{\kappa}.$
(4.3)

Similarly, one can define the _Ghys-Langevin-Walczak topological capacity of a
set $Y\subset X$_ by

$h^{GLW}(\mathbb{S},Y)=\lim_{\varepsilon\,\to\,0}\,\limsup_{n\,\to\,+\infty}\,\frac{1}{n}\,\log\,s(G,G_{1},Y,n,\varepsilon)$
(4.4)

where $s(G,G_{1},Y,n,\varepsilon)$ denotes the largest cardinality of a set
$E\subset Y$ that is $(n,\varepsilon)$-separated by $G$. For the sake of
completeness, let us mention that this notion of entropy was rediscovered
independently by Hofmann and Stoyanov [38] in the context of continuous
actions generated locally compact semigroups.

####  4.2.2. Free semigroup actions

A different approach, inspired by the context of random walks on groups, is to
code the semigroup using the free semigroup on $\kappa$ generators. In this
way one consider paths on the semigroup determined by finite words in
$\\{1,2,\dots,\kappa\\}^{\mathbb{Z}}$. More precisely, the semigroup action
$\mathbb{S}$ can be coded by the skew-product

$\begin{array}[]{rccc}F:&\\{1,2,\dots,\kappa\\}^{\mathbb{Z}}\times
X&\to&\\{1,2,\dots,\kappa\\}^{\mathbb{Z}}\times X\\\
&\quad\quad({\underline{\omega}},x)&\mapsto&(\sigma({\underline{\omega}}),f_{\omega_{0}}(x))\end{array}$

where the shift codes the possible paths in the free semigroup with $\kappa$
generators. The topological entropy of $F$-invariant subsets
$Z\subset\\{1,2,\dots,\kappa\\}^{\mathbb{Z}}\times X$ is defined as before.
However, this notion is not fitted specifically to deal with semigroup actions
as there is an artificial creation of entropy due to the shift, a fact which
suggests some alternative notions of entropy.

The first one was introduced by Bufetov in [18] in the case of symmetric
random walks and studied by Carvalho, Rodrigues and the second named author in
[21] for asymmetric random walks. Given $\underline{g}=g_{i_{n}}\dots
g_{i_{1}}\in G_{n}$, with $g_{i_{j}}\in G_{1}$, we say that a set $K\subset X$
is _$(\underline{g},n,\varepsilon)-$ separated_ if
$\max_{0\,\leqslant\,j\,\leqslant\,n-1}d(\underline{g}_{\,j}(x),\underline{g}_{\,j}(y))>\varepsilon$
for any distinct $x,y\in K$ where, for every $1\leqslant j\leqslant n-1$,
$\underline{g}_{\,j}$ stands for the concatenation $g_{i_{j}}\dots
g_{i_{2}}\,g_{i_{1}}\in G_{j}$, and $\underline{g}_{0}=id$. The topological
entropy of the semigroup action $\mathbb{S}$ with respect to a random walk
$\mathbb{P}$ is defined as

$h_{\text{top}}(\mathbb{S},\mathbb{P}):=\lim_{\varepsilon\to
0}\limsup_{n\to\infty}\frac{1}{n}\log\int_{\Sigma_{\kappa}^{+}}\,s(g_{\omega_{n}}\dots
g_{\omega_{1}},n,\varepsilon)\,d\mathbb{P}(\omega),$

where $s(g_{\omega_{n}}\dots g_{\omega_{1}},n,\varepsilon)$ is the maximum
cardinality of $(\underline{g},n,\varepsilon)-$separated sets, and it
coincides with a certain annealed topological pressure for the skew-product
$F$ (see [22, Proposition 4.1]). In the special case of the symmetric random
walk $\mathbb{P}_{sym}$ on $G$, which corresponds to the Bernoulli probability
$(\frac{1}{\kappa},\frac{1}{\kappa},\dots,\frac{1}{\kappa})^{\mathbb{Z}}$ on
the shift, this is written as

$h_{\text{top}}(\mathbb{S},\mathbb{P}_{sym}):=\lim_{\varepsilon\to
0}\limsup_{n\to\infty}\frac{1}{n}\log\Big{(}\frac{1}{\kappa^{n}}\sum_{\underline{g}\in
G_{n}}s(\underline{g},n,\varepsilon)\Big{)}.$ (4.5)

In this paper we will always consider symmetric random walks.

In analogy to the case of a single map, the complexity of non-compact subsets
$Z\subset X$ can be defined using a topological capacity or a Carathéodory
structure. The _(upper) topological capacity of the semigroup $\mathbb{S}$_ on
a subset $Z\subset X$ is defined by

$\overline{Ch_{\text{top}}}(\mathbb{S},Z,\mathbb{P}_{sym}):=\lim_{\varepsilon\to
0}\limsup_{n\to\infty}\frac{1}{n}\log\Big{(}\frac{1}{\kappa^{n}}\sum_{\underline{g}\in
G_{n}}s(\underline{g},Z,n,\varepsilon)\Big{)},$ (4.6)

where $s(\underline{g},Z,n,\varepsilon)$ denotes the largest cardinality of a
$(\underline{g},n,\varepsilon)$-separated subset of $Z$. In the particular
case that $X$ is compact, this coincides with the definition of entropy in
(4.5) (see [40] for more details). For notational simplicity, we will denote
the previous expression simply by
$h_{\operatorname{top}}(\mathbb{S},Z,\mathbb{P}_{sym})$.

The _annealed topological entropy of the semigroup_ of a subset $Z\subset X$
was introduced in [40] as follows: given $\varepsilon>0$,
$\alpha\in\mathbb{R}$, $N\geq 1$ and $\underline{g}\in G_{N}$, defining

$m_{\alpha}(\underline{g},Z,\varepsilon,N)=\inf_{\mathcal{G}_{\underline{g}}}\Big{\\{}\sum_{B_{\tilde{\underline{g}}}(x,n,\varepsilon)\in\mathcal{G}_{\underline{g}}}e^{-\alpha
n}\Big{\\}}$

where the infimum is taken over all finite or countable families
$\mathcal{G}_{\underline{g}}$ of dynamic balls
$B_{\tilde{\underline{g}}}(x,n,\varepsilon)$ determined by elements
$\tilde{\underline{g}}\in G_{n}$ with $n\geqslant N$ so that
$\bigcup_{B_{\tilde{\underline{g}}}(x,n,\varepsilon)\in\mathcal{G}}B_{\tilde{\underline{g}}}(x,n,\varepsilon)$
covers $Z$, and

$m_{\alpha}(\mathbb{S},Z,\varepsilon,N)=\frac{1}{\kappa^{N}}\sum_{\underline{g}\in
G_{N}}m_{\alpha}(\underline{g},Z,\varepsilon,N),$

the limit
$m_{\alpha}(\mathbb{S},Z,\varepsilon)=\lim_{N\to\infty}m_{\alpha}(\mathbb{S},Z,\varepsilon,N)$
does exist, and

$h^{B}(\mathbb{S},Z,\varepsilon)=\inf{\\{\alpha:m_{\alpha}(\mathbb{S},Z,\varepsilon)=0\\}}=\sup{\\{\alpha:m_{\alpha}(\mathbb{S},Z,\varepsilon)=+\infty\\}}.$

Then, _topological entropy of $Z\subset X$_ is given by
$h^{B}(\mathbb{S},Z)=\lim_{\varepsilon\to 0}h^{B}(\mathbb{S},Z,\varepsilon)$.
It holds that

$h^{B}(\mathbb{S},Z)\leqslant\overline{Ch_{\text{top}}}(\mathbb{S},Z,\mathbb{P}_{sym})\leqslant
h_{\text{top}}(\mathbb{S},\mathbb{P}_{sym})$

for every $Z\subset X$, that these three notions coincide when $Z=X$ and that
the first two notions coincide for any compact and $G$-invariant subset $Z$
(cf. [40, Theorem 3.9]) Moreover, although the original notion has been
defined using open coverings, a standard argument shows this Carathéodory
structure can be defined using dynamic balls (see e.g. [59]), an object that
suits our purposes better.

