  We study the (H\"older-)continuous behavior of the spectra belonging to a
family of linear bounded operators $(A_t)_{t\in T}$ indexed by a topological
space $T$. For the cases of self-adjoint, unitary and normal operators, a
characterization of the continuity of $\Sigma:T\to \mathcal{K}(\mathbb{R}),
t\mapsto \sigma(A_t),$ is proven while the distance of the spectra is measured
by the Hausorff metric. If $T$ is a metric space, the H\"older-continuous
behavior of $\Sigma$ is characterized for self-adjoint and unitary operators.
Here we observe interesting effects, namely the rate of convergence is bisect
whenever spectral gaps closes.
  Based on this, we provide a tool to prove the continuity of the spectra for
large classes of operators. In particular, we apply this theory to generalized
Schr\"odinger operators and show that the continuity of the spectra is
characterized by the continuous variation of the underlying dynamical systems.
  Finally, we analyze the existence of periodic dynamical systems approximating
a given dynamical system. This leads to periodic approximations of the
corresponding Schr\"odinger operators by the previously developed theory. We
prove that local symmetries of the patterns and the presence of a substitution
is a sufficient criteria for periodic approximations of subshifts in
$\mathbb{Z}^d$. For $d=1$, a characterization is proven for the existence of
periodic approximations. For these approaches, the notion of a dictionary is
further developed and defined independently of a given configuration. We prove
that the set of dictionaries equipped with the local pattern topology is
homeomorphic to the space of subshifts. This yields a useful tool to analyze
these systems. Furthermore, it delivers the connection of the existence of
periodic orbits in a subshift of finite type and the existence of periodic
approximations for subshifts.
