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Flexibility of statistical properties for smooth systems satisfying the central limit theorem

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We exhibit new classes of smooth systems which satisfy the Central Limit Theorem (CLT) and have (at least) one of the following properties:

  • Zero entropy;

  • Weak but not strong mixing;

  • (Polynomial) mixing but not K;

  • K but not Bernoulli and mixing at arbitrary fast polynomial rate.

We also give an example of a system satisfying the CLT where the normalizing sequence is regularly varying with index 1. All these examples are \(C^\infty \) except for a zero entropy diffeomorphism satisfying the CLT which can be made \(C^r\) for an arbitrary finite r.

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  1. The methods of [27] apply to more general systems in the first factor, however, they seem insufficient to produce the examples described in Theorems 1.31.5.

  2. We note that a simple interpolation argument shows that if F is mixing with exponential (respectively polynomial rate) on \(C^r\) for some \(r>0\) then it is mixing with exponential (respectively polynomial) rate on \(C^r\) for all \(r>0\), however the exponent \(\delta \) depends on r.

  3. Recall that a real valued function \(a(\cdot )\) defined on \([m, \infty )\) for some \(m\in {\mathbb {R}}\) is regularly varying in the sense of Karamata with index \(\alpha \) if for each \(s>0\), \(\displaystyle \lim _{t\rightarrow \infty } \frac{a(st)}{a(t)}=s^\alpha \). A sequence \(a_n\) is regularly varying with index \(\alpha \) if the function \(a(t)=a_{[t]}\) is regularly varying with index \(\alpha .\)

  4. CLT with normalization \(\sqrt{n\ln n}\) appears for expanding and hyperbolic maps with neutral fixed points [19, 53], as well as in several hyperbolic billiards [5, 6, 101]. In a followup paper we will show it also appears for generalized \(T, T^{-1}\) transformations with hyperbolic base and two parameter exponentially mixing flows in the fiber.

  5. We note that the requirement that the limiting distribution is Gaussian is important. If we allow other limit distributions, then there are several examples in both non-uniformly hyperbolic and parabolic settings where normalization is different, see [1, 20, 48] and references therein. If we allow our system to preserve an infinite measure then, there is an additional freedom related to the rate of return times, see e.g. [30, 96].

  6. Here, and in the sequel, \(\Rightarrow \) denotes weak convergence of random variables. Note that in contrast with Definition 1.1, we do not require \(\sigma ^2(A) >0\).

  7. Let \(\varvec{\lambda }_0\) be the smallest eigenvalue of the Laplacian on Q. According to [20, Theorem 1.2 and Corollary 1.3] (which relies on [48]) one can take \(\displaystyle \alpha =\frac{1+\sqrt{1-4\varvec{\lambda }_0}}{2}\) if \(\displaystyle \varvec{\lambda }_0<\frac{1}{4}.\) If \(\varvec{\lambda }_0\ge \frac{1}{4}\) one can take \(\displaystyle \alpha =\frac{1}{2}+{\varepsilon }\) for any \({\varepsilon }>0.\) The precise value of \(\alpha \) is not important for our purposes.

  8. Estimates such as (6.2) are often called anticoncentration inequalities since (6.2) shows that the probability that \(\tau _{(\cdot )}\) belongs to a unit interval is small no matter where this interval is located.

  9. Note that \(\Lambda \) depends on H since \(\rho \) depends on H, see (6.6).

  10. Recall that \({\varepsilon }\) is the diameter of \(\{C_l\}.\)

  11. Note that if \({\mathbf {H}}\) is given by (6.15) where H satisfies (5.1) then \(\nu ({\mathbf {H}})=0\) so (6.17) reduces to (6.16).

  12. The minimality of \({{\bar{\tau }}}\) is not essential for our argument. If \({{\bar{\tau }}}\) was not minimal the argument of Sect. 6.1 would still go through but the summations over \(k\in {\mathbb {Z}}\) would need to be replaced by the summations over \(k\in \ell {\mathbb {Z}}\) for some \(\ell >1\). What is important is that the results of [40] allow us to describe the local distribution of \(\tau _t\) in all the cases.

