  We establish a connection between the structure of a stationary symmetric
alpha-stable random field (0 < alpha < 2) and ergodic theory of non-singular
group actions, elaborating on a previous work by Rosinski (2000). With the help
of this connection, we study the extreme values of the field over increasing
boxes. Depending on the ergodic theoretical and group theoretical structures of
the underlying action, we observe different kinds of asymptotic behavior of
this sequence of extreme values.
