2 Irregular points for continuous maps on compact metric spaces

Suppose now that $(X,d)$ is a compact metric space. Let $\mathcal {P}(X)$ denote the set of 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 $({1}/{n} \sum _{j=0}^{n-1} \delta _{T^{j}x})_{n \in \mathbb {N}}$ by $V_T(x)$ . Our next result imparts new information about the irregular set without requiring any assumption about the existence of isolated points in X.

The next result is a consequence of [Reference Winkler36, Propositions 1 and 2], [Reference Hou, Lin and Tian20, Corollary 2.2 and Proposition 3.1] and Theorem 1.4.

We may ask whether the notion of $(T,\varphi )$ -sensitivity (cf. Definition 1.1) is somehow related to the classical concepts of sensitivity to initial conditions and expansiveness. Let us recall these two notions.

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.

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

(2.1) $$ \begin{align} \begin{array}{ccc} T \text{ is strongly transitive and has} & & \\ \qquad \text{ dense periodic orbits} & & \\ \Downarrow & & \\ T \text{ has dense periodic orbits} & & \\ \text{and dense pre-orbits} & & \text{Expansiveness} \\ \Downarrow & & \Downarrow \; \; \not\Uparrow\\ \text{ there exists } \,\varphi\in C(X, \mathbb R) : X \text{ is } (T,\varphi)\,- & \quad\Rightarrow \quad & \text{Sensitivity to initial conditions}. \\ \text{sensitive and} \; \text{the interior of } {\mathcal I}(T,\varphi) =\emptyset & & \end{array} \end{align} $$

When X is a compact topological manifold there is a link between expansiveness and sensitivity with respect to a well-chosen continuous map $\varphi : X \to \mathbb R$ . Indeed, Coven and Reddy (cf. [Reference Coven, Reddy, Nitecki and Robinson14]) proved that, if $T: X \to X$ is a continuous expansive map acting on a 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 $\unicode{x3bb}> 1$ and $\delta _0> 0 $ such that, for all $x, y, z\in X$ , one has

• • $\tilde d(T(x),T(y)) \geqslant \unicode{x3bb} \tilde d(x,y)$ 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. Moreover, the interior of ${\mathcal I}(T,\varphi )$ is empty by the denseness of the set of periodic points of T. Therefore, summoning Corollary 1.7, we conclude as follows.

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 $\varphi \in C(X, \mathbb R)$ .

Another consequence of Theorem 1.2 concerns the irregular sets for continuous maps satisfying the strong transitivity condition (see Definition 1.2).

We note that, as strongly transitive homeomorphisms on a compact metric space X are minimal, Corollary 2.4 extends the information in Remark 1.5 to strongly transitive continuous maps on compact metric spaces.

3 Proof of Theorem 1.2

The argument is a direct adaptation of that used to show [Reference Carvalho and Varandas9, Theorem $1$ ]. Suppose that there are dense subsets $A, B$ of Y and $\varepsilon> 0$ such that for any $(a,b) \in A \times B$ there exist $(r_{a},r_{b}) \in \{\varphi _{n}(a): n \geq 1 \}' \times \{\varphi _{n}(b): n \geq 1 \}'$ satisfying $\lvert r_{a} -r_{b}\rvert> \varepsilon $ . Fix $0 <\eta < {\varepsilon }/{3}$ . As the maps $\varphi _n$ are continuous, given an integer $N \in \mathbb {N}$ the set

$$ \begin{align*}\Lambda_{N} = \{y \in Y: \lvert \varphi_{n}(y) - \varphi_{m}(y) \rvert \leqslant \eta \text{ for all } m,n \geqslant N \}\end{align*} $$

is closed in Y. Moreover, we have the following result.

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 $a \in {\operatorname {int}} (\Lambda _{N})$ . As $\varphi _{N}$ is continuous, there exists $\delta _{N}>0$ such that $\lvert \varphi _{N}(a) - \varphi _{N}(y)\rvert < \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 $d(a,b) < \delta _{N}$ . In addition, according to the definition of $\Lambda _{N}$ , one has

$$ \begin{align*} \lvert \varphi_{n}(a) - \varphi_{m}(a) \rvert \leqslant \eta \quad\text{and}\quad \lvert \varphi_{n}(b) - \varphi_{m}(b)\rvert \leqslant \eta \quad\text{for all } m, n \geqslant N. \end{align*} $$

For the previous pair $(a,b) \in A\times B$ , choose $ (r_{a},r_{b}) \in \{\varphi _{n}(a): n \geq 1 \}' \times \{\varphi _{n}(b): n \geq 1 \}'$ satisfying $\lvert r_{a}-r_{b}\rvert> \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

$$ \begin{align*} \lvert r_{a} - \varphi_{N}(a)\rvert \leqslant \eta \quad\text{and}\quad \lvert r_{b} - \varphi_{N}(b)\rvert \leqslant \eta. \end{align*} $$

Therefore,

$$ \begin{align*}\varepsilon < \lvert r_{a} - r_{b}\rvert \leqslant \lvert \varphi_{N}(a) - r_{a} \rvert + \lvert \varphi_{N}(b) - r_{b} \rvert + \lvert \varphi_{N}(a) - \varphi_{N}(b)\rvert \leqslant 3 \eta\end{align*} $$

contradicting the choice of $\eta $ . Thus, $\Lambda _{N} $ must have empty interior.

We can now finish the proof of Theorem 1.2. Using the fact that $\limsup _{n \to +\infty } \lVert \varphi _n\rVert _\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 (that is, those points for which $\Phi $ is a convergent sequence) 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 1.2 also shows that one has

$$ \begin{align*} \{y \in Y: \limsup_{n} \varphi_{n}(y) - \liminf_{n} \varphi_{n}(y) < \eta \} \subseteq \bigcup_{N=1}^{\infty} \Lambda_{N} \end{align*} $$

and so the set

$$ \begin{align*}\{y \in Y : \limsup_{n} \varphi_{n}(y) - \liminf_{n} \varphi_{n}(y) \geqslant \eta \}\end{align*} $$

is Baire generic in Y.

4 Proof of Corollary 1.3

We start by 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.

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

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

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 1.3.

