Multifractal analysis of Birkhoff averages on "self-affine" symbolic spaces

TL;DR

This paper conducts a multifractal analysis of Birkhoff averages on self-affine symbolic spaces, revealing complex spectra that differ from self-similar cases, with implications for understanding self-affine fractals.

Contribution

It introduces a novel multifractal analysis framework for self-affine carpets, showing that their spectra are derived from distinct functions, unlike the self-similar case.

Findings

Hausdorff spectra are Legendre transforms of two different functions

Spectra cannot be obtained by simple transformations from Gibbs measures

Results extend multifractal analysis to self-affine symbolic spaces

Abstract

We achieve on self-affine Sierpinski carpets the multifractal analysis of the Birkhoff averages of potentials satisfying a Dini condition. Given such a potential, the corresponding Hausdorff spectrum cannot be deduced from that of the associated Gibbs measure by a simple transformation. Indeed, these spectra are respectively obtained as the Legendre transform of two distinct concave differentiable functions that cannot be deduced from one another by a dilation and a translation. This situation is in contrast with what is observed in the familiar self-similar case. Our results are presented in the framework of almost-multiplicative functions on products of two distinct symbolic spaces and their projection on the associated self-affine carpets.