  We construct a well-behaved transfer map from the p-local Burnside ring of
the underlying p-group S to the p-local Burnside ring of a saturated fusion
system F. Using this transfer map, we give new results on the characteristic
idempotent of F -- the unique idempotent in the p-local double Burnside ring of
S satisfying properties of Linckelmann and Webb. We describe this idempotent
explicitly both in terms of fixed points and as a linear combination of
transitive bisets. Additionally, using fixed points we determine the map for
Burnside rings given by multiplication with the characteristic idempotent, and
show that this is the transfer map previously constructed. Applying these
results, we show that for every saturated fusion system the ring generated by
all (not necessarily idempotent) characteristic elements in the p-local double
Burnside ring is isomorphic as rings to the p-local "single" Burnside ring of
the fusion system, and we disprove a conjecture by Park-Ragnarsson-Stancu on
the composition product of fusion systems.
