# Sensitivity and historic behavior for continuous
maps on Baire metric spaces

M. Carvalho  ,  V. Coelho  ,  L. Salgado  and  P. Varandas  Maria Carvalho,
CMUP & Departamento de Matemática, Faculdade de Ciências da Universidade do
Porto
Rua do Campo Alegre 687, 4169-007 Porto, Portugal<EMAIL_ADDRESS>Vinícius
Coelho, Universidade Federal do Oeste da Bahia, Centro Multidisciplinar de Bom
Jesus da Lapa
Av. Manoel Novais, 1064, Centro, 47600-000 - Bom Jesus da Lapa-BA-Brazil
<EMAIL_ADDRESS>Luciana Salgado, Universidade Federal do Rio de
Janeiro, Instituto de Matemática
Avenida Athos da Silveira Ramos 149 Cidade Universitária, P.O. Box 68530,
21941-909 Rio de Janeiro-RJ-Brazil<EMAIL_ADDRESS><EMAIL_ADDRESS>Paulo Varandas, CMUP, Universidade do Porto &
Departamento de Matemática, Universidade Federal da Bahia
Av. Ademar de Barros s/n, 40170-110 Salvador, Brazil<EMAIL_ADDRESS>
###### Abstract.

We introduce a notion of sensitivity, with respect to a continuous bounded
observable, which provides a sufficient condition for a continuous map, acting
on a Baire metric space, to exhibit a Baire generic subset of points with
historic behavior. The applications of this criterion recover, and extend,
several known theorems on the genericity of the irregular set, besides
yielding a number of new results, including information on the irregular set
of geodesic flows, in both negative and non-positive curvature, and semigroup
actions.

###### Key words and phrases:

Baire metric space; Historic behavior; Irregular set; Transitivity;
Sensitivity, Group and semigroup actions

###### 2010 Mathematics Subject Classification:

Primary: 37A30, 37C10, 37C40, 37D20.

##  1\. Introduction

###  1.1. Historic behavior

In what follows, we shall write $X$ to denote a compact metric space and $Y$
will stand for an arbitrary metric space. Given such a space $Y$ and $A\subset
Y$, denote by $A^{\prime}\subset S$ the set of non-isolated accumulation
points in $A$, that is, $y\in A^{\prime}$ if and only if $y$ belongs to the
closure $\overline{A\setminus\\{y\\}}$. Let $C(Y,\mathbb{R})$ be the set of
real valued, continuous maps on $Y$ and $C^{b}(Y,\mathbb{R})$ be its subset of
bounded elements endowed with the supremum norm $\|\cdot\|_{\infty}$.

Given a Baire metric space $(Y,d)$, a continuous map $T:Y\to Y$ and
$\varphi\in C^{b}(Y,\mathbb{R})$, the set of _$(T,\varphi)$ -irregular
points_, or points with historic behavior, is defined by

${\mathcal{I}}(T,\varphi)=\Big{\\{}x\in
X:\Big{(}\lim\limits_{n\,\to\,+\infty}\frac{1}{n}\sum\limits_{j=0}^{n-1}\varphi(T^{j}(x))\Big{)}_{n\,\in\,\mathbb{N}}$
does not converge$\Big{\\}}$.

The Birkhoff’s ergodic theorem ensures that, for any Borel $T$-invariant
probability measure $\mu$ and every $\mu$-integrable observable
$\varphi:Y\to\mathbb{R}$, the sequence of averages
$\big{(}\frac{1}{n}\sum_{j=0}^{n-1}\varphi(T^{j}(y))\big{)}_{n\,\in\,\mathbb{N}}$
converges at $\mu$-almost every point in $Y$. So, the set of
$(T,\varphi)$-irregular points is negligible with respect to any $T$-invariant
probability measure. In the last decades, though, there has been an intense
study concerning the set of points for which Cesàro averages do not converge.
Contrary to the previous measure-theoretical description, the set of the
irregular points may be Baire generic and, moreover, have full topological
pressure, full metric mean dimension or full Hausdorff dimension (see [2, 5,
3, 4, 22, 24, 31]). In [8], the first and the fourth named authors obtained a
simple and unifying criterion, using first integrals, to guarantee that
${\mathcal{I}}(T,\varphi)$ is Baire generic whenever $T:X\to X$ is a
continuous dynamics acting on a compact metric space $X$. More precisely,
given $\varphi\in C(X,\mathbb{R})$, consider the map
$L_{\varphi}:X\to\mathbb{R}$ defined by

$x\in X\quad\mapsto\quad
L_{\varphi}(x)=\limsup\limits_{n\to\infty}\frac{1}{n}\sum\limits_{j=0}^{n-1}\varphi\circ
T^{j}(x).$ (1.1)

This is a first integral with respect to the map $T$, that is,
$L_{\varphi}\circ T=L_{\varphi}$. The existence of dense sets of discontinuity
points for this first integral turns out to be a sufficient condition for the
genericity of the historic behavior.

######  Theorem 1.1.

[8, Theorem A] Let $(X,d)$ be a compact metric space, $T:X\to X$ be a
continuous map and $\varphi:X\to\mathbb{R}$ be a continuous observable. Assume
that there exist two dense subsets $A,B\subset X$ such that the restrictions
of $L_{\varphi}$ to $A$ and to $B$ are constant, though the value at $A$ is
different from the one at $B$. Then ${\mathcal{I}}(T,\varphi)$ is a Baire
generic subset of $X$.

The assumptions of the previous theorem are satisfied by a vast class of
continuous maps on compact metric spaces, including minimal non-uniquely
ergodic homeomorphisms, non-trivial homoclinic classes, continuous maps with
the specification property, Viana maps and partially hyperbolic
diffeomorphisms (cf. [8]).

In this work we establish a criterion with a wider scope than the one of
Theorem 1.1. It applies to Baire metric spaces and general sequences of
bounded continuous real-valued maps, rather than just Cesàro averages, subject
to a weaker requirement than the one demanded in the previous theorem. In
particular, one obtains new results on the irregular set of several classes of
maps and flows, which comprise geodesic flows on certain non-compact
Riemannian manifolds, countable Markov shifts and endomorphisms with two
physical measures exhibiting intermingled basins of attraction. Regarding
semigroup actions, we note that irregular points for group actions with
respect to Cesàro averages were first studied in [16]. We will provide
additional information on the Baire genericity of irregular sets for averages
that take into account the group structure. We refer the reader to Sections 10
and 11 for the precise statements.

In the next subsections we will state our main definitions and results.

###  1.2. Sensitivity and genericity of historic behavior

Given a metric space $(Y,d)$, a sequence
$\Phi=(\varphi_{n})_{n\,\in\mathbb{N}}\in C^{b}(Y,\mathbb{R})^{\mathbb{N}}$
and $y\in Y$, let
$W_{\Phi}(y):=\big{\\{}\varphi_{n}(y):\,\,n\in\mathbb{N}\big{\\}}^{\prime}$
denote the set of accumulation points of the sequence
$(\varphi_{n}(y))_{n\,\in\,\mathbb{N}}.$ The next notion is inspired by the
concept of sensitivity to initial conditions.

######  Definition 1.

_Let $(Y,d)$ be a metric space and $\Phi\in C^{b}(Y,\mathbb{R})^{\mathbb{N}}$.
We say that $Y$ is $\Phi$-_sensitive_ (or sensitive with respect to the
sequence $\Phi$) if there exist dense subsets $A,\,B\subset Y$, where $B$ can
be equal to $A$, and $\varepsilon>0$ such that for any $(a,b)\in A\times B$
one has_

$\sup_{r\,\in\,W_{\Phi}(a),\,\,s\,\in\,W_{\Phi}(b)}\,\,|r-s|>\varepsilon.$

_In the particular case of a sequence $\Phi$ of Cesàro averages
$(\varphi_{n})_{n\,\in\,\mathbb{N}}=\big{(}\frac{1}{n}\,\sum_{j=0}^{n-1}\,\varphi\circ
T^{j}\big{)}_{n\,\in\,\mathbb{N}}$ associated to a potential $\varphi\in
C^{b}(Y,\mathbb{R})$ and a continuous map $T:Y\to Y$, we say that $Y$ is
$(T,\varphi)$-_sensitive_ if the space $Y$ is $\Phi$-_sensitive_ , and write
$W_{\varphi}$ instead of $W_{\Phi}$._

We refer the reader to Example 1 as an illustration of this definition. We
observe that being $(T,\varphi)$-sensitive is a direct consequence of the
assumption on $L_{\varphi}$ stated in Theorem 1.1, though it may be strictly
weaker (cf. Example 3). Our first result concerns $\Phi$-sensitive sequences
and strengths Theorem 1.1.

######  Theorem A.

Let $(Y,d)$ be a Baire metric space and $\Phi\in
C^{b}(Y,\mathbb{R})^{\mathbb{N}}$ be a sequence of continuous, bounded maps
such that $\limsup_{n\,\to\,+\infty}\,\|\varphi_{n}\|_{\infty}<+\infty$. If
$Y$ is $\Phi$-sensitive then the set

${\mathcal{I}}(\Phi):=\Big{\\{}y\in
Y\colon\,\lim_{n\,\to\,+\infty}\,\varphi_{n}(y)\,\text{does not
exist}\,\Big{\\}}$

is a Baire generic subset of $Y$. In particular, if $T:Y\to Y$ is a continuous
map, $\varphi$ belongs to $C^{b}(Y,\mathbb{R})$ and $Y$ is
$(T,\varphi)$-sensitive, then ${\mathcal{I}}(T,\varphi)$ is a Baire generic
subset of $Y$.

###  1.3. Irregular points for continuous maps on Baire metric spaces with
dense orbits

In this subsection, building over [14, 19], we will discuss the relation
between transitivity, existence of dense orbits and the size of the set of
irregular points for continuous maps on Baire metric spaces.

######  Definition 2.

Given a continuous map $T:Y\to Y$ on a Baire metric space $(Y,d)$, one says
that:

  * •

$T$ is transitive if for every non-empty open sets $U,\,V\subset Y$ there
exists $n\in\mathbb{N}$ such that $U\cap T^{-n}(V)\neq\emptyset$.

  * •

$T$ is strongly transitive if $\bigcup_{n\,\geqslant\,0}\,T^{n}(U)=Y$ for
every non-empty open set $U\subset Y$.

  * •

$T$ has a dense orbit if there is $y\in Y$ such that
$\\{T^{j}(y):\,j\,\in\,\mathbb{N}\cup\\{0\\}\\}$ is dense in $Y$.

It is worthwhile observing that, if the metric space is compact and has no
isolated points, then the map is transitive if and only if it has a dense
orbit (see [1, Theorem 1.4] and Example 2).

Denote by $\operatorname{Trans}(Y,T)$ the set $\\{y\in Y\colon\,\text{ the
orbit of $y$ by $T$ is dense in }Y\\}$ and take on the following notation:

$\displaystyle\mathcal{H}(Y,T)\coloneqq\\{\varphi\in
C^{b}(Y,\mathbb{R})\colon\,{\mathcal{I}}(T,\varphi)\neq\emptyset\\}$
$\displaystyle\mathcal{D}(Y,T)\coloneqq\\{\varphi\in
C^{b}(Y,\mathbb{R})\colon\,{\mathcal{I}}(T,\varphi)\text{ is dense in }Y\\}$
$\displaystyle\mathcal{R}(Y,T)\coloneqq\\{\varphi\in
C^{b}(Y,\mathbb{R})\colon\,{\mathcal{I}}(T,\varphi)\text{ is Baire generic in
}Y\\}.$

The set of _completely irregular points_ with respect to $T$ is precisely the
intersection

$\bigcap\limits_{\varphi\,\in\,\mathcal{H}(Y,T)}\,{\mathcal{I}}(T,\varphi).$

Clearly $\mathcal{R}(Y,T)\subset\mathcal{D}(Y,T)\subset\mathcal{H}(Y,T)$.
Inspired by [19], we aim at finding sufficient conditions on $(Y,T)$ and
$\varphi$ under which the sets $\mathcal{R}(Y,T)$ and $\mathcal{D}(Y,T)$
coincide. We note that the set $\mathcal{H}(Y,T)$ may be uncountable and that
Tian proved in (cf. [30, Theorem 2.1]) that, if $Y$ is compact and $T$ has the
almost-product and uniform separation properties, then the set of completely
irregular points is either empty or carries full topological entropy. More
recently, in [19], Hou, Lin and Tian showed that, for each transitive
continuous map on a compact metric space $Y$, either every point with dense
orbit is contained in the basin of attraction of an invariant probability
measure (so it is regular with respect to any continuous potential), or
irregular behavior occurs on $\operatorname{Trans}(Y,T)$ and the irregular set
is Baire generic for every _typical_ $\varphi\in C(X,\mathbb{R})$. The next
consequence of Theorem A generalizes this information.

######  Corollary B.

Let $(Y,d)$ be a Baire metric space and $T:Y\to Y$ be a continuous map such
that $\operatorname{Trans}(Y,T)\neq\emptyset$. Then:

  * ($i$)

When $(Y,d)$ has an isolated point,

$\mathcal{D}(Y,T)\,\neq\,\emptyset\quad\quad\Leftrightarrow\quad\quad\operatorname{Trans}(Y,T)\,\subseteq\bigcap\limits_{\varphi\,\in\,\mathcal{D}(Y,T)}\,{\mathcal{I}}(T,\varphi).$

In particular, if $Y$ has an isolated point and
$\mathcal{D}(Y,T)\neq\emptyset$, then $\mathcal{R}(Y,T)=\mathcal{D}(Y,T)$.

  * ($ii$)

If $\mathfrak{F}\subset C^{b}(Y,\mathbb{R})$ and
$\bigcap\limits_{\varphi\,\in\,\mathfrak{F}}\,{\mathcal{I}}(T,\varphi)$ is
Baire generic, then

$\operatorname{Trans}(Y,T)\,\cap\,\bigcap\limits_{\varphi\,\in\,\mathfrak{F}}\,{\mathcal{I}}(T,\varphi)\neq\emptyset.$

###  1.4. Oscillation of the time averages

Define for any $\varphi\in C^{b}(Y,\mathbb{R})$ and $y\in Y$,

$\ell_{\varphi}(y):=\,\liminf\limits_{n\,\to\,+\infty}\,\frac{1}{n}\,\sum_{j=0}^{n-1}\,\varphi(T^{j}y)\quad\quad\text{
and }\quad\quad
L_{\varphi}(y):=\,\limsup\limits_{n\,\to\,+\infty}\,\frac{1}{n}\,\sum_{j=0}^{n-1}\,\varphi(T^{j}y)$

and consider

$\ell_{\varphi}^{*}=\,\inf\limits_{y\,\in\,\operatorname{Trans}(Y,T)}\,\ell_{\varphi}(y)\quad\quad\text{
and }\quad\quad
L_{\varphi}^{*}=\,\sup\limits_{y\,\in\,\operatorname{Trans}(Y,T)}\,L_{\varphi}(y).$
(1.2)

For each $\alpha\leqslant\beta$, take the sets

$I_{\varphi}[\alpha,\,\beta]\,:=\,\Big{\\{}y\in
Y\colon\,\ell_{\varphi}(y)=\alpha\text{ and }\beta=L_{\varphi}(y)\Big{\\}}$

and

$\widehat{I_{\varphi}[\alpha,\,\beta]}\,:=\,\Big{\\{}y\in
Y\colon\,\ell_{\varphi}(y)\leqslant\alpha\text{ and }\beta\leqslant
L_{\varphi}(y)\Big{\\}}.$

The next corollary estimates the topological size of the previous level sets
for dynamics with dense orbits.

