Abstract
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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Change history
01 August 2022
A Correction to this paper has been published: https://doi.org/10.1007/s00222022011376
Notes
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.
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 \(\alpha .\)
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.
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 nonuniformly 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].
Here, and in the sequel, \(\Rightarrow \) denotes weak convergence of random variables. Note that in contrast with Definition 1.1, we do not require \(\sigma ^2(A) >0\).
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 \(\displaystyle \alpha =\frac{1+\sqrt{14\varvec{\lambda }_0}}{2}\) 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.
Note that \(\Lambda \) depends on H since \(\rho \) depends on H, see (6.6).
Recall that \({\varepsilon }\) is the diameter of \(\{C_l\}.\)
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.
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 \(h_u.\)
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 \(\Vert u \Vert <\delta \).
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.
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
[100] refers to classical CLT, but since the time it was written several CLTs with non classical normalization has been proven, cf. footnote 4.
Realizabilty problem also makes sense and is interesting in other settings such as for symbolic or hamiltonian systems.
References
Aaronson, J., Denker, M.: Local limit theorems for partial sums of stationary sequences generated by Gibbs–Markov maps. Stoch. Dyn. 1, 193–237 (2001)
Abramov, L.M., Rokhlin, V.A.: Entropy of a skew product of mappings with invariant measure. Vestnik Leningrad. Univ. 17, 5–13 (1962)
Anosov, D.V., Katok, A.B.: New examples in smooth ergodic theory: ergodic diffeomorphisms. Trans. Moscow Math. Soc. 23, 1–35 (1970)
Arnold, V.I.: Topological and ergodic properties of closed 1forms with incommensurable periods. Funktsionalnyi Analiz i Ego Prilozheniya 25, 1–12 (1991)
Bálint, P., Chernov, N., Dolgopyat, D.: Limit theorems for dispersing billiards with cusps. Commun. Math. Phys. 308, 479–510 (2011)
Bálint, P., Gouëzel, S.: Limit theorems in the stadium billiard. Commun. Math. Phys. 263, 461–512 (2006)
Barreira, L., Pesin, Y.: Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents. Cambridge University Press, Cambridge (2007)
Beck, J.: Randomness of the square root of 2 and the giant leap. Period. Math. Hungar. Part 1: 60, 137–242 (2010); Part 2: 62, 127–246 (2011)
Björklund, M., Einsiedler, M., Gorodnik, A.: Quantitative multiple mixing. JEMS 22, 1475–1529 (2020)
Björklund, M., Gorodnik, A.: Central limit theorems for group actions which are exponentially mixing of all orders. J. d’Analyse Mathematiques 141, 457–482 (2020)
Bonatti, C., Diaz, L.J., Viana, M.: Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective. Springer, Berlin (2005)
Bolthausen, E.: A central limit theorem for twodimensional random walks in random sceneries. Ann. Probab. 17, 108–115 (1989)
Bowen, R.: Symbolic dynamics for hyperbolic flows. Am. J. Math. 95, 429–460 (1973)
Bowen, R.: Equilibrium States and Ergodic Theory of Anosov Diffeomorphisms. Springer, New York (1975)
Bressaud, X., Bufetov, A.I., Hubert, P.: Deviation of ergodic averages for substitution dynamical systems with eigenvalues of modulus 1. Proc. Lond. Math. Soc. 109, 483–522 (2014)
Brin, M.I.: The topology of group extensions of C systems. Mat. Zametki 18, 453–465 (1975)
Brin, M.I., Feldman, J., Katok, A.: Bernoulli diffeomorphisms and group extensions of dynamical systems with nonzero characteristic exponents. Ann. Math. 113, 159–179 (1981)
