  Periodic approximations of quasicrystals are a powerful tool in analyzing
spectra of Schr\"odinger operators arising from quasicrystals, given the known
theory for periodic crystals. Namely, we seek periodic operators whose spectra
approximate the spectrum of the limiting operator (of the quasicrystal). This
naturally leads to study the convergence of the underlying dynamical systems.
We treat dynamical systems which are based on one-dimensional substitutions. We
first find natural candidates of dynamical subsystems to approximate the
substitution dynamical system. Subsequently, we offer a characterization of
their convergence and provide estimates for the rate of convergence. We apply
the proposed theory to some guiding examples.