######  Corollary C.

Let $(Y,d)$ be a Baire metric space and $T:Y\to Y$ be a continuous map with a
dense orbit. Given $\varphi\in C^{b}(Y,\mathbb{R})$, one has:

  * $(i)$

$I_{\varphi}[\ell_{\varphi}^{*},\,L_{\varphi}^{*}]$ is a Baire generic subset
of $Y$. In particular, ${\mathcal{I}}(T,\varphi)$ is either Baire generic or
meagre in $Y$.

  * $(ii)$

${\mathcal{I}}(T,\varphi)$ is a meagre subset of $Y$ if and only if there
exists $C_{\varphi}\in\mathbb{R}$ such that

$I_{\varphi}[C_{\varphi},\,C_{\varphi}]\,=\,\Big{\\{}y\in
Y\colon\,\lim\limits_{n\,\to\,+\infty}\,\frac{1}{n}\,\sum\limits_{j=0}^{n-1}\,\varphi(T^{j}(y))=C_{\varphi}\Big{\\}}$

is a Baire generic set containing $\operatorname{Trans}(Y,T)$.

Consequently, if $(Y,d)$ is a Baire metric space, $T:Y\to Y$ is a continuous
map with a dense orbit and
$\bigcup_{\varphi\,\in\,C^{b}(Y,T)}\,{\mathcal{I}}(T,\varphi)$ is not Baire
generic, then $Y\setminus{\mathcal{I}}(T,\varphi)$ is Baire generic for every
$\varphi\in C^{b}(Y,\mathbb{R})$, and there exists a linear functional
$\mathcal{F}:C^{b}(Y,\mathbb{R})\to\mathbb{R}$ such that

$\operatorname{Trans}(Y,T)\subset\Big{\\{}y\in
Y:\lim_{n\,\to\,+\infty}\,\frac{1}{n}\,\sum_{j=0}^{n-1}\,\varphi(T^{j}(y))\,=\,\mathcal{F}(\varphi)\quad\quad\forall\,\varphi\in
C^{b}(Y,\mathbb{R})\Big{\\}}.$

Note that, if $Y$ is compact, $\mathcal{F}$ is represented by the space of
Borel invariant measures with the weak∗-topology. More generally, if $Y$ is a
metric space, then the dual of $C^{b}(Y,\mathbb{R})$ is represented by the
regular, bounded, finitely additive set functions with the norm of total
variation (cf. [15, Theorem 2, IV.6.2]).

######  Remark 1.2.

Corollary C implies that, if $(Y,d)$ is a Baire metric space and $T:Y\to Y$ is
a continuous minimal map (that is, $\operatorname{Trans}(Y,T)=Y$), then for
every $\varphi\in C^{b}(X,\mathbb{R})$ either
${\mathcal{I}}(T,\varphi)=\emptyset$ or ${\mathcal{I}}(T,\varphi)$ is Baire
generic.

######  Remark 1.3.

From [19, Theorem B], we know that, in the context of compact metric spaces
and transitive maps, the set ${\mathcal{I}}(T,\varphi)$ is either meagre or
Baire generic for every $\varphi\in C(X,\mathbb{R})$. It is unknown whether
there exist a Baire metric space $(Y,d)$, a continuous map $T:Y\to Y$ and
$\varphi\in C^{b}(Y,\mathbb{R})$ such that $T$ has a dense orbit and
${\mathcal{I}}(T,\varphi)$ is a non-empty meagre set.

The next corollary is a counterpart of [19, Lemma 3.1] for continuous maps on
Baire metric spaces.

######  Corollary D.

Let $(Y,d)$ be a Baire metric space and $T:Y\to Y$ be a continuous map with a
dense orbit, and take $\varphi\in C^{b}(Y,\mathbb{R})$. The following
conditions are equivalent:

  * $(i)$

$Y$ is $(T,\varphi)$-sensitive.

  * $(ii)$

${\mathcal{I}}(T,\varphi)$ is Baire generic in $X$.

  * $(iii)$

$\operatorname{Trans}(Y,T)\cap{\mathcal{I}}(T,\varphi)\neq\emptyset$.

######  Remark 1.4.

Corollary D implies that $\mathcal{R}(Y,T)=\big{\\{}\varphi\in
C^{b}(Y,\mathbb{R})\colon\,\ell_{\varphi}^{*}\,<\,L_{\varphi}^{*}\big{\\}}.$
We also note that one has $(i)\Rightarrow(ii)$ in Corollary D even without
assuming the existence of dense orbits.

######  Remark 1.5.

An immediate consequence of the proof of Corollary D is that, within the
setting of continuous maps $T:Y\to Y$ acting on a Baire metric space $Y$ and
having a dense orbit, the Definition 1 of $(T,\varphi)$-sensitivity is
equivalent to the following statement: _There exist a dense set $A\subset Y$
and $\varepsilon>0$ such that for any $a\in A$ one has_

$\sup_{r\,\in\,W_{\varphi}(a),\,\,s\,\in\,W_{\varphi}(a)}\,\,|r-s|>\varepsilon.$

Indeed, the latter statement clearly implies Definition 1. Conversely, if $X$
is $(T,\varphi)$-sensitive, then we may take
$A=\\{T^{j}(x_{0})\colon\,j\in\mathbb{N}\,\cup\,\\{0\\}\\}$ equal to a dense
orbit contained in ${\mathcal{I}}(T,\varphi)$, whose existence is guaranteed
by item $(iii)$ of Corollary D. Given two distinct accumulation points $r,\,s$
in $W_{\varphi}(x_{0})$ and $\varepsilon=|r-s|>0$, then for every $a,\,b\in A$
one has $W_{\varphi}(a)=W_{\varphi}(b)=W_{\varphi}(x_{0})$ and
$\sup_{r_{a}\,\in\,W_{\Phi}(a),\,\,r_{b}\,\in\,W_{\Phi}(b)}\,\geqslant\,|r-s|>\varepsilon.$

The remainder of the paper is organized as follows. In Section 2, we convey
the previous results to the particular case of continuous dynamics acting on
compact metric spaces. The aforementioned results are then proved in the
ensuing sections, where we also discuss their scope and compare them with
properties established in other references. In Section 10 we test our
assumptions on some examples and in Section 11 we provide some applications,
namely within the settings of semigroup actions and geodesic flows on non-
compact manifolds.

##  2\. Irregular points for continuous maps on compact metric spaces

Suppose now that $(X,d)$ is a compact metric space. Let $\mathcal{P}(X)$ stand
for the Borel probability measures on $X$ with the weak∗-topology and consider
a continuous map $T:X\to X$ with a dense orbit. For every $x\in X$, let
$\delta_{x}$ be the Dirac measure supported at $x$ and denote the set of
accumulation points in $\mathcal{P}(X)$ of the sequence of empirical measures
$\big{(}\frac{1}{n}\,\sum_{j=0}^{n-1}\,\delta_{T^{j}x}\big{)}_{n\,\in\,\mathbb{N}}$
by $V_{T}(x)$. Our next result is a refinement of [19, Theorem A], since it
imparts new information about the irregular set, besides establishing a
generalization of [19, Theorem B] inasmuch as it does not require any
assumption about isolated points.

######  Corollary 2.1.

Let $(X,d)$ be a compact metric space and $T:X\to X$ be a continuous map with
a dense orbit. Then:

  * $(a)$

$X_{\Delta}\coloneqq\,\Big{\\{}x\in
X\colon\bigcup_{t\,\in\,\operatorname{Trans}(X,T)}\,V_{T}(t)\,\subseteq\,V_{T}(x)\Big{\\}}$
is Baire generic in $X$. Moreover,
$X_{\Delta}\,\subseteq\,\bigcap_{\varphi\,\in\,C(X,\mathbb{R})}\,\widehat{I_{\varphi}[\ell_{\varphi}^{*},\,L_{\varphi}^{*}]}$,
so the latter set is Baire generic as well.

  * $(b)$

$\bigcup_{\varphi\,\in\,C(X,\mathbb{R})}\,{\mathcal{I}}(T,\varphi)$ is either
Baire generic or meagre. If it is meagre, there exists a Borel $T$-invariant
measure $\mu$ such that

$\operatorname{Trans}(X,T)\,\subset\,\Big{\\{}x\in
X:\,\lim_{n\,\to\,+\infty}\,\frac{1}{n}\,\sum_{j=0}^{n-1}\,\varphi(T^{j}(x))=\int\varphi\,d\mu\quad\forall\,\varphi\in
C(X,\mathbb{R})\Big{\\}}.$

  * $(c)$

$\bigcap_{\varphi\,\in\,\mathcal{H}(X,T)}\,{\mathcal{I}}(T,\varphi)$ is either
Baire generic or meagre. In addition,

$\bigcap_{\varphi\,\in\,\mathcal{H}(X,T)}\,{\mathcal{I}}(T,\varphi)\text{ is
Baire generic }\,\,\Leftrightarrow\,\,\Big{\\{}\varphi\in
C(X,\mathbb{R})\colon\,{\mathcal{I}}(T,\varphi)\neq\emptyset\text{ and
}\ell_{\varphi}^{*}=L_{\varphi}^{*}\Big{\\}}\,=\,\emptyset.$

######  Remark 2.2.

It follows from Corollary 2.1 that, if $(X,d)$ is a compact metric space and
$T:X\to X$ is a continuous minimal map, then either $T$ is uniquely ergodic or
the set $\bigcap_{\varphi\,\in\,\mathcal{H}(X,T)}\,{\mathcal{I}}(T,\varphi)$
is Baire generic.

We may ask whether the notion of $(T,\varphi)$-sensitivity (cf. Definition 1)
is somehow related to the classical concepts of sensitivity to initial
conditions and expansiveness. Let us recall these two notions.

######  Definition 3.

Let $T:X\to X$ be a continuous map on a compact metric space $(X,d)$. We say
that:

  * •

$T$ has _sensitivity to initial conditions_ if there exists $\varepsilon>0$
such that, for every $x\in X$ and any $\delta>0$, there is $z\in B(x,\delta)$
satisfying

$\sup_{n\,\in\,\mathbb{N}}\,d(T^{n}(x),T^{n}(z))>\varepsilon.$

  * •

$T$ is (positively) _expansive_ if there exists $\varepsilon>0$ such that, for
any points $x,z\in X$ with $x\neq z$, one has

$\sup_{n\,\in\,\mathbb{N}}\,d(T^{n}(x),T^{n}(z))>\varepsilon.$

It is clear that an expansive map has sensitivity to initial conditions. Our
next result shows that the condition of $(T,\varphi)$-sensitivity often
implies sensitivity to initial conditions.

######  Theorem 2.3.

Let $X$ be a compact metric space, $T:X\to X$ be a continuous map and
$\varphi\in C(X,\mathbb{R})$. If $X$ is $(T,\varphi)$-sensitive then either
$X$ has sensitivity to initial conditions or ${\mathcal{I}}(T,\varphi)$ has
non-empty interior. In particular, if $T$ has a dense set of periodic orbits
and $X$ is $(T,\varphi)$-sensitive, then $X$ has sensitivity to initial
conditions.

The previous discussion together with well known examples yield the following
scheme of connections:

$\begin{array}[]{ccc}\text{$T$ is strongly transitive}&&\\\ \qquad\text{ and
has dense periodic orbits}&&\\\ \Downarrow&&\\\ \text{$T$ has dense periodic
orbits}&&\\\ \text{and dense pre-orbits}&&\text{Expansiveness}\\\
\Downarrow&&\Downarrow\;\;\not\Uparrow\\\ \exists\,\varphi\in
C(X,\mathbb{R})\colon\,X\text{ is
}(T,\varphi)-{sensitive}&\quad\Rightarrow&\text{Sensitivity to initial
conditions}.\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\text{and}\;\text{int}({\mathcal{I}}(T,\varphi))=\emptyset&&\end{array}$
(2.1)

When $X$ is a compact topological manifold there is a link between
expansiveness and sensitivity regarding a well chosen continuous map
$\varphi:X\to\mathbb{R}$. Indeed, Coven and Reddy (cf. [13]) proved that, if
$T:X\to X$ is a continuous and expansive map acting on compact topological
manifold, then there exists a metric $\tilde{d}$ compatible with the topology
of $(X,d)$ such that $T:(X,\tilde{d})\to(X,\tilde{d})$ is a Ruelle-expanding
map: there are constants $\lambda>1$ and $\delta_{0}>0$ such that, for all
$x,y,z\in X$, one has

  * •

$\tilde{d}(T(x),T(y))\geqslant\lambda\tilde{d}(x,y)$ whenever
$\tilde{d}(x,y)<\delta_{0}$;

  * •

$B(x,\delta_{0})\cap T^{-1}(\\{z\\})$ is a singleton whenever
$\tilde{d}(T(x),z)<\delta_{0}$.

In particular, if $X$ is connected then $T$ is topologically mixing. Moreover,
as Ruelle-expanding maps admit finite Markov partitions and are semiconjugate
to subshifts of finite type, one can choose $\varphi\in C(X,\mathbb{R})$ such
that ${\mathcal{I}}(T,\varphi)$ is a Baire generic subset of $X$. Besides, the
interior of ${\mathcal{I}}(T,\varphi)$ is empty by the denseness of the set of
periodic points of $T$. Therefore, by Corollary D:

######  Corollary 2.4.

Let $(X,d)$ be a compact connected topological manifold and $T:X\to X$ be a
continuous and expansive map. Then there exists $\varphi\in C(X,\mathbb{R})$
such that $X$ is $(T,\varphi)$-sensitive.

