  Dynamical systems generated by $d\ge2$ commuting homeomorphisms (topological
$\mathbb{Z}^d$-actions) contain within them structures on many scales, and in
particular contain many actions of $\mathbb{Z}^k$ for $1\le k\le d$. Familiar
dynamical invariants for homeomorphisms, like entropy and periodic point data,
become more complex and permit multiple definitions. We briefly survey some of
these and other related invariants in the setting of algebraic
$\mathbb{Z}^d$-actions, showing how, even in settings where the natural entropy
as a $\mathbb{Z}^d$-action vanishes, a powerful theory of directional entropy
and periodic points can be built. An underlying theme is uniformity in
dynamical invariants as the direction changes, and the connection between this
theory and problems in number theory; we explore this for several invariants.
We also highlight Fried's notion of average entropy and its connection to
uniformities in growth properties, and prove a new relationship between this
entropy and periodic point growth in this setting.