  13. Note that the discontinuity set of \(\beta \) on Q is a finite number of geodesic arcs. Namely let \(Q={\mathbb {H}}^2/\Gamma \). If q is a discontinuity point of \(\beta \), then there is \({{\bar{\gamma }}}\in \Gamma \setminus \{ Id \}\) such that \(\displaystyle d(q, q_0)=d(q, {{\bar{\gamma }}}q_0)=\min \nolimits _{\gamma \in \Gamma } d(q, \gamma q_0).\) Since the diameter of Q is finite, the discontinuity set of the map \(x\mapsto \beta (q(x))\) on X is contained in a finite number of analytic surfaces transverse to the orbits of \(h_u.\)

  14. Although Theorem 4.1(b) in [35] only covers the discrete case, the proof is the same for continuous time, see Remark 4.11 in [35].

  15. In fact, by [9] G is exponentially mixing of all orders. The multiple exponential mixing plays important role in verifying that F satisfies the CLT if \(d\ge 3\) (see §B.1), but it is not needed in the proof of Theorem 9.1.

  16. Notice that for any \(i\in {\mathbb {N}}\) the points \(F^i(\omega ',y')\) and \(F^i(\omega ',G_u y')\) are \(\delta \) close. Indeed, they have the same first coordinate and the second one is \(G_{\tau _i(\omega )}y'\) vs \(G_{u+\tau _i(\omega )}y'\) which are \(\delta \) close since \(\Vert u \Vert <\delta \).

  17. Applying this result to \(F^{-1}\) gives that all exponents of F are in fact zero, but we do not need this fact for the proof of Lemma 2.1.

  18. Other interesting statistical properties include Large Deviations, Poisson Limit Theorem, and Local Limit Theorem. We do not include them into our list since our paper does not contain new results or counter examples pertaining to these properties

  19. [100] refers to classical CLT, but since the time it was written several CLTs with non classical normalization has been proven, cf. footnote 4.

  20. Realizabilty problem also makes sense and is interesting in other settings such as for symbolic or hamiltonian systems.

  21. One can also ask which limit distributions can appear in the limit theorems in the context of measurable dynamics and which normalizations are possible. These issues are discussed in [55, 102].


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We thank Bassam Fayad and Jean-Paul Thouvenot for useful discussions. D. D. was partially supported by the NSF grant DMS-1956049, C. D. was partially supported by AMS Simons travel grant and Nankai Zhide Foundation, A. K. was partially supported by the NSF grant DMS-1956310, P. N. was partially supported by the NSF grants DMS-1800811 and DMS-1952876.

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Part VI. Appendices

Appendix A. Entropy of skew products

Proof of Lemma 2.1

We prove the statement for \((T, T^{-1})\) diffeomorphisms, the result for flows then follows by considering the time 1 map.

By Ruelle inequality it suffices to show that all Lyapunov exponents of F are non positive.Footnote 17 Differentiating (1.3) we get that for each \((x,y)\in (X\times Y)\), \(u\in T_x X,\) \(v\in T_y Y\)

$$\begin{aligned} DF^N(x,y) \left( \begin{array}{c} u \\ v \end{array} \right) = \left( \begin{array}{c} Df^N u \\ \displaystyle \sum _{j=1}^d d(\tau _{(j)})_N(u) {\mathcal {Y}}_j+D(G_{\tau _N}) (v) \end{array}\right) (x,y), \end{aligned}$$

where \(\tau _{(j)}\) denotes the j-th component of \(\tau \), \({\mathcal {Y}}_j=\frac{d}{ds}|_{s=0} G_{s e_j} \) and \(\{e_j\}\) is the standard basis in \({\mathbb {R}}^d.\)

Since f has zero entropy, the Pesin formula shows that the Lyapunov exponents of f are zero. Hence \(\displaystyle \lim _{N\rightarrow \infty } \frac{\ln \Vert Df^N(x)\Vert }{N}=0\) for a.e. x. Also since

$$\begin{aligned} (d(\tau _{(j)})_N(v))(x) =\sum _{n=0}^{N-1} d\tau _{(j)} (f^n x)(Df^n v)\end{aligned}$$

it follows that for a.e. x and all \(j\in \{1,\dots , d\}\)\(\displaystyle \limsup _{N\rightarrow \infty } \frac{\ln \Vert d(\tau _{(j)})_N(x)\Vert }{N}\le 0\). Also for a.e. (xy)

$$\begin{aligned} \limsup _{N\rightarrow \infty } \frac{\ln \Vert DG_{\tau _N(x)}(y)\Vert }{N}\le C \lim _{N\rightarrow \infty } \frac{\Vert \tau _N(x)\Vert }{N}=0 \end{aligned}$$

where the last step follows since f is ergodic and \(\tau \) has zero mean.

