On Falconer's formula for the generalised Renyi dimension of a self-affine measure

TL;DR

This paper examines Falconer's formula for the generalized Renyi dimension of self-affine measures, revealing its discontinuities and dependence on the linear parts of affine transformations, with implications for understanding fractal dimensions.

Contribution

It demonstrates that Falconer's formula for the generalized Renyi dimension can be discontinuous, contrasting with its behavior for Hausdorff dimension, and links this to spectral radius discontinuities.

Findings

Falconer's formula for Renyi dimension is discontinuous under linear changes.

Discontinuities occur on sets of positive Lebesgue measure under certain conditions.

Spectral radius discontinuities underpin the observed irregularities.

Abstract

We investigate a formula of K. Falconer which describes the typical value of the generalised R\'enyi dimension, or generalised $q$-dimension, of a self-affine measure in terms of the linear components of the affinities. We show that in contrast to a related formula for the Hausdorff dimension of a typical self-affine set, the value of the generalised $q$-dimension predicted by Falconer's formula varies discontinuously as the linear parts of the affinities are changed. Conditionally on a conjecture of J. Bochi and B. Fayad, we show that the value predicted by this formula for pairs of two-dimensional affine transformations is discontinuous on a set of positive Lebesgue measure. These discontinuities derive from discontinuities of the lower spectral radius which were previously observed by the author and J. Bochi.