  Based on our recent work on the discretization of the radial AdS$_2$ geometry
of extremal BH horizons,we present a toy model for the chaotic unitary
evolution of infalling single particle wave packets.
  We construct explicitly the eigenstates and eigenvalues for the single
particle dynamics for an observer falling into the BH horizon, with time
evolution operator the quantum Arnol'd cat map (QACM).
  Using these results we investigate the validity of the eigenstate
thermalization hypothesis (ETH), as well as that of the fast scrambling time
bound (STB).
  We find that the QACM, while possessing a linear spectrum, has eigenstates,
which are random and satisfy the assumptions of the ETH.
  We also find that the thermalization of infalling wave packets in this
particular model is exponentially fast, thereby saturating the STB, under the
constraint that the finite dimension of the single--particle Hilbert space
takes values in the set of Fibonacci integers.