The foregoing discussion shows that for a.e. (xy), \(\displaystyle \limsup _{N\rightarrow \infty } \frac{\ln \Vert DF^N\Vert (x,y)}{N}\le 0.\) Therefore all Lyapunov exponents of F indeed non positive, and so \(ent_\zeta (F)=0.\) \(\square \)

Appendix B. Ergodic sums over subshifts of finite type

B.1 CLT for \((T, T^{-1})\) transformations with SFT in the base

Theorem B.1

Consider a generalized \((T, T^{-1})\) transformation (1.2) with (Xf) being a subshift of finite type, \(\mu \) is a Gibbs measure with a Hölder potential, and \(G_t\) is an \({\mathbb {R}}^d\) action which is exponentially mixing of all orders. Suppose that \(d\ge 3\) and \(\tau : X\rightarrow {\mathbb {R}}^d\) is an irreducible Hölder cocycle. Then F satisfies the CLT on the space of Hölder functions.

Remark B.2

As it was mentioned in Sect. 2.4, this result is a special case of Theorem 5.1 in [35]. We include the proof here to make this paper more self contained and to demonstrate the power of Theorem 3.1. We also note that in contrast to [35] the present proof does not rely on the exponential mixing of f, it just uses the the properties of the local distribution of \(\tau \) such as the anticoncentration inequality (B.2) below.


By Lemma 5.6 it suffices to show that F satisfies the quenched CLT in the sense of Definition 5.1.

We define \({\mathfrak {m}}_N\) by (5.2) and check the conditions of Proposition 4.1.

(a) is evident.

To prove property (b), let \(\ell (x, t, N)=\mathrm{Card}\{n \le N: |\tau _n(x)-t|\le 1\}\). We claim that for each p, there is a constant \(C_p\) such that for each \(t\in {\mathbb {R}}^d\) for each n

$$\begin{aligned} \mu \left( \ell ^p(\cdot , t, n)\right) \le C_p. \end{aligned}$$


$$\begin{aligned}&\mu \left( \ell ^p(\cdot , t, n)\right) \le \sum _{q=1}^p {{\hat{C}}}_p \sum _{n_1\le n_2\le \dots \le n_q\le n} \mu \left( \prod _{j=1}^q 1_{\Vert \tau _{n_j}(x)-t\Vert \le 1} \right) \\&\qquad \le \sum _{q=1}^p {{\hat{C}}}_p \sum _{n_1< n_2\le \dots < n_q\le n} \mu \left( 1_{\Vert \tau _{n_1}(x)-t\Vert \le 1} \left[ \prod _{j=2}^q 1_{\Vert \tau _{n_j-n_{j-1}}(f^{n_{j-1}} x)\Vert \le 2}\right] \right) . \end{aligned}$$

The multiple anticoncentration inequality of [35, Lemma A.4] tells us that there is a contant \({\bar{C}}\) such that for each tuple \((n_1, \dots n_q)\) we have

$$\begin{aligned}&\mu \left( 1_{\Vert \tau _{n_1}(x)-t\Vert \le 1} \left[ \prod _{j=2}^q 1_{\Vert \tau _{n_j-n_{j-1}}(f^{n_{j-1}} x)\Vert \le 2}\right] \right) \nonumber \\&\qquad \le {\bar{C}}(n_1)^{-d/2} \left[ \prod _{j=2}^q (n_j-n_{j-1})^{-d/2} \right] . \end{aligned}$$

Summing over \(n_1,...,n_q\), we obtain (B.1).

With (B.1) proven, the Markov inequality implies that for each \({\varepsilon }, t, p\) we have

$$\begin{aligned} \mu \left( x: \ell (x, t, N)\ge N^{(1/5)-{\varepsilon }}\right) \le \frac{C_p}{N^{[(1/5)-{\varepsilon }]p}}. \end{aligned}$$

It follows that

$$\begin{aligned} \mu \left( x: \exists t: \Vert t\Vert \le \Vert \tau \Vert N \text { and }\ell (x, t, N)\ge N^{(1/5)-{\varepsilon }}\right) \le \frac{C_p^* N^d}{N^{[(1/5)-{\varepsilon }]p}}. \end{aligned}$$

Taking \(p= 6d\), \({\varepsilon }=0.01\), property (b) follows.

