We establish the following model-theoretic characterization of the fragment
$I\Delta_{0}+\mathop{\rm Exp}+B\Sigma_{1}$ of Peano arithmetic in terms of fixed
points of automorphisms of models of bounded arithmetic (the fragment
$I\Delta_{0}$ of Peano arithmetic with induction limited to
$\Delta_{0}$-formulae).Theorem A.
The following two conditions are
equivalent for a countable model $\mathfrak{M}$ of the language of
arithmetic:
(a) $\mathfrak{M}$ satisfies $I\Delta_{0}+B\Sigma
_{1}+\mathop{\rm Exp}$;(b) $\mathfrak{M}=I_{{\rm fix}}(j)$ for some nontrivial
automorphism $j$ of an end extension $\mathfrak{N}$ of
$\mathfrak{M}$ that satisfies $I\Delta_{0}.$
Here $I_{{\rm fix}}(j)$ is the largest initial segment of the domain of
$j$ that is pointwise fixed by $j$, $\mathop{\rm Exp}$ is the axiom asserting the totality
of the exponential function, and $B\Sigma_{1}$ is the
$\Sigma_{1}$-collection scheme consisting of the universal closure of formulae of the
form
$$
[\forall x a\ \exists y\ \varphi (x,y)]\rightarrow [\exists
z\ \forall x a\ \exists y z\ \varphi (x,y)],
$$
where $\varphi $ is a $\Delta_{0}$-formula. Theorem A was
inspired by a theorem of Smoryński, but the method of proof of Theorem A
is quite different and yields the following strengthening of Smoryński's
result:
Theorem B. Suppose $\mathfrak{M}$ is a
countable recursively saturated model of\/ {\rm PA} and $I$ is
a proper initial segment of $\mathfrak{M}$ that is closed under
exponentiation. There is a group embedding $j\mapsto \widehat{j}$
from $\mathop{\rm Aut}\nolimits(\mathbb{Q})$ into $\mathop{\rm Aut}\nolimits(\mathfrak{M})$ such
that $I=I_{{\rm fix}}(\widehat{j})$ for every nontrivial $j\in \mathop{\rm Aut}\nolimits(\mathbb{Q}).$
Moreover, if $j$ is fixed point free, then the fixed
point set of $\widehat{j}$ is isomorphic to $\mathfrak{M}$.
Here $\mathop{\rm Aut}\nolimits(X)$ is the group of automorphisms of the structure $X$,
and $\mathbb{Q}$ is the ordered set of rationals.
