On dissonance of self-conformal measures in ℝ^d

2025/11/18 by Pyörälä, Aleksi
Mathematics · #Mathematical Dynamics and Fractals #Geometry and complex manifolds #Spectral Theory in Mathematical Physics

paper · doi:10.48550/arxiv.2511.14493

Abstract

Let μ be a self-conformal measure on ℝ^d. In this note we establish conditions for μ under which dim(μ*ν) = min\lbrace d,dimμ+dimν\rbrace holds when ν is any Ahlfors-regular or self-conformal measure on ℝ^d. Our main result states the following sufficient condition: μ is totally non-linear and not supported on a smooth hypersurface. We also establish sufficient (likely non-sharp) algebraic conditions for self-conformal measures which are not totally non-linear. The proofs combine recent results on scaling sceneries of self-conformal measures with a Marstrand-type projection theorem for product sets due to López and Moreira.

Citations

• L²-Flattening of Self-similar Measures on Non-degenerate Curves
• Spectral gaps and Fourier decay for self-conformal measures in the plane
• Van der Corput and metric theorems for geometric progressions for self-similar measures
• Scaling limits of self-conformal measures
• Spectral gaps and Fourier dimension for self-conformal sets with overlaps
• Polynomial Fourier decay and a cocycle version of Dolgopyat's method for self conformal measures
• Resonance between planar self-affine measures
• On measures that improve L^q dimension under convolution
• The Zero Set of a Real Analytic Function
• A generalization of Marstrand's theorem for projections of cartesian products
• Dynamics on fractals and fractal distributions

Related

• Pointwise normality and Fourier decay for self-conformal measures
• Scaling limits of self-conformal measures
• Quantitative recurrence properties for self-conformal sets
• Logarithmic Fourier decay for self conformal measures
• A dichotomy of self-conformal subsets of the real line with overlaps