Recall (5.5). In view of Lemma 5.5, to prove property (c) it suffices to check that (5.8) holds for some \(\beta >1.\) Using (5.7) we get

$$\begin{aligned}&\int _M \left| \sigma _{0,k} (x) \right| d\mu (x) \le C \sum _{m=0}^\infty \left[ \mu (\Vert \tau _k\Vert \in [m, m+1)) e^{-cm}\right] \\&\qquad \le C \sum _{m=0}^\infty \left[ \frac{m^{d-1}}{k^{d/2}} e^{-cm}\right] \le \frac{C}{k^{d/2}}, \end{aligned}$$

where the second inequality relies on (B.2) with \(q=1\) (noting that we can cover the set \(\{ z \in {\mathbb {R}}^d : \Vert z \Vert \in [m, m+1) \}\) with \(Cm^{d-1}\) unit cubes). This shows that (5.8) holds with \(\beta =d/2\). This completes the verification of conditions of Proposition 4.1. \(\square \)

B.2 Visits to cones

Proof of Lemma 11.4

We only prove the result for the forward orbits, the proof for the backward orbits is similar.

Set \(n_1 = 2\), \(n_{k+1}=n_k^3,\) \(m_k=n_k-n_{k-1}\) and consider the sets \(\displaystyle A_k=\{\omega : \tau _{n_k} (\omega )\in {\mathcal {C}}\}.\)

Let \({\mathcal {F}}_{a,b}\) denote the \(\sigma \)-algebra generated by \(\{\omega _j\}_{a\le j\le b}.\) Since \(\tau \) only depends on the past, \(A_k\) is measurable with respect to \({\mathcal {F}}_{-\infty , n_k}\).

Therefore by Lévy’s extension of the Borel–Cantelli Lemma (see e.g. [107, §12.15]) it is enough to show that for almost all \(\omega \)

$$\begin{aligned} \sum _k \mu (A_{k+1}|{\mathcal {F}}_{-\infty , n_k})=\infty . \end{aligned}$$

Let \(\displaystyle {\hat{{\mathcal {C}}}} =\{v\in {\mathcal {C}}: \text {dist}(v, \partial {\mathcal {C}})\ge 1 \},\)  \(\displaystyle {{\hat{A}}}_k=\left\{ \omega : \frac{\tau _{m_k}(\sigma ^{n_{k-1}} \omega )}{\sqrt{m_k}} \in {\hat{{\mathcal {C}}}}\right\} ,\)

\(\displaystyle A_k^*=\{\omega : \exists {{\hat{\omega }}}\in {{\hat{A}}}_k:\; \omega _j={{\hat{\omega }}}_j \text { for } j\in [n_{k-1}, n_k] \}. \) Note that \(A_k^* \subset A_k\) because for any \(\omega \in A_k^*\) and for the corresponding \(\hat{\omega }\), \(\tau _{m_k}(\sigma ^{n_{k-1}} {{\hat{\omega }}})\) is inside \({\mathcal {C}}\) and is at least \( \frac{1}{2} \sqrt{m_k}\) away from the boundary whereas

$$\begin{aligned} \tau _{n_k}(\omega )-\tau _{m_k}(\sigma ^{n_{k-1}}{{\hat{\omega }}})= & {} [\tau _{n_k}(\omega )-\tau _{n_k}({{\hat{\omega }}})]+[\tau _{n_k}({{\hat{\omega }}})-\tau _{m_k}(\sigma ^{n_{k-1}} {{\hat{\omega }}})]\\= & {} O(n_{k-1}) \ll \sqrt{m}_k.\end{aligned}$$


$$\begin{aligned} \mu (A_{k+1}|{\mathcal {F}}_{-\infty , n_k})\ge \mu (A_{k+1}^*|{\mathcal {F}}_{-\infty , n_k})\ge \frac{\mu (A_{k+1}^*|{\mathcal {F}}_{n_k, n_k})}{{{\hat{K}}}}\ge \frac{\mu ({{\hat{A}}}_{k+1}|{\mathcal {F}}_{n_k, n_k})}{{{\hat{K}}}}, \end{aligned}$$

