Resonance between planar self-affine measures

Abstract

We show that if {ϕi}i∈Γ and {ψj}j∈Λ are self-affine iterated function systems on the plane that satisfy strong separation, domination and irreducibility, then for any associated self-affine measures µ and ν, the inequality dimH(µ ∗ ν) < min{2, dimH µ + dimH ν} implies that there is algebraic resonance between the eigenvalues of the linear parts of ϕi and ψj . This extends to planar non-conformal setting the existing analogous results for self-conformal measures on the line.