On the L^q Dimensions of Measures on Hueter-Lalley Type Self-Affine Sets

TL;DR

This paper investigates the L^q-dimensions of self-affine measures on certain sets, providing conditions under which these dimensions match theoretical predictions, thereby enabling explicit Hausdorff dimension calculations.

Contribution

It introduces checkable conditions for L^q-dimensions to align with Falconer's predictions on a broad class of self-affine sets.

Findings

L^q-dimensions equal Falconer's predictions under new conditions

Extended class of self-affine sets with explicitly calculable Hausdorff dimension

Combined potential theoretic and dynamical methods for analysis

Abstract

We study the L^q -dimensions of self-affine measures and the Kaenmaki measure on a class of self-affine sets similar to the class considered by Hueter and Lalley. We give simple, checkable conditions under which the Lq -dimensions are equal to the value predicted by Falconer for a range of q. As a corollary this gives a wider class of self-affine sets for which the Hausdorff dimension can be explicitly calculated. Our proof combines the potential theoretic approach developed by Hunt and Kaloshin with recent advances in the dynamics of self-affine sets.