Proof. Let f be a non-statistical map and let $x\in X$ be a non-statistical point. Since the sequence of empirical measures of this point does not converge,

$$ \begin{align*}d_x:=\inf_{N>0}\sup_{N\leqslant n,m}d_w(e_{n}^f(x),e_{m}^f(x))>0.\end{align*} $$

By definition, the set of non-statistical points has positive measure and $x\mapsto d_x$ is measurable, thus,

$$ \begin{align*}\Delta_{N}(f,f)=\int_{X} \sup_{N\leqslant n,m}d_w(e_{n}^f(x),e_{m}^f(x))\,d\mu \ge \int_X d_x\,d\mu>0.\end{align*} $$

To prove the other side, let f be a map for which the sequence of empirical measures of almost every point converges. So, for almost every $x\in X$ , we have

$$ \begin{align*}\lim_{N\to\infty} \sup_{N\leqslant n,m}d_w(e_{n}^f(x),e_{m}^f(x))=0.\end{align*} $$

Since the distance between empirical measures is bounded, we can then use Lebesgue dominated convergence theorem to conclude

$$ \begin{align*} \lim_{N\to \infty} \Delta_{N}(f,f)&=\lim_{N\to \infty}\int_{X}\sup_{N\leqslant n,m}d_w(e_{n}^f(x),e_{m}^f(x))\,d\mu\\ &=\int_{X}\lim_{N\to\infty} \sup_{N\leqslant n,m}d_w(e_{n}^f(x),e_{m}^f(x))\,d\mu=0. \end{align*} $$

This finishes the proof.

Proof. Let $\{f_k\}_k $ be a sequence of maps converging to f. For each k, we can find a natural number $N_k$ , and two maps $g_k$ and $h_k$ near $f_k$ such that

$$ \begin{align*}|\Delta(f_k)-\Delta_{N_k}(g_k,h_k)|<\frac{1}{k}.\end{align*} $$

Note that we can choose the sequence $\{N_k\}_k$ such that it converges to infinity, and also the sequence of maps $\{g_k\}_k$ and $\{h_k\}_k$ such that both converge to f. So, we obtain

$$ \begin{align*} \Delta(f)\geqslant \limsup_{k}\Delta(f_k), \end{align*} $$

and this implies the upper semi-continuity of $\Delta $ .

Proof of Theorem B.

Since by Proposition 2.10 the map $\Delta $ is an upper semi-continuous map, there is a Baire generic set $\mathcal {G}$ on which the map $\Delta $ is continuous. For a map $f \in \mathcal {G}$ , which is also in the interior of statistically unstable maps, there exists a neighborhood $\mathcal {U}_f$ around f on which $\Delta $ is uniformly positive:

$$ \begin{align*} \text{ there exists } d>0 \quad \text{such that} \text{ for all } g \in \mathcal{U}_f,\quad \Delta(g)>d. \end{align*} $$

Now, we construct a sequence of open and dense subsets in ${\mathcal U}_f$ such that any map in the intersection of these sets is non-statistical. Then, we can conclude that non-statistical maps are Baire generic in ${\mathcal U}_f$ .

To construct such open and dense sets, we need to show a semi-continuity property of the map $\Delta _N$ .

Proof. We recall that for $h,g\in \Lambda $ ,

$$ \begin{align*} \Delta_{N}(h,g):=\int_{X}{\ \sup_{N\leqslant n,m}d_w(e_{n}^{h}(x),e_{m}^{g}(x)})\, d\mu.\end{align*} $$

Now, note that

$$ \begin{align*} \Delta_{N}(h,g)= \sup_{N\leqslant M}\int_{X} {\ \sup_{N\leqslant n,m \leqslant M}d_w(e_{n}^{h}(x),e_{m}^{g}(x))\, d\mu}, \end{align*} $$

and we know that $\int _{X} {\ \sup _{N\leqslant n,m \leqslant M}d_w(e_{n}^{h}(x),e_{m}^{g}(x))\, d\mu }$ is continuous with respect to h and g, and supremum of a sequence of continuous functions is lower semi-continuous, so we are done.

