Probing quantum gravity through chaotic orbits and strong-field effects in Kerr black holes embedded in perfect fluid dark matter

We study the nonlinear photon dynamics in quantum improved rotating black hole surrounded by perfect fluid dark matter (PFDM) using several methods of analysis, including Poincaré sections, Lyapunov exponents, Kolmogorov–Sinai (KS) entropy and weighted Birkhoff averages (WBA). In particular, we examine the effect of quantum-improved parameter (ω~) and PFDM parameter (ζ) on the null geodesic motion hence stability of circular orbit. Poincaré sections illustrate the transition from regular to chaotic motion as these parameters increase, characterized by the deformation and fragmentation of invariant tori and the emergence of scattered chaotic regions in phase space. The stability properties of null circular orbits are quantified through the Lyapunov indicators, revealing that both the quantum improvement parameter and the PFDM parameter enhance the sensitivity of photon trajectories to initial conditions. The KS entropy provides an independent measure of dynamical complexity and confirms the growth of chaotic behavior with increasing quantum and PFDM corrections. Additionally, the WBA method offers a robust quantitative criterion for distinguishing regular and chaotic orbits and allows a detailed mapping of the phase-space structure. The results demonstrate that the combined effects of quantum gravity corrections and PFDM significantly modify the effective potential governing photon motion, leading to a rich mixed phase-space structure with coexisting regular and chaotic regions. These findings underscore the crucial role of quantum and dark matter contributions in shaping photon dynamics near rotating black holes and suggest possible observational implications on black hole shadows and gravitational lensing in strong-field regimes.