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

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 posed by Fraser [On the $L^q$ -spectrum of planar self-affine measures. Trans. Amer. Math. Soc. 368 (8) (2016), 5579–5620] concerning the $L^q$ -spectra of planar self-affine measures generated by diagonal matrices. An interesting observation of the closed form expression is that it is possible to calculate the $L^q$ -spectrum of a measure without involving the exact $L^q$ -spectra of its projections to the axes.