Saturation of chaos bound in a phenomenological rotating black hole effective matter system at low-temperature limit

We study the chaotic behavior of a phenomenological system of rotating black holes (BHs) surrounded by radiation, dust, and dark matter as effective matter fluids. We show that the chaos bound is saturated -- but never violated -- when the density parameter $k \rightarrow 1$, which corresponds to a high dark matter concentration in the halo. This saturation is independent of radiation and dust density, occurring in the low-temperature regime where the BH's surface gravity and Hawking temperature decrease. Our results suggest that rotating BHs subjected to a phenomenological representation of a more realistic matter environment can scramble information as efficiently as allowed by physical laws in this limit. This work links quantum chaos research with phenomenological BH-matter configurations, providing a motivation for studies of chaos in astrophysical BHs.