It is still an open question whether for each expansive map $T$ on a compact
metric space $X$ there exists $\varphi\in C(X,\mathbb{R})$ such that $X$ is
$(T,\varphi)$-sensitive. Although we have no examples, it is likely to exist
continuous maps on compact metric spaces which have sensitivity to initial
conditions but for which $X$ is not $(T,\varphi)$-sensitive for every
$\varphi\in C(X,\mathbb{R})$.

A further consequence of our results concerns the irregular sets for
continuous maps satisfying the strong transitivity condition (see Definition
2). equivalent to say that $T$ is minimal.

######  Corollary 2.5.

Let $T:X\to X$ be a continuous map on a compact metric space $X$. If $T$ is
strongly transitive and $\varphi\in C(X,\mathbb{R})$, then either
$\mathcal{I}(T,\varphi)=\emptyset$ or $\mathcal{I}(T,\varphi)$ is a Baire
generic subset of $X$.

We note that, as strongly transitive homeomorphisms on a compact metric space
$X$ are minimal, Corollary 2.5 extends the information in Remark 1.2 to
strongly transitive continuous maps on compact metric spaces.

##  3\. Proof of Theorem A

The argument is a direct adaptation of the one in [8, Theorem A]. Suppose that
there exist dense subsets $A,B$ of $Y$ and $\varepsilon>0$ such that for any
$(a,b)\in A\times B$ there exist $(r_{a},r_{b})\in\\{\varphi_{n}(a):n\geq
1\\}^{\prime}\times\\{\varphi_{n}(b):n\geq 1\\}^{\prime}$ satisfying
$|r_{a}-r_{b}|>\varepsilon$. Fix $0<\eta<\frac{\varepsilon}{3}$. Since the
maps $\varphi_{n}$ are continuous, given an integer $N\in\mathbb{N}$ the set

$\Lambda_{N}=\Big{\\{}y\in
Y:\,|\varphi_{n}(y)-\varphi_{m}(y)|\leqslant\eta\quad\forall m,n\geqslant
N\Big{\\}}$

is closed in $Y$. Moreover:

######  Lemma 3.1.

$\Lambda_{N}$ has empty interior for every $N\in\mathbb{N}$.

###### Proof.

Assume that there exists $N\in\mathbb{N}$ such that $\Lambda_{N}$ has non-
empty interior (which we abbreviate into
$\operatorname{int}(\Lambda_{N})\neq\emptyset$). Hence there exists $a\in A$
such that $a\in\operatorname{int}(\Lambda_{N})$. Since $\varphi_{N}$ is
continuous, there exists $\delta_{N}>0$ such that
$|\varphi_{N}(a)-\varphi_{N}(y)|<\eta$ for every $y\in Y$ satisfying
$d(a,y)<\delta_{N}$. By the denseness of $B$, one can choose $b\in B$ such
that $b\in\operatorname{int}(\Lambda_{N})$ and $d(a,b)<\delta_{N}$. Besides,
according to the the definition of $\Lambda_{N}$ one has

$|\varphi_{n}(a)-\varphi_{m}(a)|\leqslant\eta\quad\quad\text{ and
}\quad\quad|\varphi_{n}(b)-\varphi_{m}(b)|\leqslant\eta\quad\forall,m,n\geqslant
N.$

Given $(a,b)\in A\times B$, choose $(r_{a},r_{b})\in\\{\varphi_{n}(a):n\geq
1\\}^{\prime}\times\\{\varphi_{n}(b):n\geq 1\\}^{\prime}$ satisfying
$|r_{a}-r_{b}|>\varepsilon$. Fixing $m=N$, taking the limit as $n$ goes to
$+\infty$ in the first inequality along a subsequence converging to $r_{a}$
and taking the limit as $n$ tends to $+\infty$ in the second inequality along
a subsequence convergent to $r_{b}$, we conclude that

$|r_{a}-\varphi_{N}(a)|\leqslant\eta\quad\quad\text{ and
}\quad\quad|r_{b}-\varphi_{N}(b)|\leqslant\eta.$

Therefore,

$\varepsilon\,<\,|r_{a}-r_{b}|\,\leqslant\,|\varphi_{N}(a)-r_{a}|+|\varphi_{N}(b)-r_{b}|+|\varphi_{N}(a)-\varphi_{N}(b)|\,\leqslant\,3\eta$

contradicting the choice of $\eta$. Thus $\Lambda_{N}$ must have empty
interior. ∎

We can now finish the proof of Theorem A. Using the fact that
$\limsup_{n\,\to\,+\infty}\|\varphi_{n}\|_{\infty}<+\infty$, one deduces that
$Y\setminus{\mathcal{I}}(\Phi)\subset\bigcup_{N=1}^{\infty}\Lambda_{N}$. Thus,
by Lemma 3.1, the set of $\Phi$-regular points is contained in a countable
union of closed sets with empty interior. This shows that
${\mathcal{I}}(\Phi)$ is Baire generic, as claimed. The second statement in
the theorem is a direct consequence of the first one. $\square$

######  Remark 3.2.

It is worth mentioning that the proof of Theorem A also shows that one has
$\\{y\in
Y:\limsup_{n}\varphi_{n}(y)-\liminf_{n}\varphi_{n}(y)<\eta\\}\subseteq\bigcup_{N=1}^{\infty}\Lambda_{N}$,
and so the set $\\{y\in
Y:\limsup_{n}\varphi_{n}(y)-\liminf_{n}\varphi_{n}(y)\geqslant\eta\\}$ is
Baire generic in $Y$.

######  Remark 3.3.

The argument used in the proof of Theorem A adapts naturally to the context of
continuous-time dynamical systems. This fact will be used later, when applying
Theorem A to geodesic flows on non-positive curvature (see Example 6).

##  4\. Proof of Corollary B

We start showing that the existence of an irregular point with respect to an
observable $\varphi$ whose orbit by $T$ is dense is enough to ensure that $Y$
is $(T,\varphi)$-sensitive.

######  Lemma 4.1.

Let $(Y,d)$ be a Baire metric space, $T:Y\to Y$ be a continuous map such that
$(Y,T)$ has a dense orbit, and $\varphi\in C^{b}(Y,\mathbb{R})$. If
${\mathcal{I}}(T,\varphi)\cap\operatorname{Trans}(Y,T)\neq\emptyset$ then $Y$
is $(T,\varphi)$-sensitive.

###### Proof.

Suppose that $\varphi\in C^{b}(Y,\mathbb{R})$ and
${\mathcal{I}}(T,\varphi)\cap\operatorname{Trans}(Y,T)\neq\emptyset$. Let
$y\in Y$ be a point in this intersection. Then there is $\varepsilon>0$ such
that
$\varepsilon<\limsup_{n\,\to\,+\infty}\varphi_{n}(y)-\liminf_{n\,\to\,+\infty}\varphi_{n}(y)$.
Since $\varphi$ is a bounded function, the values

$\liminf_{n\,\to\,+\infty}\,\frac{1}{n}\,\sum_{j=0}^{n-1}\,\varphi(T^{j}(z))\quad\text{
and
}\quad\limsup_{n\,\to\,+\infty}\,\frac{1}{n}\,\sum\limits_{j=0}^{n-1}\,\varphi(T^{j}(z))$

are constant for every $z\in\\{T^{j}(y):j\in\mathbb{N}\cup\\{0\\}\\}$. This
invariance, combined with the fact that
$\\{T^{n}(y):n\in\mathbb{N}\cup\\{0\\}\\}$ is a dense subset of $Y$, implies
that $Y$ is $(T,\varphi)$-sensitive. ∎

Let us resume the proof of Corollary B.

$(i)\,$ We will adapt the argument in the proof of [19, Lemma 3.4]. Take
$y\in\operatorname{Trans}(Y,T)$. Assume that $Y$ has an isolated point and
$\mathcal{D}(Y,T)\neq\emptyset$. Then there exists $N\in\mathbb{N}\cup\\{0\\}$
such that $T^{N}(y)$ is an isolated point of $Y$. Take
$\psi\in\mathcal{D}(Y,T)$ whose set ${\mathcal{I}}(T,\psi)$ is dense in $Y$.
Since $\\{T^{N}(y)\\}$ is an open subset of $Y$, one has
$\\{T^{N}(y)\\}\cap{\mathcal{I}}(T,\psi)\neq\emptyset$, so $T^{N}(y)$ belongs
to ${\mathcal{I}}(T,\psi)$. Therefore,
$T^{N}(y)\in\bigcap_{\psi\,\in\,\mathcal{D}(Y,T)}{\mathcal{I}}(T,\psi).$

In fact, more is true:
$y\in\bigcap_{\psi\,\in\,\mathcal{D}(Y,T)}{\mathcal{I}}(T,\psi).$ Indeed,
suppose that there exists $\psi_{0}\in D(Y,T)$ such that $y\in
Y\setminus{\mathcal{I}}(T,\psi_{0})$. Consider the aforementioned integer
$N\in\mathbb{N}\cup\\{0\\}$ such that
$T^{N}(y)\in\bigcap_{\psi\,\in\,\mathcal{D}(Y,T)}{\mathcal{I}}(T,\psi)$. As
$Y\setminus I(T,\psi)$ is $T$-invariant and $y\in
Y\setminus{\mathcal{I}}(T,\psi_{0})$, we have $T^{N}(y)\in
Y\setminus{\mathcal{I}}(T,\psi_{0})$. This contradicts the choice of $N$. This
way we have shown that
$\operatorname{Trans}(Y,T)\subseteq\bigcap_{\psi\,\in\,\mathcal{D}(Y,T)}{\mathcal{I}}(T,\psi)$.

We are left to prove that, if $(Y,d)$ has an isolated point and
$\mathcal{D}(Y,T)\neq\emptyset$, then $\mathcal{R}(Y,T)=\mathcal{D}(Y,T)$.
Suppose that $\mathcal{D}(Y,T)\neq\emptyset$. As we have just proved,
$\operatorname{Trans}(Y,T)\subseteq\bigcap_{\psi\,\in\,\mathcal{D}(Y,T)}{\mathcal{I}}(T,\psi)$.
This implies that for each $\psi\in\mathcal{D}(Y,T)$ one has
${\mathcal{I}}(T,\psi)\cap\operatorname{Trans}(Y,T)\neq\emptyset$. By Lemma
4.1 and Theorem A we conclude that ${\mathcal{I}}(T,\psi)$ is Baire generic,
so $\psi\in\mathcal{R}(Y,T)$.

$(ii)\,$ Suppose that
$\bigcap_{\varphi\,\in\,\mathfrak{F}}{\mathcal{I}}(T,\varphi)$ is Baire
generic. By hypothesis, there exists $y$ in $Y$ such that
$\Omega_{y}=\overline{\\{T^{n}(y):n\in\mathbb{N}\cup\\{0\\}\\}}=Y$, so $Y$ is
separable. This implies that, if $\omega(y)=\Omega_{y}$, then $Y$ does not
have isolated points, and so $T$ is transitive. Since $Y$ is a Baire separable
metric space, $\operatorname{Trans}(Y,T)$ is Baire generic as well. Therefore,
$\operatorname{Trans}(Y,T)\cap\bigcap_{\varphi\,\in\,\mathfrak{F}}{\mathcal{I}}(T,\varphi)$
is also Baire generic. Thus
$\operatorname{Trans}(Y,T)\cap\bigcap_{\varphi\,\in\,\mathfrak{F}}{\mathcal{I}}(T,\varphi)$
is not empty.

Assume now that $\omega(y)\subsetneq\Omega_{y}$. Then $Y$ has an isolated
point. As $\emptyset\neq\mathfrak{F}\subseteq\mathcal{D}(Y,T)$ and $Y$ has an
isolated point, by item $(i)$ we know that
$\operatorname{Trans}(Y,T)\subseteq\bigcap_{\varphi\,\in\,\mathcal{D}(Y,T)}{\mathcal{I}}(T,\varphi)$.
Moreover, as $\mathfrak{F}\subseteq\mathcal{D}(Y,T)$, one has
$\bigcap_{\varphi\,\in\,\mathcal{D}(Y,T)}{\mathcal{I}}(T,\varphi)\subseteq\bigcap_{\varphi\,\in\,\mathfrak{F}}{\mathcal{I}}(T,\varphi)$.
Thus,
$\operatorname{Trans}(Y,T)\subseteq\bigcap_{\varphi\,\in\,\mathfrak{F}}{\mathcal{I}}(T,\varphi)$.
$\square$

##  5\. Proof of Corollary C

We will adapt the proof of [19, Theorem B]. Consider the sets

$A_{\ell_{\varphi}^{*}}=\Big{\\{}y\in
Y\colon\liminf\limits_{n\,\to\,+\infty}\,\frac{1}{n}\,\sum\limits_{j=0}^{n-1}\,\varphi(T^{j}y)=\ell_{\varphi}^{*}\Big{\\}}$
$A_{L_{\varphi}^{*}}=\\{y\in
Y\colon\limsup\limits_{n\,\to\,+\infty}\,\frac{1}{n}\,\sum\limits_{j=0}^{n-1}\,\varphi(T^{j}y)=L_{\varphi}^{*}\Big{\\}}$

and, for every $\alpha>\ell_{\varphi}^{*}$,

$B_{\alpha}=\bigcap_{N=1}^{+\infty}\bigcup_{n=N}^{+\infty}\,\Big{\\{}y\in
Y\colon\frac{1}{n}\,\sum_{i=0}^{n-1}\,\varphi(T^{i}y)<\alpha\Big{\\}}.$

As $\alpha>\ell_{\varphi}^{*}$, there exists
$y_{0}\in\operatorname{Trans}(Y,T)$ such that $y_{0}\in B_{\alpha}$.
Consequently, for every $N\in\mathbb{N}$ the set
$\bigcup_{n=N}^{+\infty}\Big{\\{}y\in
Y\colon\frac{1}{n}\,\sum_{i=0}^{n-1}\,\varphi(T^{i}y)<\alpha\\}$ is an open,
dense subset of $Y$. Therefore, $B_{\alpha}$ is Baire generic in $Y$ for any
$\alpha>\ell_{\varphi}^{*}$.

Take a convergent sequence $\\{\alpha_{n}\\}_{n=1}^{+\infty}$ in $\mathbb{R}$
such that $\lim_{n\,\to\,+\infty}\alpha_{n}=\ell_{\varphi}^{*}$ and
$\alpha_{n}>\ell_{\varphi}^{*}$ for every $n\in\mathbb{N}$. Then
$\bigcap_{n=1}^{+\infty}B_{\alpha_{n}}\subseteq A_{\ell_{\varphi}^{*}}$. This
implies that $A_{\ell_{\varphi}^{*}}$ is Baire generic in $Y$. We deduce
similarly that $A_{L_{\varphi}^{*}}$ is Baire generic in $Y$. Thus
$A_{\ell_{\varphi}^{*}}\cap
A_{L_{\varphi}^{*}}=I_{\varphi}[\ell_{\varphi}^{*},L_{\varphi}^{*}]$ is Baire
generic in $Y$.

