Explicit formulae for limit periodic flows on networks

In recent papers it was shown that, under certain conditions, the C0-semigroup describing a flow on a network (metric graph) that contains terminal strong (ergodic) components converges to the direct sum of periodic semigroups generated by the flows on these components. In this note we shall provide an explicit description of these limit semigroups in terms of the components of the adjacency matrix of the line graph of the network. The result is based on the Frobenius-Perron theory and the estimates of long term behaviour of iterates of reducible matrices.