  We present the first kinematic study of an $\alpha\Omega$-dynamo in the
General Relativistic Magneto-HydroDynamics (GRMHD) regime, applied to thick
disks orbiting around Kerr black holes and using a fully covariant mean field
dynamo closure for the Ohm law. We show that the $\alpha\Omega$-dynamo
mechanism leads to a continuous exponential growth of the magnetic field within
the disk and to the formation of dynamo waves drifting away or toward the
equatorial plane. Since the evolution of the magnetic field occurs
qualitatively in the same fashion as in the Sun, we present also butterfly
diagrams that characterize our models and show the establishment of an
additional timescale, which depends on the microscopic properties of the
turbulent motions, possibly providing an alternative explanation to
periodicities observed in many high-energy astrophysical sources where
accretion onto a rotating black hole is believed to operate.
