  We explore a new class of general properties of thermal holographic Green's
functions that can be deduced from the near-horizon behaviour of classical
perturbations in asymptotically anti-de Sitter spacetimes. We show that at
negative imaginary Matsubara frequencies and appropriate complex values of the
wavenumber the retarded Green's functions of generic operators are not uniquely
defined, due to the lack of a unique ingoing solution for the bulk
perturbations. From a boundary perspective these `pole-skipping' points
correspond to locations in the complex frequency and momentum planes at which a
line of poles of the retarded Green's function intersects with a line of
zeroes. As a consequence the dispersion relations of collective modes in the
boundary theory at energy scales $\omega\sim T$ are directly constrained by the
bulk dynamics near the black-brane horizon. For the case of conserved $U(1)$
current and energy-momentum tensor operators we give examples where the
dispersion relations of hydrodynamic modes pass through a succession of
pole-skipping points as real wavenumber is increased. We discuss implications
of our results for transport, hydrodynamics and quantum chaos in holographic
systems.