(i) 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. As $\{T^{N}(y)\}$ is an open subset of Y, one has $\{T^{N}(y) \} \cap {\mathcal I}(T,\psi ) \neq \emptyset $ , hence $T^{N}(y)$ 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. Thus, 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 1.2 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

$$ \begin{align*}\Omega_{y} = \overline{\{T^{n}(y): n \in \mathbb{N}\cup\{0\}\}} = Y\end{align*} $$

so Y is separable. This implies that, if $\omega (y) = \Omega _{y}$ , then Y does not have isolated points, and so T is transitive. As Y is a Baire separable metric space, ${\operatorname {Trans}} (Y,T)$ is Baire generic as well (cf. [Reference Akin and Carlson1, Proposition 4.7]). 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 $\bigcap _{\varphi \in \mathcal {D}(Y,T)} {\mathcal I}(T,\varphi ) \subseteq \bigcap _{\varphi \in \mathfrak {F}} {\mathcal I}(T,\varphi )$ . Consequently,

$$ \begin{align*}{\operatorname{Trans}} (Y,T) \subseteq \bigcap_{\varphi \in \mathfrak{F}} {\mathcal I}(T,\varphi).\end{align*} $$

5 Proof of Theorem 1.4

We adapt the proof of [Reference Hou, Lin and Tian20, Lemma 4.1]. Consider the sets

$$ \begin{align*}A_{\ell_\varphi^{*}} = \bigg\{y \in Y : \liminf\limits_{n \to +\infty} \frac{1}{n} \sum\limits_{j=0}^{n-1} \varphi (T^{j} y) = \ell_\varphi^{*} \bigg\},\end{align*} $$

$$ \begin{align*}A_{L_\varphi^{*}} = \bigg\{y \in Y : \limsup\limits_{n \to +\infty} \frac{1}{n} \sum\limits_{j=0}^{n-1} \varphi (T^{j} y) = L_\varphi^{*} \bigg\}\end{align*} $$

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

$$ \begin{align*} B_\alpha = \bigcap_{N = 1}^{+\infty} \bigcup_{n = N}^{+\infty} \bigg\{y \in Y : \frac1n \sum_{i=0}^{n-1} \varphi(T^i y) < \alpha\bigg\}. \end{align*} $$

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 $\bigcup _{n=N}^{+\infty }\{y \in Y : (1/n) \sum _{i=0}^{n-1} \varphi (T^i y) < \alpha \}$ 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

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

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 ${\mathcal I}(T,\varphi )$ .

Remark 5.1. According to [Reference Pinheiro30, 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

$$ \begin{align*}\ell_\varphi(y) = c_{-}^{\varphi} \quad\text{and}\quad L_\varphi(y) = c_{+}^{\varphi} \quad \text{for all } y \in \mathfrak{R}.\end{align*} $$

6 Proof of Corollary 1.7

$\mathrm {(i)} \Rightarrow \mathrm {(ii)}$ 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, hence dense, subset of Y. Note that, for any z and w in $\mathcal {D}$ , one has either $\lvert L_{\varphi }(z) - \ell _{\varphi }(w)\rvert> {\eta }/{2}$ or $\lvert L_{\varphi }(w) - \ell _{\varphi }(z)\rvert> {\eta }/{2}$ . This implies that Y is $(T,\varphi )$ -sensitive, and so, by Theorem 1.2, the set ${\mathcal I}(T,\varphi )$ is Baire generic.

$\mathrm {(ii)} \Rightarrow \mathrm {(iii)}$ Apply Corollary 1.3(ii) with $\mathfrak {F}= \{\varphi \}$ .

$\mathrm {(iii)} \Rightarrow \mathrm {(i)}$ 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 $(\varphi _{m_{k}}(y_1))_{k \in \mathbb {N}}$ . These are enough data to confirm that Y is $(T,\varphi )$ -sensitive.

7 Proof of Corollary 2.1

(a) Suppose that X is a compact metric space that has a dense orbit. From [Reference Winkler36, Propositions 1 and 2] or [Reference Hou, Lin and Tian20, Corollary 2.2 and Proposition 3.1], 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, in contrast, 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

$$ \begin{align*} \ell_\varphi(x)> \ell_\varphi(x_0) = \lim\limits_{k \to +\infty} \frac{1}{n_{k}} \sum\limits_{j=0}^{n_{k}-1} \varphi (T^{j}(x_0)). \end{align*} $$

Reducing to a subsequence if necessary, we find $\mu \in V_{T}(x_0)$ such that $({1}/{n_{k}}) \sum _{j=0}^{n_{k}-1} \delta _{T^{j}(y)}$ 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 $ . As, by hypothesis, $\mu $ belongs to $V_{T}(x)$ , there exists an infinite sequence $(n_{q})_{q \in \mathbb {N}}$ such that $({1}/{n_{q}}) \sum _{j=0}^{n_{q}-1} \delta _{T^{j}(x)}$ converges to $\mu $ as q goes to $+\infty $ . This yields

$$ \begin{align*} \int \varphi \,d\mu = \lim\limits_{q \to +\infty} \frac{1}{n_{q}} \sum\limits_{j=0}^{n_{q}-1} \varphi (T^{j}(x)) \geqslant \ell_\varphi(x)>\ell_\varphi(x_0) = \int \varphi \,d\mu, \end{align*} $$

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 by establishing the following auxiliary result.

Proof. As $\mathcal {R}(X,T) = \emptyset $ , from Theorem 1.4 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 $C(X, \mathbb {R})$ , which exists because X is compact (cf. [Reference Kelley22]). Then $\bigcap _{\psi \in S} X \setminus \mathcal {I}(T,\psi )$ is Baire generic in X. We are left to prove that

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

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 $\lVert \varphi - \psi \rVert _\infty < {\varepsilon }/{3}$ . Since $x \in X \setminus \mathcal {I}(T,\psi )$ , there is $N \in \mathbb {N}\cup \{0\}$ such that, for every $n, m \geqslant N$ , we have $\lVert \psi _{n}(x)- \psi _{m}(x)\rVert < {\varepsilon }/{3}$ . Therefore,

$$ \begin{align*} \lVert \varphi_{n}(x) - \varphi_{m}(x)\rVert \leqslant \lVert \varphi_{n}(x) - \psi_{n}(x) \rVert + \lVert \psi_{n}(x) - \psi_{m}(x) \rVert + \lVert \psi_{m}(x) - \varphi_{m}(x) \rVert < \varepsilon \end{align*} $$

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, due to the compactness of X, Riesz representation theorem and Theorem 1.4, there exists a Borel invariant probability measure $\mu $ such that

$$ \begin{align*} {\operatorname{Trans}}(X,T) \subset \bigg\{x \in X: \lim_{n \to +\infty} \frac{1}{n} \sum_{j=0}^{n-1} \varphi (T^{j}(x)) = \int \varphi \, d\mu \text{ for all } \varphi \in C(X,\mathbb{R})\bigg\}. \end{align*} $$

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

$$ \begin{align*} X_{\Delta} \cap \bigcap_{\varphi \in C(X, \mathbb{R})} X \setminus {\mathcal I}(T,\varphi) &= \bigg\{x \in X: \lim_{n \to +\infty} \frac{1}{n} \sum\limits_{j=0}^{n-1} \varphi (T^{j}(x))\\ &= \int \varphi \,d\mu \text{ for all } \varphi \in C(X,\mathbb{R})\bigg\} \end{align*} $$

is Baire generic in X.

(c) First, we note that, because Corollary 1.7 shows that

$$ \begin{align*} \mathcal{R}(X,T) = \{\varphi \in C(X,\mathbb{R}) : \ell_\varphi^{*} < L_\varphi^{*}\}, \end{align*} $$

then one has

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

Proof. We prove that $\mathrm {(i)} \Leftrightarrow \mathrm {(ii)}$ , $\mathrm {(ii)} \Rightarrow \cdots \Rightarrow \mathrm {(vii)}$ and $\mathrm {(vii)} \Rightarrow \mathrm {(ii)}$ .

