Collusions in Teichmuller 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 Teichmuller expansion of $alpha$ at $mathfrak{p}$. We observe three peculiar and seemingly unrelated patterns that frequently appear in the computation of Teichmuller expansions, then develop a unifying theory to explain these patterns in terms of the dynamics of an affine group action on $mathbb{Z}[zeta]$.