  We consider dynamical systems on a finite measure space fulfilling a spectral
gap property and Birkhoff sums of a non-negative, non-integrable observable.
For such systems we generalize strong laws of large numbers for intermediately
trimmed sums only known for independent random variables. The results split up
in trimming statements for general distribution functions and for regularly
varying tail distributions. In both cases the trimming rate can be chosen in
the same or almost the same way as in the i.i.d. case. As an example we show
that piecewise expanding interval maps fulfill the necessary conditions for our
limit laws. As a side result we obtain strong laws of large numbers for
truncated Birkhoff sums.
