Lower bounds on mixing rates for a class of mixing flows on surfaces

Abstract

We study mixing rates for locally Hamiltonian flows on compact surfaces with asymmetric logarithmic singularities. For a full measure set of such flows, we show that the decay of correlations of smooth observables cannot be uniformly faster than a power of $\log t$. In particular, there exist sequences of times and observables for which correlations admit lower bounds of order $(\log t)^{-2-\nu}$ for any $\nu>0$. We further show that for a typical Arnol'd flow on $\mathbb T^2$, the self-correlation of every box in the minimal component is bounded below by $(\log t)^{-1}$ along an unbounded sequence of times. Motivated by questions in spectral theory, we also construct examples of such flows for which the self-correlation of a box fails to be square-integrable. These results complement previous upper bounds for correlations in both settings, which are also of polynomial order in $\log t$, and show that logarithmic decay rates are essentially sharp along sequences of times.