Fourier duality for fractal measures with affine scales

Abstract:

For a family of fractal measures, we find an explicit Fourier duality. The measures in the pair have compact support in $\mathbb {R}^d$, and they both have the same matrix scaling; but the two use different translation vectors, one by a subset $B$ in $\mathbb {R}^d$, and the other by a related subset $L$. Among other things, we show that there is then a pair of infinite discrete sets $\Gamma (L)$ and $\Gamma (B)$ in $\mathbb {R}^d$ such that the $\Gamma (L)$-Fourier exponentials are orthogonal in $L^2(\mu _B)$, and the $\Gamma (B)$-Fourier exponentials are orthogonal in $L^2(\mu _L)$. These sets of orthogonal “frequencies” are typically lacunary, and they will be obtained by scaling in the large. The nature of our duality is explored below both in higher dimensions and for examples on the real line.

Our duality pairs do not always yield orthonormal Fourier bases in the respective $L^2(\mu )$-Hilbert spaces, but depending on the geometry of certain finite orbits, we show that they do in some cases. We further show that there are new and surprising scaling symmetries of relevance for the ergodic theory of these affine fractal measures.