###  4.3. Fibered topological entropies

Given $\varepsilon>0$ and
${\underline{\omega}}\in\\{1,2,\dots,\kappa\\}^{\mathbb{Z}}$ consider the
metric in $X_{\underline{\omega}}$ given by
$d_{{\underline{\omega}}}^{n}(x,y)=\max_{0\leqslant j\leqslant
n-1}d(g_{\underline{\omega}}^{j}(x),g_{\underline{\omega}}^{j}(y))$ for
$x,y\in X_{\underline{\omega}}$ and $n\geqslant 1$. The dynamic ball
determined by $x\in X_{\omega}$, $n\geqslant 1$ and $\varepsilon>0$ is the set

$B_{{\underline{\omega}}}(x,n,\varepsilon)=\Big{\\{}y\in
X_{\underline{\omega}}:d_{{\underline{\omega}}}^{n}(x,y)<\varepsilon\;\;\mbox{for
all}\;\;0\leqslant j<n.\Big{\\}}$

A subset $K\subset X_{{\underline{\omega}}}$ is called
_$({\underline{\omega}},n,\varepsilon)$ -separated_ if
$d_{{\underline{\omega}}}^{n}(x,y)>\varepsilon$ for any distinct $x,y\in K$.

We define the _fibered topological entropy for the semigroup_ on the compact
metric space $X$ as

$h^{path}(\mathbb{S})=\sup_{{\underline{\omega}}\in\\{1,2,\dots,\kappa\\}^{\mathbb{Z}}}\,h_{\operatorname{top}}(\mathcal{F}_{\underline{\omega}}),F_{\underline{\omega}})$
(4.7)

where
$h_{\operatorname{top}}(\mathcal{F}_{\underline{\omega}})=\lim_{\varepsilon\to
0}\limsup_{n\to\infty}\frac{1}{n}\log s(g_{\underline{\omega}},n,\varepsilon)$
denotes the topological pressure of sequential dynamical system
$\mathcal{F}_{\underline{\omega}}=(g_{{\underline{\omega}}}^{j})_{j\geqslant
1}$ introduced in [45]. Related measure theoretical notions of fibered entropy
appeared in early studies by Pinsker [60], Abramov and Rokhlin [1, 64],
Ledrappier and Walters [52] on the entropy of skew-products and later
associated to the context of random dynamical systems (see e.g. [44]). Indeed,
it is known that

$h_{\mu}(F)=h_{\pi_{*}\mu}(\sigma)+h_{\mu}(F\\!\mid\\!\sigma)$ (4.8)

for each $F$-invariant probability measure $\mu$, where the relative entropy
is defined by the supremum
$h_{\mu}(F\\!\mid\\!\sigma)=\sup_{\xi}h_{\mu}(F\\!\mid\\!\sigma,\xi)$ over
finite partitions $\xi$ of $\\{1,2,\dots,\kappa\\}^{\mathbb{Z}}\times X$,
where

$h_{\mu}(F\\!\mid\\!\sigma,\xi)=\lim_{n\to\infty}\frac{1}{n}H_{\mu}\Big{(}\bigvee_{j=0}^{n-1}F^{-j}\xi\mid\pi^{-1}(\mathcal{E})\Big{)}$

and $\mathcal{E}$ denotes the partition of
$\\{1,2,\dots,\kappa\\}^{\mathbb{Z}}$ into points.

######  Remark 4.2.

The expression (4.8) is known as the Abramov-Rokhlin formula, although it is
sometimes referred as Pinsker formula in the special case that $\sigma$ is
invertible (cf. [60, 64]). It was proven for skew-products with non-invertible
fiber maps by Bogenschütz and Crauel [11].

######  Remark 4.3.

Whenever $\mathbb{P}$ is a $\sigma$-invariant probability and $\mu$ is an
$F$-invariant probability whose marginal $\pi_{*}\mu$ in $\Sigma_{\kappa}$
coincides with $\mathbb{P}$, the value $\int
h_{\operatorname{top}}(\mathcal{F}_{\underline{\omega}})\,d\mathbb{P}$ is the
(quenched) topological entropy of $\mu$ for the random dynamical system driven
by $\mathbb{P}$ (see [44]). Moreover,
$h_{\mu}(F\\!\mid\\!\sigma)=h_{\mu}(\mathcal{F})$ where
$h_{\mu}(\mathcal{F})=\sup_{\xi}h_{\mu}(\mathcal{F},\xi)$, the supremum is
taken over all finite Borel partitions $\xi$ of $X$,

$h_{\mu}(\mathcal{F},\xi)=\lim_{n\to+\infty}\frac{1}{n}\int
H_{\mu_{\underline{\omega}}}\Big{(}\bigvee_{k=0}^{n-1}(f^{k}_{\underline{\omega}})^{-1}\xi\Big{)}d\mathbb{P}({\underline{\omega}}),$

and $\mu=(\mu_{\underline{\omega}})_{\underline{\omega}}$ is the almost
everywhere defined Rokhlin disintegration of $\mu$ on its sample measures (see
e.g. [10, 11]). The variational principle for entropy states that

$\int
h_{\operatorname{top}}(\mathcal{F}_{\underline{\omega}})\,d\mathbb{P}({\underline{\omega}})=\sup\Big{\\{}h_{\mu}(\mathcal{F})\colon\mu\in\mathcal{M}_{inv}(F),\pi_{*}\mu=\mathbb{P}\Big{\\}}$

(cf. [50, Theorem 3.1]). We will consider the previous equality when
$\mathbb{P}=\pi_{*}\mu$ in order to compare the fibered entropies of
$F$-invariant probabilities $\mu$. In particular, when the $F$-invariant
measure $\mu$ changes then the entropies $h_{\mu}(F\mid\sigma)$ are not
related to any fixed random dynamical system. Nevertheless, $\int
h_{\operatorname{top}}(\mathcal{F}_{\underline{\omega}})\,d\pi_{*}\mu\geqslant
h_{\mu}(F\\!\mid\\!\sigma)\;\text{for every}\,\mu\in\mathcal{M}_{inv}(F).$
This, together with the ergodic decomposition theorem, ensures that

$h^{path}(\mathbb{S})\geqslant\sup_{\mu\in\mathcal{M}_{inv}(F)}\,h_{\mu}(F\mid\sigma)=\sup_{\mu\in\mathcal{M}_{erg}(F)}\,h_{\mu}(F\mid\sigma)=:h_{\operatorname{top}}(F\mid\sigma),$
(4.9)

an expression which relates the fibered entropy of the semigroup with that
generated by $F$-invariant probabilities.

Let us now recall another notion of measure theoretical entropy for non-
autonomous dynamical systems, which is adaptable to general probabilities, but
for which we will be consider with respect to the sample measures of
$F$-invariant probabilities. Let us be more precise. Assume that $\mu$ is an
$F$-invariant and ergodic probability, that $0<\delta<1$, $\varepsilon>0$ and
${\underline{\omega}}\in\\{1,2,\dots,\kappa\\}^{\mathbb{Z}}$. By compactness
of $X_{\underline{\omega}}$, for each $Y\subset X_{\underline{\omega}}$ the
largest cardinality $s({\underline{\omega}},n,\varepsilon,Y)$ of an
$({\underline{\omega}},n,\varepsilon)$-separated subset of $Y$ is finite. Now,
take
$s({\underline{\omega}},n,\varepsilon,\delta):=\min\\{s({\underline{\omega}},n,\varepsilon,Y):\mu(Y)\geqslant
1-\delta\\}$ and define

$h_{\mu}^{\delta}(\underline{\omega}):=\lim_{\varepsilon\rightarrow
0}\liminf_{n\rightarrow\infty}\frac{1}{n}\log
s({\underline{\omega}},n,\varepsilon,\delta),$ (4.10)

which is clearly bounded above by
$h_{\operatorname{top}}(\mathcal{F}_{\underline{\omega}})$. A related notion
of entropy, using a Shannon-McMillan-Breiman type formula appeared in [9]. The
_Katok $\delta$-entropy_ of $\mu\in\mathcal{M}_{inv}(F)$ is defined by

