Researchers have developed a systematic analytical method to study high-order pole-skipping in near-extremal holographic black holes. This advancement provides a deeper understanding of the structure of holographic Green's functions in low-temperature regimes. Pole-skipping is a specific behavior of poles in the Green's functions of a boundary field theory in an AdS spacetime, related to the diffusion of energy and momentum in strongly coupled systems.

The study focuses on the near-extremal regime, where the temperature T approaches zero. In this limit, the near-horizon geometry of the black holes develops an approximate $\mathrm{AdS}_2 \times \mathbb{R}^{d-1}$ structure. The authors identify the mode index $q$ labeling pole-skipping points with the infrared conformal dimension $\Delta_{\mathrm{IR}} = q$ in the emergent $\mathrm{AdS}_2/\mathrm{CFT}_1$ correspondence. This provides a concrete physical interpretation for the subleading pole-skipping tower.

The methodology reorganizes the near-horizon Frobenius expansion according to powers of temperature, revealing a temperature-graded hierarchical structure. This reduces the n-th-order pole-skipping condition to a factorized algebraic equation, where each pole-skipping momentum depends only on the mode index $q$, not on the order $n$. This n-independence produces a high degeneracy as $T \to 0$, where pole-skipping momenta at all orders collapse onto a discrete set of values determined by the near-horizon geometry and the scalar field mass. These values can be expressed in terms of thermodynamic quantities such as entropy density and specific heat. In the limit $n \gg 1$ (with $nT$ remaining small), the leading pole-skipping momenta grow asymptotically as $k_{n,n} \propto n$. The results were verified through numerical analysis of the Dyonic Gubser-Rocha model, confirming that near-horizon physics governs high-order pole-skipping at low temperature.