Collusions in Teichm\"uller expansions

If $\mathfrak{p} \subseteq \mathbb{Z}[\zeta]$ is a prime ideal over $p$ in the $(p^d - 1)$th cyclotomic extension of $\mathbb{Z}$, then every element $\alpha$ of the completion $\mathbb{Z}[\zeta]_\mathfrak{p}$ has a unique expansion as a power series in $p$ with coefficients in $\mu_{p^d -1} \cup \{0\}$ called the Teichm\"uller expansion of $\alpha$ at $\mathfrak{p}$. We observe three peculiar and seemingly unrelated patterns that frequently appear in the computation of Teichm\"uller expansions, then develop a unifying theory to explain these patterns in terms of the dynamics of an affine group action on $\mathbb{Z}[\zeta]$.