Bromberg, M., Ulcigrai, C.: A temporal central limit theorem for realvalued cocycles over rotations. Ann. Inst. Henri Poincare Probab. Stat. 54, 2304–2334 (2018)
Bruin, H.: On volume preserving almost Anosov flows. arXiv:1908.05675
Bufetov, A., Forni, G.: Limit theorems for horocycle flows. Ann. Sci. Ec. Norm. 47, 851–903 (2014)
Burton, R., Denker, M.: On the central limit theorem for dynamical systems. Trans. AMS 302, 715–726 (1987)
Burton, R., Shields, P.: A skewproduct which is Bernoulli. Monatsh. Math. 86, 155–165 (1978/79)
Butzer, P.L., Westphal, W.: The mean ergodic theorem and saturation. Indiana Univ. Math. J. 20, 1163–1174 (1970/71)
Chazottes, J.R., Gouëzel, S.: Optimal concentration inequalities for dynamical systems. Commun. Math. Phys. 316, 843–889 (2012)
Chernov, N.I.: Limit theorems and Markov approximations for chaotic dynamical systems. Probab. Theory Rel. Fields 101, 321–362 (1995)
Chernov, N., Markarian, R.: Chaotic billiards. In: AMS Mathematical Surveys and Monographs, vol. 127 (2006)
Cohen, G., Conze, J.P.: The CLT for rotated ergodic sums and related processes. Discrete Contin. Dyn. Syst. 33, 3981–4002 (2013)
Cohen, G., Conze, J.P.: CLT for random walks of commuting endomorphisms on compact abelian groups. J. Theor. Probab. 30, 143–195 (2017)
Conze, J.P., Isola, S., Le Borgne, S.: Diffusive behavior of ergodic sums over rotations. Stoch. Dyn. 19, 1950016 (2019)
Darling, D.A., Kac, M.: On occupation times for Markoff processes. Trans. AMS 84, 444–458 (1957)
den Hollander, F., Keane, M.S., Serafin, J., Steif, J.E.: Weak Bernoullicity of random walk in random scenery. Jpn. J. Math. 29, 389–406 (2003)
den Hollander, F., Steif, J.E.: Mixing properties of the generalized \(T, T^{1}\)process. J. Anal. Math. 72, 165–202 (1997)
Dolgopyat, D.: On mixing properties of compact group extensions of hyperbolic systems. Isr. Math. J. 130, 157–205 (2002)
Dolgopyat, D.: Limit theorems for partially hyperbolic systems. Trans. AMS 356, 1637–1689 (2004)
Dolgopyat, D., Dong, C., Kanigowski, A., Nándori, P.: Mixing properties of generalized \(T, T^{1}\) transformations. Isr. J. Math. 2022, 1–53 (2022)
Dolgopyat, D., Fayad, B., Liu, S.: Multiple Borel Cantelli lemma in dynamics and multilog law for recurrence. J. Mod. Dyn. arXiv:2103.08382
Dolgopyat, D., Kanigowski, A., Rodriguez, H,F.: Exponential mixing implies Bernoulli. arXiv:2106.03147
Dolgopyat, D., Nándori, P.: Non equilibrium density profiles in Lorentz tubes with thermostated boundaries. Commun. Pure Appl. Math. 69, 649–692 (2016)
Dolgopyat, D., Nándori, P.: Infinite measure renewal theorem and related results. Bull. LMS 51, 145–167 (2019)
Dolgopyat, D., Nándori, P.: On mixing and the local central limit theorem for hyperbolic flows. Ergod. Theory Dyn. Syst. 40, 142–174 (2020)
Dolgopyat, D., Pesin, Y.: Every compact manifold carries a completely hyperbolic diffeomorphism. Ergod. Theory Dyn. Syst. 22, 409–435 (2002)
Dolgopyat, D., Sarig, O.: Temporal distributional limit theorems for dynamical systems. J. Stat. Phys. 166, 680–713 (2017)
Dolgopyat, D., Sarig, O.: Asymptotic windings of horocycles. Isr. J. Math. 228, 119–176 (2018)
Eagleson, G.K.: Some simple conditions for limit theorems to be mixing. Teor. Verojatnost. i Primenen. 21, 653–660 (1976)
Einsiedler, M., Lind, D.: Algebraic \(\mathbb{Z}^d\)actions of entropy rank one. Trans. AMS 356, 1799–1831 (2004)
Erdős, P., Kac, M.: The Gaussian law of errors in the theory of additive number theoretic functions. Am. J. Math. 62, 738–742 (1940)
Fayad, B.: Polynomial decay of correlations for a class of smooth flows on the two torus. Bull. SMF 129, 487–503 (2001)