We are now going to show that either
$I_{\varphi}[\ell_{\varphi}^{*},L_{\varphi}^{*}]\subseteq{\mathcal{I}}(T,\varphi)$
or $I_{\varphi}[\ell_{\varphi}^{*},L_{\varphi}^{*}]\subseteq
Y\setminus{\mathcal{I}}(T,\varphi)$. For any $y\in
I_{\varphi}[\ell_{\varphi}^{*},L_{\varphi}^{*}]$, we have

$\liminf\limits_{n\,\to\,+\infty}\,\frac{1}{n}\,\sum\limits_{j=0}^{n-1}\,\varphi(T^{j}(y))=\ell_{\varphi}^{*}\leqslant
L_{\varphi}^{*}=\limsup\limits_{n\,\to\,+\infty}\,\frac{1}{n}\,\sum\limits_{j=0}^{n-1}\,\varphi(T^{j}(y)).$

Therefore, if $\ell_{\varphi}^{*}<L_{\varphi}^{*}$, then
$I_{\varphi}[\ell_{\varphi}^{*},L_{\varphi}^{*}]\subseteq{\mathcal{I}}(T,\varphi)$
and ${\mathcal{I}}(T,\varphi)$ is Baire generic in $Y$. Otherwise, if
$\ell_{\varphi}^{*}=L_{\varphi}^{*}$, then
$I_{\varphi}[\ell_{\varphi}^{*},L_{\varphi}^{*}]\subseteq
Y\setminus{\mathcal{I}}(T,\varphi)$. These inclusions imply that
${\mathcal{I}}(T,\varphi)$ is either Baire generic or meagre, and characterize
the case of a meagre ${\mathcal{I}}(T,\varphi)$. $\square$

######  Remark 5.1.

According to [28, Proposition 3.11], in the context of Baire ergodic maps $T$,
for any Baire measurable function $\varphi:Y\to\mathbb{R}$ there exist a Baire
generic subset $\mathfrak{R}$ of $Y$ and constants $c_{-}^{\varphi}$ and
$c_{+}^{\varphi}$ such that

$\ell_{\varphi}(y)\,=\,c_{-}^{\varphi}\quad\quad\text{ and }\quad\quad
L_{\varphi}(y)\,=\,c_{+}^{\varphi}\quad\quad\forall\,y\in\mathfrak{R}.$

##  6\. Proof of Corollary D

$(i)\Rightarrow(ii)$ By Remark 3.2 there exists $\eta>0$ such that
$\mathcal{D}=\\{y\in Y:L_{\varphi}(y)-\ell_{\varphi}(y)\geq\eta\\}$ is a Baire
generic, and so dense, subset of $Y$. Note that, for any $z$ and $w$ in
$\mathcal{D}$, one has either
$|L_{\varphi}(z)-\ell_{\varphi}(w)|>\frac{\eta}{2}$ or
$|L_{\varphi}(w)-\ell_{\varphi}(z)|>\frac{\eta}{2}$. This implies that $Y$ is
$(T,\varphi)$-sensitive, and so, by Theorem A, the set
${\mathcal{I}}(T,\varphi)$ is Baire generic.

$(ii)\Rightarrow(iii)$ Apply item ($ii$) of Corollary B with
$\mathfrak{F}=\\{\varphi\\}$.

$(iii)\Rightarrow(i)$ Assume that
$\operatorname{Trans}(Y,T)\cap{\mathcal{I}}(T,\varphi)\neq\emptyset$. Then,
there are $y_{0},\,y_{1}\in\operatorname{Trans}(Y,T)$ (possibly equal) and
convergent subsequences $(\varphi_{n_{k}}(y_{0}))_{k\,\in\,\mathbb{N}}$ and
$(\varphi_{m_{k}}(y_{1}))_{k\,\in\,\mathbb{N}}$ whose limits are distinct. Let
$A$ be the (dense) orbit of $y_{0}$, $B$ the (dense) orbit of $y_{1}$ and
$\varepsilon$ equal to half the distance between the limits of the convergent
subsequences $(\varphi_{n_{k}}(y_{0}))_{k\,\in\,\mathbb{N}}$ and
$(\varphi_{m_{k}}(y_{1}))_{k\,\in\,\mathbb{N}}$. This is the data needed to
confirm that $Y$ is $(T,\varphi)$-sensitive. $\square$

##  7\. Proof of Corollary 2.1

(a) Suppose that $X$ is a compact metric space that has a dense orbit. From
[32, Proposition 1 and Proposition 2] or [19, Proposition 2.1 and Proposition
2.2], we know that $X_{\Delta}$ is Baire generic in $X$.

We proceed by showing that
$X_{\Delta}\subseteq\bigcap_{\varphi\,\in\,C(X,\mathbb{R})}\widehat{I_{\varphi}[\ell_{\varphi}^{*},L_{\varphi}^{*}]}$.
Take $x$ in $X_{\Delta}$ and $\varphi\in C(X,\mathbb{R})$. We claim that
$\ell_{\varphi}(x)\leqslant\ell_{\varphi}^{*}$ and $L_{\varphi}^{*}\leqslant
L_{\varphi}(x)$. Assume, on the contrary, that
$\ell_{\varphi}(x)>\ell_{\varphi}^{*}$. Then there exists a transitive point
$x_{0}$ such that $\ell_{\varphi}(x)>\ell_{\varphi}(x_{0})$. Therefore, there
is a subsequence $(n_{k})_{k\,\in\,\mathbb{N}}$ such that

$\ell_{\varphi}(x)>\ell_{\varphi}(x_{0})=\lim\limits_{k\,\to\,+\infty}\,\frac{1}{n_{k}}\,\sum\limits_{j=0}^{n_{k}-1}\,\varphi(T^{j}(x_{0})).$

Reducing to a subsequence if necessary, we find $\mu\in V_{T}(x_{0})$ such
that $\frac{1}{n_{k}}\sum_{j=0}^{n_{k}-1}\delta_{T^{j}(y)}$ converges to $\mu$
in the weak∗-topology as $k$ goes to $+\infty$, and so
$\ell_{\varphi}(x)>\ell_{\varphi}(x_{0})=\int\varphi\,d\mu$. Since, by
hypothesis, $\mu$ belongs to $V_{T}(x)$, there exists an infinite sequence
$(n_{q})_{q\,\in\,\mathbb{N}}$ such that
$\frac{1}{n_{q}}\sum_{j=0}^{n_{q}-1}\delta_{T^{j}(x)}$ converges to $\mu$ as
$q$ goes to $+\infty$. This yields to

$\int\varphi\,d\mu=\lim\limits_{q\,\to\,+\infty}\,\frac{1}{n_{q}}\,\sum\limits_{j=0}^{n_{q}-1}\,\varphi(T^{j}(x))\geqslant\ell_{\varphi}(x)>\ell_{\varphi}(x_{0})=\int\varphi\,d\mu$

which is a contradiction. Therefore, we must have
$\ell_{\varphi}(x)\leqslant\ell_{\varphi}^{*}$.

We prove similarly that $L_{\varphi}^{*}\leqslant L_{\varphi}(x)$ for every
$x\in X_{\Delta}$. Thus
$x\in\widehat{I_{\varphi}[\ell_{\varphi}^{*},L_{\varphi}^{*}]}$.

(b) We start establishing the following auxiliary result.

######  Lemma 7.1.

Let $(X,d)$ be a compact metric space, $T:X\to X$ be a continuous map with a
dense orbit. If $\mathcal{R}(X,T)=\emptyset$, then
$\bigcap_{\varphi\,\in\,C(X,\mathbb{R})}\,X\setminus{\mathcal{I}}(T,\varphi)$
is Baire generic in $X$.

###### Proof.

Since $\mathcal{R}(X,T)=\emptyset$, from Corollary C we know that
${\mathcal{I}}(T,\varphi)$ is meagre for every
$\varphi\,\in\,C(X,\mathbb{R})$. Hence $X\setminus\mathcal{I}(T,\varphi)$ is
Baire generic for all $\varphi\in C(X,\mathbb{R})$. Let $S$ be a countable,
dense subset of $C(X,\mathbb{R})$. Then
$\bigcap_{\psi\,\in\,S}X\setminus\mathcal{I}(T,\psi)$ is Baire generic. We are
left to prove that

$\bigcap\limits_{\psi\,\in\,S}\,X\setminus\mathcal{I}(T,\psi)\,\subseteq\,\bigcap_{\varphi\,\in\,C(X,\mathbb{R})}\,X\setminus{\mathcal{I}}(T,\varphi).$

Take $x\in\bigcap_{\psi\,\in\,S}X\setminus{\mathcal{I}}(T,\psi)$ and
$\varphi\in C(X,\mathbb{R})$; we need to show that $x$ belongs to
$X\setminus{\mathcal{I}}(T,\varphi)$. As $S$ is dense, given $\varepsilon>0$
there exists $\psi\in S$ such that
$\|\varphi-\psi\|_{\infty}<\frac{\varepsilon}{3}$. Since $x$ is in
$X\setminus\mathcal{I}(T,\psi)$, there is $N\in\mathbb{N}\cup\\{0\\}$ such
that, for every $n,m\geqslant N$, we have
$\|\psi_{n}(x)-\psi_{m}(x)\|<\frac{\varepsilon}{3}$. Therefore,

$\|\varphi_{n}(x)-\varphi_{m}(x)\|\leqslant\|\varphi_{n}(x)-\psi_{n}(x)\|+\|\psi_{n}(x)-\psi_{m}(x)\|+\|\psi_{m}(x)-\varphi_{m}(x)\|<\varepsilon$

so $x$ is in $X\setminus{\mathcal{I}}(T,\varphi)$. ∎

Let us go back to the proof of item (b). We begin by showing that
$\bigcup_{\varphi\,\in\,C(X,\mathbb{R})}\mathcal{I}(T,\varphi)$ is either
Baire generic or meagre. Suppose that
$\bigcup_{\varphi\,\in\,C(X,\mathbb{R})}\mathcal{I}(T,\varphi)$ is not Baire
generic, so $\mathcal{R}(X,T)$ is empty. By Lemma 7.1, the set
$\bigcap_{\varphi\,\in\,C(X,\mathbb{R})}X\setminus{\mathcal{I}}(T,\varphi)$ is
Baire generic in $X$, and so
$\bigcup_{\varphi\,\in\,C(X,\mathbb{R})}\mathcal{I}(T,\varphi)$ is a meagre
set. Moreover, using compactness of $X$ and Corollary C, there exists a Borel
invariant probability measure $\mu$ such that

$\operatorname{Trans}(X,T)\,\subset\,\Big{\\{}x\in
X:\lim_{n\,\to\,+\infty}\,\frac{1}{n}\,\sum_{j=0}^{n-1}\,\varphi(T^{j}(x))\,=\,\int\varphi
d\mu\quad\quad\forall\,\varphi\in C(X,\mathbb{R})\Big{\\}}.$

By item (a), $X_{\Delta}=\\{x\in X\colon\mu\in V_{T}(x)\\}$ is Baire generic
in $X$. This implies that the set

$X_{\Delta}\cap\bigcap_{\varphi\,\in\,C(X,\mathbb{R})}X\setminus{\mathcal{I}}(T,\varphi)=\Big{\\{}x\in
X:\lim_{n\,\to\,+\infty}\,\frac{1}{n}\,\sum\limits_{j=0}^{n-1}\varphi(T^{j}(x))=\int\varphi\,d\mu\quad\forall\,\varphi\in
C(X,\mathbb{R})\Big{\\}}$

is Baire generic in $X$.

######  Proposition 7.2.

Let $(X,d)$ be a compact metric space and $T:X\to X$ be a continuous map with
a dense orbit. The following statements are equivalent:

  * $(i)$

$\\#\Big{(}\bigcup\limits_{t\,\in\,\operatorname{Trans}(X,T)}\,V_{T}(t)\Big{)}>1$.

  * $(ii)$

$\mathcal{R}(X,T)\neq\emptyset$.

  * $(iii)$

$\mathcal{R}(X,T)$ is open and dense in $C(X,\mathbb{R})$.

  * $(iv)$

$X_{\Delta}\subseteq\bigcap_{\varphi\,\in\,\mathcal{R}(X,T)}\,{\mathcal{I}}(T,\varphi)$.

  * $(v)$

$\bigcap_{\varphi\,\in\,\mathcal{R}(X,T)}\,{\mathcal{I}}(T,\varphi)$ is Baire
generic.

  * $(vi)$

$\operatorname{Trans}(X,T)\cap\bigcap_{\varphi\,\in\,\mathcal{R}(X,T)}\,{\mathcal{I}}(T,\varphi)\neq\emptyset$.

  * $(vii)$

$\bigcup_{\varphi\,\in\,C(X,\mathbb{R})}\,{\mathcal{I}}(T,\varphi)$ is Baire
generic.

###### Proof.

We will prove that $(i)\Leftrightarrow(ii)$,
$(ii)\Rightarrow\cdots\Rightarrow(vii)$ and $(vii)\Rightarrow(ii)$.

$(i)\Rightarrow(ii)$ Suppose that
$\\#\left(\cup_{x\,\in\,\operatorname{Trans}(X,T)}V_{T}(x)\right)>1$. Then
there exist two distinct Borel probability measures $\mu$, $\nu$ in
$\cup_{x\,\in\,\operatorname{Trans}(X,T)}V_{T}(x)$ and $\varphi\in
C(X,\mathbb{R})$ such that $\int\varphi\,d\mu\neq\int\varphi\,d\nu$. This
implies that $\\{\mu,\nu\\}\subseteq V_{T}(x)$ for all $x\in X_{\Delta}$, and
so $X_{\Delta}\subseteq{\mathcal{I}}(T,\varphi)$. Therefore,
${\mathcal{I}}(T,\varphi)$ is Baire generic since, by item (a), the set
$X_{\Delta}$ is Baire generic.

$(ii)\Rightarrow(i)$ Suppose that that there exists $\varphi\in
C(X,\mathbb{R})$ such that ${\mathcal{I}}(T,\varphi)$ is Baire generic. By
Corollary D, one has
$\operatorname{Trans}(X,T)\cap{\mathcal{I}}(T,\varphi)\neq\emptyset$. This
implies that $\ell_{\varphi}(y)<L_{\varphi}(y)$, and so $\\#V_{T}(y)>1$.