where the second inequality is due to (9.2) (note that \(A_k^*\) is \({\mathcal {F}}_{n_{k-1}, n_k}\)–measurable, and hence \({\mathcal {F}}_{-\infty , n_k}\)–measurable and so (9.2) can be applied), and the third one holds because \(A_{k+1}^*\supset {{\hat{A}}}_{k+1}.\) Since \(\mu \) is shift invariant

$$\begin{aligned} \mu ({{\hat{A}}}_{k+1}|{\mathcal {F}}_{n_k, n_k})(\omega )= \mu \left( \frac{\tau _{m_{k+1}}}{\sqrt{m_{k+1}}}\in {{\hat{C}}}\Big |{\mathcal {F}}_{0,0}\right) (\sigma ^{-n_k} \omega )\end{aligned}$$

By the mixing CLT ( [44, 92]) if \({\varvec{\omega }}\) is any symbol in the alphabet of \(\Sigma _A\)

$$\begin{aligned} \lim _{m\rightarrow \infty } \mu \left( \frac{\tau _m(\omega )}{\sqrt{m}}\in {\hat{{\mathcal {C}}}}|\omega _0={\varvec{\omega }}\right) = {{\mathbb {P}}}({\mathcal {N}}\in {\hat{{\mathcal {C}}}}) \end{aligned}$$

uniformly in \({\varvec{\omega }},\) where \({\mathcal {N}}\) is the normal random variable with zero mean and variance \(D^2(\tau )\) given by (2.9). By the assumptions of Lemma 11.4 and Proposition 2.8, we see that \(D^2(\tau )\) is non degenerate. Thus \({{\mathbb {P}}}({\mathcal {N}}\in {\hat{{\mathcal {C}}}})>0\) for any cone \({\mathcal {C}}\). It follows that there exists \({\varepsilon }={\varepsilon }({\mathcal {C}})\) such that for all sufficiently large k and all \(\omega \)

$$\begin{aligned} \mu (A_{k+1}|{\mathcal {F}}_{-\infty , n_k})(\omega ) \ge {\varepsilon }. \end{aligned}$$

(B.3) follows competing the proof of the lemma. \(\square \)

B.3 Separation estimates for cocycles

Proof of Lemma 14.1

(m2) follows from the fact that there exists a constant \(C_\tau \) such that if \(\omega '\) and \(\omega ''\) belong to the same cylinder of length N, then

$$\begin{aligned} |\tau _N(\omega ')-\tau _N(\omega '')|\le C_\tau . \end{aligned}$$

To prove (m1) let

$$\begin{aligned}&N_A(\omega , k)\!=\!\#\left\{ (i,j) \in [0,(10k)^{100}]\times [0,(10k)^{100}], \right. \\&\quad \left. i\ne j:\,\frac{\Vert \tau _{(j-i)n_{k-1}}(\sigma ^{in_{k-1}}\omega )\Vert }{(|j-i|n_{k-1})^{1/2}}< k^{-20}\right\} . \end{aligned}$$

Denote \(m_{ij}=|i-j| n_{k-1}.\) Covering the ball with center at the origin and radius \(\displaystyle \frac{\sqrt{m_{ij}}}{k^{20}}\) in \({\mathbb {R}}^d\) by unit cubes and applying the anticoncentration inequality (B.2) with \(q=1\) (or [35, formula (A.4)]) to each cube, we obtain that

$$\begin{aligned} \mu \left( \Vert \tau _{m_{ij}}(\omega )\Vert \le \frac{ \sqrt{m_{ij}}}{k^{20}} \right) \le C k^{-20 d}. \end{aligned}$$

Since \(\mu \) is shift invariant we conclude that

$$\begin{aligned}\mu \left( \frac{\Vert \tau _{m_{ij}}(\sigma ^{in_{k-1}}\omega )\Vert }{m_{ij}^{1/2}} < \frac{1}{k^{20}}\right) \le C k^{-20 d}.\end{aligned}$$

Summing over i and j we obtain

$$\begin{aligned} \mu \left( N_A(\cdot , k)\right) \le C (10k)^{200-20 d}. \end{aligned}$$

