  Hamiltonian tridiagonal matrices characterized by multi-fractal spectral
measures in the family of Iterated Function Systems can be constructed by a
recursive technique here described. We prove that these Hamiltonians are
almost-periodic. They are suited to describe quantum lattice systems with
nearest neighbours coupling, as well as chains of linear classical oscillators,
and electrical transmission lines.
  We investigate numerically and theoretically the time dynamics of the systems
so constructed. We derive a relation linking the long-time, power-law behaviour
of the moments of the position operator, expressed by a scaling function
$\beta$ of the moment order $\alpha$, and spectral multi-fractal dimensions,
D_q, via $\beta(\alpha) = D_{1-\alpha}$. We show cases in which this relation
is exact, and cases where it is only approximate, unveiling the reasons for the
discrepancies.
