The packing spectrum for Birkhoff averages on a self-affine repeller

TL;DR

This paper analyzes the multifractal structure of Birkhoff averages on a self-affine Sierpiński sponge, establishing a variational principle for the packing dimension of level sets and exploring the spectrum's properties.

Contribution

It provides a variational principle for the packing dimension, characterizes the spectrum's concavity and continuity, and offers conditions for analyticity and full dimension attainment.

Findings

Packing spectrum is concave and continuous.

A variational principle for the packing dimension is established.

Conditions for the spectrum to be real analytic and to attain full dimension are provided.

Abstract

We consider the multifractal analysis for Birkhoff averages of continuous potentials on a self-affine Sierpi\'{n}ski sponge. In particular, we give a variational principal for the packing dimension of the level sets. Furthermore, we prove that the packing spectrum is concave and continuous. We give a sufficient condition for the packing spectrum to be real analytic, but show that for general H\"{o}lder continuous potentials, this need not be the case. We also give a precise criterion for when the packing spectrum attains the full packing dimension of the repeller. Again, we present an example showing that this is not always the case.