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We consider a holographic thermal state and perturb it by a scalar operatorwhose associated real-time Green's function has only gapped poles. These gappedpoles correspond to the non-hydrodynamic quasinormal modes of a massive scalarperturbation around a Schwarzschild black brane. Relations betweenpole-skipping points, critical points and quasinormal modes in general emergewhen the mass of the scalar and hence the dual operator dimension is varied.First, this novel analysis reveals a relation between the location of a mode inthe infinite tower of quasinormal modes and the number of pole-skipping pointsconstraining its dispersion relation at imaginary momenta. Second, for thefirst time, we consider the radii of convergence of the derivative expansionsabout the gapped quasinormal modes. These convergence radii turn out to bebounded from above by the set of all pole-skipping points. Furthermore, atransition between two distinct classes of critical points occurs at aparticular value for the conformal dimension, implying close relations betweencritical points and pole-skipping points in one of those two classes. We shownumerically that all of our results are also true for gapped modes of vectorand tensor operators.

Constraints on quasinormal modes and bounds for critical points from pole-skipping