$h_{\mu}^{\delta}({F})=\int
h_{\mu}^{\delta}(\underline{\omega})\,d(\pi_{*}\mu)({\underline{\omega}}).$

In the special case that $\mu$ is ergodic one has that

$h_{\mu}^{\delta}({F})=h_{\mu}^{\delta}(\underline{\omega})=h_{\mu}(\mathcal{F}_{\underline{\omega}})=h_{\mu}(F\\!\mid\\!\sigma)\quad\text{for
$(\pi_{*}\mu)$-a.e. ${\underline{\omega}}$}$ (4.11)

coincides with the quenched topological entropy of the random dynamical system
driven by $\mathbb{P}=\pi_{*}\mu$ (cf. [47, Theorem A]). In particular we
conclude that

$H^{\text{Pinsker}}(\psi)=H^{Katok}(\psi),$ (4.12)

where the left hand-side was defined in (2.7) and the second term is defined
by

$\displaystyle H^{Katok}(\psi):=\sup\Big{\\{}c\geqslant 0\colon$
$\displaystyle\forall\varepsilon>0\,\text{there
exist}\,\mu_{1},\mu_{2}\in\mathcal{M}_{erg}(F)\,\text{so that}\,$
$\displaystyle h_{\mu_{i}}^{\delta}(F)\geqslant
c-\varepsilon\;\text{and}\,\int\varphi_{\psi}\,d\mu_{1}<\int\varphi_{\psi}\,d\mu_{2}\Big{\\}}.$
(4.13)

In the case of non-compact subsets $Y\subset X_{\underline{\omega}}$,
Stadlbauer, Suzuki and the second named author [65, Appendix A] introduced a
notion of relative entropy using Carathéodory structures in the realm of
random dynamical systems. Let us recall this notion. Fix $\varepsilon>0$ and
${\underline{\omega}}\in\Omega$ and consider the sets
${\mathcal{I}}_{\underline{\omega}}^{n}=X_{\underline{\omega}}\times\\{n\\}$
and
${\mathcal{I}}_{\underline{\omega}}=X_{\underline{\omega}}\times\mathbb{N}$.
For every $\alpha\in\mathbb{R}$ and $N\geqslant 1$, define

$m_{\alpha}({\underline{\omega}},f,\phi,\Lambda,\varepsilon,N)=\inf_{\mathcal{G}_{\underline{\omega}}}\Big{\\{}\sum_{{(x,n)}\in\mathcal{G}_{\underline{\omega}}}e^{-\alpha
n(x)+S_{n(x)}\phi_{\underline{\omega}}({\mathrm{B}_{\underline{\omega}}(x,n(x),\varepsilon)})}\Big{\\}},$
(4.14)

where the infimum is taken over all finite or countable families $\mathcal{G}$
of the form $\mathcal{G}_{\underline{\omega}}\subset\cup_{n\geqslant
N}{\mathcal{I}}_{\underline{\omega}}^{n}$ such that the collection of sets
$\\{B_{\underline{\omega}}(x,n,\varepsilon):(x,n)\in\mathcal{G}_{\underline{\omega}}\\}$
cover $\Lambda_{\underline{\omega}}$ and such that (4.14) is measurable in
$\omega$. As before,
$S_{n}\phi_{\underline{\omega}}({\mathrm{B}_{\underline{\omega}}(x,n,\varepsilon)})=\sup\\{\sum_{j=0}^{n-1}\phi_{\theta^{j}({\underline{\omega}})}(g_{\underline{\omega}}^{j}(y)):y\in
B_{\underline{\omega}}(x,n,\varepsilon)\\}.$ Using that the previous sequence
is increasing on $N$, the limit

$m_{\alpha}({\underline{\omega}},f,\phi,\Lambda,\varepsilon)=\lim_{N\to+\infty}m_{\alpha}({\underline{\omega}},f,\phi,\Lambda,\varepsilon,N)$
(4.15)

does exist. If $\alpha$ is such that (4.15) is finite and $\beta>\alpha$ then
$m_{\beta}(f,\phi,\Lambda,\varepsilon,N)\leqslant
e^{-(\beta-\alpha)N}m_{\alpha}(f,\phi,\Lambda,\varepsilon,N)$ tends to zero as
$N\to\infty$, hence $m_{\beta}(f,\phi,\Lambda,\varepsilon)=0$. For that reason
we define

$\pi_{\phi}({\underline{\omega}},f,\Lambda,\varepsilon)=\inf{\\{\alpha:m_{\alpha}({\underline{\omega}},f,\phi,\Lambda,\varepsilon)=0\\}},$

which is decreasing on $\varepsilon$. The fibered _relative pressure_ of the
set $\Lambda$ with respect to $(f,\phi)$ is defined by

$\pi_{\phi}({\underline{\omega}},f,\Lambda)=\lim_{\varepsilon\to
0}\pi_{\phi}({\underline{\omega}},f,\Lambda,\varepsilon).$ (4.16)

The previous limit exists as
$\varepsilon\mapsto\pi_{\phi}({\underline{\omega}},f,\Lambda,\varepsilon)$ is
non-increasing. In the special case that $\phi\equiv 0$ we write simply

$h_{\Lambda_{{\underline{\omega}}}}(\mathcal{F}_{\omega}):=\pi_{0}({\underline{\omega}},f,\Lambda_{\underline{\omega}}).$

It is not hard to check that
$h_{\Lambda_{{\underline{\omega}}}}(\mathcal{F}_{\omega})\leqslant
h_{\operatorname{top}}(\mathcal{F}_{\underline{\omega}})$ for every
${\underline{\omega}}\in\\{1,2,\dots,\kappa\\}^{\mathbb{Z}}$.

In general, upper bounds are much easier to obtain as these rely on choices of
certain coverings of $\Lambda$ using dynamic balls at a certain scale. The
next result provides an entropy distribution principle for fibered sets, which
is a simple modification of [71, Theorem 3.6] and [76, Proposition 2.6] for
sequential dynamical systems and serves as a criterion to provide lower bounds
on these fibered entropies.

######  Proposition 4.4 (Fibered entropy distribution principle).

Let ${\underline{\omega}}\in\\{1,2,\dots,\kappa\\}^{\mathbb{Z}}$ and
$\Lambda_{\underline{\omega}}\subset X_{\underline{\omega}}$ be an arbitrary
Borel set. Suppose there exist $\varepsilon>0$, $\alpha\geqslant 0$,
$K_{\underline{\omega}}(\varepsilon)>0$ and a sequence of probability measures
$(\mu_{{\underline{\omega}},k})_{k\geqslant 1}$ on $X_{\underline{\omega}}$
such that

$\limsup_{k\rightarrow\infty}\mu_{\underline{\omega},k}(B_{\underline{\omega}}(x,n,\varepsilon))\leqslant
K_{\underline{\omega}}(\varepsilon)\,e^{-\alpha n}$ (4.17)

for sufficiently large $n$ and every ball
$B_{\underline{\omega}}(x,n,\varepsilon)$ which has non-empty intersection
with $\Lambda_{{\underline{\omega}}}$. If there exists a weak∗ accumulation
point $\nu_{\underline{\omega}}$ of the probability measures
$(\mu_{\underline{\omega},k})_{k\geqslant 1}$ so that
$\nu_{\underline{\omega}}(\Lambda_{\underline{\omega}})>0$ then
$h_{\Lambda_{{\underline{\omega}}}}(\mathcal{F}_{\omega})\geqslant\alpha$.

