  We propose a finite discretization for the black hole geometry and dynamics.
We realize our proposal, in the case of extremal black holes, for which the
radial and temporal near horizon geometry is known to be
AdS$_2=SL(2,\mathbb{R})/SO(1,1,\mathbb{R})$. We implement its discretization by
replacing the set of real numbers $\mathbb{R}$ with the set of integers modulo
$N$, with AdS$_2$ going over to the finite geometry
AdS$_2[N]=SL(2,\mathbb{Z}_N)/SO(1,1,\mathbb{Z}_N)$.
  We model the dynamics of the microscopic degrees of freedom by generalized
Arnol'd cat maps, ${\sf A}\in SL(2,\mathbb{Z}_N)$, which are isometries of the
geometry at both the classical and quantum levels.
  These exhibit well studied properties of strong arithmetic chaos, dynamical
entropy, nonlocality and factorization in the cutoff discretization $N$, which
are crucial for fast quantum information processing.
  We construct, finally, a new kind of unitary and holographic correspondence,
for AdS$_2[N]$/CFT$_1[N]$, via coherent states of both the bulk and boundary
geometries.