$(ii)\Rightarrow(iii)$ For every $x\in X$, denote by $\mathcal{U}_{x}$ the set
$\\{\varphi\in C(X,\mathbb{R}):x\in{\mathcal{I}}(T,\varphi)\\}$. If
$\mathcal{U}_{x}\neq\emptyset$, then $\mathcal{U}_{x}$ is an open dense subset
of $C(X,\mathbb{R})$ (see [19, Lemma 3.2]) and, by Corollary D,
$\mathcal{R}(X,T)=\bigcup_{x\,\in\,\operatorname{Trans}(X,T)}\mathcal{U}_{x}$.
Therefore, if $\mathcal{R}(X,T)\neq\emptyset$, then $\mathcal{R}(X,T)$ is open
and dense in $C(X,\mathbb{R})$.

$(iii)\Rightarrow(iv)$ Suppose that $\mathcal{R}(X,T)$ is open and dense in
$C(X,\mathbb{R})$. In particular, $\mathcal{R}(X,T)$ is not empty. From
Corollary D, $\mathcal{R}(X,T)=\\{\varphi\in
C(X,\mathbb{R})\colon\ell_{\varphi}^{*}<L_{\varphi}^{*}\\}$. Using item $(a)$,
we know that
$X_{\Delta}\subseteq\bigcap_{\varphi\,\in\,C(X,\mathbb{R})}\widehat{I_{\varphi}[\ell_{\varphi}^{*},L_{\varphi}^{*}]}$.
Moreover,

$X_{\Delta}\subseteq\bigcap\limits_{\varphi\,\in\,C(X,\mathbb{R})}\widehat{I_{\varphi}[\ell_{\varphi}^{*},L_{\varphi}^{*}]}=\left(\bigcap\limits_{\ell_{\varphi}^{*}<L_{\varphi}^{*}}\widehat{I_{\varphi}[\ell_{\varphi}^{*},L_{\varphi}^{*}]}\right)\cap\left(\bigcap\limits_{\ell_{\varphi}^{*}=L_{\varphi}^{*}}\widehat{I_{\varphi}[\ell_{\varphi}^{*},L_{\varphi}^{*}]}\right)$

and

$X_{\Delta}\subseteq\bigcap\limits_{\ell_{\varphi}^{*}<L_{\varphi}^{*}}\widehat{I_{\varphi}[\ell_{\varphi}^{*},L_{\varphi}^{*}]}.$

Consequently,

$X_{\Delta}\subseteq\bigcap\limits_{\ell_{\varphi}^{*}<L_{\varphi}^{*}}{\mathcal{I}}(T,\varphi)=\bigcap\limits_{\varphi\,\in\,\mathcal{R}(X,T)}\mathcal{I}(T,\varphi).$

$(iv)\Rightarrow(v)$ Suppose that
$X_{\Delta}\subseteq\bigcap_{\varphi\,\in\,\mathcal{R}(X,T)}\mathcal{I}(T,\varphi)$.
By item $(a)$, $X_{\Delta}$ is Baire generic; hence
$\bigcap_{\varphi\,\in\,\mathcal{R}(X,T)}\mathcal{I}(T,\varphi)$ is Baire
generic in $X$.

$(v)\Rightarrow(vi)$ Suppose that
$\bigcap_{\varphi\,\in\,\mathcal{R}(X,T)}{\mathcal{I}}(T,\varphi)$ is Baire
generic. By item $(ii)$ of Corollary B, we know that
$\operatorname{Trans}(X,T)\cap\bigcap_{\varphi\,\in\,\mathcal{R}(X,T)}\mathcal{I}(T,\varphi)\neq\emptyset$.

$(vi)\Rightarrow(vii)$ This is clear from Corollary D.

$(vii)\Rightarrow(ii)$ Suppose that
$\bigcup_{\varphi\,\in\,C(X,\mathbb{R})}\mathcal{I}(T,\varphi)$ is Baire
generic. If $\mathcal{R}(X,T)=\emptyset$, using Lemma 7.1 one deduces that
$\bigcap_{\varphi\,\in\,C(X,\mathbb{R})}X\setminus{\mathcal{I}}(T,\varphi)$ is
Baire generic. Then

$\emptyset\,=\,\Big{(}\bigcup\limits_{\varphi\,\in\,C(X,\mathbb{R})}\mathcal{I}(T,\varphi)\Big{)}\cap\Big{(}\bigcap\limits_{\varphi\,\in\,C(X,\mathbb{R})}X\setminus{\mathcal{I}}(T,\varphi)\Big{)}$

is Baire generic as well, so it is not empty. This contradiction ensures that
$\mathcal{R}(X,T)\neq\emptyset$.

∎

$(c)$ Firstly, we note that, by Corollary D, one has

$\mathcal{H}(X,T)=\\{\varphi\in
C(X,\mathbb{R})\colon\,\ell_{\varphi}^{*}<L_{\varphi}^{*}\\}\,\cup\,\\{\varphi\in
C(X,\mathbb{R}):\,\mathcal{I}(T,\varphi)\neq\emptyset\,\text{ and
}\,\ell_{\varphi}^{*}=L_{\varphi}^{*}\\}$

$\mathcal{H}(X,T)=\mathcal{R}(X,T)\,\cup\,\\{\varphi\in
C(X,\mathbb{R}):\,{\mathcal{I}}(T,\varphi)\neq\emptyset\,\text{ and
}\,\ell_{\varphi}^{*}=L_{\varphi}^{*}\\}.$

Suppose that $\mathcal{H}(X,T)$ is not empty. If $\\{\varphi\in
C(X,\mathbb{R}):\mathcal{I}(T,\varphi)\neq\emptyset$ and
$\ell_{\varphi}^{*}=L_{\varphi}^{*}\\}$ is empty, then
$\mathcal{H}(X,T)=\mathcal{R}(X,T)$ is not empty. Therefore, by Proposition
7.2, we conclude that
$\bigcap_{\varphi\,\in\,\mathcal{H}(X,T)}\mathcal{I}(T,\varphi)$ is Baire
generic.

Assume, otherwise, that $\\{\varphi\in
C(X,\mathbb{R}):{\mathcal{I}}(T,\varphi)\neq\emptyset$ and
$\ell_{\varphi}^{*}=L_{\varphi}^{*}\\}$ is not empty. Then there exists
$\varphi\in C(X,\mathbb{R})$ such that $\mathcal{I}(T,\varphi)\neq\emptyset$
and $\ell_{\varphi}^{*}=L_{\varphi}^{*}$. Thus $\mathcal{I}(T,\varphi)$ is
meagre by Corollary C. As
$\bigcap_{\varphi\,\in\,\mathcal{H}(X,T)}\mathcal{I}(T,\varphi)\subseteq\mathcal{I}(T,\varphi)$,
the set $\bigcap_{\varphi\,\in\,\mathcal{H}(X,T)}\mathcal{I}(T,\varphi)$ is
meagre as well.

This way, we have shown that
$\bigcap_{\varphi\,\in\,\mathcal{H}(X,T)}\mathcal{I}(T,\varphi)$ is either
Baire generic or meagre. Actually, we have proved more: that the set
$\bigcap_{\varphi\,\in\,\mathcal{H}(X,T)}\mathcal{I}(T,\varphi)$ is Baire
generic if and only if $\\{\varphi\in
C(X,\mathbb{R}):\mathcal{I}(T,\varphi)\neq\emptyset$ and
$\ell_{\varphi}^{*}=L_{\varphi}^{*}\\}$ is empty. $\square$

##  8\. Proof of Theorem 2.3

Let $(X,d)$ be a compact metric space and $\varphi\in C(X,\mathbb{R})$ such
that $X$ is $(T,\varphi)$-sensitive. Recall that this means that there exist
dense sets $A,B\subset X$ and $\varepsilon>0$ such that, for any pair
$(a,b)\in A\times B$ there is
$(r_{a},\,r_{b})\in\\{\varphi_{n}(a):n\in\mathbb{N}\\}^{\prime}\times\\{\varphi_{n}(b):n\in\mathbb{N}\\}^{\prime}$
satisfying $|r_{a}-r_{b}|>\varepsilon$. Using the uniform continuity of
$\varphi$, one can choose $\delta>0$ such that
$|\varphi(z)-\varphi(w)|<\frac{\varepsilon}{2}$ for every $z,w\in X$ with
$d(z,w)<\delta$.

Assume, by contradiction, that ${\mathcal{I}}(T,\varphi)$ has empty interior
and $X$ has no sensitivity to initial conditions. The latter implies that for
each $\theta>0$ there exist $x_{\theta}\in X$ and an open neighborhood
$U_{x_{\theta}}$ of $x_{\theta}$ such that
$d(T^{n}(x_{\theta}),\,T^{n}(x))\leqslant\theta$ for every $x\in
U_{x_{\theta}}$ and $n\in\mathbb{N}\cup\\{0\\}$. Choose
$\theta=\frac{\delta}{3}$ and take $x_{\theta}$ and $U_{x_{\theta}}$ as above.

As ${\mathcal{I}}(T,\varphi)$ has empty interior, the set
$X\setminus{\mathcal{I}}(T,\varphi)$ is dense in $X$. Therefore, there are
$a\in A\cap U_{x_{\theta}}$, $b\in B\cap U_{x_{\theta}}$ and a
$\varphi$-regular point $c\in U_{x_{\theta}}$ satisfying

$|\varphi(T^{n}(a))-\varphi(T^{n}(c))|<\frac{\varepsilon}{2}\quad$ and
$\quad|\varphi(T^{m}(b))-\varphi(T^{m}(c))|<\frac{\xi}{2}\quad\quad\forall\,n,m\in\mathbb{N}\cup\\{0\\}.$

Consequently, for every $n,m\in\mathbb{N}\cup\\{0\\}$, one has

$\left|\frac{1}{n}\sum\limits_{j=0}^{n-1}\varphi(T^{j}(a))-\frac{1}{n}\sum\limits_{j=0}^{n-1}\varphi(T^{j}(c))\right|<\frac{\varepsilon}{2}\quad$
and
$\quad\left|\frac{1}{m}\sum\limits_{j=0}^{m-1}\varphi(T^{j}(b))-\frac{1}{m}\sum\limits_{j=0}^{m-1}\varphi(T^{j}(c))\right|<\frac{\varepsilon}{2}.$

Taking the limit as $n$ goes to $+\infty$ in the first inequality along a
subsequence $(n_{k})_{k\,\in\,\mathbb{N}}$ converging to $r_{a}$ and taking
the limit as $m$ tends to $+\infty$ in the second inequality along a
subsequence $(m_{k})_{k\,\in\,\mathbb{N}}$ convergent to $r_{b}$, and using
that $c$ is a $\varphi$-regular point, we obtain

$\left|r_{a}-\lim\limits_{k\,\to\,+\infty}\,\frac{1}{n_{k}}\,\sum\limits_{j=0}^{n_{k}-1}\,\varphi(T^{j}(c))\right|\leqslant\frac{\varepsilon}{2}\quad\text{and}\quad\left|r_{b}-\lim\limits_{k\,\to\,\infty}\,\frac{1}{m_{k}}\,\sum\limits_{j=0}^{m_{k}-1}\,\varphi(T^{j}(c))\right|\leqslant\frac{\varepsilon}{2}$

and so $\left|r_{a}-r_{b}\right|\leqslant\varepsilon$. We have reached a
contradiction, thus proving that either $X$ has sensitivity on initial
conditions or the irregular set ${\mathcal{I}}(T,\varphi)$ has non-empty
interior, as claimed.

Item (a) is a direct consequence of the previous statement and the fact that
periodic points of $T$ are $\varphi$-regular. Item (b) follows from item (a).
$\square$

##  9\. Proof of Corollary 2.5

Let $T:X\to X$ be a strongly transitive continuous endomorphism of a compact
metric space $X$. Given $\varphi\in C(X,\mathbb{R})$ satisfying
$\mathcal{I}(T,\varphi)\neq\emptyset$, let us show that
$\mathcal{I}(T,\varphi)$ is a Baire generic subset of $X$. Fix
$x_{0}\in\mathcal{I}(T,\varphi)$ and let
$\varepsilon:=L_{\varphi}(x_{0})-\ell_{\varphi}(x_{0})>0.$ The strong
transitivity assumption ensures that, for every non-empty open subset $U$ of
$X$, there is $N\in\mathbb{N}\cup\\{0\\}$ such that $x_{0}\in T^{N}(U)$. Thus,
the pre-orbit $\mathcal{O}_{T}^{-}(x_{0}):=\,\big{\\{}x\in
X\colon\,T^{n}(x)=x_{0}\,\text{ for some $n\in\mathbb{N}$}\big{\\}}$ of
$x_{0}$ is dense in $X$. Moreover,

$\sup_{r,\,s\,\in\,W_{\varphi}(x)}\,\,|r-s|>\varepsilon\,\qquad\forall\,x\in\mathcal{O}_{T}^{-}(x_{0}).$

This proves that $X$ is $(T,\varphi)$-sensitive and so, by Theorem A, the set
$\mathcal{I}(T,\varphi)$ is Baire generic in $X$. $\square$

##  10\. Examples

In this section we discuss the hypothesis, and derive some consequences, of
the main results. The first example illustrates the Definition 1.