$\mathrm {(i)} \Rightarrow \mathrm {(ii)}$ Suppose that $\# (\bigcup _{x \in {\operatorname {Trans}}(X,T)} V_{T}(x))> 1$ . Then there exist two distinct Borel probability measures $\mu $ , $\nu $ in $\bigcup _{x \in {\operatorname {Trans}}(X,T)} V_{T}(x)$ 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 because, by item (a), the set $X_{\Delta }$ is Baire generic.

$\mathrm {(ii)} \Rightarrow \mathrm {(i)}$ Suppose that there exists $\varphi \in C(X,\mathbb {R})$ such that ${\mathcal I}(T,\varphi )$ is Baire generic. By Corollary 1.7, one has ${\operatorname {Trans}} (X,T) \cap {\mathcal I}(T,\varphi ) \neq \emptyset $ . Given $y \in {\operatorname {Trans}} (X,T) \cap {\mathcal I}(T,\varphi )$ , one has $\ell _\varphi (y) < L_\varphi (y)$ , and so $\#V_{T}(y)> 1$ .

$\mathrm {(ii)} \Rightarrow \mathrm {(iii)}$ 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 [Reference Hou, Lin and Tian20, Lemma 3.2]) and, by Corollary 1.7, $\mathcal {R}(X,T) = \bigcup _{x \in {\operatorname {Trans}}(X,T)} \mathcal {U}_{x}$ . Therefore, if $\mathcal {R}(X,T) \neq \emptyset $ , then $\mathcal {R}(X,T)$ is open and dense in $C(X,\mathbb {R})$ .

$\mathrm {(iii)} \Rightarrow \mathrm {(iv)}$ 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 1.7, $\mathcal {R}(X,T) = \{\varphi \in C(X,\mathbb {R}): \ell _\varphi ^{*} < L_\varphi ^{*}\}$ . Using item (a), we know that $X_{\Delta } \subseteq \bigcap _{\varphi \in C(X,\mathbb {R})} \widehat {I_{\varphi }[\ell _\varphi ^{*},L_\varphi ^{*}]}$ . Moreover,

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

and

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

Consequently,

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

$\mathrm {(iv)} \Rightarrow \mathrm {(v)}$ 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.

$\mathrm {(v)} \Rightarrow \mathrm {(vi)}$ Suppose that $\bigcap _{\varphi \in \mathcal {R}(X,T)} {\mathcal I}(T,\varphi )$ is Baire generic. By Corollary 1.3(ii), we know that ${\operatorname {Trans}} (X,T) \cap \bigcap _{\varphi \in \mathcal {R}(X,T)} \mathcal {I}(T,\varphi ) \neq \emptyset $ .

$\mathrm {(vi)} \Rightarrow \mathrm {(vii)}$ This is clear from Corollary 1.7.

$\mathrm {(vii)} \Rightarrow \mathrm {(ii)}$ 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

$$ \begin{align*} \emptyset = \bigg(\bigcup\limits_{\varphi \in C(X,\mathbb{R})} \mathcal{I}(T,\varphi)\bigg) \cap \bigg(\bigcap\limits_{\varphi \in C(X, \mathbb{R})} X \setminus {\mathcal I}(T,\varphi) \bigg) \end{align*} $$

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

We now resume the proof of item (c). 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 $\ell _\varphi ^{*} = L_\varphi ^{*}$ .

In fact, suppose that $\mathcal {I}(T,\varphi ) \neq \emptyset $ and $\ell _\varphi ^{*} = L_\varphi ^{*}$ . Denote this common value by $\gamma $ . Then by Theorem 1.4(i), the set $I_{\varphi }[\gamma , \gamma ]$ is Baire generic in Y. Moreover, we have ${\operatorname {Trans}}(X,T) \subseteq I_{\varphi }[\gamma , \gamma ]$ . Therefore, by Theorem 1.4(ii), $\mathcal {I}(T,\varphi )$ is meagre.

As $\bigcap _{\varphi \in \mathcal {H}(X,T)} \mathcal {I}(T,\varphi ) \subseteq \mathcal {I}(T,\varphi )$ , we conclude from the previous claim that the set $\bigcap _{\varphi \in \mathcal {H}(X,T)} \mathcal {I}(T,\varphi )$ is meagre as well. Thus, 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: 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.

As a consequence of Corollary 2.1 we obtain the following dichotomy between uniquely ergodicity and Baire genericity of historic behavior.

Proof. Let $T : X \to X$ be a continuous minimal map which is not uniquely ergodic. Suppose, by contradiction, that there is $\varphi $ in $C(X, \mathbb {R})$ such that ${\mathcal I}(T,\varphi ) \neq \emptyset $ and $\ell _{\varphi }^{*} = L_{\varphi }^{*}$ . Then there exists $x_0 \in X$ such that $ \ell _{\varphi }(x_0) < L_{\varphi }(x_0)$ . As T is minimal, one has ${\operatorname {Trans}} (X,T) = X$ ; in particular, the orbit of $x_0$ is dense in X. Consequently,

$$ \begin{align*} \ell_{\varphi} ^{*} \leqslant \ell_{\varphi}(x_0) < L_{\varphi} (x_0) \leqslant L_{\varphi}^{*}. \end{align*} $$

However, this contradicts the assumption $\ell _{\varphi }^{*} = L_{\varphi }^{*}$ . Thus,

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

and therefore, by Corollary 2.1(c), the set $\bigcap _{\varphi \in \mathcal {H}(X,T)} {\mathcal I}(T,\varphi )$ is Baire generic.

8 Proof of Theorem 2.2

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 $(r_{a}, r_{b}) \in \{\varphi _{n}(a): n \in \mathbb {N}\}' \times \{\varphi _{n}(b): n \in \mathbb {N}\}'$ satisfying $\lvert r_{a} -r_{b}\rvert> \varepsilon $ . Using the uniform continuity of $\varphi $ , one can choose $\delta> 0$ such that $\lvert \varphi (z) - \varphi (w)\rvert < {\varepsilon }/{2}$ for every $z,w \in X$ with $d(z,w) < \delta $ .

Assume, by contradiction, that ${\mathcal I}(T,\varphi )$ has empty interior and T has no sensitivity to initial conditions. The latter condition 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 $\theta = {\delta }/{3}$ and take $x_\theta $ and $U_{x_\theta }$ as previously.

