Packing spectra for Bernoulli measures supported on Bedford-McMullen carpets

TL;DR

This paper investigates the packing spectra of Bernoulli measures on Bedford-McMullen carpets, revealing typical dimension disparities, exact calculations for specific measures, and discontinuities in the spectrum.

Contribution

It provides the first explicit calculations of packing spectra for these measures and demonstrates the typical and special behaviors of their dimensions.

Findings

Typically, the packing dimension of the regular set is smaller than that of the attractor.

Exact packing spectrum calculations are possible for certain Bernoulli measures.

The packing spectrum exhibits discontinuities as a function on the measure space.

Abstract

In this paper we consider the packing spectra for local dimension of Bernoulli measures supported on Bedford-McMullen carpets. We show that typically the packing dimension of the regular set is smaller than the packing dimension of the attractor. We also consider a specific class of measures for which we are able to calculate the packing spectrum exactly and we show that the packing spectrum is discontinuous as a function on the space of Bernoulli measures.