A study of three butterflies: entanglement wedge method, OTOC and pole-skipping

In this work, we investigate two salient chaotic features, namely Lyapunov exponent and butterfly velocity, in the context of an asymptotically Lifshitz black hole background with an arbitrary critical exponent. These features are computed using three methods: entanglement wedge method, out-of-time-ordered correlator computation and pole-skipping. We provide a comparative analysis of the results from these methods, where the features derived from the entanglement wedge and OTOC methods are equivalent. On the contrary, the butterfly velocity obtained from pole-skipping matches only when the critical exponent becomes unity, i.e., the usual asymptotically AdS black hole. Furthermore, we evaluate the chaos at the classical level by computing the eikonal phase and Lyapunov exponent from the bulk gravity. These quantities emerge as nontrivial functions of the anisotropy index. By examining the classical eikonal phase, we uncover different scattering scenarios in the near-horizon and near-boundary regimes. We also discuss potential limitations regarding the choice of the turning point of the null geodesic in our approach.