###### Proof.

Let $\varepsilon>0$, $\alpha\geqslant 0$,
$K_{\underline{\omega}}(\varepsilon)>0$ be as above, and assume that
$(\mu_{\underline{\omega},k_{j}})_{j\geqslant 1}$ is weak∗ convergent to
$\nu_{{\underline{\omega}}}$. Assume also that
$\nu_{\underline{\omega}}(\Lambda_{\underline{\omega}})>0$. Fix
$N_{\underline{\omega}}\geqslant 1$ large so that (4.17) holds for all
$n\geqslant N_{\underline{\omega}}$. Consider an arbitrary countable family
$\mathcal{G}_{\underline{\omega}}\subset\cup_{n\geqslant
N_{\underline{\omega}}}{\mathcal{I}}_{\underline{\omega}}^{n}$ such that the
collection
$\\{B_{\underline{\omega}}(x,n(x),\varepsilon):(x,n(x))\in\mathcal{G}_{\underline{\omega}}\\}$
covers $\Lambda_{\underline{\omega}}$. Extracting a subcollection of sets, if
necessary, one may assume without loss of generality that
$B_{{\underline{\omega}}}(x,n(x),\varepsilon)\cap\Lambda_{{\underline{\omega}}}\neq\emptyset$
for every $(x,n(x))\in\mathcal{G}_{\underline{\omega}}$. Therefore, by the
Portmanteau theorem on the characterization of weak∗ convergence one concludes
that

$\displaystyle\sum_{{(x,n)}\in\mathcal{G}_{\underline{\omega}}}e^{-\alpha
n(x)}$ $\displaystyle\geqslant
K_{{\underline{\omega}}}(\varepsilon)^{-1}\sum_{i}\limsup_{k\rightarrow\infty}\mu_{{\underline{\omega}},k}(B_{{\underline{\omega}}}(x,n(x),\varepsilon))$
$\displaystyle\geqslant
K_{\omega}(\varepsilon)^{-1}\sum_{i}\nu_{{\underline{\omega}}}(B_{{\underline{\omega}}}(x,n(x),\varepsilon))$
$\displaystyle\geqslant
K_{{\underline{\omega}}}(\varepsilon)^{-1}\nu(\Lambda_{{\underline{\omega}}})>0$

As $\mathcal{G}_{\underline{\omega}}$ was chosen arbitrary this implies that
$m_{\alpha}({\underline{\omega}},f,0,\Lambda_{\underline{\omega}},\varepsilon,N)\geqslant
K_{{\underline{\omega}}}(\varepsilon)^{-1}\nu(\Lambda_{{\underline{\omega}}})>0$
for every $N\geqslant N_{\underline{\omega}}$ and, consequently,
$h_{\Lambda_{{\underline{\omega}}}}(\mathcal{F}_{\omega})\geqslant\alpha$. ∎

######  Remark 4.5.

We observe that Proposition 4.4 is of independent interest in the special case
of random dynamical systems, modeled by the skew-product $F$. Indeed, it
allows to estimate the entropy of equivariant sets contained in fibers
$X_{\underline{\omega}}$ associated to a $\mathbb{P}$-generic point
${\underline{\omega}}$ provided the existence of $F$-invariant probabilities
$(\mu_{k})_{k\geqslant 1}$ whose disintegrations
$(\mu_{{\underline{\omega}},k})_{k\geqslant 1}$, defined almost everywhere by
Rokhlin theorem as a measurable family
${\underline{\omega}}\mapsto\mu_{{\underline{\omega}},k}$ of probabilities so
that $\operatorname{supp}\mu_{{\underline{\omega}},k}\subset
X_{\underline{\omega}}$ and
$\mu=\int\mu_{{\underline{\omega}},k}\,d\mathbb{P}({\underline{\omega}})$,
satisfy (4.17) at a certain scale $\varepsilon$ and every large $n$.

###  4.4. Some examples

In the remainder of this section we compare the quantities
$h_{*}(\varphi_{\psi})$, $H^{\text{KS}}(\psi)$ $H^{\text{Pinsker}}(\psi)$ and
$H^{\sigma}(\psi)$ defined by (2.5)-(2.8), using a couple of examples. For the
sake of simplicity we shall consider semigroups generated by certain
continuous maps on the interval $[0,1]$.

######  Example 4.6.

Let $G$ be the semigroup action generated by two continuous and Markov
interval maps $f_{1},f_{2}:[0,1]\to[0,1]$ as described in the following
figure. Let $\psi:[0,1]\to\mathbb{R}$ be a continuous observable.

Figure 4.1. Generators $G_{1}=\\{f_{1},f_{2}\\}$

Assume further that $\psi$ is not cohomologous to a constant neither with
respect to $f_{1}$ or $f_{2}$ (in particular $\varphi_{\psi}$ is not
cohomologous to a constant with respect to the step skew-product $F$ on
$\Sigma_{2}^{+}\times[0,1]$ determined by $f_{1}$ and $f_{2}$. Assume,
additionally, that $\psi\mid_{[a,1]}\equiv 1$.

Firstly we compute the entropy of the skew-product $F$. The map
$F\mid_{\Sigma_{2}^{+}\times[0,a]}$ is a distance expanding map and is
semiconjugate to a full shift with $4$ symbols, hence
$h_{\operatorname{top}}(F\mid_{\Sigma_{2}^{+}\times[0,a]})=\log 4$. Proceeding
with a similar description of the three transitive components and Markov
property of the skew-product it is easy to check that
$h_{\operatorname{top}}(F)=\log 4$. As $F\mid_{\Sigma_{2}^{+}\times[0,a]}$
satisfies the specification property then the space of ergodic probability
measures is entropy dense, hence the assumptions on $\psi$ guarantee that
there exist $F$-invariant and ergodic probability measures $\mu_{1},\mu_{2}$
with entropy arbitrarily close to $\log 4$ so that
$\int\varphi_{\psi}d\mu_{1}\neq\int\varphi_{\psi}d\mu_{2}$, hence

$h_{\operatorname{top}}(F)=\log 4=h_{*}(\varphi_{\psi}).$

Secondly, let us observe now the non-autonomous dynamical systems generated by
concatenations of maps $f_{1}$ and $f_{2}$. The restriction of each of these
maps to any of its invariant intervals $\\{[0,a],[a,b],[b,1]\\}$ defines a
continuous, piecewise affine distance expanding and Markov map with degree at
most three. Hence,

$h_{\operatorname{top}}(\mathcal{F}_{\underline{\omega}})\leqslant\log
3\quad\text{for every }{\underline{\omega}}\in\Sigma_{2}^{+}.$

Actually, $H^{KS}(\psi)=\log 3,$ where the maximum is attained for any
${\underline{\omega}}$ in the pre-orbit of the fixed point
$(2,2,2,\dots)\in\Sigma_{2}^{+}$.

Finally, let us now draw our attention to the Pinsker entropy and the
contribution on the shift of measures with large fibered entropy. Let
$\mu_{1},\mu_{2}$ be two $F$-invariant and ergodic probabilities so that
$\int\varphi_{\psi}\,d\mu_{1}<\int\varphi_{\psi}\,d\mu_{2}$. As $\psi$ is
constant on the interval $[a,b]$, both probabilities cannot be simultaneously
supported on the invariant domain $\Sigma_{2}^{+}\times[a,b]$ and, using
(4.11), we deduce that $h_{\mu_{i}}(F\mid\sigma)\leqslant\log 2$ for either
$i=1$ or $i=2$. We claim that $H^{\text{Pinsker}}(\psi)=\log 2$. Indeed, this
follows as a standard consequence of the specification property for
$F\mid_{\Sigma_{2}^{+}\times[0,a]}$ (which implies that ergodic measures are
entropy dense) combined with the fact that the unique maximal entropy
$\mu^{1}_{max}$ measure for $F\mid_{\Sigma_{2}^{+}\times[0,a]}$ projects over
the Bernoulli measure $(\frac{1}{2},\frac{1}{2})^{\mathbb{N}}$ on
$\Sigma_{2}^{+}$ and, consequently,
$h_{\mu^{1}_{max}}(F\mid\sigma)=h_{\mu^{1}_{max}}(F)-h_{\pi_{*}\mu^{1}_{max}}(\sigma)=\log
4-\log 2=\log 2.$ Altogether,

