Finite-order exponential mixing does not imply the central limit theorem

Abstract

We show that finite-order exponential mixing does not imply the central limit theorem for smooth diffeomorphisms. For every integer $R\geq2$, we construct a $\mathcal{C}^{\infty}$ diffeomorphism $T:M\to M$ of a compact smooth manifold $M$ and a $T$-invariant probability measure $μ$ such that $(T,μ)$ is exponentially mixing of order $R$ for $\mathcal{C}^1(M)$ observables, but not of order $2R+2$. For some centred observable $f_0\in\mathcal{C}^{\infty}(M)$, no subsequence of the normalised sums $\frac{1}{\sqrt n}\sum_{i=0}^{n-1}f_0\circ T^i$ converges in distribution to a Gaussian law. In particular, exponential decay of two-point correlations implies neither exponential mixing of all orders nor the central limit theorem. We realise $(T,μ)$ in two ways. In the first, $T$ is a fixed linear Anosov diffeomorphism of $\mathbb T^2$ and $μ$ is singular with respect to Lebesgue measure. In the second, $M$ can be any compact smooth surface without boundary and $μ$ is a physical measure whose statistical basin contains a nonempty open set. The main theorems have been formalised in Lean 4.