  Symmetries are known to dictate important physical properties and can be used
as a design principle in particular in wave physics, including wave structures
and the resulting propagation dynamics. Local symmetries, in the sense of a
symmetry that holds only in a finite domain of space, can be either the result
of a self-organization process or a structural ingredient into a synthetically
prepared physical system. Applying local symmetry operations to extend a given
finite chain we show that the resulting one-dimensional lattice consists of a
transient followed by a subsequent periodic behaviour. Due to the fact that, by
construction, the implanted local symmetries strongly overlap the resulting
lattice possesses a dense skeleton of such symmetries. We proof this behaviour
on the basis of a class of local symmetry operations allowing us to conclude
upon the 'asymptotic' properties such as the final period, decomposition of the
unit-cell and the length and decomposition of the transient. As an example
case, we explore the corresponding tight-binding Hamiltonians. Their energy
eigenvalue spectra and eigenstates are analyzed in some detail, showing in
particular the strong variability of the localization properties of the
eigenstates due to the presence of a plethora of local symmetries.
