More refined multi-fractal spectra for Bedford-McMullen carpets

Abstract

For a self-affine measure μ on a Bedford McMullen carpet, the multifractal spectrum is the map
f(α) = dimH Xα,
where Xα = {x ∈ X : locμ(x) = α} and locμ(x) is the local dimension at x. The spectrum was computed by King for carpets with the very strong separation condition, and this was later generalised to all carpets by Jordan and Rams. This spectrum coincides exactly with the spectrum obtained where locμ(x) is replaced by a symbolic version of local dimension.
We define refined level sets that take into account points for which the (symbolic) local dimension does not necessarily exist by considering upper and lower symbolic local dimension level sets. When the refined spectra are computed for self-similar sets satisfying the open set condition, they all coincide with the ordinary spectrum. For carpets on a n×2 grid, and the measure that assigns mass uniformly on rectangles, we exhibit different behaviour from that seen in the self-similar setting. In particular there are values of α for which the refined spectra are strictly greater than f(α), and we compute the spectra for all values of α.

Date of Award	3 Oct 2023
Original language	English
Awarding Institution	University of Bristol
Supervisor	Thomas M Jordan (Supervisor)