D e d i c a t e d t o R a i n e r N a g e l o n t h e o c c a s i o n o f h i s 65 t h b i r t h d a y, w i t h s i n c e r e t h a n k s f o r h i s a d v i c e a n d s u p p o r t Abstract. Generalizing a recent result of E.B. Davies [4], we show that generators of bounded positive C 0 -semigroups on atomic Banach lattices with order continuous norm have trivial peripheral point spectrum. Moreover, we give examples that the peripheral spectrum can be any closed cyclic subset of iR.

We consider a transport process on an infinite network and, using the corresponding flow semigroup as in Dorn (Semigroup Forum 76:341-356, 2008), investigate its long term behavior. Combining methods from functional analysis, graph theory and stochastics, we are able to characterize the networks for which the flow semigroup converges strongly to a periodic group.

We use the full range of the Perron-Frobenius-Schaefer spectral theory and some results from harmonic analysis in order to characterize the asymptotic behavior of positive irreducible C0-semigroups on Banach lattices.

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