Next, by the Markov inequality,

$$\begin{aligned} \mu \left( \omega : N_A(\omega , k)\ge (10k)^{191}\right) \le \frac{C}{k^{20d-9}}. \end{aligned}$$

This shows that the measure of the complement of \(A_k\) is small. The estimate of measure of \(B_k\) is similar except we replace (B.5) by

$$\begin{aligned} \mu \left( \max _{n\le m} \Vert \tau _n(\omega )\Vert \ge k^{20}\sqrt{m} \right) \le c_1 e^{-c_2 k^{40}}. \end{aligned}$$

To prove (B.6) it is sufficient to consider the case \(d=1\) since for higher dimensions we can consider each coordinate separately. Thus it suffices to show that

$$\begin{aligned} \mu \left( \max _{n\le m} \tau _n(\omega )\ge k^{20}\sqrt{m} \right) \le c_1 e^{-c_2 k^{40}} \end{aligned}$$

(the bound on \(\displaystyle \mu \left( \min _{n\le m} \tau _n(\omega ) \le - k^{20}\sqrt{m} \right) \) is obtained by replacing \(\tau \) by \(-\tau \)).

To prove (B.7) with \(d=1\) we use the reflection principle. Namely, [35, formula (A.3)] shows that for each L

$$\begin{aligned} \mu \left( |\tau _m(\omega )|\ge L \sqrt{m} \right) \le {\bar{c}}_1 e^{-{\bar{c}}_2 L^2}. \end{aligned}$$


$$\begin{aligned} D_m(k)=\left\{ \omega : \exists n\le m, \text { and }{{\bar{\omega }}}: {{\bar{\omega }}}_j=\omega _j\quad \text { for } j\in 0, \dots n-1 \text { and } \tau _n(\omega )\ge k^{20}\sqrt{m} \right\} . \end{aligned}$$

Note that \(D_{m}(k)\) contains the LHS of (B.7) and that \(D_m(k)\) is a disjoint union of the cylinders of length at most m, \(\displaystyle D_m=\bigcup \nolimits _j {\mathcal {D}}_j\) (to see this, take for each \(\omega \) the smallest n such that the last display holds and recall that \(\tau \) only depends on the past). Next, similarly to (B.4) (since \(d=1\) the relevant cone is the cone of positive numbers) there exists \({\varepsilon }>0\) such that for each cylinder \({\mathcal {D}}\) of length \(n=n({\mathcal {D}})\) and for each \(m \ge n\),

$$\begin{aligned} \mu \left( \tau _{m-n}(\omega ) \ge { 0}|\omega \in \sigma ^{-n}{\mathcal {D}}\right) \ge { {\varepsilon }} . \end{aligned}$$

Combining this with (B.8), we obtain

$$\begin{aligned}&{\bar{c}}_1 e^{-{\bar{c}}_2 k^{40}/4}\ge \mu \left( \tau _m\ge \frac{k^{20} \sqrt{m}}{2}\right) \ge \sum _j \mu \left( \omega \in {\mathcal {D}}_j,\; \tau _m\ge \frac{k^{20} \sqrt{m}}{2}\right) \\&\quad \ge \sum _j \mu ({\mathcal {D}}_j) \mu \left( \tau _m\ge \frac{k^{20} \sqrt{m}}{2}\Big |\omega \in {\mathcal {D}}_j\right) \ge {\varepsilon }\sum _j \mu ({\mathcal {D}}_j)={\varepsilon }\mu (D_m) \end{aligned}$$

proving (B.7) and completing the proof of the lemma. \(\square \)

Appendix C. The main results in general context

Here we put our results into a general context of flexibility of statistical properties in smooth dynamics.

There is a vast literature on statistical properties of dynamical systems. A survey by Sinai [100] lists the following hierarchy of chaotic properties for dynamical systems preserving a smooth measure (the properties marked with * are not on the list in [100] but we added them to obtain a more complete listFootnote 18).

(1) (Erg) Ergodicity; (2*) (WM) Weak Mixing (3) (M) Mixing; (4*) (PE) Positive entropy; (5) (K) K property; (6) (B) Bernoulli property; (7) (CLT) Central Limit TheoremFootnote 19; (8) (PM) Polynomial mixing; (9) (EM) Exponential mixing.