######  Example 1.

Consider the shift space $X=\\{0,1\\}^{\mathbb{N}}$ endowed with the metric
defined by

$d((a_{n})_{n\,\in\,\mathbb{N}},\,(b_{n})_{n\,\in\,\mathbb{N}})\,=\,\sum_{n=1}^{+\infty}\,\frac{|a_{n}-b_{n}|}{2^{n}}$

and take the shift map $\sigma:X\to X$ given by
$\sigma(\,(a_{n})_{n\,\in\mathbb{N}}\,)=(a_{n+1})_{n\,\in\mathbb{N}}$. Then
the sets

$\displaystyle A$ $\displaystyle=$
$\displaystyle\big{\\{}(a_{n})_{n\,\in\,\mathbb{N}}\,|\,\,\exists\,N\in\,\mathbb{N}\colon\,a_{n}=0\quad\forall\,n\geqslant
N\big{\\}}$ $\displaystyle B$ $\displaystyle=$
$\displaystyle\big{\\{}(a_{n})_{n\,\in\,\mathbb{N}}\,|\,\,\exists\,N\in\,\mathbb{N}\colon\,a_{n}=1\quad\forall\,n\geqslant
N\big{\\}}$

are the stable sets of the fixed points $\overline{0}$ and $\overline{1}$, and
are dense subsets of $X$. Besides, if $\varphi\in C(X,\mathbb{R})$, then for
every $a\in A$ and $b\in B$ one has

$\lim_{n\,\to\,+\infty}\,\varphi(\sigma^{n}(a))=\varphi(\overline{0})\quad\quad\text{and}\quad\quad\lim_{n\,\to\,+\infty}\,\varphi(\sigma^{n}(a))=\varphi(\overline{1}).$

Therefore,

$\lim_{n\,\to\,+\infty}\,\frac{1}{n}\,\sum_{j=1}^{n-1}\,\varphi(\sigma^{j}(a))=\varphi(\overline{0})\quad\quad\text{and}\quad\quad\lim_{n\,\to\,+\infty}\,\frac{1}{n}\,\sum_{j=1}^{n-1}\,\varphi(\sigma^{j}(b))=\varphi(\overline{1}).$

And so, if $\varphi(\overline{0})\neq\varphi(\overline{1})$ and one chooses
$\varepsilon=|\varphi(\overline{0})-\varphi(\overline{1})|/2$, then we
conclude that for every $(a,b)\in A\times B$ there are $r_{a}\in
W_{\varphi}(a)$ and $r_{b}\in W_{\varphi}(b)$ such that
$|r_{a}-r_{b}|>\varepsilon$. Consequently, $X$ is $(\sigma,\varphi)$-sensitive
with respect to any $\varphi\in C(X,\mathbb{R})$ satisfying
$\varphi(\overline{0})\neq\varphi(\overline{1})$. So, by Theorem A, for every
such maps $\varphi$ the set ${\mathcal{I}}(T,\varphi)$ is Baire generic in
$X$.

######  Remark 10.1.

We note that a similar reasoning shows that, if $T:Y\to Y$ is a continuous map
acting on a Baire metric space such that $T$ has two periodic points with
dense pre-orbits, then there exists $\varphi\in C^{b}(Y,\mathbb{R})$ whose set
${\mathcal{I}}(T,\varphi)$ is Baire generic in $Y$.

The second example helps to clarify the requirements in Theorem A and
Corollary B, besides calling our attention to the difference between
transitivity and the existence of a dense orbit.

######  Example 2.

Consider the space $X=\\{1/n:n\in\mathbb{N}\\}\cup\\{0\\}$ endowed with the
Euclidean metric. Let $T:X\to X$ be the continuous map given by $T(0)=0$ and
$T(1/n)=1/(n+1)$ for every $n\in\mathbb{N}$. Notice that $X$ has isolated
points and $T$ has a dense orbit, though $\operatorname{Trans}(X,T)=\\{1\\}$.
However, $T$ is not transitive. Besides, ${\mathcal{I}}(T,\varphi)=\emptyset$
for every $\varphi\in C(X,\mathbb{R})$.

The next example shows that the existence of a discontinuous first integral
$L_{\varphi}$ with two dense level sets for a continuous map $T:Y\to Y$ acting
on a metric space $Y$ may be indeed stronger than requiring
$(T,\varphi)$-sensitivity.

######  Example 3.

Let $(\Psi_{t})_{t\,\in\,\mathbb{R}}$ be a smooth Morse-Smale flow on
$\mathbb{S}^{2}$ with hyperbolic singularities
$\\{\sigma_{1},\sigma_{2},\sigma_{3},\sigma_{4},S,N\\}$ and displaying an
attracting union of four separatrices, as illustrated in Figure 1. More
precisely, there exist four separatrices $\gamma_{1},\gamma_{2},\gamma_{3}$
and $\gamma_{4}$ associated to hyperbolic singularities
$\sigma_{1},\sigma_{2}$ of saddle type, while all the other singularities are
repellers.

Figure 1. Flow with historic behavior.

Let $\varphi:\mathbb{S}^{2}\to\mathbb{R}$ be a continuous observable
satisfying $\varphi(x)\in[0,1]$ for every $x\in\mathbb{S}^{2}$,
$\varphi(\sigma_{1})=1$ and $\varphi(\sigma_{2})=0$. Consider the time-one map
$T:Y\to Y$ of the flow $(\Psi_{t})_{t\,\in\,\mathbb{R}}$. As the orbits by $T$
of all the points in
$Y:=\,\mathbb{S}^{2}\setminus(\bigcup_{i=1}^{4}\gamma_{i})$ accumulate on the
closure of the union of the separatrices, it is immediate that the first
integral $L_{\varphi}$, defined by (1.1), is everywhere constant in $Y$:
$L_{\varphi}(y)=1$ for every $y\in Y$. However, as $W_{\varphi}(y)=[0,1]$ for
every $y\in Y$, the space $Y$ is $(T,\varphi)$-sensitive.

In the following example we will address the irregular set in the context of
countable Markov shifts. These symbolic systems appear naturally as models for
non-uniformly hyperbolic dynamical systems on compact manifolds (see [27] and
references therein), hyperbolic systems with singularities [9], including
Sinai dispersing billiards, and certain classes of piecewise monotone interval
maps [18], which encompass the piecewise expanding Lorenz interval maps, just
to mention a few.

######  Example 4.

Let $\mathcal{A}$ be a countable set,
$\mathbb{A}=(a_{i,j})_{i,j\,\in\,\mathcal{A}}$ be a matrix of zeroes and ones
and $\Sigma_{\mathbb{A}}\subset\mathcal{A}^{\mathbb{N}}$ be the subset

$\Sigma_{\mathbb{A}}=\big{\\{}(x_{n})_{n\,\in\mathbb{N}}\in\mathcal{A}^{\mathbb{N}}\colon\,a_{{x_{n}},{x_{n+1}}}=1\quad\forall\,n\in\mathbb{N}\big{\\}}.$

Endow $\Sigma_{\mathbb{A}}$ with the metric

$d((x_{n})_{n\,\in\mathbb{N}},\,(y_{n})_{n\,\in\mathbb{N}})=\left\\{\begin{array}[]{ll}2^{-\min\,\\{k\,\in\,\mathbb{N}\colon\,x_{k}\,\neq\,y_{k}\\}}&\text{
if $\\{k\,\in\mathbb{N}\colon\,x_{k}\neq y_{k}\\}\neq\emptyset$}\\\ 0&\text{
otherwise}.\end{array}\right.$

We note that $a_{i,j}=0$ for all but finitely many values of
$(i,j)\in\mathcal{A}\times\mathcal{A}$ if and only if $\Sigma_{\mathbb{A}}$ is
a compact metric space. Besides, the metric space $(\Sigma_{\mathbb{A}},d)$
has a countable basis of the topology, generated by the countably many
cylinders, and it is invariant by the shift map
$\sigma\colon\mathcal{A}^{\mathbb{N}}\to\mathcal{A}^{\mathbb{N}}$.

Observe also that the set of periodic points of
$\sigma_{|\Sigma_{\mathbb{A}}}$ is dense in $\Sigma_{\mathbb{A}}$. Thus, if
$\sigma_{|\Sigma_{\mathbb{A}}}$ is transitive, then all points have dense pre-
orbits and there exists $\varphi\in C^{b}(\Sigma_{\mathbb{A}},\mathbb{R})$
such that $\Sigma_{\mathbb{A}}$ is $(\sigma,\varphi)$-sensitive. In fact, for
each $\varphi\in C^{b}(\Sigma_{\mathbb{A}},\mathbb{R})$ either
$\Sigma_{\mathbb{A}}$ is $(\sigma,\varphi)$-sensitive or $\varphi$ complies
with the following rigid condition: there exists $c_{\varphi}\in\mathbb{R}$
satisfying

$\frac{1}{\pi(p)}\,\sum_{j=0}^{\pi(p)-1}\,\varphi(\sigma^{j}(p))=c_{\varphi}\quad\text{for
every periodic point $p$}$

where $\pi(p)\in\mathbb{N}$ denotes the minimal period of $p$.

Our next example concerns a skew-product admitting two invariant probability
measures whose basins of attraction have positive Lebesgue measure and are
dense.

######  Example 5.

Consider the annulus $\mathbb{A}=\mathbb{S}^{1}\times[0,1]$ and the map
$T:\mathbb{A}\to\mathbb{A}$ given by

$T(x,t)\,=\,\Big{(}\,3x\,(\mbox{mod}~{}1),\,t+\frac{t(1-t)}{32}\cos(2\pi
x)\Big{)}\quad\quad\forall\,(x,t)\in\mathbb{S}^{1}\times[0,1].$

In [20], Kan proved that $T$ admits two physical measures, namely
$\mu_{0}=\mbox{Leb}_{\mathbb{S}^{1}}\times\delta_{0}$ and
$\mu_{1}=\mbox{Leb}_{\mathbb{S}^{1}}\times\delta_{1}$, whose basins of
attraction are intermingled, that is, for every non-empty open set
$\mathcal{U}\subset\mathbb{A}$

$\mbox{Leb}_{\mathbb{A}}\big{(}\mathcal{U}\cap B(\mu_{1})\big{)}>0\quad\text{
and }\quad\mbox{Leb}_{\mathbb{A}}\big{(}\mathcal{U}\cap B(\mu_{2})\big{)}>0.$

Later, Bonatti, Díaz and Viana introduced in [6] the concept of Kan-like map
and proved that any such map robustly admits two physical measures. More
recently, Gan and Shi [17] showed that, in the space of $C^{2}$
diffeomorphisms of $\mathbb{A}$ preserving the boundary, every $C^{2}$ Kan-
like map $T_{0}$ admits a $C^{2}$-open neighborhood $\mathscr{V}$ such that
the following holds: for each $T\in\mathscr{V}$ and every non-empty open set
$\mathcal{U}\subset\mathbb{A}$,

$\text{the interior of
}(\mathbb{A})\,\subset\bigcup_{n\,\geqslant\,0}T^{n}(\mathcal{U}).$

Using Theorem A we conclude that, if $T$ is a Kan-like map and $\varphi\in
C(\mathbb{A},\mathbb{R})$ satisfies
$\int\varphi\,d\mu_{0}\neq\int\varphi\,d\mu_{1}$, then
$\mathcal{I}(T,\varphi)$ is a Baire generic subset of $\mathbb{A}$. More
generally, the argument that established Corollary 2.5 also ensures that, for
any $\varphi\in C(\mathbb{A},\mathbb{R})$, one has

  * (a)

either $\mathcal{I}(T,\varphi)\cap\text{interior }(\mathbb{A})=\emptyset$

  * (b)

or $\mathcal{I}(T,\varphi)$ is a Baire generic subset of $\mathbb{A}$.

##  11\. Applications

As it will become clear in the remainder of this section, Theorem A has a wide
range of applications according to the class of sequences
$\Phi=(\varphi_{n})_{n\geq 1}$ of observables one considers. Let us provide
two such applications, one with a geometric motivation and another in the
context of semigroup actions.

Application 1. The irregular sets of uniformly hyperbolic maps and flows on
compact Riemannian manifolds have been extensively studied. One of the reasons
for this success is that these dynamical systems can be modeled by symbolic
dynamical systems which satisfy the so-called specification property.
Irregular sets for continuous maps acting on compact metric spaces and
satisfying the specification property have been studied in [21]. Many
difficulties arise, though, if one drops the compactness assumption. An
important example of a hyperbolic dynamical system with non-compact phase
space is given by the geodesic flow on a complete connected negatively curved
manifold. The next example applies Theorem A precisely to this setting.

######  Example 6.

Let $(M,g)$ be a connected, complete Riemannian manifold. We will discuss the
Baire genericity of points with historic behavior in the following two cases:

  * (I)

$(M,g)$ is negatively curved and the non-wandering set of the geodesic flow
$(\Psi^{g}_{t})_{t\,\in\,\mathbb{R}}$ contains more than two periodic orbits.

  * (II)

$(M,g)$ has non-positive curvature, its universal curvature has no flat strips
and the geodesic flow $(\Psi^{g}_{t})_{t\,\in\,\mathbb{R}}$ has at least three
periodic orbits.

Regarding (I), by [11, Theorem 1.1] it is known that the space $\mathcal{E}$
of Borel ergodic probability measures fully supported on the non-wandering set
$\Omega$ is a $G_{\delta}$-dense subset of all Borel probability measures on
$T^{1}M$ which are invariant by the geodesic flow. In particular, one can
choose a continuous observable $\varphi:T^{1}M\to\mathbb{R}$ such that

$\inf_{\mu\,\in\,\mathcal{E}}\,\int\varphi\,d\mu<\sup_{\mu\,\in\,\mathcal{E}}\,\int\varphi\,d\mu.$

As the ergodic basins of attraction of the probability measures in
$\mathcal{E}$ are dense in $\Omega$, one concludes that

$L_{\varphi}(\cdot):=\,\limsup_{t\,\to\,+\infty}\,\frac{1}{t}\,\int_{0}^{t}\varphi(\Psi^{g}_{s}(\cdot))\,ds$

is a first integral for the geodesic flow
$(\Psi^{g}_{t})_{t\,\in\,\mathbb{R}}$. Moreover, there are subsets $A,B\subset
T^{1}M$ which are dense in $\Omega$ and whose $L_{\varphi}$ value is constant,
though the value in $A$ is different from the one in $B$. The existence of $A$
and $B$ means that $\Omega$ is $(T,\Phi)$-sensitive, where $T=\Psi^{g}_{1}$ is
the time-1 map of the geodesic flow and $\Phi$ is defined by
$\Phi=\int_{0}^{1}(\varphi\circ\Psi^{g}_{s})\,ds$. Then Theorem A ensures that
$\mathcal{I}(T,\Phi)$ is Baire generic. Consequently,

$\mathcal{I}((\Psi^{g}_{t})_{t\,\in\,\mathbb{R}},\,\varphi):=\Big{\\{}y\in\Omega:\,\lim\limits_{t\,\to\,+\infty}\,\frac{1}{t}\,\int_{0}^{t}\varphi(\Psi^{g}_{s}(y))\,ds\quad\text{does
not exist}\Big{\\}}$

is a Baire generic subset of $\Omega$ as well.

The previous argument can be repeated in the case (II) to yield the conclusion
that $I((\Psi^{g}_{t})_{t\,\in\,\mathbb{R}},\,\varphi)$ is Baire generic
since, according to [12, Theorem 1.1], the space $\mathcal{E}$ of Borel
ergodic probability measures with full support on the non-wandering set
$\Omega$ also form a $G_{\delta}$-dense set.