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

$$ \begin{align*} \lvert \varphi(T^{n}(a)) - \varphi(T^{n}(c))\rvert < \frac{\varepsilon}{2}\quad \text{and} \quad \lvert \varphi(T^{m}(b)) - \varphi(T^{m}(c))\rvert < \frac{\varepsilon}{2} \quad \text{for all } n,m \in \mathbb{N} \cup \{0\}. \end{align*} $$

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

$$ \begin{align*} \bigg|\frac{1}{n} \sum\limits_{j=0}^{n-1} \varphi(T^{j}(a)) - \frac{1}{n} \sum\limits_{j=0}^{n-1}\varphi(T^{j}(c)) \bigg| < \frac{\varepsilon}{2}\quad \text{and} \quad \bigg|\frac{1}{m} \sum\limits_{j=0}^{m-1} \varphi(T^{j}(b)) - \frac{1}{m} \sum\limits_{j=0}^{m-1} \varphi(T^{j}(c)) \bigg| < \frac{\varepsilon}{2}. \end{align*} $$

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

$$ \begin{align*}\bigg|r_{a} - \lim\limits_{k \to +\infty} \frac{1}{n_k} \sum\limits_{j=0}^{n_k-1} \varphi(T^{j}(c)) \bigg| \leqslant \frac{\varepsilon}{2} \quad\text{and}\quad \bigg|r_{b} - \lim\limits_{k \to \infty} \frac{1}{m_k} \sum\limits_{j=0}^{m_k-1} \varphi(T^{j}(c)) \bigg| \leqslant \frac{\varepsilon}{2} \end{align*} $$

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

Assume now that X is $(T,\varphi )$ -sensitive and T has a dense set of periodic orbits. As the periodic points of T are $\varphi $ -regular and form a dense subset of X, the set ${\mathcal I}(T,\varphi )$ must have empty interior. Thus, by the previous statement, T has sensitivity on initial conditions.

9 Proof of Corollary 2.4

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 $\mathcal O_T^-(x_0) = \{x \in X : T^n(x)=x_0 \text { for some } n \in \mathbb {N}\}$ of $x_0$ is dense in X. Moreover,

$$ \begin{align*} \sup_{r, s \in W_{\varphi}(x)} \lvert r-s\rvert> \varepsilon \quad \text{for all } x \in \mathcal O_T^-(x_0). \end{align*} $$

This proves that X is $(T,\varphi )$ -sensitive and so, by Theorem 1.2, the set $\mathcal I(T,\varphi )$ is Baire generic in X.

10 Examples

In this section we discuss the hypothesis and derive some consequences of the main results. The first example helps to clarify the requirements in Theorem 1.2 and Corollary 1.3, in addition to calling our attention to the difference between transitivity and the existence of a dense orbit.

The second example illustrates Definition 1.1.

Example 10.2. Consider the shift space $X = \{0,1\}^{\mathbb {N}}$ endowed with the metric defined by

$$ \begin{align*} d((a_n)_{n \in \mathbb{N}}, (b_n)_{n \in \mathbb{N}}) = \sum_{n=1}^{+\infty} \frac{\lvert a_n - b_n\rvert }{2^n} \end{align*} $$

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

$$ \begin{align*} A &= \{(a_n)_{n \in \mathbb{N}} \mid \text{there exists } N \in \mathbb{N} : a_n = 0 \text{ for all } n \geqslant N\}, \\ B &= \{(a_n)_{n \in \mathbb{N}} \mid \text{there exists } N \in \mathbb{N} : a_n = 1 \text{ for all } n \geqslant N\} \end{align*} $$

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

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

Therefore,

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

Thus, if $\varphi (\overline {0}) \neq \varphi (\overline {1})$ and one chooses $\varepsilon = \lvert \varphi (\overline {0}) - \varphi (\overline {1})\rvert /2$ , 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 $\lvert r_a - r_b\rvert> \varepsilon $ . Consequently, X is $(\sigma , \varphi )$ -sensitive with respect to any $\varphi \in C(X, \mathbb {R})$ satisfying $\varphi (\overline {0}) \neq \varphi (\overline {1})$ . Hence, by Theorem 1.2, for every such map $\varphi $ the set ${\mathcal I}(T,\varphi )$ is Baire generic in X.

Remark 10.3. 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.

In the following example we 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 [Reference Pinheiro29] and references therein), hyperbolic systems with singularities [Reference Chen, Wang and Zhang10], including Sinai dispersing billiards, and certain classes of piecewise monotone interval maps [Reference Hofbauer19], which encompass the piecewise expanding Lorenz interval maps, just to mention a few.

Example 10.4. Let $\mathcal {A}$ be a countable set, $\mathbb A = (a_{i,j})_{i,j \in \mathcal {A}}$ be a matrix of zeros and ones and $\Sigma _{\mathbb A} \subset \mathcal {A}^{\mathbb N}$ be the subset

$$ \begin{align*} \Sigma_{\mathbb A} = \{(x_n)_{n \in \mathbb{N}} \in \mathcal{A}^{\mathbb N} : a_{{x_n},{x_{n+1}}} = 1 \text{ for all } n \in \mathbb{N}\}. \end{align*} $$

Endow $\Sigma _{\mathbb A}$ with the metric

$$ \begin{align*}d((x_n)_{n \in \mathbb{N}}, (y_n)_{n \in \mathbb{N}}) = \left\{ \begin{array}{ll} 2^{-\min \{k \in \mathbb{N} : x_k \neq y_k\}} & \text{ if } \{k \in \mathbb{N} : x_k \neq y_k\} \neq \emptyset, \\ 0 & \text{ otherwise}. \end{array} \right.\end{align*} $$

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. In addition, the metric space $(\Sigma _{\mathbb A},d)$ has a countable basis for the topology, generated by the countably many cylinders, and it is invariant by the shift map $\sigma : \mathcal {A}^{\mathbb N} \to \mathcal {A}^{\mathbb N}$ .

If $\sigma _{|\Sigma _{\mathbb A}}$ has the periodic specification property (see the examples in [Reference Sarig32]), then the set of periodic points of $\sigma _{|\Sigma _{\mathbb A}}$ is dense in $\Sigma _{\mathbb A}$ and all the points in $\Sigma _{\mathbb A}$ have dense pre-orbits. Therefore, there exists $\varphi \in C^b(\Sigma _{\mathbb A}, \mathbb R)$ such that $\Sigma _{\mathbb A}$ is $(\sigma ,\varphi )$ -sensitive, whose $\mathcal {I}(\sigma ,\varphi )$ is Baire generic by Theorem 1.2. Indeed, for each $\varphi \in C^b(\Sigma _{\mathbb A},\mathbb R)$ , either:

1. (a) there exists a constant $c_\varphi \in \mathbb R$ such that, for every periodic point p,

   $$ \begin{align*} \frac1{\pi(p)} \sum_{j=0}^{\pi(p)-1} \varphi(\sigma^j(p)) = c_\varphi, \end{align*} $$
   where $\pi (p)\in \mathbb {N}$ denotes the minimal period of p; or

2. (b) there are two periodic points $p, q \in \Sigma _{\mathbb A}$ such that

   $$ \begin{align*} \frac1{\pi(p)} \sum_{j=0}^{\pi(p)-1} \varphi(\sigma^j(p)) \neq \frac1{\pi(q)} \sum_{j=0}^{\pi(q)-1} \varphi(\sigma^j(q)) \end{align*} $$
   and so $\Sigma _{\mathbb A}$ is $(\sigma ,\varphi )$ -sensitive, because the pre-orbits of p and q are dense and, therefore, provide two sets A and B complying with the Definition 1.1 (see Remark 10.3).

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 10.5. Let $(\Psi _t)_{t \in \mathbb R}$ be the Bowen’s example, that is, 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, whereas all the other singularities are repellers. 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 orbit by T of every point in $Y = \mathbb S^2\setminus (\bigcup _{i=1}^4 \gamma _i)$ accumulates 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.

Figure 1

Flow with historic behavior.

