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