Flaminio, L., Forni, G.: Invariant distributions and time averages for horocycle flows. Duke Math. J. 119, 465–526 (2003)
Fraczek, K., Lemańczyk, M.: On symmetric logarithm and some old examples in smooth ergodic theory. Fund. Math. 180, 241–255 (2003)
Galatolo, S.: Dimension and hitting time in rapidly mixing systems. Math. Res. Lett. 14, 797–805 (2007)
Gordin, M.I.: The central limit theorem for stationary processes. Dokl. Akad. Nauk SSSR 188, 739–741 (1969)
Gorodnik, A., Spatzier, R.: Mixing properties of commuting nilmanifold automorphisms. Acta Math. 215, 127–159 (2015)
Gouëzel, S.: Central limit theorem and stable laws for intermittent maps. Probab. Theory Rel. Fields 128, 82–122 (2004)
Gouëzel, S.: Limit theorems in dynamical systems using the spectral method. Proc. Symp. Pure Math. 89, 161–193 (2015)
Gouëzel, S.: Growth of normalizing sequences in limit theorems for conservative maps. Electron. Commun. Prob. 23, 99 (2018)
Gouëzel, S., Melbourne, I.: Moment bounds and concentration inequalities for slowly mixing dynamical systems. Electron. J. Probab. 19, 30 (2014)
Guivarc’h, Y.: Propriétés ergodiques, en mesure infinie, de certains systemes dynamiques fibrés. Ergod. Theory Dyn. Syst. 9, 433–453 (1989)
Hall, P., Heyde, C.: Martingale Limit Theory and Its Application. Academic Press, New York (1980)
Halmos, P.R.: Lectures on Ergodic Theory, Chelsea Publishing, New York, vii+101 pp (1960)
Kalikow, S.A.: \(T, T^{1}\) transformation is not loosely Bernoulli. Ann. Math. 115, 393–409 (1982)
Kanigowski, A.: Bernoulli property for homogeneous systems. arXiv:1812.03209
Kanigowski, A., Rodriguez, H.F., Vinhage, K.: On the nonequivalence of the Bernoulli and K properties in dimension four. J. Mod. Dyn. 13, 221–250 (2018)
Katok, A.: Smooth nonBernoulli Kautomorphisms. Invent. Math. 61, 291–299 (1980)
Katok, A., Kononenko, A.: Cocycles’ stability for partially hyperbolic systems. Math. Res. Lett. 3, 191–210 (1996)
Katok, A., Katok, S., Schmidt, K.: Rigidity of measurable structure for \(\mathbb{Z}^d\) actions by automorphisms of a torus. Comment. Math. Helv. 77, 718–745 (2002)
Katok, A., Spatzier, R.: First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity. Publ. IHES 79, 131–156 (1994)
Katsuda, A., Sunada, T.: Homology and closed geodesics in a compact Riemann surface. Am. J. Math. 110, 145–155 (1988)
Kesten, H., Spitzer, F.: A limit theorem related to a new class of selfsimilar processes. Z. Wahrsch. Verw. Gebiete 50, 5–25 (1979)
Khanin, K.M., Sinai, Y.G.: Mixing for some classes of special flows over rotations of the circle. Funct. Anal. Appl. 26, 155–169 (1992)
Khintchine, A.: Zur metrischen Theorie der diophantischen Approximationen. Math. Zeitschr. 24, 706–714 (1926)
Kifer, Y.: Limit theorems for random transformations and processes in random environments. Trans. AMS 350, 1481–1518 (1998)
Kleinbock, D.Y., Margulis, G.A.: Bounded orbits of nonquasiunipotent flows on homogeneous spaces. AMS Transl. 171, 141–172 (1996)
Kochergin, A.V.: Nondegenerate saddles and the absence of mixing. Math. Notes 19, 277–286 (1976)
Kochergin, A.V.: Mixing in special flows over a shifting of segments and in smooth flows on surfaces. Mat. Sb. 96, 471–502 (1975)
Korepanov, A., Kosloff, Z., Melbourne, I.: Martingalecoboundary decomposition for families of dynamical systems. Ann. Inst. H. Poincare 35, 859–885 (2018)
Kosloff, Z., Volný, D.: Local limit theorem in deterministic systems. Annales de l’Institut Henri Poincare Prob. Stat. 58, 548–566 (2022)
Lacey, M.: On central limit theorems, modulus of continuity and Diophantine type for irrational rotations. J. Anal. Math. 61, 47–59 (1993)
Ledrappier, F., Sarig, O.: Unique ergodicity for nonuniquely ergodic horocycle flows. Discrete Contin. Dyn. Syst. 16, 411–433 (2006)