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

Example 10.6. Consider the annulus $\mathbb A = {\mathbb S}^1\times [0,1]$ and the map $T: \mathbb A \to \mathbb A$ given by

$$ \begin{align*} T(x,t) = \bigg( 3x (\mbox{mod}~1), t+ \frac{t(1-t)}{32} \cos (2\pi x) \bigg) \quad\text{for all } (x,t) \in \mathbb S^1 \times [0,1]. \end{align*} $$

In [Reference Kan21], Kan proved that T admits two physical measures (that is, T-invariant probability measures whose basins of attraction have positive Lebesgue measure), 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 $B(\mu _0)$ and $B(\mu _1)$ are intermingled, that is, for every non-empty open set $\mathcal U \subset \mathbb A$

$$ \begin{align*} \mbox{Leb}_{\mathbb A} ( \mathcal U \cap B(\mu_0) )> 0 \quad \text{and} \quad \mbox{Leb}_{\mathbb A}( \mathcal U \cap B(\mu_1) ) > 0, \end{align*} $$

where

$$ \begin{align*} B(\mu) = \bigg\{x \in \mathbb A : \bigg(\frac1n \sum_{j=0}^{n-1}\varphi (T^j(y))\bigg)_{n \in \mathbb{N}} \text{ converges to } \int \varphi \, d\mu, \quad \text{for all } \varphi \in C(\mathbb A, \mathbb{R})\bigg\}. \end{align*} $$

Later, Bonatti, Díaz and Viana introduced in [Reference Bonatti, Díaz and Viana6] the concept of a Kan-like map and proved that any such map robustly admits two physical measures. More recently, Gan and Shi [Reference Gan and Shi18] 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 $\mathcal U \subset \mathbb A$ ,

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

Using Theorem 1.2 we conclude that, if T is a Kan-like map and $\varphi \in C(\mathbb A, \mathbb R)$ satisfies the inequality $\int \varphi \,d\mu _0 \neq \int \varphi \,d\mu _1$ , then $\mathcal I(T,\varphi )$ is a Baire generic subset of $\mathbb A$ . More generally, the argument that established Corollary 2.4 also ensures that, for any $\varphi \in C(\mathbb A, \mathbb R)$ , one has either:

1. (a) $ \mathcal I(T,\varphi ) \cap \text {interior of }\mathbb A = \emptyset $ ; or

2. (b) $ \mathcal I(T,\varphi )$ is a Baire generic subset of $\mathbb A$ .

In particular, when $\mathcal I(T,\varphi )$ is dense, then it is Baire generic; thus, $\mathcal {D}(\mathbb A,T) = \mathcal {R}(\mathbb A,T)$ .

11 Applications

As will become clear in the remainder of this section, Theorem 1.2 has a wide range of applications according to the class of sequences $\Phi =(\varphi _n)_{n\ge 1}$ of observables one considers. Let us provide two such applications, one with a geometric motivation and another in the context of semigroup actions.

11.1 Application 1

The irregular sets of uniformly hyperbolic maps and flows on compact Riemannian manifolds have been studied extensively. 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 [Reference Li and Wu23]. 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 1.2 precisely to this setting.

Example 11.1. Let $(M,g)$ be a connected, complete Riemannian manifold. We discuss the Baire genericity of points with historic behavior in the following two cases (we refer the reader to [Reference Coudène and Schapira12] for precise definitions and more information):

1. (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 hyperbolic periodic orbits;

2. (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 case (I), by [Reference Coudène and Schapira12, Theorem 1.1] it is known that the space $\mathcal E$ of Borel ergodic probability measures fully supported on the non-wandering set $\Omega $ (that is, every point in $\Omega $ belongs to their support) is a $G_\delta $ dense subset of all Borel probability measures on $T^1 M$ which are invariant by the geodesic flow. In particular, as $\# \mathcal E \geqslant 2$ due to the assumption that the geodesic flow has at least two distinct periodic orbits, one can choose a continuous observable $\varphi : T^1 M \to \mathbb R$ such that

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

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

$$ \begin{align*} L_\varphi(\cdot) = \limsup_{t \to +\infty} \frac1t \int_0^t \varphi(\Psi^g_s(\cdot)) \, ds \end{align*} $$

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 that 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 1.2 ensures that $\mathcal {I}(T,\Phi )$ is Baire generic. Consequently,

$$ \begin{align*} \mathcal{I}((\Psi^g_t)_{t \in \mathbb R}, \varphi) = \bigg\{y \in \Omega : \lim\limits_{t \to + \infty} \frac1t \int_0^t \varphi(\Psi^g_s(y))\, ds \text{ does not exist}\bigg\} \end{align*} $$

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

The previous argument can be easily adapted to case (II) to yield the conclusion that $I((\Psi ^g_t)_{t \in \mathbb R}, \varphi )$ is Baire generic because, according to [Reference Coudène and Schapira13, 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.

11.2 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 $m(F \Delta K F) < \delta m(F)$ , 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 [Reference Pfister and Sullivan27] for alternative formulations of this concept. A sequence $(F_n)_n$ of compact subsets of G is a Følner sequence if, for every compact $K \subset G$ and every $\delta> 0$ , 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

$$ \begin{align*} m\bigg(\bigcup_{1 \leqslant k < n} F_k^{-1} F_n \bigg) \leqslant C m( F_n ) \quad \text{for all } n \in \mathbb{N}. \end{align*} $$

It is known that every Følner sequence has a tempered subsequence and that every amenable group has a tempered Følner sequence (cf. [Reference Lindenstrauss25, 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

$$ \begin{align*} \lim_{n \to +\infty} \frac1{m(F_n)} \int_{F_n} \varphi(g(x)) \, dm(g) \end{align*} $$

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. [Reference Lindenstrauss25, Theorem 1.2]).

Let $(X,d)$ be a compact metric space and G be a 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 $\mu (g^{-1}(A)) = \mu (A)$ for every measurable set A and every $g \in G$ . We denote the space of G-invariant probability measures by $\mathcal M_G(X)$ , whose subset of ergodic elements is $\mathcal {E}_G(X)$ . A group action of G on X is said to be uniquely ergodic if it admits a unique G-invariant ergodic probability measure (a property equivalent to the existence of a unique G-invariant probability measure if G is a countable amenable group, due to the ergodic decomposition; see [Reference Ornstein and Weiss26]). Given $\mu \in \mathcal M_G(X)$ , the basin of attraction of $\mu $ is defined by

$$ \begin{align*} B(\mu) = \bigg\{x \in X : \frac1{\lvert F_n\rvert} \sum_{g \in F_n} \delta_{g(x)} \to \mu (\text{convergence in the weak}^* \text{ topology})\bigg\} \end{align*} $$

where $\delta _{g(x)}$ denotes the Dirac probability measure supported on the point $g(x)$ .

