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 inequalitydimH⁡(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.