Lesigne, E.: Almost sure central limit theorem for strictly stationary processes. Proc. AMS 128, 1751–1759 (2000)
Le Borgne, S.: Exemples de systemes dynamiques quasihyperboliques a decorrelations lentes (preprint); research announcement. CRAS 343, 125–128 (2006)
Le Jan, Y.: The central limit theorem for the geodesic flow on noncompact manifolds of constant negative curvature. Duke Math. J. 74, 159–175 (1994)
Livsic, A.N.: Cohomology of dynamical systems. Math. USSR Izvestija 6, 1278–1301 (1972)
Marcus, B., Newhouse, S.: Measures of maximal entropy for a class of skew products. Lect. Notes Math. 729, 105–125 (1979)
Margulis, G.A.: On Some Aspects of the Theory of Anosov Systems. Springer, Berlin (2004)
Meilijson, I.: Mixing properties of a class of skewproducts. Isr. J. Math. 19, 266–270 (1974)
Melbourne, I.: Large and moderate deviations for slowly mixing dynamical systems. Proc. AMS 137, 1735–1741 (2009)
Moore, C.C.: Ergodicity of flows on homogeneous spaces. Am. J. Math. 88, 154–178 (1966)
Ornstein, D.: Bernoulli shifts with the same entropy are isomorphic. Adv. Math. 4, 337–352 (1970)
Ornstein, D., Weiss, B.: Entropy and isomorphism theorems for actions of amenable groups. J. d’Anal. Math. 8, 1–141 (1987)
Ornstein, D., Weiss, B.: The Shannon–McMillan–Breiman theorem for a class of amenable groups. Isr. J. Math. 44, 53–60 (1983)
Paquette, E., Son, Y.: Birkhoff sum fluctuations in substitution dynamical systems. Ergod. Theory Dyn. Syst. 39, 1971–2005 (2019)
Parry, W., Pollicott, M.: Zeta functions and the periodic orbit structure of hyperbolic dynamics. Asterisque 187–188, 268 (1990)
Peligrad, M., Wu, W.B.: Central limit theorem for Fourier transforms of stationary processes. Ann. Probab. 38, 2009–2022 (2010)
Pene, F.: Planar Lorentz process in random scenery. Ann. Inst. H. Poincare 45, 818–839 (2009)
Pene, F.: Random Walks in random sceneries and related models. ESAIM Proc. Surv. 68, 35–51 (2020)
Pene, F., Thomine, D.: Central limit theorems for the \(\mathbb{Z}^2\)periodic Lorentz gas. Isr. J. Math. 241, 539–582 (2021)
Pesin, Y.B., Senti, S., Shahidi, F.: Area preserving surface diffeomorphisms with polynomial decay of correlations are ubiquitous. arXiv: 2003.08503
Rokhlin, V.A., Sinai, Y.G.: The structure and properties of invariant measurable partitions (in Russian). Dokl. Akad. Nauk SSSR 141, 1038–1041 (1961)
Rudolph, D.: Asymptotically Brownian skew products give nonloosely Bernoulli Kautomorphisms. Invent. Math. 91, 105–128 (1988)
Sinai, Y.G.: The hierarchy of stochastic properties of deterministic systems. Encycl. Math. Sci. 100, 106–108 (2000)
Szász, D., Varjú, T.: Limit laws and recurrence for the planar Lorentz process with infinite horizon. J. Stat. Phys. 129, 59–80 (2007)
Thouvenot, J.P., Weiss, B.: Limit laws for ergodic processes. Stoch. Dyn. 12, 9 (2012)
Ulcigrai, C.: Weak mixing for logarithmic flows over interval exchange transformations. J. Mod. Dyn. 3, 35–49 (2009)
Ulcigrai, C.: Absence of mixing in areapreserving flows on surfaces. Ann. Math. 173, 1743–1778 (2011)
Volný, D.: Invariance principles and Gaussian approximation for strictly stationary processes. Trans. AMS 351, 3351–3371 (1999)
Weiss, B.: The isomorphism problem in ergodic theory. Bull. AMS 78, 668–684 (1972)
Williams, D.: Probability with Martingales, Cambridge Mathematical Textbooks (1991)
Wilkinson, A.: The cohomological equation for partially hyperbolic diffeomorphisms. Asterisque 358, 75–165 (2013)
Young, L.S.: Statistical properties of dynamical systems with some hyperbolicity. Ann. Math. 147, 585–650 (1998)
Acknowledgements
We thank Bassam Fayad and JeanPaul Thouvenot for useful discussions. D. D. was partially supported by the NSF grant DMS1956049, C. D. was partially supported by AMS Simons travel grant and Nankai Zhide Foundation, A. K. was partially supported by the NSF grant DMS1956310, P. N. was partially supported by the NSF grants DMS1800811 and DMS1952876.