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

(11.1) $$ \begin{align} \varphi_n(x) = \frac1{m(F_n)} \int_{F_n} \varphi(g(x)) \, dm(g). \end{align} $$

Clearly, $\lVert \varphi _n\rVert _\infty \leqslant \lVert \varphi \rVert _\infty $ for every $n \in \mathbb {N}$ . Moreover, if we assume that there are fully supported G-invariant ergodic Borel probability measures $\mu _1 \neq \mu _2$ , then there is $\varphi \in C(X,\mathbb R)$ such that X is $\Phi $ -sensitive, because the basins of attraction of $\mu _1$ and $\mu _2$ are disjoint and both dense in X. Thus, under these assumptions, Theorem 1.2 ensures that, for the map $\varphi $ , the irregular set

$$ \begin{align*} \bigg\{x \in X : \lim_{n \to +\infty} \frac1{m(F_n)} \int_{F_n} \varphi(g(x)) \, dm(g) \text{ does not exist} \bigg\} \end{align*} $$

is a Baire generic subset of X.

In what follows, we introduce a requirement weaker than the previous version, which is satisfied by countable amenable group actions with the specification property and also ensures that the irregular set of some potential is Baire generic. Afterwards, we check it on an example (cf. Example 11.4).

Definition 11.1. Following [Reference Chung and Li11, Reference Ren, Tian and Zhou31], we say that a 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: for any finite sample of points $x_0, x_1, x_2, \ldots , x_\kappa $ in X and any collection of finite subsets $\hat F_0, \hat F_1, \hat F_2, \ldots , \hat F_\kappa $ of G satisfying the condition

(11.2) $$ \begin{align} K_\varepsilon \hat F_i \cap \hat F_j = \emptyset \quad \text{for every distinct } 0\leqslant i,j \leqslant \kappa, \end{align} $$

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

(11.3) $$ \begin{align} d(g(x), g(x_i)) < \varepsilon \quad \text{for every } g\in \bigcup_{0 \leqslant j \leqslant \kappa} \hat F_j. \end{align} $$

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 11.1 coincides with the classical notion of specification for g (cf. [Reference Sigmund33]).

It is known that, if X is a compact metric space and $T : X \to X$ is a continuous map with the specification property, then the basin of attraction of any T-invariant ergodic probability measure is dense in X (see [Reference Sigmund33, Proof of Theorem 4]). To extend this information to countable amenable group actions with the specification property (and the counting measure m, which we denote by $\lvert \cdot \rvert $ ), consider the generalized basin of attraction of any G-invariant ergodic probability measure $\mu $ , defined by

$$ \begin{align*} C(\mu) = \{x \in X : \mu \in V(x)\}, \end{align*} $$

where $V(x)$ denotes the set of accumulation points, in the weak $^*$ -topology, of the sequence

$$ \begin{align*} \bigg(\frac1{\lvert F_n\rvert} \sum_{g \in F_n} \delta_{g(x)}\bigg)_{n \in \mathbb{N}}. \end{align*} $$

Lemma 11.2. 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 ergodic probability measure on X, then $C(\mu )$ 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 countable amenable group actions [Reference Lindenstrauss25] there is a full $\mu $ -measure subset $X_{\ell }\subset X$ such that, for every $x \in X_\ell $ ,

$$ \begin{align*} \lim_{n \to +\infty} \frac1{\lvert F_n\rvert} \sum_{g \in F_n} \varphi_\ell(g(x)) = \int \varphi_\ell\, d\mu. \end{align*} $$

Therefore,

$$ \begin{align*} \bigcap_{\ell \in \mathbb{N}} X_\ell \subseteq B(\mu) \subseteq C(\mu) \end{align*} $$

and so

$$ \begin{align*} \mu(B(\mu)) = 1 = \mu(C(\mu)). \end{align*} $$

Moreover, given $z \in \bigcap _{\ell \in \mathbb {N}} X_\ell $ , a continuous map $\varphi \in C(X,\mathbb R)$ and a compact set $F\subset G$ , as $\lim _{n \to +\infty } \lvert F_n\rvert = +\infty $ one also has

$$ \begin{align*} \lim_{n \to +\infty} \; \frac1{\lvert F_n \setminus F\rvert } \sum_{g \in F_n\setminus F} \varphi(g(z)) = \int \varphi\, d\mu. \end{align*} $$

We are left to show that $C(\mu )$ is dense in X. The following argument is inspired by [Reference Sigmund33, Theorem 4].

We start by noting that, given $\varepsilon>0$ , let $K_\varepsilon \subset G$ be given by the specification property. Then, for every $k \in \mathbb {N}$ , one can choose a positive integer $n_k$ such that, as k goes to $+\infty $ one obtains

$$ \begin{align*}\frac{K_{\delta/2^k}}{\lvert F_{n_k}\rvert } \to 0 \quad\text{and}\quad \frac{\lvert F_{n_{k-1}}\rvert}{\lvert F_{n_k}\rvert} \to 0. \end{align*} $$

Recall that, by the compactness of X, given a continuous map $\varphi \in C(X,\mathbb R)$ , its modulus of uniform continuity, defined by

$$ \begin{align*} \zeta_\varepsilon (\varphi) = \sup \{ \lvert \varphi(u)-\varphi(v)\rvert : v\in B(u,\varepsilon), \; u\in X \}, \end{align*} $$

where $B(u,\varepsilon )$ denotes the ball in X centered at u with radius $\varepsilon $ , converges to zero as $\varepsilon $ goes to $0^+$ .

Fix $x\in X$ and $\delta>0$ . We claim that $C(\mu )$ 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 $C(\mu )$ which start at the ball $B(x,\delta )$ . More precisely, take $z \in \bigcap _{\ell \in \mathbb {N}} X_\ell $ and consider $x_0=x$ , $x_1=z$ and the finite sets (which satisfy (11.2))

$$ \begin{align*} \hat F_0 &= \{\mathrm{id}\}, \\ \hat F_1 &= (K_{\delta/2}^{-1} [F_{n_1} \setminus \hat F_0]) \setminus K_{\delta/2}, \end{align*} $$

where $K_\alpha ^{-1} = \{g^{-1} : g\in K_\alpha \}$ for every $\alpha> 0$ . By the specification property, there is $y_1 \in B(x,\delta /2)$ such that $d(g(y_1), g(z)) < \varepsilon $ for every $g \in \hat F_1.$ In particular, given a continuous map $\varphi \in C(X,\mathbb R)$ one has

$$ \begin{align*} \bigg| \frac1{\lvert \hat F_1\rvert } \sum_{g \in \hat F_1} \varphi({g(y_1)}) - \frac1{\lvert \hat F_1\rvert } \sum_{g \in \hat F_1} \varphi({g(z)}) \bigg| < \zeta_{{\delta}/{2}}(\varphi). \end{align*} $$

Consider now $x_2 = y_1$ and $x_3=z$ and the finite sets (which satisfy (11.2))

$$ \begin{align*} \hat F_2 &= \hat F_0 \cup \hat F_1 , \\ \hat F_3 &= (K_{\delta/2^2}^{-1} [F_{n_2} \setminus \hat F_2]) \setminus (K_{\delta/2^2} \hat F_2). \end{align*} $$

