Resonance between planar self-affine measures

TL;DR

This paper demonstrates that in the plane, self-affine measures exhibit algebraic resonance when the Hausdorff dimension of their convolution is strictly less than the sum of their individual dimensions, extending known results from one-dimensional conformal cases.

Contribution

It establishes a link between dimension drop in convolutions of self-affine measures and algebraic resonance of eigenvalues in a planar non-conformal setting.

Findings

Dimension drop implies algebraic resonance of eigenvalues.

Extends one-dimensional results to planar non-conformal measures.

Shows resonance condition under strong separation, domination, and irreducibility.

Abstract

We show that if $\lbrace \varphi_i\rbrace_{i\in \Gamma}$ and $\lbrace \psi_j\rbrace_{j\in\Lambda}$ 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 $$\dim_{\rm H}(\mu*\nu) < \min \lbrace 2, \dim_{\rm H} \mu + \dim_{\rm H} \nu \rbrace$$ implies that there is algebraic resonance between the eigenvalues of the linear parts of $\varphi_i$ and $\psi_j$. This extends to planar non-conformal setting the existing analogous results for self-conformal measures on the line.