Next, we show that for any $N\in \mathbb N$ , the set

$$ \begin{align*}E(N):=\bigg\{h\in {\mathcal U}_f \mid \Delta_N(h,h)>\frac{d}{3}\bigg\}\end{align*} $$

is an open and dense subset of ${\mathcal U}_f$ , and moreover, every map in the intersection $\bigcap _N E(N)$ is non-statistical.

The openness of $E(N)$ is guaranteed by lower semi-continuity of $\Delta _N.$ Now, we prove the denseness of $E(N)$ . For any arbitrary map $h\in {\mathcal U}_f\subset \Lambda $ , take a neighborhood $V_h$ such that for any map $g\in V_h$ and any $x\in X$ ,

(3.1) $$ \begin{align} d_w(e^h_{N}(x),e^g_{N}(x))<\frac{d}{6}. \end{align} $$

This is possible because N is fixed, $e^g_{N}$ depends continuously on g, and X is compact. Now, since $\Delta (h)>d$ , we can choose $g_1,g_2\in V_h$ such that for some integer $M>N$ , it holds true that

(3.2) $$ \begin{align} \Delta_M(g_1,g_2)>d. \end{align} $$

However, since $\Delta _N(g_1,g_2)$ is decreasing in N, we obtain

(3.3) $$ \begin{align} \Delta_N(g_1,g_2)>d. \end{align} $$

Now, note that for each $x\in X$ , we have

$$ \begin{align*} & \sup_{N\leqslant n,m} d_w(e_{n}^{g_1}(x),e_{m}^{g_2}(x)) \\ & \quad \leqslant \sup_{N\leqslant n,m}d_w(e_{n}^{g_1}(x),e_{m}^{g_1}(x)) + \sup_{N\leqslant n,m}d_w(e_{n}^{g_2}(x),e_{m}^{g_2}(x))+ d_w(e_{N}^{g_1}(x),e_{N}^{g_2}(x)), \end{align*} $$

and hence, after integrating with respect to $\mu $ and using inequality (3.1), we obtain

$$ \begin{align*}\Delta_N(g_1,g_2)\leqslant \Delta_N(g_1,g_1)+\Delta_N(g_2,g_2)+ \frac{d}{3}.\end{align*} $$

Now, using inequality (3.3), we conclude that at least one of the maps $g_1$ and $g_2$ is inside $E(N)$ , and then recalling that h was chosen arbitrarily in ${\mathcal U}_f$ and $V_h$ arbitrary small, we conclude that $E(N)$ is dense in ${\mathcal U}_f$ .

Observe that Lemma 2.4 implies that any map h in the set $\bigcap _{N=1}^\infty E(N)$ , which is a Baire generic set inside ${\mathcal U}_f$ , is non-statistical. So, the non-statistical maps are generic in some open neighborhoods of f and f is an arbitrary map in the interior of statistically unstable maps intersected by the generic set $\mathcal G$ . This implies that non-statistical maps are indeed a generic subset of the interior of statistically unstable maps as well.

Proof of Theorem C.

By definition, every map in $\Lambda =\{F_x\}_{x\in X}$ is statistically unstable. Using Theorem B, we conclude that a Baire generic map $F_x$ is non-statistical. This indeed implies that the point x is non-statistical for the map f.

Proof of Corollary 1.8.

As the map f is not uniquely ergodic, it has at least two different ergodic measures $\nu _1$ and $\nu _2$ . As the map is minimal, the support of any invariant measure is the whole space. So, for any $x\in X$ and any neighborhood U of x, there is two points $y_1$ and $y_2$ in U such that

$$ \begin{align*}\lim_{n\rightarrow \infty} e_n^f(y_i)=\nu_i, \quad i\in \{1,2\}.\end{align*} $$

This will imply that for the family of maps $\Lambda =\{F_x \mid x\in X\}$ defined as above, we have

$$ \begin{align*}\Delta_\Lambda(F_x)\geqslant d_w(\nu_1,\nu_2),\end{align*} $$

and hence, $F_x$ is statistically unstable. This will imply that x is statistically unstable, so we can use Theorem C.

