Self-embeddings of models of arithmetic; fixed points, small submodels, and extendability

In this paper we will show that for every cut $I$ of any countable nonstandard model $\mathcal{M}$ of $\mathrm{I}\Sigma_{1}$, each $I$-small $\Sigma_{1}$-elementary submodel of $\mathcal{M}$ is of the form of the set of fixed points of some proper initial self-embedding of $\mathcal{M}$ iff $I$ is a strong cut of $\mathcal{M}$. Especially, this feature will provide us with some equivalent conditions with the strongness of the standard cut in a given countable model $\mathcal{M}$ of $\mathrm{I}\Sigma_{1}$. In addition, we will find some criteria for extendability of initial self-embeddings of countable nonstandard models of $\mathrm{I}\Sigma_{1}$ to larger models.