Resonance between planar self-affine measures

Abstract

We show that if $\{{\phi }_i{\}}_{i\in \Gamma }$ and $\{{\psi }_j{\}}_{j\in \Lambda }$ are self-affine iterated function systems on the plane that satisfy strong separation, domination and irreducibility, then for any associated self-affine measures $\mu$ and $\nu$, the inequality $${\dim }_{\mathrm{H}}(\mu \ast \nu )<\min \{2,{\dim }_{\mathrm{H}}\mu +{\dim }_{\mathrm{H}}\nu \}$$ implies that there is algebraic resonance between the eigenvalues of the linear parts of ${\phi }_i$ and ${\psi }_j$. This extends to planar non-conformal setting the existing analogous results for self-conformal measures on the line.

55 pages. Proof of Proposition 6.1 contained an error and its statement has been slightly modified. Main results unchanged

References in corpus (3)

• Ledrappier-Young formula and exact dimensionality of self-affine measures
• On the Ledrappier-Young formula for self-affine measures
• Self-affine sets with fibered tangents