$H^{\text{Pinsker}}(\psi)=\log 2<\log 3=H^{KS}(\psi).$

Furthermore, assume that the unique maximal entropy $\mu^{2}_{max}$ measure
for $F\mid_{\Sigma_{2}^{+}\times[a,b]}$ is such that
$\int\varphi_{\psi}\,d\mu^{1}_{max}<\int\varphi_{\psi}\,d\mu^{2}_{max}$
(otherwise use entropy denseness of ergodic measures to produce an ergodic
measure arbitrarily close to $\mu^{2}_{max}$ with fibered entropy arbitrary
close to $\log 3$). As this measure projects over the Bernoulli measure
$(\frac{1}{2},\frac{1}{2})^{\mathbb{N}}$ as well we conclude that

$H^{\text{Pinsker}}_{\sigma}(\psi)=\log 2=h_{\operatorname{top}}(\sigma).$

######  Example 4.7.

Let $G$ be the semigroup generated by the continuous and Markov interval maps
$f_{2},f_{3}:[0,1]\to[0,1]$ described in the previous figure, and let
$\hat{F}:\Sigma_{2}^{+}\times[0,1]\to\Sigma_{2}^{+}\times[0,1]$ denote the
step skew-product determined by them. Let $\psi:[0,1]\to\mathbb{R}$ be a
continuous observable so that $\psi\mid_{[a,b]}\equiv 1$ and that is not
cohomologous to a constant neither with respect to $f_{2}$ or $f_{3}$ (in
particular $\varphi_{\psi}$ is not cohomologous to a constant with respect to
$\hat{F}$. It is easy to check that $h_{\operatorname{top}}(\hat{F})=\log 6$.
Moreover, as $\psi$ is constant over the interval $[a,b]$, given two
$\hat{F}$-invariant and ergodic probabilities $\mu_{1},\mu_{2}$ so that
$\int\varphi_{\psi}d\mu_{1}\neq\int\varphi_{\psi}d\mu_{2}$ at least one of
them is supported either on $\Sigma_{2}^{+}\times[0,a]$ or
$\Sigma_{2}^{+}\times[b,1]$. Using the specification property for the
restriction of $\hat{F}$ to any of these invariant domains, one gets that

$h_{\operatorname{top}}(\hat{F})=\log 6>\log 4\geqslant
h_{*}(\varphi_{\psi}).$

and $H^{\text{Pinsker}}_{\sigma}(\psi)=\log 2=h_{\operatorname{top}}(\sigma).$

Finally, it is worthwhile to point out that the entropy estimates in Examples
4.6 and 4.7 involve counting elements in a Markov partition and the fact that
all compositions are also Markov. Hence, it is likely that similar examples
can be produced involving higher dimensional piecewise expanding maps or even
Axiom A diffeomorphisms on a compact Riemannian manifold.

##  5\. The frequent hitting times property

In this section we not only provide sufficient conditions to ensure that
single dynamics, semigroup actions and skew-products satisfy the notion of
frequent hitting times, as we discuss the relation between it, some weak
versions of specification and hyperbolicity.

###  5.1. Sufficient condition for the frequent hitting times property

Our aim here is to provide a checkable condition which implies on the frequent
hitting times property, and to provide several examples that illustrate its
applicability. Assume that the semigroup action generated by a collection
$G_{1}=\\{id,f_{1},f_{2},\dots,f_{\kappa}\\}$ of continuous maps on a compact
metric space $X$ satisfying the following strong transitivity property: for
every $\varepsilon>0$ there exists $K(\varepsilon)\geqslant 1$ such that:

$\forall x\in
X\,\exists\,{\underline{\omega}}={\underline{\omega}}_{x}\in\Sigma_{\kappa}\qquad\bigcup_{j=0}^{K(\varepsilon)}\,f_{\underline{\omega}}^{j}(B(x,\varepsilon))=X.$
(C)

In the special case that the continuous maps $f_{i}$ are actually
homeomorphisms it will be often more convenient to express the previous
condition (C) using their inverses: for every $\varepsilon>0$ there exists
$K(\varepsilon)\geqslant 1$, and for each $x\in X$ there exists
${\underline{\omega}}\in\Sigma_{\kappa}$ such that

$\forall x\in
X\,\exists\,{\underline{\omega}}={\underline{\omega}}_{x}\,\in\Sigma_{\kappa}\qquad\bigcup_{j=0}^{K(\varepsilon)}\,f_{{\underline{\omega}}_{0}}^{-1}\circ\dots\circ
f_{{\underline{\omega}}_{1}}^{-1}\circ
f_{{\underline{\omega}}_{K(\varepsilon)}}^{-1}(B(x,\varepsilon))=X.$ (C’)

Conditions (C) and (C’) are clearly equivalent for semigroup actions generated
by isometries. Actually, we have the following:

######  Lemma 5.1.

Let $X$ be a compact metric space and consider the semigroup action
$\mathbb{S}$ generated by a collection of homeomorphisms
$G_{1}=\\{id,f_{1},f_{2},\dots,f_{\kappa}\\}$ satisfying property (C). If the
smallest sub-semigroup action satisfying property (C) is generated by a
collection $\widetilde{G_{1}}\subset G_{1}$ of isometries then the semigroup
action $\mathbb{S}$ has frequent hitting times.

###### Proof.

Assume there exists a collection $\widetilde{G_{1}}\subset G_{1}$ of
isometries generating a sub-semigroup action satisfying (C), and for any
$\varepsilon>0$ let $K(\varepsilon)\geqslant 1$ be given according to this
property. We claim that the semigroup action generated by $G_{1}$ has frequent
hitting times. Fix arbitrary balls $(B_{1},B_{2})$ with
$B_{1}=B(x_{1},\varepsilon)$ and $B_{2}$ of radius $0<r_{2}<\varepsilon/2$. If
$\hat{B}_{1}\subset B_{1}$ is the ball of radius $\varepsilon/3$ centered at
$x_{1}$ then there exists $0\leqslant p\leqslant K(\varepsilon/3)$, a finite
word ${\underline{\omega}}\in\\{1,2,\dots,\kappa\\}^{p}$ and elements
$f_{\omega_{i}}\in\widetilde{G_{1}}$ for every $0\leqslant i\leqslant p$ so
that $f_{\underline{\omega}}^{p}(\hat{B}_{1})\cap B_{2}\neq\emptyset$. By the
hypothesis, the map $f_{\underline{\omega}}^{p}$ is an isometry. Therefore, if
$r_{2}<\varepsilon/3$ then $f_{\underline{\omega}}^{n}(B_{1})\supset B_{2}$.
Otherwise, $\varepsilon/3\leqslant r_{2}<\varepsilon/2$ and
$f_{\underline{\omega}}^{p}(B_{1})\cap B_{2}$ contains a ball of radius
$\frac{\varepsilon}{3}=\frac{\varepsilon}{2}\frac{2}{3}>r_{2}\frac{2}{3}>\frac{r_{2}}{2}$.
Altogether, this shows that $G_{1}$ satisfies the frequent hitting time
property with transition time $K(\varepsilon/3)$, and proves the lemma. ∎

In other words, the previous lemma ensures that the frequent hitting times
property hold for semigroup actions which contain some minimal action by
isometries. Recall that a homeomorphism $f:X\to X$ is _minimal_ if all points
have a dense orbit, and a semigroup action generated by a collection of
homeomorphisms $G_{1}$ is _minimal_ if for every $x\in X$ there exists
${\underline{\omega}}\in\Sigma_{\kappa}$ so that
$X\,=\,\overline{\\{f_{\underline{\omega}}^{n}(x)\colon n\geqslant 0\\}}$ (cf.
[39]). The previous lemma can be reformulated to provide some classes of
examples, defined in terms of minimality, which satisfy the frequent hitting
times property.