Application 2. Recall that a locally compact group $G$ is _amenable_ if for
every compact set $K\subset G$ and $\delta>0$ there is a compact set $F\subset
G$, called $(K,\delta)$-invariant, such that $m(F\Delta KF)<\delta\,m(F)$,
where $m$ denotes the counting measure on $G$ if $G$ is discrete, and stands
for the Haar measure in $G$ otherwise. We refer the reader to [26] for
alternative formulations of this concept. A sequence $(F_{n})_{n}$ of compact
subsets of $G$ is a _Følner sequence_ if, for every compact $K\subset G$, the
set $F_{n}$ is $(K,\delta)$-invariant for every sufficiently large
$n\in\mathbb{N}$ (whose estimate depends on $K$). A Følner sequence
$(F_{n})_{n}$ is _tempered_ if there exists $C>0$ such that

$m\Big{(}\bigcup_{1\,\leqslant\,k\,<\,n}\,F_{k}^{-1}\,F_{n}\,\Big{)}\leqslant
C\,m\big{(}\,F_{n}\,\big{)}\qquad\forall\,n\in\mathbb{N}.$

It is known that every Følner sequence has a tempered subsequence and that
every amenable group has a tempered Følner sequence (cf. [23, Proposition
1.4]). Furthermore, if $G$ is an amenable group acting on a probability space
$(X,\mu)$ by measure preserving maps and $(F_{n})_{n}$ is a tempered Følner
sequence, then for every $\varphi\in L^{1}(\mu)$ the limit

$\lim_{n\,\to\,+\infty}\,\frac{1}{m(F_{n})}\,\int_{F_{n}}\varphi(g(x))\,dm(g)$

exists for $\mu$-almost every $x\in X$; if, in addition, the $G$-action is
ergodic, the previous limit is $\mu$-almost everywhere constant and coincides
with $\int\varphi\,d\mu$ (cf. [23, Theorem 1.2]).

Let $(X,d)$ be a compact metric space, $G$ be an amenable group. We say that a
Borel probability measure $\mu$ on $X$ is _$G$ -invariant_ (or invariant by
the action $\Gamma:G\times X\to X$ of $G$ on $X$) if
$\mu({\Gamma_{g}}^{-1}(A))=\mu(A)$ for every measurable set $A$ and every
$g\in G$. We denote the space of $G$-invariant probability measures by
$\mathcal{M}_{G}(X)$. A group action of $G$ on $X$ is said to be _uniquely
ergodic_ if it admits a unique $G$-invariant probability measure.

Consider $\varphi\in C(X,\mathbb{R})$ and the sequence
$\Phi=(\varphi_{n})_{n\,\in\,\mathbb{N}}$ of continuous and bounded maps
defined by

$\varphi_{n}(x)=\,\frac{1}{m(F_{n})}\,\int_{F_{n}}\,\varphi(g(x))\,dm(g).$
(11.1)

Clearly, $\|\varphi_{n}\|_{\infty}\leqslant\|\varphi\|_{\infty}$ for every
$n\in\mathbb{N}$. Besides, if there are fully supported $G$-invariant and
ergodic Borel probability measures $\mu_{1}\neq\mu_{2}$, then $X$ is
$\Phi$-sensitive. Thus, Theorem A ensures that

$\Big{\\{}x\in
X\colon\,\lim_{n\,\to\,+\infty}\,\frac{1}{m(F_{n})}\,\int_{F_{n}}\varphi(g(x))\,dm(g)\,\text{
does not exist}\Big{\\}}$

is a Baire generic subset of $X$. In what follows, we will show that the
previous requirement is satisfied by countable amenable group actions with the
specification property.

######  Definition 4.

Following [10, 29], we say that the continuous group action $\Gamma:G\times
X\to X$ has the _specification property_ if for every $\varepsilon>0$ there
exists a finite set $K_{\varepsilon}\subset G$ (depending on $\varepsilon$)
such that the following holds: for any finite sample of points
$x_{0},x_{1},x_{2},\dots,x_{\kappa}$ in $X$ and any collection of finite
subsets $\hat{F}_{0},\hat{F}_{1},\hat{F}_{2},\dots,\hat{F}_{\kappa}$ of $G$
satisfying the condition

$K_{\varepsilon}\hat{F}_{i}\cap\hat{F}_{j}=\emptyset\qquad\text{for every
distinct\; }0\leqslant i,j\leqslant\kappa$ (11.2)

there exists a point $x\in X$ such that

$d(\Gamma_{g}(x),\Gamma_{g}(x_{i}))<\varepsilon\qquad\text{for every\;
}g\in\bigcup_{0\,\leqslant\,j\,\leqslant\,\kappa}\hat{F}_{j}.$ (11.3)

In rough terms, the previous property asserts that any finite collection of
pieces of orbits can be shadowed by a true orbit provided that there is no
overlap of the (translated) group elements that parameterize the orbits. We
note that, if $G$ is generated by a single map $g$, then Definition 4
coincides with the classical notion of specification for $g$.

As a first consequence of this property one can show that the _basin of
attraction_ of a $G$-invariant probability measure $\mu$, defined by

$B(\mu)=\Big{\\{}x\in
X\colon\,\frac{1}{|F_{n}|}\,\sum_{g\,\in\,F_{n}}\,\delta_{g(x)}\to\mu\;\;\text{(in
the weak${}^{*}$ topology)}\Big{\\}}$

is topologically large.

######  Lemma 11.1.

Let $G$ be a countable amenable group, $(F_{n})_{n\,\in\,\mathbb{N}}$ be a
tempered Følner sequence, $(X,d)$ be a compact metric space and
$\Gamma:G\times X\to X$ be a continuous group action satisfying the
specification property. If $\mu$ is a $G$-invariant and ergodic probability
measure on $X$ then $B(\mu)$ is a dense subset of $X$.

###### Proof.

As $X$ is compact, the space $C(X,\mathbb{R})$ is separable. Given a dense
sequence $(\varphi_{\ell})_{\ell\,\in\,\mathbb{N}}$ in $C(X,\mathbb{R})$ and
$\ell\in\mathbb{N}$, by the ergodic theorem for amenable group actions [23]
there is a full $\mu$-measure subset $X_{\ell}\subset X$ such that, for every
$x\in X_{\ell}$,

$\lim_{n\,\to\,+\infty}\,\frac{1}{|F_{n}|}\,\sum_{g\,\in\,F_{n}}\,\varphi(g(x))=\int\varphi\,d\mu.$

Therefore, $\bigcap_{\ell\,\in\,\mathbb{N}}X_{\ell}\subset B(\mu)$ and
$\mu(B(\mu))=1$. Take $z\in B(\mu)$ and observe that, as
$\lim_{n\to\infty}|F_{n}|=+\infty$, for each $\varphi\in C(X,\mathbb{R})$ and
any compact set $F\subset G$ one has

$\lim_{n\,\to\,+\infty}\;\frac{1}{|F_{n}\setminus
F|}\,\sum_{g\,\in\,F_{n}\setminus F}\,\varphi(g(z))=\int\varphi\,d\mu.$

Now, for each $\varepsilon>0$, let $K_{\varepsilon}\subset G$ be given by the
specification property. Consider the set
$K_{\varepsilon}^{-1}:=\\{g^{-1}\colon g\in K_{\varepsilon}\\}$ and notice
that, by the compactness of $X$, the modulus of uniform continuity
$\zeta_{\varepsilon}:=\sup\\{|\varphi(u)-\varphi(v)|\colon\,v\in
B(u,\varepsilon),\;u\in X\\}$, where $B(u,\varepsilon)$ stands for the ball in
$X$ centered in $u$ with radius $\varepsilon$, tends to zero as $\varepsilon$
goes to $0$.

Fix an arbitrary point $x\in X$ and $\delta>0$. For each $k\in\mathbb{N}$ one
can choose a positive integer $n_{k}\gg 1$ such that, as $k$ goes to
$+\infty$,

$\frac{K_{\delta/2^{k}}}{|F_{n_{k}}|}\to
0\quad\quad\text{and}\quad\quad\frac{|F_{n_{k-1}}|}{|F_{n_{k}}|}\to 0.$

We claim that $B(\mu)$ intersects the closed ball $\overline{B(x,\delta)}$ of
radius $\delta$ around $x$. The idea to prove this assertion is to shadow
pieces of orbits of increasing size in the basin of $\mu$ which start in the
ball $B(x,\delta)$. More precisely, taking $x_{0}=x$, $x_{1}=z$ and the finite
sets $\hat{F}_{0}=\\{id\\}$ and
$\hat{F}_{1}=(K_{\delta/2}^{-1}[F_{n_{1}}\setminus\hat{F}_{0}])\setminus
K_{\delta/2}$, which satisfy (11.2), by the specification property there is
$y_{1}\in B(x,\delta/2)$ such that
$d(\Gamma_{g}(y_{1}),\Gamma_{g}(z))<\varepsilon$ for every $g\in\hat{F}_{1}.$
In particular,

$\Big{|}\frac{1}{|\hat{F}_{1}|}\sum_{g\,\in\,\hat{F}_{1}}\varphi({g(y_{1})})-\frac{1}{|\hat{F}_{1}|}\sum_{g\,\in\,\hat{F}_{1}}\varphi({g(z)})\Big{|}<\zeta_{\delta/2}.$

Now set $x_{2}=y_{1}$, $x_{3}=z$, $\hat{F}_{2}:=\hat{F}_{0}\cup\hat{F}_{1}$
and
$\hat{F}_{3}=(K_{\delta/2^{2}}^{-1}[F_{n_{2}}\setminus\hat{F}_{2}])\setminus(K_{\delta/2^{2}}\hat{F}_{2})$.
Hence, if $\hat{F}_{4}:=\hat{F}_{2}\cup\hat{F}_{3}$, using the specification
property once more one obtains a point $y_{2}\in B(y_{1},\delta/2^{2})$ such
that $d(\Gamma_{g}(y_{2}),\Gamma_{g}(z))<\varepsilon$ for every
$g\in\hat{F}_{3}$, from which it is immediate that

$\Big{|}\frac{1}{|\hat{F}_{3}|}\sum_{g\,\in\,\hat{F}_{3}}\varphi({g(y_{1})})-\frac{1}{|\hat{F}_{3}|}\sum_{g\,\in\,\hat{F}_{3}}\varphi({g(z)})\Big{|}<\zeta_{\delta/2^{2}}.$

Proceeding recursively, given $y_{j}\in B(y_{j-1},\delta/2^{j})$ and a finite
set $\hat{F}_{j}\subset G$ containing $\\{id\\}$ we take

$\hat{F}_{j+1}=(K_{\delta/2^{j}}^{-1}[F_{n_{j+1}}\setminus\hat{F}_{j}])\setminus(K_{\delta/2^{j}}\hat{F}_{j})$
(11.4)

and, by the specification property, we find $y_{j+1}\in
B(y_{j},\delta/2^{j+1})$ satisfying

$d(\Gamma_{g}(y_{j+1}),\Gamma_{g}(y_{j}))<\varepsilon\quad\text{for every
}g\in\hat{F}_{j}$

and

$d(\Gamma_{g}(y_{j+1}),\Gamma_{g}(z))<\varepsilon\quad\text{for every
}g\in\hat{F}_{j+1}.$

Thus, by construction, $(y_{k})_{k\in\mathbb{N}}$ is a Cauchy sequence in
$\overline{B(x,\delta)}$, hence convergent to some point
$y_{\infty}\in\overline{B(x,\delta)}$. Moreover, the choice of the sets
$\hat{F}_{k}$ implies that

$\lim_{k\,\to\,+\infty}\,\frac{|\hat{F}_{k}\,\Delta\,F_{n_{k}}|}{|F_{n_{k}}|}=0$

and

$\Big{|}\frac{1}{|\hat{F}_{j+1}|}\sum_{g\,\in\,\hat{F}_{j+1}}\varphi({g(y_{j+1})})-\frac{1}{|\hat{F}_{j+1}|}\sum_{g\,\in\,\hat{F}_{j+1}}\varphi({g(z)})\Big{|}<\zeta_{\delta/2^{j}}$

from which we conclude that $y_{\infty}$ belongs to the basin of $\mu$. ∎

The following dichotomy is a direct consequence of Lemma 11.1 and Theorem A.

######  Corollary 11.2.

Let $G$ be a countable amenable group, $(F_{n})_{n\,\in\,\mathbb{N}}$ be a
tempered Følner sequence, $X$ be a compact metric space and $\Gamma:G\times
X\to X$ be a continuous group action satisfying the specification property.
Then for every $\varphi\in C(X,\mathbb{R})$ either

$\int\varphi\,d\mu_{1}=\int\varphi\,d\mu_{2}\quad\quad\forall\,\mu_{1},\,\mu_{2}\in\mathcal{M}_{G}(X)$

or

$\Big{\\{}x\in
X\colon\,\lim_{n\,\to\,+\infty}\,\frac{1}{|F_{n}|}\,\sum_{g\,\in\,F_{n}}\,\varphi(g(x))\,\text{
does not exist}\Big{\\}}$

is a Baire generic subset of $X$.

We observe that Corollary 11.2 has an immediate consequence regarding the
empirical measures distributed along elements of a tempered Følner sequence:
under the assumptions of this corollary, one has that either the amenable
group action is uniquely ergodic or the set

$\big{\\{}x\in
X\colon\,\Big{(}\frac{1}{|F_{n}|}\,\sum_{g\,\in\,F_{n}}\,\delta_{g(x)}\Big{)}_{n\,\in\,\mathbb{N}}\,\text{
does not converge in the weak${}^{*}$ topology}\big{\\}}$

is Baire generic in $X$. This extends Furstenberg’s theorem (see [25, Theorem
3.2.7]), according to which an amenable group action by homeomorphisms on a
compact metric space is uniquely ergodic if and only if the averages (11.1) of
every continuous function converge to a constant.

######  Example 7.

Consider the $2$-torus $\mathbb{T}^{2}$, with a Riemannian metric $d$, and the
linear Anosov diffeomorphisms $g_{1}$ and $g_{2}$ of $\mathbb{T}^{2}$ induced
by the matrices $A_{1}=\left(\begin{array}[]{cc}2&1\\\ 1&1\\\
\end{array}\right)$ and $A_{2}=\left(\begin{array}[]{cc}1&1\\\ 1&0\\\
\end{array}\right)$. As $A_{1}=A_{2}^{2}$, the maps $g_{1}$ and $g_{2}$
commute and induce a $\mathbb{Z}^{2}$-action on the 2-torus given by

$\displaystyle\Gamma:\mathbb{Z}^{2}\times\mathbb{T}^{2}\quad$
$\displaystyle\to$ $\displaystyle\quad\mathbb{T}^{2}$
$\displaystyle\big{(}(m,n),\,x\big{)}\quad$ $\displaystyle\mapsto$
$\displaystyle\quad\big{(}g_{1}^{m}\circ g_{2}^{n}\big{)}(x).$

Moreover, this action satisfies the specification property. Let us see why.

