This is the abstract of the paper "On the spectral mapping theorem for perturbed strongly continuous semigroups" by Simon Brendle and Rainer Nagel and Jan Poland. For the whole paper send the command get evolve-l 98-00045 to: "listserv@rz.uni-karlsruhe.de". Instead you may get the paper at "http://ma1serv.mathematik.uni-karlsruhe.de/evolve-l/index.html" too. Abstract: We consider a strongly continuous semigroup $(T(t))_{t \geq 0}$ with generator $A$ on a Banach space $X$, an $A$-bounded perturbation $B$, and the semigroup $(S(t))_{t \geq 0}$ generated by $A+B$. Using the critical spectrum introduced recently, we improve existing spectral mapping theorems for the perturbed semigroup $(S(t))_{t \geq 0}$. The results are applied to a cell equation with age structure and spatial distribution.