For a CrC^r diffeomorphism of a compact manifold preserving a smooth measure, exponential decay of two-point correlations (exponential mixing) is one of the strongest statistical properties commonly established in practice. The paper Flexibility of statistical properties for smooth systems satisfying the central limit theorem (Inventiones mathematicae, 2022) isolates, as its Problem 5.2, the question of whether exponential mixing implies multiple exponential mixing — exponential decay, in the minimal time gap, of correlations of products of any number of observables. This is a quantitative version of Rokhlin's classical problem (does mixing imply mixing of all orders?), and since the classical central limit theorem is known to follow from multiple exponential mixing, an affirmative answer would make the CLT an automatic consequence of exponential mixing for smooth volume-preserving systems. A 2024 Annals of Mathematics result resolves a neighboring cell (exponential mixing implies Bernoulli), which yields qualitative mixing of all orders but no rate and no CLT; the exponential-rate implication asked here remains open, with only class-specific multiple-mixing results known.

Background

A volume-preserving diffeomorphism is an invertible smooth map FF of a compact orientable manifold MM, together with a measure μ\mu absolutely continuous with respect to a fixed volume form and invariant under FF. The system is mixing if correlations of two observables vanish as time grows: ∫MA(Fnx)B(x) dμ−μ(A)μ(B)→0\int_M A(F^n x)B(x)\,d\mu - \mu(A)\mu(B) \to 0 for smooth A,BA,B. It is exponentially mixing if this convergence is exponential, that is, there exist C,δ>0C,\delta>0 such that the left side is bounded by C∥A∥Cr∥B∥Cre−δnC\|A\|_{C^r}\|B\|_{C^r}e^{-\delta n} for all nn; exponential mixing is the property most spectral and transfer-operator methods directly establish (for instance for Anosov systems, which are uniformly hyperbolic diffeomorphisms, and for many homogeneous systems).

Mixing of all orders (multiple mixing) is the analogous higher-order condition: for every m≥2m \ge 2 and sets A1,…,AmA_1,\dots,A_m, one has μ(⋂j=1mF−njAj)→∏jμ(Aj)\mu\big(\bigcap_{j=1}^m F^{-n_j}A_j\big) \to \prod_j \mu(A_j) whenever the times n1<n2<⋯<nmn_1 < n_2 < \dots < n_m separate, i.e. min⁡i≠j∣ni−nj∣→∞\min_{i \ne j}|n_i - n_j| \to \infty. Rokhlin's famous 1949 problem asks whether ordinary mixing implies mixing of all orders; it remains open even for abstract measure-preserving transformations. Multiple exponential mixing is the quantitative strengthening in which, for each order mm, there are constants Cm,δm>0C_m,\delta_m > 0 such that ∣∫M∏j=1mAj(Fnjx) dμ−∏j=1mμ(Aj)∣≤Cm∏j∥Aj∥Cre−δmmin⁡i≠j∣ni−nj∣\Big|\int_M \prod_{j=1}^m A_j(F^{n_j}x)\,d\mu - \prod_{j=1}^m \mu(A_j)\Big| \le C_m \prod_j \|A_j\|_{C^r} e^{-\delta_m \min_{i \ne j}|n_i - n_j|} (equation (5.1) of the source paper), so higher-order correlations decay exponentially in the minimal gap, with constants allowed to depend on the order.

The central limit theorem (CLT) for a dynamical system says that ergodic sums AN(x)=∑n<NA(Fnx)A_N(x) = \sum_{n<N} A(F^n x) of a smooth observable AA with μ(A)=0\mu(A) = 0 satisfy AN/N⇒N(0,σ2(A))A_N/\sqrt{N} \Rightarrow N(0,\sigma^2(A)), the normal distribution, with σ2\sigma^2 not identically zero. A classical Markov-approximation argument of Chernov (1995), and a generalization by Björklund–Gorodnik (2020) for group actions, shows that multiple exponential mixing implies the classical CLT. Hence the question does exponential mixing imply the CLT? reduces, on this route, to whether exponential mixing implies multiple exponential mixing.

