  In this work we present a formulation of the Teukolsky equation for generic
spin perturbations on the hyperboloidal and horizon penetrating foliation of
Kerr recently proposed by Racz and Toth. An additional, spin-dependent
rescaling of the field variable can be used to achieve stable, long-term, and
accurate time-domain evolutions of generic spin perturbations. As an
application (and a severe numerical test), we investigate the late-time decays
of electromagnetic and gravitational perturbations at the horizon and future
null infinity by means of 2+1 evolutions. As initial data we consider four
combinations of (non-)stationary and (non-)compact-support initial data with a
pure spin-weighted spherical harmonic profile. We present an extensive study of
late time decays of axisymmetric perturbations. We verify the power-law decay
rates predicted analytically, together with a certain "splitting" behaviour of
the power-law exponent. We also present results for non-axisymmetric
perturbations. In particular, our approach allows to study the behaviour of the
late time decays of gravitational fields for nearly extremal and extremal black
holes. For rapid rotation we observe a very prolonged, weakly damped,
quasi-normal-mode phase. For extremal rotation the field at future null
infinity shows an oscillatory behaviour decaying as the inverse power of time,
while at the horizon it is amplified by several orders of magnitude over long
time scales. This behaviour can be understood in terms of the superradiance
cavity argument.
