The decay rate of correlations is an important characterization of the statistical properties of dynamical systems. More precisely, given a dynamical system \(T:X\to X\) preserving an invariant measure \(\mu\), the correlation of two functions \(\phi,\psi:X\to\mathbb{R}\), for each \(n\geq 0\), is defined by
\[ C_n(\phi,\psi,T,\mu)=\int_X \phi\circ T^n\cdot \psi d\mu-\int_X\phi d\mu \cdot \int_X \psi d\mu. \]
For example, the correlations of uniformly hyperbolic systems are known to have exponentially fast decay rate. It is not easy to determine the decay rates of general systems. The linked-twist maps can be viewed as generalizations of the linear hyperbolic map on \(\mathbb{T}^2\) induced by the matrix \(\binom{2 1}{1 1}\), and are known to be non-uniformly hyperbolic due to the existence of singularities.
In this paper, the authors study these linked-twist maps and prove that the correlations of the standard linked-twist map decays as \(\mathcal{O}(1/n)\). The authors remark that their proof can be applied to a pair of general annuli used to define the linked-twist map, and point out that there is a strong transition from polynomial decay to exponential decay if the two defining annuli are both thickened to the whole torus.