The source paper of Dolgopyat, Dong, Kanigowski and Nándori (arXiv 2020, Inventiones mathematicae 2022) studies the flexibility of smooth systems: which combinations of qualitative properties (ergodicity, weak mixing, the K property — meaning the remote future is trivial, i.e. the intersection over all nn of the sigma-algebras generated by the iterates FkF^k for k≥nk \ge n contains only sets of measure zero or one — and the Bernoulli property, meaning isomorphism to a sequence of independent identically distributed random variables) and quantitative properties (CLT, exponential large deviations, polynomial mixing, exponential mixing) can coexist. Their table of implications between these ten properties contains open cells precisely in the row of exponential mixing. A companion result of Dolgopyat–Kanigowski–Rodriguez Hertz (Annals of Mathematics, 2024), announced as forthcoming work in the source paper, proves that a C1+αC^{1+\alpha} volume-preserving exponentially mixing diffeomorphism is Bernoulli — thereby filling the combined K/Bernoulli cell of the table, and incidentally giving qualitative mixing of all orders (Bernoulli shifts are mixing of all orders). The cells concerning exponential large deviations and the CLT, however, remain open, and Problem 5.2 isolates the quantitative mechanism behind the latter two.

Problem Statement

Problem 5.2 of Dolgopyat–Dong–Kanigowski–Nándori. Let FF be a CrC^r diffeomorphism of a smooth compact orientable manifold MM with a fixed volume form, preserving a measure μ\mu absolutely continuous with respect to volume (the C2C^2 case is the one singled out in the paper's discussion). Suppose FF is exponentially mixing: there exist C,δ>0C,\delta>0 such that for all observables A,B∈Cr(M)A,B \in C^r(M) and all nn, ∣∫MA(Fnx)B(x) dμ−μ(A)μ(B)∣≤C ∥A∥Cr∥B∥Cr e−δn.\Big|\int_M A(F^n x)B(x)\,d\mu - \mu(A)\mu(B)\Big| \le C\,\|A\|_{C^r}\|B\|_{C^r}\,e^{-\delta n}. Does it follow that FF is multiple exponentially mixing, in the sense of equation (5.1) of the source paper: for every order m≥2m \ge 2 there exist Cm,δm>0C_m,\delta_m>0 such that for all A1,…,Am∈Cr(M)A_1,\dots,A_m \in C^r(M) and all times n1,…,nmn_1,\dots,n_m, ∣∫M∏j=1mAj(Fnjx) dμ−∏j=1mμ(Aj)∣≤Cm∏j=1m∥Aj∥Cr  e−δm min⁡i≠j∣ni−nj∣ ?\Big|\int_M \prod_{j=1}^{m} A_j(F^{n_j}x)\,d\mu - \prod_{j=1}^{m}\mu(A_j)\Big| \le C_m \prod_{j=1}^{m}\|A_j\|_{C^r}\; e^{-\delta_m\,\min_{i \ne j}|n_i-n_j|}\,? More generally, the authors ask whether mixing at a given rate implies higher-order mixing at the same rate — a quantitative version of Rokhlin's problem on mixing of all orders. Because the classical central limit theorem is a known consequence of multiple exponential mixing (Chernov 1995; Björklund–Gorodnik 2020), an affirmative answer would in particular establish that every exponentially mixing smooth measure-preserving system satisfies the CLT, one of the implications still marked unknown in the source paper's table.

References

  1. Dmitry Dolgopyat, Changguang Dong, Adam Kanigowski, Péter Nándori, Flexibility of statistical properties for smooth systems satisfying the central limit theorem, Inventiones mathematicae 230 (2022), 31–120, DOI 10.1007/s00222-022-01121-0, https://doi.org/10.1007/s00222-022-01121-0; preprint arXiv:2006.02191, https://arxiv.org/abs/2006.02191 (source of Problem 5.2)
  2. Dmitry Dolgopyat, Adam Kanigowski, Federico Rodriguez Hertz, Exponential mixing implies Bernoulli, Annals of Mathematics 199 (2024), no. 3, DOI 10.4007/annals.2024.199.3.5, https://doi.org/10.4007/annals.2024.199.3.5; preprint arXiv:2106.03147, https://arxiv.org/abs/2106.03147
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  9. Jonathan DeWitt, Dmitry Dolgopyat, Conservative Coexpanding on Average Diffeomorphisms, preprint arXiv:2503.06855 (2025), https://arxiv.org/abs/2503.06855