Classifying pole-skipping points

A bstract We clarify general mathematical and physical properties of pole-skipping points. For this purpose, we analyse scalar vector fields in hyperbolic space. This setup is chosen because it simple enough to allow us obtain analytical expressions for the Green’s function check everything explicitly, while contains all essential features classify points three types (type-I, II, III). Type-I Type-II are distinguished by (limiting) behavior near Type-III can arise at non-integer i? values, which due a specific UV condition, contrary I related non-unique horizon boundary condition. also relation between structure analysis. point out that there subtle cases where analysis alone may not be able capture existence