Complexity of conjugacy classes of \(A(\mathbb Q)\) (Q1873303)

Let \(\text{Sym}(\mathbb Q)\) be the group of permutations and \(A(\mathbb Q)\) the group of order preserving transformations of the set of rational numbers. The notion of \textit{H. Becker} [J. Am. Math. Soc. 11, 397--449 (1998; Zbl 0894.03027)] of \(\iota\)-embeddability of an element \(f\) to an element \(g\) of a Polish space \(X\) defined in terms of a Polish group \(P\) acting on \(X\) is applied to the conjugacy actions acting on the groups \(\text{Sym}(\mathbb Q)\) and \(A(\mathbb Q)\). The conjugacy class of \(\sigma\) can be separated from a conjugacy class \(\rho\) by a \(G_{\delta}\) set in \(\text{Sym} (\mathbb Q)\) if and only if \(\sigma\) is not \(\iota\)-embeddable into \(\sigma\). The notions of \(\iota\)-embeddability in \(A(\mathbb Q)\) and of \(G_{\delta}\) separability in \(A(\mathbb Q)\) are studied and a conclusion is that there are \(f,g\in A(\mathbb Q)\) such that \(f\) is not \(\iota\)-embedded into \(g\) and still the conjugacy class of \(f\) is not \(G_{\delta}\) separated from the conjugacy class of \(g\) in \(A(\mathbb Q)\). The property that the conjugacy class of \(f\) is \(G_{\delta}\) in \(A(\mathbb Q)\) is characterized in terms of the family of orbitals of \(f\), i.e., the family of the sets of the form \(\{q\in\mathbb Q: (\exists m,n\in\mathbb Z)(f^n(x)\leq q \leq f^m(x))\}\).