One of the key challenges in the dimension theory of smooth dynamical systems lies
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 nonconformal 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 says
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.
Keywords
dimension theory, box dimension, dynamical systems,
invariant set