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Appendices
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 \((x,y)\in (X\times Y)\), \(u\in T_x X,\) \(v\in T_y Y\)
where \(\tau _{(j)}\) denotes the jth 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
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. (x, y)
where the last step follows since f is ergodic and \(\tau \) has zero mean.
The foregoing discussion shows that for a.e. (x, y), \(\displaystyle \limsup _{N\rightarrow \infty } \frac{\ln \Vert DF^N\Vert (x,y)}{N}\le 0.\) 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 (X, f) 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.
Proof
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
Indeed,
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
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
It follows that
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
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 \(Cm^{d1}\) 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 \(n_1 = 2\), \(n_{k+1}=n_k^3,\) \(m_k=n_kn_{k1}\) 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 \(\omega \)
Let \(\displaystyle {\hat{{\mathcal {C}}}} =\{v\in {\mathcal {C}}: \text {dist}(v, \partial {\mathcal {C}})\ge 1 \},\) \(\displaystyle {{\hat{A}}}_k=\left\{ \omega : \frac{\tau _{m_k}(\sigma ^{n_{k1}} \omega )}{\sqrt{m_k}} \in {\hat{{\mathcal {C}}}}\right\} ,\)
\(\displaystyle A_k^*=\{\omega : \exists {{\hat{\omega }}}\in {{\hat{A}}}_k:\; \omega _j={{\hat{\omega }}}_j \text { for } j\in [n_{k1}, n_k] \}. \) Note that \(A_k^* \subset A_k\) because for any \(\omega \in A_k^*\) and for the corresponding \(\hat{\omega }\), \(\tau _{m_k}(\sigma ^{n_{k1}} {{\hat{\omega }}})\) is inside \({\mathcal {C}}\) and is at least \( \frac{1}{2} \sqrt{m_k}\) away from the boundary whereas
Next
where the second inequality is due to (9.2) (note that \(A_k^*\) is \({\mathcal {F}}_{n_{k1}, n_k}\)–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
By the mixing CLT ( [44, 92]) if \({\varvec{\omega }}\) is any symbol in the alphabet of \(\Sigma _A\)
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 \(\omega \)
(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
To prove (m1) let
Denote \(m_{ij}=ij n_{k1}.\) 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
Since \(\mu \) is shift invariant we conclude that
Summing over i and j we obtain
Next, by the Markov inequality,
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
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
(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
Let
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 \(m \ge n\),
Combining this with (B.8), we obtain
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 list^{Footnote 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 Theorem^{Footnote 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 nonuniform 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 smooth^{Footnote 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 (i, j) means that the property in row i implies the property in the column j. (k) in cell (i, j) 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  WM/M  PE  K/B  CLT  PM  EM  

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 (i, j) 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  M  PE  K  B  PM  

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 CLT^{Footnote 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). https://doi.org/10.1007/s00222022011210
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DOI: https://doi.org/10.1007/s00222022011210