  Mean dimension is a topological invariant for dynamical systems that is
meaningful for systems with infinite dimension and infinite entropy. Given a
$\mathbb{Z}^k$-action on a compact metric space $X$, we study the following
three problems closely related to mean dimension.
  (1) When is $X$ isomorphic to the inverse limit of finite entropy systems?
  (2) Suppose the topological entropy $h_{\mathrm{top}}(X)$ is infinite. How
much topological entropy can be detected if one considers $X$ only up to a
given level of accuracy? How fast does this amount of entropy grow as the level
of resolution becomes finer and finer?
  (3) When can we embed $X$ into the $\mathbb{Z}^k$-shift on the infinite
dimensional cube $([0,1]^D)^{\mathbb{Z}^k}$?
  These were investigated for $\mathbb{Z}$-actions in [Lindenstrauss, Mean
dimension, small entropy factors and an embedding theorem, Inst. Hautes
\'Etudes Sci. Publ. Math. \textbf{89} (1999) 227-262], but the generalization
to $\mathbb{Z}^k$ remained an open problem. When $X$ has the marker property,
in particular when $X$ has a completely aperiodic minimal factor, we completely
solve (1) and a natural interpretation of (2), and give a reasonably
satisfactory answer to (3).
  A key ingredient is a new method to continuously partition every orbit into
good pieces.
