  One of the key challenges in the dimension theory of smooth dynamical systems
is in establishing whether or not the Hausdorff, lower and upper box dimensions
coincide for invariant sets. For sets invariant under conformal dynamics, these
three dimensions always coincide. On the other hand, considerable attention has
been given to examples of sets invariant under non-conformal dynamics whose
Hausdorff and box dimensions do not coincide. These constructions exploit the
fact that the Hausdorff and box dimensions quantify size in fundamentally
different ways, the former in terms of covers by sets of varying diameters and
the latter in terms of covers by sets of fixed diameters. In this article we
construct the first example of a dynamically invariant set with distinct lower
and upper box dimensions. Heuristically, this describes that if size is
quantified in terms of covers by sets of equal diameters, a dynamically
invariant set can appear bigger when viewed at certain resolutions than at
others.
