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Dynamical system induced by quantum walk

We consider the Grover walk model on a connected finite graph with two infinite length tails and we set an $\ell^\infty$-infinite external source from one of the tails as the initial state. We show that for any connected internal graph, a stationary state exists, moreover a perfect transmission to the opposite tail always occurs in the long time limit. We also show that the lower bound of the norm of the stationary measure restricted to the internal graph is proportion to the number of edges of this graph. Furthermore when we add more tails (e.g., $r$-tails) to the internal graph, then we find that from the temporal and spatial global view point, the scattering to each tail in the long time limit coincides with the local one-step scattering manner of the Grover walk at a vertex whose degree is $(r+1)$.