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 Arnold 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.
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