Hence, setting $\hat F_4 = \hat F_2 \cup \hat F_3$ and using the specification property once more, one obtains a point $y_2 \in B(y_1, \delta /2^2)$ such that $d(g(y_2), g(z)) < \varepsilon $ for every $g\in \hat F_3$ , from which it is immediate that for every continuous map $\varphi \in C(X,\mathbb R)$ one has

$$ \begin{align*} \bigg| \frac1{\rvert \hat F_3\rvert } \sum_{g \in \hat F_3} \varphi({g(y_1)}) - \frac1{\lvert \hat F_3\rvert } \sum_{g \in \hat F_3} \varphi({g(z)}) \bigg| < \zeta_{{\delta}/{2^2}}(\varphi). \end{align*} $$

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

(11.4) $$ \begin{align} \hat F_{j+1}= (K_{\delta/2^j}^{-1} [F_{n_{j+1}} \setminus \hat F_j]) \setminus (K_{\delta/2^j} \hat F_j) \end{align} $$

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

$$ \begin{align*} d(g(y_{j+1}), g(y_j)) < \varepsilon \quad\text{for every } g\in \hat F_{j}, \end{align*} $$

$$ \begin{align*} d(g(y_{j+1}), g(z)) < \varepsilon \quad\text{for every } g\in \hat F_{j+1} \end{align*} $$

and, for every continuous map $\varphi \in C(X,\mathbb R)$ ,

$$ \begin{align*} \bigg| \frac1{\lvert \hat F_{j+1}\rvert} \sum_{g \in \hat F_{j+1}} \varphi({g(y_{j+1})}) - \frac1{\lvert \hat F_{j+1}\rvert } \sum_{g \in \hat F_{j+1}} \varphi({g(z)}) \bigg| < \zeta_{{\delta}/{2^j}}(\varphi). \end{align*} $$

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 $\hat F_k$ ensures that

$$ \begin{align*} \lim_{k \to +\infty} \frac{\lvert \hat F_k \Delta F_{n_k}\rvert }{\lvert F_{n_k}\rvert } = 0. \end{align*} $$

In addition, from the initial selection of z and the previous estimates we conclude that, for every continuous map $\varphi \in C(X,\mathbb R)$ , the subsequence of averages given by

$$ \begin{align*} \bigg(\frac1{\lvert \hat F_{n}\rvert} \sum_{g \in \hat F_{n}} \varphi(g(y_\infty))\bigg)_{n \in \mathbb{N}} \end{align*} $$

converges to $\int \varphi \, d\mu $ . Thus $y_\infty $ belongs to $C(\mu )$ .

The following dichotomy is a direct consequence of Lemma 11.2 and Theorem 1.2.

Corollary 11.3. Let G be a countable amenable group, let $(F_n)_{n \in \mathbb {N}}$ be a tempered Følner sequence, let X be a compact metric space and let $\Gamma : G\times X \to X$ be a continuous group action satisfying the specification property. Then, for every $\varphi \in C(X, \mathbb {R})$ , either

$$ \begin{align*} \int \varphi \, d\mu_1 = \int \varphi \, d\mu_2 \quad\text{for all } \mu_1, \mu_2 \in \mathcal M_G(X) \end{align*} $$

or

$$ \begin{align*} \bigg\{x \in X : \lim_{n \to +\infty} \frac1{\lvert F_n\rvert} \sum_{g \in F_n} \varphi(g(x)) \text{ does not exist} \bigg\} \end{align*} $$

is a Baire generic subset of X.

Proof. Fix $\varphi \in C(X, \mathbb {R})$ and suppose that there are two probability measures $\mu _1, \mu _2 \in \mathcal M_G(X)$ , which we may assume to be ergodic (using the ergodic decomposition [Reference Ornstein and Weiss26]), such that

$$ \begin{align*} \int \varphi \, d\mu_1 \neq \int \varphi \, d\mu_2. \end{align*} $$

Their generalized basins of attraction $C(\mu _1)$ and $C(\mu _2)$ are both dense in X by Lemma 11.2. Define $\varepsilon = \tfrac 12 \lvert \int \varphi \,d\mu _1 - \int \varphi \,d\mu _2\rvert> 0$ . Using the sets $A = C(\mu _1)$ , $B = C(\mu _2)$ and $\varepsilon $ we confirm that X is $\Phi $ -sensitive, where $\Phi = (\varphi _n)_{n \in \mathbb {N}}$ and

$$ \begin{align*} \varphi_n(x) = \frac1{\lvert F_n\rvert } \sum_{g \in F_n} \varphi(g(x)). \end{align*} $$

Consequently, by Theorem 1.2, the irregular set

$$ \begin{align*} \bigg\{x \in X : \lim_{n \to +\infty} \frac1{\lvert F_n\rvert} \sum_{g \in F_n} \varphi(g(x)) \text{ does not exist} \bigg\} \end{align*} $$

is Baire generic in X.

We observe that Corollary 11.3 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

$$ \begin{align*} \bigg\{x \in X : \bigg(\frac1{\lvert F_n\rvert } \sum_{g \in F_n} \delta_{g(x)}\bigg)_{n \in \mathbb{N}} \text{ does not converge in the weak}^* \text{ topology} \bigg\} \end{align*} $$

is Baire generic in X. This extends Furstenberg’s theorem (see [Reference Ollagnier26, 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 there is a constant c such that the sequence of averages (11.1) of every continuous function converges to c.

Example 11.4. 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 $A_1 = (\!\begin {smallmatrix} 2 & 1 \\ 1 & 1 \\ \end {smallmatrix} \!)$ and $A_2 = (\!\begin {smallmatrix} 1 & 1 \\ 1 & 0 \\ \end {smallmatrix} \!)$ . 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

$$ \begin{align*} \Gamma : \mathbb Z^2 \times \mathbb T^2 &\to \mathbb T^2 \\ ((m,n), x) &\mapsto (g_1^m \circ g_2^n)(x). \end{align*} $$

Moreover, this action has 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, \ldots , x_\kappa $ in $\mathbb {T}^2$ and any choice of finite subsets $\hat F_0, \hat F_1, \hat F_2, \ldots , \hat F_\kappa $ of $\mathbb Z^2$ satisfying the condition

(11.5) $$ \begin{align} K_\varepsilon \hat F_i \cap \hat F_j = \emptyset \quad \text{for every distinct } 0\leqslant i,j \leqslant \kappa, \end{align} $$

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

$$ \begin{align*} d(\Gamma_{(m,n)}(x), \Gamma_{(m,n)}(x_i)) < \varepsilon \quad \text{for every } (m,n) \in \bigcup_{0 \leqslant j \leqslant \kappa} \hat F_j. \end{align*} $$

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

(11.6) $$ \begin{align} \Gamma((m,n), \cdot)= g_2^{\Theta((m,n))}(\cdot) \quad\text{for all } m,n\in \mathbb Z. \end{align} $$

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 $i\neq j$ . Consequently,

$$ \begin{align*} \inf \{\lvert u-v\rvert : u \in \Theta(\hat F_i), v \in \Theta(\hat F_j)\} \geqslant k_\varepsilon. \end{align*} $$

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, that 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 11.3 asserts that

$$ \begin{align*} \bigg\{x \in \mathbb{T}^2 : \lim_{n \to +\infty} \frac1{\lvert F_n\rvert } \sum_{g \in F_n} \varphi(g(x)) \text{ does not exist} \bigg\} \end{align*} $$

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

11.3 Application 3

Let G be a free semigroup, finitely generated by a finite set of self-maps $G_1 =\{\mathrm {Id}, g_1, \ldots , g_p\}$ of a probability measure space $(X,\mathfrak {B}, \mu )$ . Assume that $\mu $ is invariant by $g_i: X \to X$ for every $1 \leqslant i \leqslant p$ . The choice of $G_1$ endows G with a norm $\lvert \cdot \rvert $ defined as follows: given $g \in G$ , then $\lvert g\rvert $ stands for the length of the shortest word over the alphabet $G_1$ representing g. Denote by $G_k$ the set $\{g \in G : \lvert g\rvert = k\}$ .

