Polynomial decay of correlations in linked-twist maps (Q2928258)

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 NEWLINENEWLINE\[NEWLINEC_n(\phi,\psi,T,\mu)=\int_X \phi\circ T^n\cdot \psi d\mu-\int_X\phi d\mu \cdot \int_X \psi d\mu.NEWLINE\]NEWLINE NEWLINEFor 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.NEWLINENEWLINEIn 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.