Recall that a formal definition of (CLT), (PM), and (EM) were given in Sect. 1. The definitions of the other properties are standard.

Properties (1)–(6) are qualitative. They make sense for any measure preserving dynamical system. Properties (7)–(9) are quantitative. They require smooth structure but provide quantitative estimates. Currently there are many examples of systems enjoying a full array of chaotic properties which follow from either uniform hyperbolicity or non-uniform hyperbolicity, in case there is a control on the region where hyperbolicity is weak [11, 14, 26, 109]. Systems which satisfy only some of the above properties are less understood. In fact, it is desirable to have more examples of such systems in order to understand the full range of possible behaviors of partially chaotic systems.

Thus we have the following list of statistical properties of dynamical systems.

(Erg), (WM), (M), (PE), (K), (M), (CLT), (PM), (EM).

While properties on the bottom of the list are often more difficult to establish especially in the context of nonuniformly hyperbolic systems discussed in [100], property (j) of the list in general does not imply property i for \(i\le j.\) Thus it is desirable to study the following realizability problem: given two disjoint subsets \({\mathcal {A}}_1, {\mathcal {A}}_2\subset \{1, \dots , 9\}\), is there a smoothFootnote 20 map preserving a smooth probability measure that satisfies all properties in \({\mathcal {A}}_1\) and does not have any of the properties in \({\mathcal {A}}_2\)?

The simplest version of the realizability problem is when \(|{\mathcal {A}}_1| = |{\mathcal {A}}_2| = 1\), which case is presented in the following table. Here Y in cell (ij) means that the property in row i implies the property in the column j. (k) in cell (ij) means that a diffeo number (k) on the list below has property (i) but not property (j).

The examples in the table below are the following (the papers cited in the list contain results needed to verify some properties in the table):

(1) irrational rotation; (2) horocycle flow ( [20]); (3) Anosov diffeo \(\times \) identity; (4) maps from Theorem 1.3; (5) skew products on \({\mathbb {T}}^2\times {\mathbb {T}}^2\) of the form \((Ax, y+\alpha \tau (x))\) where A is linear Anosov map, \(\alpha \) is Liouvillian and \(\tau \) is not a coboundary [33]; (6) Anosov diffeo\(\times \)Diophantine rotation (see [27, 71] and Theorem 3.1).

Erg \(\clubsuit \) (1) (1) (1) (1) (1) (1)
WM/M Y \(\clubsuit \) (2) (2) (5) (5) (5)
PE (3) (3) \(\clubsuit \) (3) (3) (3) (3)
K/B Y Y Y \(\clubsuit \) (5) (5) (5)
CLT Y (6) (4) (6) \(\clubsuit \) (6) (6)
PM Y Y (2) (2) (2) \(\clubsuit \) (2)
EM Y Y Y Y ?? Y \(\clubsuit \)

We combined (WM) and (M) (as well as (K) and (B)) together since the same counter examples work for both properties. It is well known that weak mixing does not imply mixing (see Sect. 8.3) and that K does not imply Bernoulli (see Part V).

The positive implications in the top left \(4\times 4\) corner are standard and can be found in most textbooks on ergodic theory. It is also clear that Exponential Mixing \(\Rightarrow \) Polynomial Mixing \(\Rightarrow \) Mixing and that CLT implies the weak law of large numbers which in turn entails ergodicity. The fact that the exponential mixing implies the Bernoulli property (and hence both K property and positive entropy) is more recent [37].

The only open problem in the above table, namely the existence of a system satisfying (EM) but not (CLT) seems hard. Recall from Sect. 4 that the classical CLT follows if the system enjoys exponential mixing of all orders. Therefore the problem whether (EM) implies (CLT) is related to the question whether exponential mixing implies multiple exponential mixing which can be thought of as a quantitative version of the famous open problem of Rokhlin. Except for this specific question, the realizability problem is well understood in case \(|{\mathcal {A}}_1|=|{\mathcal {A}}_2|=1\).

Next, we study the realizability problem with \(|{\mathcal {A}}_1|=2,\) \(|{\mathcal {A}}_2|=1\) and CLT\(\in {\mathcal {A}}_1.\) The table below lists in cell (ij) a map which has both property (i) and satisfies CLT but does not have property j. Clearly the question makes sense only if we have an example of a system which has property (i) but not property (j).