######  Corollary 5.2.

Let $X$ be a compact metric space. The semigroup action generated by the
collection $G_{1}$ of continuous maps has frequent hitting times in the
following cases:

  1. (1)

$G_{1}=\\{f\\}$, where $f:X\to X$ is a minimal isometry;

  2. (2)

$G_{1}=\\{f\\}$, where $f:\mathbf{S}^{1}\to\mathbf{S}^{1}$ is an irrational
rotation;

  3. (3)

$G_{1}=\\{id,f_{1},f_{2},\dots,f_{\kappa}\\}$ and there exists a sub-
collection $\widetilde{G_{1}}\subset G_{1}$ formed by isometries which
generate a minimal semigroup action. .

###### Proof.

Items (1) and (2) are immediate consequences of item (3), hence we are left to
prove the latter. Moreover, using Lemma 5.1 it is enough to show that the
collection $\widetilde{G_{1}}$ generates a sub-semigroup satisfying property
(C). The argument is identical to the one used in the proof of [68, Lemma 5.1]
adapted to the semigroup actions context, which we include for completeness.
Indeed, by minimality of the sub-semigroup action, for any $x\in X$ there
exists ${\underline{\omega}}\in\Sigma_{\kappa}$ so that
$f_{\omega_{i}}\in\widetilde{G_{1}}$ for every $i\geqslant 0$ and
$X\,=\,\overline{\\{f_{\underline{\omega}}^{n}(x)\colon n\geqslant 0\\}}.$
Thus, for every $\varepsilon>0$ we obtain that $\bigcup_{n\geqslant
0}(f_{\underline{\omega}}^{n})^{-1}(B(x,\varepsilon))\,=\,X.$ Moreover, by
compactness of $X$, there exists $K_{x}(\varepsilon)\geqslant 1$ so that
$\bigcup_{n=0}^{K_{x}(\varepsilon)}(f_{\underline{\omega}}^{n})^{-1}(B(x,\varepsilon))\,=\,X.$
Now, using the uniform continuity of the finite collection of maps
$f_{\omega_{i}}$, there exists $\delta_{x}>0$ so that

$\bigcup_{n=0}^{K_{x}(\varepsilon)}(f_{\underline{\omega}}^{n})^{-1}(B(y,\varepsilon))\,=\,X\quad\text{for
every $y\in B(x,\delta_{x})$}.$

Extracting a finite sub cover $\\{B(x_{i},\delta_{i}):1\leqslant i\leqslant
s\\}\subset\\{B(x,\delta_{x}):x\in X\\}$, we conclude that (C’) holds with
$K(\varepsilon)=\max_{1\leqslant i\leqslant s}K_{x_{i}}(\varepsilon)$.
Furthermore, as each elements $f_{\omega_{i}}$ is an isometry, we conclude
that property (C) holds for the sub-semigroup action generated by
$\widetilde{G_{1}}$, which completes the proof of the corollary. ∎

######  Remark 5.3.

In the special case that $f:\mathbf{S}^{1}\to\mathbf{S}^{1}$ is an irrational
rotation it is easy to check from the proof of Lemma 5.1 and simple estimates
on the minimal covering time (cf. [13]) that the frequent hitting times
property holds with $K(\varepsilon)=[\frac{3}{\varepsilon}]+1$ . In
particular, the transition time may be exponentially large on balls with
exponentially small radius.

######  Remark 5.4.

In the case of a single homeomorphism, one can replace the assumption that the
dynamics $f$ is an isometry by equicontinuity of the family
$\\{f^{n}\\}_{n\in\mathbb{Z}}$, as in this context
$d_{f}(x,y):=\sup_{n\in\mathbb{Z}}d(f^{n}(x),f^{n}(y))$ defines a metric on
$X$ with respect to which $f$ becomes an isometry.

######  Remark 5.5.

The frequent hitting times property can be rephrased as follows: for any
$\varepsilon>0$ there exists $K(\varepsilon)\geqslant 1$ so that for any balls
$(B_{1},B_{2})$ such that $|B_{1}|=\varepsilon$ there exists $0\leqslant
p\leqslant K(\varepsilon)$,
${\underline{\omega}}\in\\{1,2,\dots,\kappa\\}^{p}$ and a ball
$B_{2}^{\prime}\subset B_{2}$ of radius
$r\geqslant\min\\{|B_{1}|/4,|B_{2}|/2\\}$ so that
$g_{\underline{\omega}}^{p}(B_{1})\supset B_{2}^{\prime}$. Indeed, if
$|B_{2}|>\varepsilon/2$ and one takes $\hat{B}_{2}\subset B_{2}$ a ball of
radius $|B_{1}|/2$ then there exists $B_{2}^{\prime}\subset\hat{B}_{2}$ of
radius $|\hat{B}_{2}|/2=|B_{1}|/4$ so that
$g_{\underline{\omega}}^{p}(B_{1})\supset B_{2}^{\prime}$.

Let us provide a simple example which fits in the context of item (3) in the
corollary.

######  Example 5.6.

Consider the torus $\mathbb{T}^{3}=\mathbb{R}^{3}/\mathbb{Z}^{3}$, let
$v_{1},v_{2},v_{3}\in\mathbb{R}^{3}$ be a base of $\mathbb{R}^{3}$ such that
the components of each vector $v_{i}$ are rationally independent. For each
$i=1,2,3$ consider the translation $f_{i}(x)=x+v_{i}(mod\,\mathbb{Z}^{3})$,
$x\in\mathbb{T}^{3}$, which is clearly an isometry. Even though each
individual translation may fail to be transitive, the choice of the vectors
$v_{i}$ guarantees that the semigroup action generated by
$\\{f_{1},f_{2},f_{3}\\}$ is minimal. According to Corollary 5.2, the
semigroup action generated by $G_{1}=\\{f_{1},f_{2},f_{3}\\}$ has frequent
hitting times.

###  5.2. Projective linear maps

In opposition to the case of the torus, where a single irrational translation
have dense orbits, minimal semigroup action in arbitrary manifolds often
require some minimum number of generators (see [39] and references therein for
a discussion on this topic). This is precisely the case for continuous maps on
spheres, which appear naturally in applications to linear cocycles as these
are double covering of real projective Euclidean spaces.

######  Example 5.7.

Let ${\mathbf{S}}^{2}\subset\mathbb{R}^{3}$ be the unit sphere an d, given
$\alpha,\beta\in\mathbb{R}\setminus\mathbb{Q}$ consider the semigroup action
on ${\mathbf{S}}^{2}$ generated by the collection $G_{1}=\\{f_{0},f_{1}\\}$ of
elements in $SO(3)$ defined as

$f_{1}=\left(\begin{array}[]{ccc}\cos\alpha&-\sin\alpha&0\\\
\sin\alpha&\cos\alpha&0\\\ 0&0&1\end{array}\right)\quad\text{and}\quad
f_{2}=\left(\begin{array}[]{ccc}1&0&0\\\ 0&\cos\beta&-\sin\beta\\\
0&\sin\beta&\cos\beta\end{array}\right)$

The maps $f_{1}$ and $f_{2}$ are irrational rotations around the $z$-axis and
the $x$-axis, respectively. It is clear from the use of polar coordinates and
the fact that the rotations are irrational that the semigroup action generated
by the collection $G_{1}=\\{f_{1},f_{2}\\}$ of isometries is minimal.
Corollary 5.2 implies that such semigroup action has frequent hitting times.

