  We propose a new example of discrete holography that provides a new step
towards establishing the AdS/CFT duality for discrete spaces. A class of
boundary Hamiltonians is obtained in a natural way from regular tilings of the
hyperbolic Poincar\'e disk, via an inflation rule that allows to construct the
tiling using concentric layers of tiles. The models in this class are aperiodic
spin chains, whose sequences of couplings are obtained from the bulk inflation
rule. We explicitly choose the aperiodic XXZ spin chain with spin 1/2 degrees
of freedom as an example. The properties of this model are studied by using
strong disorder renormalization group techniques, which provide a tensor
network construction for the ground state of this spin chain. This can be
regarded as discrete bulk reconstruction. Moreover we compute the entanglement
entropy in this setup in two different ways: a discretization of the
Ryu-Takayanagi formula and a generalization of the standard computation for the
boundary aperiodic Hamiltonian. For both approaches, a logarithmic growth of
the entanglement entropy in the subsystem size is identified. The coefficients,
i.e. the effective central charges, depend on the bulk discretization
parameters in both cases, albeit in a different way.
