$L^q$-spectra of box-like graph-directed self-affine measures: closed forms, with rotation

TL;DR

This paper derives closed-form formulas for the $L^q$-spectra and box dimensions of planar graph-directed self-affine measures generated by diagonal or anti-diagonal matrices, addressing a question from Fraser (2016).

Contribution

It provides a general closed-form expression for the $L^q$-spectra of such measures under certain conditions, including a solution to Fraser's 2016 question.

Findings

Derived closed-form expressions for $L^q$-spectra of self-affine measures.

Obtained formulas for box dimensions of associated self-affine sets.

Provided a precise answer to Fraser's 2016 question.

Abstract

We consider $L^q$-spectra of planar graph-directed self-affine measures generated by diagonal or anti-diagonal matrices. Assuming the directed graph is strongly connected and the system satisfies the rectangular open set condition, we obtain a general closed form expression for the $L^q$-spectra. Consequently, we obtain a closed form expression for box dimensions of associated planar graph-directed box-like self-affine sets. We also provide a precise answer to a question of Fraser in 2016 concerning the $L^q$-spectra of planar self-affine measures generated by diagonal matrices.