Proof. Since f can be approximated by maps that are conjugate to rotation, there is a rational number ${p}/{q}$ and a $C^r$ diffeomorphism h such that the map $h^{-1}R_{{p}/{q}}h$ is close to f. Let

$$ \begin{align*}B:=[r_1,r_2]\times \mathbb R/ \mathbb Z\end{align*} $$

for some distinct real numbers $r_1,r_2\in (0,1)$ . Take a real number $\theta>0$ and define

$$ \begin{align*}B_1:= [r_1,r_2]\times [0,\theta)\end{align*} $$

and

$$ \begin{align*}B_2:= [r_1,r_2]\times [\theta,1). \end{align*} $$

Now, let $\hat {g}$ be a map found in the Sublemma. This map can be lifted using the covering map $\pi :\mathbb {A}\to \mathbb {A}$ , $\pi (r,\theta )=(r,q\theta )$ . Let g be the lift of $\hat {g}$ that is identity around the set C. The diffeomorphism g has similar properties:

• • ${g}$ is identity on a neighborhood of the set C;

• • $\mathrm{Leb}({g}(\pi ^{-1}(B_{1})))>\sigma $ ;

• • ${g}(\pi ^{-1}(B_{2}))\subset N_\epsilon (C)$ .

Now note that g commutes with $R_{{p}/{q}}$ , so

$$ \begin{align*} h\circ g \circ R_{{p}/{q}} \circ g^{-1} \circ h^{-1}=h\circ R_{{p}/{q}} \circ h^{-1}. \end{align*} $$

Choose $\alpha '$ irrational and small enough so that $h\circ g \circ R_{\alpha '} \circ g^{-1} \circ h^{-1}$ is arbitrary close to $h\circ R_{{p}/{q}} \circ h^{-1} $ . Indeed, $h\circ g$ is $C^r$ and the map sending $\alpha $ to $h\circ g \circ R_{\alpha } \circ g^{-1} \circ h^{-1}$ is hence continuous. Since $\alpha '$ is irrational, the orbit closure of any point in $\mathbb {A}$ under iterating the map $h\circ g \circ R_{\alpha '} \circ g^{-1} \circ h^{-1}=:f'$ is an $h\circ g$ -image of the orbit closure of a point under iterating the map $R_{\alpha '}$ , which is a circle in $\mathbb {A} $ . So, for any point $x\in h\circ g(\pi ^{-1}(B))$ , the orbit closure of x is the $h\circ g$ image of a vertical circle $C'$ contained in $\pi ^{-1}(B)$ . The map $h\circ g \circ R_{\alpha '} \circ g^{-1} \circ h^{-1}|_{h\circ g(C')}$ is conjugate to $R_{\alpha '}|_{C'}$ . Now, note that if $R_{\alpha '}^{n}\circ g^{-1}\circ h^{-1}(x) \in \pi ^{-1}(B_{i})$ , then $(f')^{n}(x)\in h\circ g(\pi ^{-1}(B_{i}))$ for $i\in \{1,2\}$ . The orbit of each point in $C'$ , on average, spends ${\theta }$ portion of times in $\pi ^{-1}(B_1)$ and ${1-\theta }$ portion of times in $\pi ^{-1}(B_2)$ . So, the orbit of the point x spends ${\theta }$ portion of times in $h\circ g(\pi ^{-1}(B_1))$ and ${1-\theta }$ portion of times in $h\circ g(\pi ^{-1}(B_2))$ . By choosing $\theta $ and $\epsilon $ sufficiently small, we can guarantee that the asymptotic averages of any point in $h\circ g(\pi ^{-1}(B_1))$ is arbitrarily close to a measure $\nu _{f'}$ supported on C, which is the pushforward of the Lebesgue measure on the boundary component C by the map $h\circ g$ . This is because the image of $\pi ^{-1}(B_1)$ under the map g can be so close to C that after iteration by h, it still remains arbitrarily close to C. Since h is smooth and sends positive measure sets to positive measure sets, it can be shown that the sets with large measure are sent to large measure sets. Hence, if $\sigma $ is chosen sufficiently close to one, then map g is so that $\mathrm{Leb}(h\circ g(\pi ^{-1}(B_1)))$ is sufficiently close to one and so since for a large number n, the nth empirical measures of points in $h\circ g(\pi ^{-1}(B_1))$ is close to $\nu _{h\circ g}$ , the measure $\hat {e}^{f'}_n$ is close to $\hat {\delta }_{\nu _{f'}}$ . Now, taking $\nu $ as any accumulation point of measures like $\nu _{f'}$ , where $f'$ approaches f, according to the definition, we can see that f statistically bifurcates toward $\hat {\delta }_{\nu }$ and so we are done.

