Non-existence of the box dimension for dynamically invariant sets

TL;DR

This paper demonstrates the first example of a dynamically invariant set with different lower and upper box dimensions, highlighting that box dimensions can vary with resolution in non-conformal dynamical systems.

Contribution

It constructs the first known example of an invariant set where lower and upper box dimensions differ, revealing new complexity in the dimension theory of dynamical systems.

Findings

First example of invariant set with distinct lower and upper box dimensions

Box dimensions can vary with resolution in non-conformal dynamics

Highlights differences between Hausdorff and box dimensions in invariant sets

Abstract

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.