  We investigate the structure of fixed point sets of self-embeddings of models
of arithmetic. In particular, given a countable nonstandard model M of a modest
fragment of Peano arithimetic, we provide complete characterizations of (a) the
initial segments of M that can be realized as the longest initial segment of
fixed points of a nontrivial self-embedding of M onto a proper initial segment
of M; and (b) the initial segments of M that can be realized as the fixed point
set of some nontrivial self-embedding of M onto a proper initial segement of M.
Moreover, we demonstrate the the standard cut is strong in M iff there is a
self-embedding of M onto a proper initial segment of itself that moves every
element that is not definable in M by an existential formula.