######  Example 5.8.

Let $Sp(2\ell,\mathbb{R})$ denote the Lie group of $2\ell\times
2\ell$-symplectic matrices. Assume that $A\in sp(4,\mathbb{R})$ is a $4\times
4$ symplectic matrix corresponding to a generic center, i.e. $A$ has two pairs
of non-real, complex conjugate eigenvalues with norm one. By continuity of the
isolated eigenvalues, there exists an open neighborhood $\mathscr{U}\subset
sp(4,\mathbb{R})$ of the matrix $A$ formed by symplectic matrices having
distinct unitary complex eigenvalues and, up to conjugacy by symplectic
matrices, each $B\in\mathscr{U}$ is determined by their eigenvalues in the
canonical symplectic basis where the $2$-form can be written as
$e_{1}\wedge\hat{e}_{1}+e_{2}\wedge\hat{e}_{2}$, one can write

$B=\begin{pmatrix}a&0&-b&0\\\ 0&c&0&-d\\\ b&0&a&0\\\ 0&d&0&c\end{pmatrix}$

where $a^{2}+b^{2}=1$, $c^{2}+d^{2}=1$ and $b,d\neq 0$. Up to conjugacy by a
symplectic change of basis matrix, the previous matrices are in one-to-one
correspondance with the previous parameterization of its eigenvalues $a\pm{\bf
i}\,b$ and $c\pm{\bf i}\,d$. The projectivization of $B$ is an isometry.
Moreover, there exists a Baire generic and full Haar measure subset
$\mathcal{R}\subset\mathscr{U}$ so that the eigenvalues of each
$B\in\mathcal{R}$ do not satisfy any of the equations
$(a+bi)^{m}\,(c+di)^{n}=1,$ with $m,n\in\mathbb{Z}$ and, consequently, the
projectivization of $B$ is a minimal isometry on $\mathbf{P}\mathbb{R}^{4}$.
The previous reasoning can also be action on the central direction of to
partially hyperbolic $sl(2\ell,\mathbb{R})$ matrices ($\ell>2$) having a
generic center in its 4-dimensional center subbundle.

###  5.3. Relation with other gluing orbit properties and hyperbolicity

Here we provide some examples to illustrate that the frequent hitting times
property is significantly different from the gluing orbit and specification
properties. Indeed, The specification property is frequently associated to
hyperbolic dynamics and the gluing orbit property can arise in both
hyperbolic, partially hyperbolic and non-uniformly hyperbolic dynamics (we
refer the reader [15, 14] for definitions and precise statements). However, as
observed in Subsection 5.1, the frequent hitting times property is guaranteed
for minimal semigroup actions by isometries. The next examples illustrate that
it may occur even if all others notions fail.

We say that the semigroup action generated by
$G_{1}=\\{id,f_{1},...,f_{\kappa}\\}$ satisfies the _gluing orbit property_ if
for any $\varepsilon>0$ there exists $K(\varepsilon)\geqslant 1$ so that for
any points
$(x_{1},{\underline{\omega}}_{1}),\dots(x_{r},{\underline{\omega}}_{r})$ in
$X\times\Sigma_{\kappa}$ there exist $x\in X$, $0\leqslant p_{i}\leqslant
K(\varepsilon)$ and $\theta_{i}\in\\{1,2,\dots,p\\}^{p_{i}}$ ($i=1\dots r-1$)
so that
$d(f^{j}_{{\underline{\omega}}_{1}}(x),f^{j}_{{\underline{\omega}}_{1}}(x_{1}))<\varepsilon$
for every $j=1,...,n_{1}$ and

$d(f^{j}_{{\underline{\omega}}_{\ell}}\dots
f_{\theta_{2}}^{p_{2}}{f}^{n_{2}}_{{\underline{\omega}}_{2}}(x)f_{\theta_{1}}^{p_{1}}{f}^{n_{1}}_{{\underline{\omega}}_{1}}(x),\,f^{j}_{{\underline{\omega}}_{\ell}}(x_{j}))<\varepsilon$

for every $j=2,...,n_{\ell}$ and $\ell=1,...,k$. The first example shows that
the frequent hitting times property may fail among hyperbolic dynamics.

######  Example 5.9.

Consider the linear Anosov automorphism $f_{A}$ on
$\mathbb{T}^{2}=\mathbb{R}^{2}/\mathbb{Z}^{2}$ induced by

$A=\left(\begin{array}[]{cc}2&1\\\ 1&1\end{array}\right)$

and let $q:\mathbb{R}^{2}\to\mathbb{T}^{2}$ be the quotient map. In
particular, $f_{A}$ satisfies the gluing orbit property [16]. Note that
$p=(0,0)$ is a hyperbolic fixed point for $f$, and that the $A$-invariant
hyperbolic splitting $\mathbb{R}^{2}=E^{u}\oplus E^{s}$ is such that the
invariant manifolds at $p$ are simply $W^{*}(p)=q(E^{*})$ where
$*\in\\{s,u\\}$. Let $\lambda>1$ be the leading eigenvalue for $A$ (and, since
$A$ is volume preserving, $\lambda^{-1}$ is the contracting eigenvalue for
$A$). Given $\varepsilon>0$, the set $f_{A}^{n}(B(p,\varepsilon))$ is the
projection by $q$ of an ellipsis in $\mathbb{R}^{2}$ whose larger and smaller
axis have length $\lambda^{n}\varepsilon$ and $\lambda^{-n}\varepsilon$,
respectively, wrapped in the torus. In consequence, the inner diameter of the
set $f_{A}^{n}(B(p,\varepsilon))$ decreases exponentially fast with $n$, hence
the frequent hitting times property does not hold.

######  Example 5.10.

Consider the semigroup action $\mathbb{S}$ generated by the homeomorphisms
$G_{1}=\\{id,f_{1},f_{2}\\}$ on the circle
$\mathbf{S}^{1}=\mathbb{R}/\mathbb{Z}$ where $f_{0}$ is a north-pole south-
pole diffeomorphism and $f_{1}$ a irrational rotation (see Figure 5.1 below).
By [13] and Corollary 5.2, the map $f_{1}$ satisfies both the gluing orbit and
the frequent hitting times properties. In particular the semigroup action
$\mathbb{S}$ has frequent hitting times.

Figure 5.1. Iterated function system with two generators $f_{0},f_{1}$ on the
circle $\mathbf{S}^{1}$: a north-pole south-pole homeomorphism and an
irrational rotation $f_{1}$

We proceed to show that such semigroup action does not satisfy a gluing orbit
property nor any non-uniform specification property. First, since $f_{1}$ is
an irrational rotation then the orbit of $\frac{1}{2}$ cannot contain $0$
(otherwise a composition of these rotations would be a rotation of rational
angle $1/2$). For any $\delta>0$ let $k(\delta)\geqslant 1$ be the largest
integer so that $d(f_{1}^{j}(\frac{1}{2}),1)>2\delta$ for every $0\leqslant
j\leqslant k(\delta)$. Clearly $k(\delta)\to+\infty$ as $\delta\to 0$. Fix
$\varepsilon>0$, $x_{1}=\frac{1}{2}$ and $x_{2}=0$. By the uniformly
contracting (resp. expanding) behavior in a neighborhood of $x_{1}$ (resp.
$x_{2}$), there exists $\lambda\in(0,1)$ so that
$f_{0}^{n}(B_{f_{0}}(x_{1},n,\varepsilon))\subset
B(x_{1},\lambda^{n}\varepsilon)$ and $B_{f_{0}}(x_{2},n,\varepsilon)\subset
B(x_{2},\lambda^{n}\varepsilon)$ for every small $\varepsilon>0$ and every
$n\geqslant 1$. Observe that the contracting behavior in a neighborhood of
$x_{1}$ guarantees that
$f_{0}^{(j+1)+n}(B_{f_{0}}(x_{1},n,\varepsilon))\subset
f_{0}^{j+n}(B_{f_{0}}(x_{1},n,\varepsilon))$ and
$f_{0}^{j+n}(B_{f_{0}}(x_{1},n,\varepsilon))\cap
B_{f_{0}}(x_{2},n,\varepsilon)=\emptyset$, for every $j\geqslant 0$. Hence,
any possible orbit between the sets $B_{f_{0}}(x_{1},n,\varepsilon)$ and
$B_{f_{0}}(x_{2},n,\varepsilon)$ need to arise from a concatenation of the map
$f_{1}$ for a certain number of times. By construction,

