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On automorphisms groups of structures of countable cofinality

In [2] Su Gao proves that the following are equivalent for a countable $M$ (cf. theorem 1.2 too): (I)There is an uncountable model of the Scott sentence of $M$. (II) There exists some $j\in \overline{Aut(M)}\setminus Aut(M)$, where $\overline{Aut(M)}$ is the closure of $Aut(M)$ under the product topology in $ω^ω$. (III) There is an $L_{ω_1,ω}$- elementary embedding $j$ from $M$ to itself such that $range(j)\subset M$. We generalize his theorem to all cardinals $κ$ of of cofinality $ω$ (cf. theorem 4.2). The following are equivalent: (I$^*$) There is a model of the Scott sentence of $M$ of size $κ^+$. (II$^*$) For all $α<β<κ^+$, there exist functions $j_{β,α}$ in $\overline{Aut(M)}^{T}\setminus Aut(M)$, such that for $α< β<γ<κ^+$, \begin{equation}(*) j_{γ,β}\circ j_{β,α}=j_{γ,α},\end{equation} where $\overline{Aut(M)}^{T}$ is the closure of $Aut(M)$ under the product topology in $κ^κ$. (III$^*$) For every $β<κ^+$, there exist $L_{\infty,κ}^{fin}$- elementary embeddings (cf. definition 2.5) $(j_α)_{α<β}$ from $M$ to itself such that $α_1<α_2\Rightarrow range(j_{α_1})\subset range(j_{α_2})$. Theorem 4.2 holds both for countable and uncountable $κ$. Condition (*) in (II$^*$), which does not appear in the countable case, can not be removed when $κ$ is uncountable (cf. theorem 4.5). Condition (II$^*$) imply the existence of at least $κ^ω$ automorphisms of $M$ (cf. corollary 4.6). It is unknown to the author whether a purely topological proof of corollary 4.6 exists.