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$L^q$-spectra of box-like graph-directed self-affine measures: closed forms, with rotation
- Part of
-
- Journal:
- Ergodic Theory and Dynamical Systems , First View
- Published online by Cambridge University Press:
- 26 January 2026, pp. 1-58
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- Article
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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.