Now take $\varphi \in L^\infty (X, \mu )$ and consider the sequence of its spherical averages

$$ \begin{align*}k \in \mathbb{N} \mapsto s_k(\varphi) = \frac{1}{\# G_k} \sum_{g \in G_k} \varphi \circ g, \end{align*} $$

where $\#$ denotes the cardinal of a finite set (if $G_k = \emptyset $ , we set $s_k(\varphi ) = 0$ ). Next consider the Cesàro averages of the previous spherical averages, that is,

(11.7) $$ \begin{align} n \in \mathbb{N} \mapsto \Phi_n = \frac{1}{n} \sum_{k = 0}^{n-1} s_k(\varphi). \end{align} $$

The main results of [Reference Bufetov7, Reference Bufetov, Khristoforov and Klimenko8] establish the pointwise convergence of the sequence $(\Phi _n)_{n \in \mathbb {N}}$ at $\mu $ -almost every point $x \in X$ .

As a consequence of Theorem 1.2, 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 11.5. Consider the unit circle $\mathbb S^1=\{z \in \mathbb {C} : \lvert z\rvert =1\}$ and the self-maps of $\mathbb S^1$ given by $g_1(z) = z^4$ and $g_2(z)=z^7$ . 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 semigroup G generated by $G_1 = \{\mathrm {Id}, g_1, g_2\}$ , and let $\varphi $ be in $C(\mathbb S^1, \mathbb {R})$ .

Regarding the averages (11.7) of $\varphi $ , in this case they are given by

$$ \begin{align*}\Phi_n = \frac{1}{n} \sum_{k = 0}^{n-1} \frac{1}{2^k} \sum_{j = 0}^k \binom kj (\varphi \circ g_1^j \circ g_2^{k-j}). \end{align*} $$

Let $z_0 \in \mathbb S^1$ be a common fixed point for $g_1$ and $g_2$ . The sequence $(\Phi _n(z_0))_{n \in \mathbb {N}}$ 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 $(\Phi _n(x))_{n \in \mathbb {N}}$ converges to $\varphi (z_0)$ as well. Let us show this claim.

Given $x \in \mathcal O^-(z_0)$ , there exists $g=g_{i_n}\circ \cdots \circ g_{i_2}\circ g_{i_1}\in G$ , 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

(11.8) $$ \begin{align} \frac{1}{2^k} \sum_{j = 0}^k \binom kj (\varphi \circ g_1^j \circ g_2^{k-j})(x) \end{align} $$

may be rewritten as

$$ \begin{align*}\frac{1}{2^k} \bigg[\sum_{j = 0}^{a-1} \binom kj (\varphi \circ g_1^j \circ g_2^{k-j})(x) &+ \sum_{j = a}^{k-b} \binom kj (\varphi \circ g_1^{j} \circ g_2^{k-j})(x)\\ &+ \sum_{j = k-b+1}^{k} \binom kj (\varphi \circ g_1^j \circ g_2^{k-j})(x)\bigg].\end{align*} $$

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

$$ \begin{align*} a \lVert \varphi\rVert_\infty \max\bigg\{\binom k0, \ldots, \binom ka \bigg\} 2^{-k} \end{align*} $$

and

$$ \begin{align*} b \lVert\varphi\rVert_\infty \max\bigg\{\binom k{k-b+1}, \ldots, \binom kk \bigg\} 2^{-k} \end{align*} $$

respectively, and both estimates converge to zero as k goes to $+\infty $ . Thus, their Cesàro averages also converge to zero. Regarding the middle term, as $g_1$ and $g_2$ commute and $z_0$ is fixed by every element of G, one has

$$ \begin{align*} \frac{1}{2^k} & \sum_{j = a}^{k-b} \binom kj (\varphi \circ g_1^{j} \circ g_2^{k-j})(x) = \frac{1}{2^k} \sum_{j = a}^{k-b} \binom kj (\varphi \circ g_1^{j-a} \circ g_2^{k-j-b})(g_1^a \circ g_2^b)(x) \\ & = \frac{1}{2^k} \sum_{j = a}^{k-b} \binom kj (\varphi \circ g_1^{j-a} \circ g_2^{k-j-b})(z_0) = \bigg[ \frac{1}{2^k} \sum_{j = a}^{k-b} \binom kj \bigg] \cdot \varphi (z_0), \end{align*} $$

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 by $g_1$ and $g_2$ , and we take $x \in \mathbb S^1$ in the pre-orbit by the semigroup action of $z_0$ (which is dense, because the pre-orbit of $z_0$ by $g_1$ is dense) and $y \in \mathbb S^1$ belongs to the (also dense) pre-orbit by the semigroup action of $z_1$ , then the sequence $(\Phi _n(x))_n$ converges to $\varphi (z_0)$ and the sequence $(\Phi _n(y))_n$ converges to $\varphi (z_1)$ . Thus, if we choose $\varphi \in C(\mathbb S^1,\mathbb R)$ such that $\varphi (z_0) \neq \varphi (z_1)$ then, by Theorem 1.2, we conclude that the irregular set ${\mathcal I}(\Phi )$ is Baire generic in $\mathbb S^1$ .

Remark 11.6. As the generators of the semigroup action described in Example 11.5 commute, one could consider, alternatively, the sequence $(\Psi _n)_n$ where

$$ \begin{align*} \Psi_n(\cdot) = \frac{1}{n^2} \sum_{k, \ell = 0}^{n-1} (\varphi \circ g_1^k \circ g_2^{\ell})(\cdot). \end{align*} $$

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

$$ \begin{align*} \frac{1}{n^2} \bigg[\sum_{(k, \ell) \in [a,n-1] \times [b,n-1]} (\varphi \circ g_1^\ell \circ g_2^{k-\ell})(x) + \sum_{(k, \ell) \notin [a,n-1]\times [b,n-1]} (\varphi \circ g_1^\ell \circ g_2^{k-\ell})(x) \bigg], \end{align*} $$

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

$$ \begin{align*} \frac{1}{n^2} \sum_{(k, \ell) \in [a,n-1]\times [b,n-1]} (\varphi \circ g_1^{\ell-a} \circ g_2^{k-b-\ell})(z_0) = \frac{(n-a)(n-b)}{n^2}\; \varphi (z_0) \end{align*} $$

and converges to $\varphi (z_0)$ as n goes to $+\infty $ . The second term has absolute value bounded above by $({a+b})/n \lVert \varphi \rVert _\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 1.2, one concludes that ${\mathcal I}(\Psi )$ is Baire generic in $\mathbb S^1$ .