A monic polynomial $f\left(x\right)\in \mathrm{\mathbb{Z}}\left[x\right]$ is called stable if $f^n\left(x\right)$ is irreducible over $\mathrm{\mathbb{Q}}$ for all $n\ge 1$, where $f^n\left(x\right)$ denotes the $n$th iterate of $f\left(x\right)$. Regardless of whether $f\left(x\right)$ is irreducible over $\mathrm{\mathbb{Q}}$, if there exists some monic $g\left(x\right)\in \mathrm{\mathbb{Z}}\left[x\right]$ such that $g\left(f^n\right(x\left)\right)$ is irreducible over $\mathrm{\mathbb{Q}}$ for all $n\ge 1$, we say that $f\left(x\right)$ is $g$-stable. Many authors have studied such polynomials since Odoni first introduced this concept of stability in 1985. We extend these concepts here by adding the additional restriction of monogeneity. A monic polynomial $f\left(x\right)\in \mathrm{\mathbb{Z}}\left[x\right]$ is defined to be monogenic if $f\left(x\right)$ is irreducible over $\mathrm{\mathbb{Q}}$ and $\left\{1,\theta ,{\theta }^2,\dots ,{\theta }^{\mathrm{deg}\left(f\right)-1}\right\}$ is a basis for the ring of integers of $\mathrm{\mathbb{Q}}\left(\theta \right)$, where $f\left(\theta \right)=0$. We say that $f\left(x\right)$ is $g$-monogenically stable, if $g\left(f^n\right(x\left)\right)$ is monogenic for all $n\ge 1$, for some monic $g\left(x\right)\in \mathrm{\mathbb{Z}}\left[x\right]$. When $g\left(x\right)=x$, we simply say that $f\left(x\right)$ is monogenically stable. In this article, we provide methods for constructing $g$-monogenically stable polynomials $f\left(x\right)$, for various polynomials $f\left(x\right)$ and $g\left(x\right)$.

Double coset spaces of adelic points on linear algebraic groups arise in the study of global fields; e.g., concerning local-global principles and torsors. A different type of double coset space plays a role in the study of semi-global fields such as $p$-adic function fields. This paper relates the two, by establishing adelic double coset spaces over semi-global fields; relating them to local-global principles and torsors; and providing explicit examples.