A dynamical system approach to Heisenberg Uniqueness Pairs

Abstract

Let $Λ$ be a set of lines in $\mathbb{R}^2$ that intersect at the origin. For $Γ\subset\mathbb{R}^2$ a smooth curve, we denote by $\mathcal{A}\mathcal{C}(Γ)$ the subset of finite measures on $Γ$ that are absolutely continuous with respect to arc length on $Γ$. For such a $μ$, $\widehatμ$ denotes the Fourier transform of $μ$. Following Hendenmalm and Montes-Rodríguez, we will say that $(Γ,Λ)$ is a Heisenberg Uniqueness Pair if $μ\in\mathcal{A}\mathcal{C}(Γ)$ is such that $\widehatμ=0$ on $Λ$, then $μ=0$. The aim of this paper is to provide new tools to establish this property. To do so, we will reformulate the fact that $\widehatμ$ vanishes on $Λ$ in terms of an invariance property of $μ$ induced by $Λ$. This leads us to a dynamical system on $Γ$ generated by $Λ$. The investigation of this dynamical system allows us to establish that $(Γ,Λ)$ is a Heisenberg Uniqueness Pair. This way we both unify proofs of known cases (circle, parabola, hyperbola) and obtain many new examples. This method also allows to have a better geometric intuition on why $(Γ,Λ)$ is a Heisenberg Uniqueness Pair.