Limit Models in Strictly Stable Abstract Elementary Classes

Abstract: In this paper, we examine the locality condition for non-splitting and determine the level of uniqueness of limit models that can be recovered in some stable, but not superstable, abstract elementary classes. In particular we prove (note that no tameness is assumed): Suppose that $\mathcal{K}$ is an abstract elementary class satisfying 1. the joint embedding and amalgamation properties with no maximal model of cardinality $\mu$. 2. stability in $\mu$. 3. $\kappa^*_\mu(\mathcal{K})<\mu^+$. 4. continuity for non-$\mu$-splitting (i.e. if $p\in\text{ga-S}(M)$ and $M$ is a limit model witnessed by $\langle M_i\mid i<\alpha\rangle$ for some limit ordinal $\alpha<\mu^+$ and there exists $N \prec M_0$ so that $p\restriction M_i$ does not $\mu$-split over $N$ for all $i<\alpha$, then $p$ does not $\mu$-split over $N$). For $\theta$ and $\delta$ limit ordinals $<\mu^+$ both with cofinality $\geq\kappa^*_\mu(\mathcal{K})$, if $\mathcal{K}$ satisfies symmetry for non-$\mu$-splitting (or just $(\mu,\delta)$-symmetry), then, for any $M_1$ and $M_2$ that are $(\mu,\theta)$ and $(\mu,\delta)$-limit models over $M_0$, respectively, we have that $M_1$ and $M_2$ are isomorphic over $M_0$.