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

In this paper we study the evolution of the wave function with the system size in a locally periodic structure. In particular we analyse the dependence of the wave function with the number of unit cells, which also reflects information about its spatial behaviour in the system. We reduce the problem to a nonlinear map and find an equivalence of its energy regions of single periodicity and of weak chaos, with the forbidden and allowed bands of the fully periodic system, respectively. At finite size the wave function decays exponentially with system size, as well as in space, when the energy lies inside a region of single periodicity, while for energies in the weak chaotic region never decays. At the transition between those regions the wave function still decays but in a $q$-exponential form; we found that the decay length is a half of the mean free path, which is larger than the lattice constant.