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 $R\ge 2$, we construct a ${\mathcal{C}}^{\infty }$ diffeomorphism $T:M\to M$ of a compact smooth manifold $M$ and a $T$-invariant probability measure $\mu$ such that $(T,\mu )$ 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 $(T,\mu )$ in two ways. In the first, $T$ is a fixed linear Anosov diffeomorphism of ${\mathbb{T}}^2$ and $\mu$ is singular with respect to Lebesgue measure. In the second, $M$ can be any compact smooth surface without boundary and $\mu$ is a physical measure whose statistical basin contains a nonempty open set. The main theorems have been formalised in Lean 4.

arXiv comment46 pages, 6 figures