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In this paper we study the evolution of the wave function with the systemsize in a locally periodic structure. In particular we analyse the dependenceof the wave function with the number of unit cells, which also reflectsinformation about its spatial behaviour in the system. We reduce the problem toa nonlinear map and find an equivalence of its energy regions of singleperiodicity and of weak chaos, with the forbidden and allowed bands of thefully periodic system, respectively. At finite size the wave function decaysexponentially with system size, as well as in space, when the energy liesinside a region of single periodicity, while for energies in the weak chaoticregion never decays. At the transition between those regions the wave functionstill decays but in a $q$-exponential form; we found that the decay length is ahalf of the mean free path, which is larger than the lattice constant.

Evolution with size in a locally periodic system: Scattering and deterministic maps