Given $\varepsilon>0$, let $k_{\varepsilon}\in\mathbb{N}$ be provided by the
specification property for the Anosov diffeomorphism $g_{2}$ and
$K_{\varepsilon}=[-k_{\varepsilon},\,k_{\varepsilon}]^{2}\subset\mathbb{Z}^{2}$.
For any finite collection of points $x_{0},x_{1},x_{2},\dots,x_{\kappa}$ in
$\mathbb{T}^{2}$ and any choice of finite subsets
$\hat{F}_{0},\hat{F}_{1},\hat{F}_{2},\dots,\hat{F}_{\kappa}$ of
$\mathbb{Z}^{2}$ satisfying the condition

$K_{\varepsilon}\hat{F}_{i}\cap\hat{F}_{j}=\emptyset\qquad\text{for every
distinct\; }0\leqslant i,j\leqslant\kappa$ (11.5)

we claim that there exists a point $x\in\mathbb{T}^{2}$ such that

$d(\Gamma_{(m,n)}(x),\Gamma_{(m,n)}(x_{i}))<\varepsilon\qquad\text{for every\;
}(m,n)\in\bigcup_{0\,\leqslant\,j\,\leqslant\,\kappa}\hat{F}_{j}.$

Indeed, consider the map $\Theta:\mathbb{Z}^{2}\to\mathbb{Z}$ given by
$\Theta((m,n))=2m+n$ and notice that

$\Gamma((m,n),\cdot)=g_{2}^{\Theta((m,n))}(\cdot)\quad\quad\forall\,m,n\in\mathbb{Z}.$
(11.6)

The choice of the sets $K_{\varepsilon}$ together with assumption (11.5) imply
that $\Theta(K_{\varepsilon}\hat{F}_{i})\cap\Theta(\hat{F}_{j})=\emptyset$ for
every $i\neq j$, and consequently

$\inf\,\Big{\\{}|u-v|\colon
u\in\Theta(\hat{F}_{i}),\,\,v\in\Theta(\hat{F}_{j})\Big{\\}}\geqslant
k_{\varepsilon}.$

To find $x\in\mathbb{T}^{2}$ as claimed, we are left to apply the
specification property of $g_{2}$ (valid since $g_{2}$ is an Anosov
diffeomorphism) and the equality (11.6).

Consider now a tempered Følner sequence $(F_{n})_{n\,\in\,\mathbb{N}}$ on
$\mathbb{Z}^{2}$; for instance, the one defined by
$F_{n}=[-n,n]^{2}\subset\mathbb{Z}^{2}$. If $P$ is the common fixed point by
both $g_{1}$ and $g_{2}$, then the probability measure $\mu_{1}=\delta_{P}$
belongs to $\mathcal{M}_{G}(X)$, and the same happens with the Lebesgue
measure (say $\mu_{2}$) on $\mathbb{T}^{2}$. Thus, given $\varphi\in
C(\mathbb{T}^{2},\mathbb{R})\setminus\\{0\\}$ such that $\varphi\geqslant 0$
and $\varphi(P)=0$, then $\int\varphi\,d\mu_{1}=0\neq\int\varphi\,d\mu_{2}$,
and therefore Corollary 11.2 asserts that

$\Big{\\{}x\in\mathbb{T}^{2}\colon\,\lim_{n\,\to\,+\infty}\,\frac{1}{|F_{n}|}\,\sum_{g\,\in\,F_{n}}\,\varphi(g(x))\,\text{
does not exist}\Big{\\}}$

is a Baire generic subset of $\mathbb{T}^{2}$.

Application 3. A different setting concerns free Markov semigroups. Let
$\Gamma$ be a free semigroup, finitely generated by
$G=\\{Id,g_{1},g_{2},\cdots,g_{p}\\}$, where for every $1\leqslant i\leqslant
p$ the map $g_{i}:X\to X$ is a $\mu$-preserving transformation acting on a
probability measure space $(X,\mathfrak{B},\mu)$. The choice of $G$ endows
$\Gamma$ with a norm $|\cdot|$ defined as follows: given $g\in\Gamma$, $|g|$
is the length of the shortest word over the alphabet $G$ representing $g$.
Denote by $G_{k}$ the set $\\{g\in\Gamma\colon\,|g|=k\\}$. Now take
$\varphi\in L^{\infty}(X,\mu)$ and consider the sequence of its spherical
averages

$k\in\mathbb{N}\quad\mapsto\quad
s_{k}(\varphi)\,=\,\frac{1}{\\#G_{k}}\,\sum_{g\,\in\,G_{k}}\,\varphi\circ g$

where $\\#$ stands for the cardinal of a finite set. Next consider the Cesàro
averages of the previous spherical averages, that is,

$n\in\mathbb{N}\quad\mapsto\quad\Phi_{n}\,=\,\frac{1}{n}\,\sum_{k=0}^{n-1}\,\frac{1}{\\#G_{k}}\,\sum_{g\,\in\,G_{k}}\,\varphi\circ
g.$ (11.7)

The main result of Bufetov _et al_ in [7] establishes the pointwise
convergence at $\mu$-almost everywhere of the sequence
$\big{(}\Phi_{n}\big{)}_{n\,\in\,\mathbb{N}}$, as $n$ goes to $+\infty$,
whenever $\Gamma$ is a Markov semigroup with respect to the generating set
$G$. As a consequence of Theorem A, if there exists a dense set of points
$x\in X$ such that $W_{\Phi}(x)$ is not a singleton, then the set of
$\Phi$-irregular points is Baire generic in $X$. Let us check this information
through an example.

######  Example 8.

Consider the unit circle $\mathbb{S}^{1}=\\{z\in\mathbb{C}\colon\,|z|=1\\}$
and the self-maps of $\mathbb{S}^{1}$ given by $g_{1}(z)=z^{4}$ and
$g_{2}(z)=z^{6}$. These transformations commute and have two fixed points in
common, whose pre-orbits by $g_{1}$ and $g_{2}$ are dense in $\mathbb{S}^{1}$.
Take the free (Markov) semigroup $G$ generated by
$G_{1}=\\{Id,g_{1},g_{2}\\}$, and let $\varphi$ be in
$C(\mathbb{S}^{1},\mathbb{R})$. It is known (cf. [16]) that, for a Baire
generic subset of sequences $\omega=\omega_{1}\omega_{2}\cdots$ in
$\\{1,2\\}^{\mathbb{N}}$, the non-autonomous dynamics of
$(g_{\omega_{n}}\circ\cdots\circ g_{\omega_{1}})_{n\,\in\,\mathbb{N}}$ has a
Baire generic set of irregular points for the Cesàro averages

$\varphi_{n}\,=\,\frac{1}{n}\,\sum_{j=0}^{n-1}\,\varphi\circ
g_{\omega_{j}}\circ\cdots\circ g_{\omega_{1}}$

of well chosen continuous potentials $\varphi$. Regarding the averages (11.7)
of $\varphi\in C(\mathbb{S}^{1},\mathbb{R})$, in this case they are given by

$\Phi_{n}\,=\,\frac{1}{n}\,\sum_{k=0}^{n-1}\,\frac{1}{2^{k}}\,\sum_{j=0}^{k}\,\binom{k}{j}\,(\varphi\circ
g_{1}^{j}\circ g_{2}^{k-j}).$

Let $z_{0}\in\mathbb{S}^{1}$ be a common fixed point for $g_{1}$ and $g_{2}$.
The sequence $\big{(}\Phi_{n}(z_{0})\big{)}_{n\,\in\,\mathbb{N}}$ converges to
$\varphi(z_{0})$ as $n$ goes to $+\infty$. We claim that, for every $x$ in the
pre-orbit $\mathcal{O}^{-}(z_{0})$ of $z_{0}$ by the semigroup action (made up
by the pre-images of $z_{0}$ by all the elements of the semigroup $G$), the
sequence $\big{(}\Phi_{n}(x)\big{)}_{n\,\in\,\mathbb{N}}$ converges to
$\varphi(z_{0})$ as well.

Given $x\in\mathcal{O}^{-}(z_{0})$, there exists $g=g_{i_{n}}\circ\dots\circ
g_{i_{2}}\circ g_{i_{1}}\in G$, where $i_{j}\in\\{1,2\\}$, such that
$g(x)=z_{0}$. Yet, as $g_{1}$ and $g_{2}$ commute, one can simply write
$g=g_{1}^{a}\circ g_{2}^{b}$ for some non-negative integers $a$ and $b$.
Assume that $a,b\geqslant 1$ (the remaining cases are identical). If
$k\geqslant a+b$, the sum

$\frac{1}{2^{k}}\,\sum_{j=0}^{k}\,\binom{k}{j}\,(\varphi\circ g_{1}^{j}\circ
g_{2}^{k-j})(x)$ (11.8)

may be rewritten as

$\frac{1}{2^{k}}\left[\sum_{j=0}^{a-1}\binom{k}{j}(\varphi\circ g_{1}^{j}\circ
g_{2}^{k-j})(x)+\sum_{j=a}^{k-b}\binom{k}{j}(\varphi\circ g_{1}^{j}\circ
g_{2}^{k-j})(x)+\sum_{j=k-b+1}^{k}\binom{k}{j}(\varphi\circ g_{1}^{j}\circ
g_{2}^{k-j})(x)\right].$

The absolute values of the first and third terms in the previous sum are
bounded above by

$a\,\|\varphi\|_{\infty}\,\max\Big{\\{}\binom{k}{0},\cdots,\binom{k}{a}\Big{\\}}\,2^{-k}$

and

$b\,\|\varphi\|_{\infty}\,\max\Big{\\{}\binom{k}{k-b+1},\cdots,\binom{k}{k}\Big{\\}}\,2^{-k}$

respectively, and both estimates converge to zero as $k$ goes to $+\infty$.
Thus, their Cesàro averages also converge to $0$. Regarding the middle term,
as $g_{1}$ and $g_{2}$ commute and $z_{0}$ is fixed by $G$, one has

$\displaystyle\frac{1}{2^{k}}$
$\displaystyle\,\sum_{j=a}^{k-b}\,\binom{k}{j}\,(\varphi\circ g_{1}^{j}\circ
g_{2}^{k-j})(x)=\frac{1}{2^{k}}\,\sum_{j=a}^{k-b}\,\binom{k}{j}\,\big{(}\varphi\circ
g_{1}^{j-a}\circ g_{2}^{k-j-b}\big{)}(g_{1}^{a}\circ g_{2}^{b})(x)$
$\displaystyle=\frac{1}{2^{k}}\,\sum_{j=a}^{k-b}\,\binom{k}{j}\,(\varphi\circ
g_{1}^{j-a}\circ
g_{2}^{k-j-b})(z_{0})=\Big{[}\frac{1}{2^{k}}\,\sum_{j=a}^{k-b}\,\binom{k}{j}\Big{]}\cdot\varphi(z_{0})$

whose limit as $k$ goes to $+\infty$ is precisely $\varphi(z_{0})$. This
proves that the sequence (11.8) converges to $\varphi(z_{0})$, hence the same
happens with its Cesàro averages.

Therefore, if $z_{0}\in\mathbb{S}^{1}$ and $z_{1}\in\mathbb{S}^{1}$ are the
two common fixed points, $x\in\mathbb{S}^{1}$ belongs to the dense pre-orbit
by the semigroup action of $z_{0}$ and $y\in\mathbb{S}^{1}$ belongs to the
dense pre-orbit by the semigroup action of $z_{1}$, then the sequence
$\big{(}\Phi_{n}(x)\big{)}_{n}$ converges to $\varphi(z_{0})$ and the sequence
$\big{(}\Phi_{n}(y)\big{)}_{n}$ converges to $\varphi(z_{1})$. So, taking into
account that $\mathcal{O}^{-}(z_{0})$ is dense in $\mathbb{S}^{1}$ (since the
pre-orbit by $g_{1}$ is dense), if we choose $\varphi\in
C(\mathbb{S}^{1},\mathbb{R})$ such that $\varphi(z_{0})\neq\varphi(z_{1})$
then, by Theorem A, we conclude that ${\mathcal{I}}(\Phi)$ is Baire generic in
$\mathbb{S}^{1}$.

######  Remark 11.3.

As the generators of the semigroup action described in Example 8 commute, one
could consider, alternatively, the sequence $(\Psi_{n})_{n}$ where

$\Psi_{n}(\cdot)\,=\,\frac{1}{n^{2}}\,\sum_{k,\,\ell=0}^{n-1}\,\,(\varphi\circ
g_{1}^{k}\circ g_{2}^{\ell})(\cdot).$

If $z_{0}\in\mathbb{S}^{1}$ is a common fixed point for $G$ and
$x\in\mathcal{O}^{-}(z_{0})$, then this sequence can be rewritten as

$\frac{1}{n^{2}}\left[\sum_{(k,\,\ell)\,\in\,[a,n-1]\times[b,n-1]}(\varphi\circ
g_{1}^{\ell}\circ
g_{2}^{k-\ell})(x)+\sum_{(k,\,\ell)\,\notin\,[a,n-1]\times[b,n-1]}(\varphi\circ
g_{1}^{\ell}\circ g_{2}^{k-\ell})(x)\right]$

where the sum is taken over pairs of integers $(k,\ell)$. The first term is
equal to

$\frac{1}{n^{2}}\sum_{(k,\,\ell)\,\in\,[a,n-1]\times[b,n-1]}(\varphi\circ
g_{1}^{\ell-a}\circ
g_{2}^{k-b-\ell})(z_{0})=\frac{(n-a)(n-b)}{n^{2}}\;\varphi(z_{0})$

and converges to $\varphi(z_{0})$ as $n$ goes to $+\infty$; meanwhile, the
second term has absolute value bounded above by
$\frac{a+b}{n}\,\|\varphi\|_{\infty}$, which goes to zero as $n$ tends to
$+\infty$. Thus, if $z_{0}\in\mathbb{S}^{1}$ and $z_{1}\in\mathbb{S}^{1}$ are
two common fixed points and $\varphi(z_{0})\neq\varphi(z_{1})$ then, by
Theorem A, one concludes that ${\mathcal{I}}(\Psi)$ is Baire generic in
$\mathbb{S}^{1}$.

### Acknowledgments

LS is partially supported by Faperj, Fapesb-JCB0053/2013 and CNPq. MC and PV
are supported by CMUP, which is financed by national funds through FCT-
Fundação para a Ciência e a Tecnologia, I.P., under the project
UIDB/00144/2020. PV also acknowledges financial support from Fundação para a
Ciência e Tecnologia (FCT) - Portugal through the grant CEECIND/03721/2017 of
the Stimulus of Scientific Employment, Individual Support 2017 Call.

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