WM \(\clubsuit \) (8) (9) (9) (9) (10)
M \(\clubsuit \) \(\clubsuit \) (9) (9) (9) (10)
PE (6) (6) \(\clubsuit \) (6) (6) (6)
K \(\clubsuit \) \(\clubsuit \) \(\clubsuit \) \(\clubsuit \) (7) ??
B \(\clubsuit \) \(\clubsuit \) \(\clubsuit \) \(\clubsuit \) \(\clubsuit \) ??
PM \(\clubsuit \) \(\clubsuit \) (9) (9) (9) \(\clubsuit \)

Here, (6) refers to the diffeomorphisms from the previous table, while (7), (8), (9), and (10) and refer to the maps from Theorems 1.5, 1.4(a), (b) and 1.3(a). To see that the example of Theorem 1.3(a) is not polynomially mixing we note that for polynomially mixing systems the growth of ergodic integrals can not be regularly varying with index one. Namely (see e.g. [35, §8.1]), for polynomially mixing systems there exists \(\delta >0\) such that the ergodic averages of smooth functions H satisfy \(\displaystyle \lim _{T\rightarrow \infty } \frac{H_T}{T^{1-\delta }}=0\) almost surely, and hence, in law.

Appendix D. Open problems

Here we list some open problems related to our results that we believe should be studied in the future.

In the examples in Theorem 1.3(b), \(\dim (M_r)\) grows with r which leads to the following natural problem:

Problem D.1

(a) Construct a \(C^\infty \) diffeomorphism with zero entropy satisfying the classical CLT.

(b) Construct a \(C^\infty \) flow with zero entropy satisfying the classical CLT.

The next problem is also motivated by Theorem 1.3:

Problem D.2

For which \(\alpha \) does there exist a smooth system satisfying the CLT with normalization which is regularly varying of index \(\alpha ?\)

We mention that several authors [8, 18, 29, 43] obtained the Central Limit Theorem for circle rotations where normalization is a slowly varying function. However, firstly, the functions considered in those papers are only piecewise smooth and, secondly, they require an additional randomness or remove zero density subset of times. Similar results in the context of substitutions are obtained in [15, 91].

In the examples in Theorem 1.4(b) the rate of polynomial mixing is rather slow (slower than linear). This motivates the following problem:

Problem D.3

Given \(m\in {\mathbb {N}}\) construct a diffeomorphism which is mixing at rate \(n^{-m}\) and satisfies at least one of the following: (a) is not K; (b) has zero entropy; (c) does not satisfy the CLT.

Theorem 1.5 motivates the following problems:

Problem D.4

Construct an example of K (or even Bernoulli) diffeomorphism which satisfies the CLT but is not polynomially mixing.

Problem D.5

Let M a compact manifold of dimension at least two. Does there exists a \(C^\infty \) diffeomorphism of M preserving a smooth measure satisfying a Central Limit Theorem?

Currently it is known that any compact manifold of dimension at least two admits an ergodic diffeomorphism of zero entropy [3], a Bernoulli diffeomorphism [17], and, moreover, a nonuniformly hyperbolic diffeomorphism [41]. We note that a recent preprint [97] constructs area preserving diffeomorphisms on any surface of class \(C^{1+\beta }\) (with \(\beta \) small) which satisfy (CLT). It seems likely that similar constructions could be made in higher dimensions. However, the method of [97] requires low regularity to have degenerate saddles where a typical orbit does not spent too much time, and so those methods do not work in higher smoothness such as \(C^2.\) We also note that [21] shows that for any aperiodic dynamical system there exists some measurable observable satisfying the CLTFootnote 21 (see [76, 77, 79, 105] for related results). In contrast Problem D.5 asks to construct a system where the CLT holds for most smooth functions.

Problem D.6

Let M be a compact manifold of dimension at least three. Does there exist a diffeomorphism of M preserving a smooth measure which is K but not Bernoulli?

We note that in case of dimension two, the answer is negative due to Pesin theory [7]. At present there are no example of K but not Bernoulli maps in dimension three. We refer the reader to [62] for more discussion on this problem.

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Dolgopyat, D., Dong, C., Kanigowski, A. et al. Flexibility of statistical properties for smooth systems satisfying the central limit theorem. Invent. math. 230, 31–120 (2022).

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