Proof of Theorem A.

Lemma 5.2 implies that for any map $f\in \mathcal {AK}^r$ , there are two measures $\nu _{1,f}$ and $\nu _{2,f}$ that are supported on different connected components of the boundary of $\mathbb {A}$ such that f statistically bifurcates toward both $\hat {\delta }_{\nu _{1,f}}$ and $\hat {\delta }_{\nu _{2,f}}$ . Now, using Theorem B.16, we conclude that for a generic map $f\in \mathcal {AK}^r$ and for almost every point x in the phase space X, the set of accumulation points of the sequence $\{{e}_n^f(x)\}_n$ contains at least two measures ${\nu _{1,f}}$ and ${\nu _{2,f}}$ . We are going to show that, generically, these two measures are the only ergodic invariant measures of the map f and the empirical measures of almost every point accumulate to any convex combination of these two ergodic measures (which is indeed the whole space of invariant measures), and hence, f is maximally oscillating. Now, approximate f by a map like $h\circ g\circ R_{\alpha '}\circ g^{-1}\circ h^{-1}$ coming from Lemma 5.2. Using the theorem of Fayad and Katok, we know that the map $R_{\alpha '}$ can be approximated in $C^\infty $ -topology (and hence in $C^r$ -topology) by a map $e\in \mathcal {AK}_{\mathrm{vol}}^\infty $ that has only three ergodic measures: two one-dimensional Lebesgue measures on the boundaries and the volume measure of the annulus. The map $h\circ g\circ e \circ g^{-1} \circ h^{-1}$ is close to the initial map f and has only three ergodic invariant measures, which are the push forward of three ergodic measures of e by the map $h\circ g$ . Note that if in the Sublemma, the numbers $r_1$ and $r_2$ are chosen close to $0$ and $1$ , then the set $B_2$ has measure close to one. In this case, observe that the pushforward of the volume measure by the map $h\circ g$ is a measure that is close to the pushforward of the one-dimensional Lebesgue measure of one of the boundary components (which is denoted by C in the lemma). Hence, the set of invariant measures for the map $h\circ g\circ e\circ g^{-1}\circ h^{-1}$ is a triangle that has two of its vertices very close to each other. We know that the map sending the dynamics to its set of invariant measures is upper semi-continuous (see Lemma B.12) and, hence, it is continuous for maps in a Baire generic set. Therefore, we can assume that f is a continuity point of this mapping and the set of invariant measures of f is in an arbitrary neighborhood of a triangle that is arbitrarily close to a segment. So, the set of invariant measures of f is a segment. However, this means that f has exactly two ergodic invariant measures. Note that these measures are supported by different boundary components of the annulus. So, on each boundary component of the annulus, there is only one ergodic measure and, hence, any invariant measures on one of the boundary components are equal to the corresponding ergodic measure on that component. So, f statistically bifurcates toward two measures $\hat {\delta }_{\nu _{1,f}}$ and $\hat {\delta }_{\nu _{2,f}}$ , and $\nu _{1,f},\nu _{2,f}$ are exactly two ergodic measures of f. Moreover, since these two measures are in the accumulation points of the sequence of empirical measures for almost every point and the set of invariant measures is the line segment between these two measures, the sequence of empirical measures of almost every point has to accumulate to every point in this line segment and this finishes the proof.