The existence and construction of periodic approximations with convergent
spectra is crucial in solid state physics for the spectral study of
corresponding Schr\"odinger operators. In a forthcoming work [9]
(arXiv:1709.00975) this task was boiled down to the existence and construction
of periodic approximations of the underlying dynamical systems in the Hausdorff
topology. As a result the one-dimensional systems admitting such approximations
are completely classified in the present work. In addition explicit
constructions are provided for dynamical systems defined by primitive
substitutions covering all studied examples such as the Fibonacci sequence or
the Golay-Rudin-Shapiro sequence. One main tool is the description of the
Hausdorff topology by the local pattern topology on the dictionaries as well as
the GAP-graphs describing the local structure. The connection of branching
vertices in the GAP-graphs and defects is discussed.Comment: 30 pages, 5 figure
