Pole skipping and zero temperature

We study the pole-skipping phenomenon of scalar retarded Green's function in rotating BTZ black hole background. In static case, points are typically located at negative imaginary Matsubara frequencies $\omega=-(2\pi T)ni$ with appropriate values complex wave number $q$. But, a $(1+1)$-dimensional CFT, one can introduce temperatures for left-moving and right-moving sectors independently. As result, $\omega$ depend both on left right extreme limit, does not occur general. But special even given by frequencies.