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In this paper, we provide an elementary, unified treatment of two distinctblue-shift instabilities for the scalar wave equation on a fixed Kerr blackhole background: the celebrated blue-shift at the Cauchy horizon (familiar fromthe strong cosmic censorship conjecture) and the time-reversed red-shift at theevent horizon (relevant in classical scattering theory).  Our first theorem concerns the latter and constructs solutions to the waveequation on Kerr spacetimes such that the radiation field along the futureevent horizon vanishes and the radiation field along future null infinitydecays at an arbitrarily fast polynomial rate, yet, the local energy of thesolution is infinite near any point on the future event horizon. Our secondtheorem constructs solutions to the wave equation on rotating Kerr spacetimessuch that the radiation field along the past event horizon (extended into theblack hole) vanishes and the radiation field along past null infinity decays atan arbitrarily fast polynomial rate, yet, the local energy of the solution isinfinite near any point on the Cauchy horizon.  The results make essential use of the scattering theory developed in [M.Dafermos, I. Rodnianski and Y. Shlapentokh-Rothman, A scattering theory for thewave equation on Kerr black hole exteriors, preprint (2014) available at\url{http://arxiv.org/abs/1412.8379}] and exploit directly the time-translationinvariance of the scattering map and the non-triviality of the transmissionmap.

Time-Translation Invariance of Scattering Maps and Blue-Shift Instabilities on Kerr Black Hole Spacetimes