$f_{1}^{j}f_{0}^{n}(B_{f_{0}}(x_{1},n,\varepsilon))\cap
B_{f_{0}}(x_{2},n,\varepsilon)=\emptyset\quad\text{for every $0\leqslant
j\leqslant k(\lambda^{n}\varepsilon)$}$

and, consequently, not only any such transition time depends on the size $n$
of the orbit of $x_{1}$ as it grows exponentially fast with $n$. Indeed, by
[13] the minimal integer $\lfloor\frac{1}{\delta}\rfloor-1\leqslant
K_{0}(\delta)\leqslant\lfloor\frac{1}{\delta}\rfloor+1$ so that
$\bigcup_{j=0}^{K_{0}(\delta)}f_{1}^{j}(B(z,\delta))=\mathbf{S}^{1}$ for every
$z\in\mathbf{S}^{1}$ is bounded above by $2K_{1}(\delta)$ where
$K_{1}(\delta)\geqslant 1$ is the smallest integer so that
$f_{1}^{j}(B(z,\delta))\cap B(z,\delta)=\emptyset$ for every
$z\in\mathbf{S}^{1}$ (as $f_{1}$ is a minimal isometry the integers
$K_{0}(\delta),K_{1}(\delta)$ are independent of the point $z$). As $0$ and
$\frac{1}{2}$ are in diametrically opposite positions one has that
$k(\delta)\geqslant\lfloor\frac{1}{2}K_{1}(\frac{\delta}{2})\rfloor\geqslant\lfloor\frac{1}{\delta}\rfloor-1$
for every small $\delta>0$, which ensures that
$\liminf_{n\to\infty}\frac{1}{n}\log
k(\lambda^{n}\varepsilon)\geqslant-\log\lambda>0$. In particular the semigroup
does not satisfy a non-uniform specification property, ie, the transition
times cannot be taken assumed to grow in a sub-linear way in comparison to the
size of shadowed orbits.

######  Remark 5.11.

The previous example has similar implications for the corresponding skew-
product maps. Indeed, assume that
$F:\mathbb{T}^{2}\times\mathbf{S}^{1}\to\mathbb{T}^{2}\times\mathbf{S}^{1}$ is
a $C^{1}$ partially hyperbolic diffeomorphism of the form
$F(x,y)=(A(x),g(x,y))$ where $A:\mathbb{T}^{2}\to\mathbb{T}^{2}$ is an Anosov
diffeomorphism and that there exists $p_{j}\in Per(A)$ of period
$\pi(p_{j})\geqslant 1$ such that
$F^{\pi(p_{j})}(p_{j},\cdot):\mathbf{S}^{1}\to\mathbf{S}^{1}$ coincides with
the map $f_{j}$ ($j=0,1$). Since $F$ has periodic points of different index
then it does not satisfy the gluing orbit property [14]. While non-uniform
specification holds for typical points of hyperbolic measures [56], the
previous example suggests that a global non-uniform specification property
fails dramatically.

In the remainder of this subsection we discuss the strength of the frequent
hitting times in the process of shadowing finite pieces of orbits, with the
specification, gluing orbit property and non-uniform specification. In fact,
instead of being bounded, or even having sublinear growth with respect to the
size of the finite pieces of orbits, transition times may be much larger than
the size of the finite pieces of orbits: given a finite collection
$(x_{i},n_{i})_{1\leqslant i\leqslant\ell}$, the frequent hitting times
property guarantees that that there exist $p_{i}\geqslant 1$, depending on
$\varepsilon$, ($1\leqslant i\leqslant\ell-1$) and $x\in X$ so that

$0<n_{1}\leqslant n_{1}+p_{1}\leqslant
n_{2}+(n_{1}+p_{1})\leqslant\dots\leqslant
n_{\ell}+\sum_{i=0}^{\ell-1}(n_{i}+p_{i})$ (5.1)

where

  1. (1)

during the shadowing intervals
$[\sum_{i=0}^{j-1}(n_{i}+p_{i}),n_{j}+\sum_{i=0}^{j-1}(n_{i}+p_{i})]$ the
orbit of $x$ remains $\varepsilon$-close to the orbit of the point $x_{j}$

  2. (2)

the transition interval
$[n_{j}+\sum_{i=0}^{j-1}(n_{i}+p_{i}),n_{j}+p_{j}+\sum_{i=0}^{j-1}(n_{i}+p_{i})]$
has size $p_{j}$

  3. (3)

$p_{j}$ may have exponential growth with respect to $n_{j}$.

Figure 5.2. Time lags and oscillating behavior between shadowing finite pieces
of orbits under: (1) specification, (2) gluing orbit property, (3) non-uniform
specification, and (4) frequent hitting time property.

Yet, while $p_{i}$ may be much larger than $n_{i}$ the crucial fact is that it
has memory loss, meaning it depends exlusively on it and not on the size of
future pieces of orbits. This makes possible to choose an adequate size
$n_{j+1}$ of an orbit and bookkeeping the previous transition times. The
effect obtained in the construction of points with oscillating time averages
is illustrated in Figures 5.2 and 5.3. We refer the reader to Section 6 and
expressions (6.4) for more details.

Figure 5.3. Illustration of the oscillating behavior of time averages along
the shadowing of finite pieces of orbits for maps satisfying: (1)
specification, (2) gluing orbit property, (3) non-uniform specification, and
(4) frequent hitting time property.

##  6\. The set of irregular points for semigroup actions is Baire generic

This section is devoted to the proof of Theorem C. In order to ease the
presentation, we first explain how the assumptions in the theorem allow one to
construct points in $X$ and an infinite path on the semigroup along which such
point has divergent Birkhoff averages for the continuous observable $\psi$
(Subsection 6.1). Exploring these ideas, we then prove that the set of non-
typical points for the skew-product $F$ and the observable $\varphi_{\psi}$
forms a Baire generic subset of $\Sigma_{\kappa}\times X$ (Subsection 6.2).
The proof of the theorem is completed in Subsection 6.3.

###  6.1. The set of irregular points is non-empty

Assume that $G_{1}=\\{id,f_{1},f_{2},\dots,f_{\kappa}\\}$ is a collection of
bi-Lipschitz homeomorphisms on a compact metric space $X$ and that
$\psi:X\to\mathbb{R}$ is a continuous observable which is not a coboundary
with respect to some of the elements in $G_{1}$. Assume, without loss of
generality, that

$\inf_{\mu\in\mathcal{M}_{1}(f_{\kappa})}\int\psi\,d\mu<\sup_{\mu\in\mathcal{M}_{1}(f_{\kappa})}\int\psi\,d\mu.$

Let $\mu_{1},\mu_{2}$ be $f_{\kappa}$-invariant and ergodic probability
measures such that $\int\psi\,d\mu_{1}<\int\psi\,d\mu_{2}$ and take $x_{i}\in
B(\mu_{i})$ be points in the ergodic basin of attraction with respect to
$f_{\kappa}$, for $i=1,2$. In particular, there are integers $1\leqslant
n_{1}<n_{2}$ so that

$\Big{|}\frac{1}{n}\sum_{j=0}^{n-1}\psi(f_{\kappa}^{j}(x_{i}))-\int\psi\,d\mu_{i}\Big{|}<\frac{\int\psi\,d\mu_{2}-\int\psi\,d\mu_{1}}{8}\quad\text{for
every}\,n\geqslant n_{i}$ (6.1)

for $i=1,2$. By uniform continuity of $\psi$, there one may choose
$\varepsilon_{0}>0$ such that

$|\psi(x)-\psi(y)|<\frac{\int\psi\,d\mu_{2}-\int\psi\,d\mu_{1}}{8}\quad\text{for
every $x,y\in X$ so that $d(x,y)<\varepsilon_{0}$.}$ (6.2)

To ease the notation, for each $x\in X$, $\varepsilon>0$ and $n\geqslant 1$
denote by $B_{f_{\kappa}}(x,n,\varepsilon)$ the dynamic ball taken with
respect to $f_{\kappa}$, that is,

$B_{f_{\kappa}}(x,n,\varepsilon):=\big{\\{}y\in X\colon
d(f_{\kappa}^{j}(y),f_{\kappa}^{j}(x))\leqslant\varepsilon\;\text{for
every}\;0\leqslant j\leqslant n-1\big{\\}}.$

Since the semigroup action satisfies the frequent hitting time property, by