A dynamical system approach to Heisenberg Uniqueness Pairs

Let $\Lambda$ be a set of lines in $\mathbb{R}^2$ that intersect at the origin. For $\Gamma\subset\mathbb{R}^2$ a smooth curve, we denote by $\mathcal{A}\mathcal{C}(\Gamma)$ the subset of finite measures on $\Gamma$ that are absolutely continuous with respect to arc length on $\Gamma$. For such a $\mu$, $\widehat{\mu}$ denotes the Fourier transform of $\mu$. Following Hendenmalm and Montes-Rodr\'iguez, we will say that $(\Gamma,\Lambda)$ is a Heisenberg Uniqueness Pair if $\mu\in\mathcal{A}\mathcal{C}(\Gamma)$ is such that $\widehat{\mu}=0$ on $\Lambda$, then $\mu=0$. The aim of this paper is to provide new tools to establish this property. To do so, we will reformulate the fact that $\widehat{\mu}$ vanishes on $\Lambda$ in terms of an invariance property of $\mu$ induced by $\Lambda$. This leads us to a dynamical system on $\Gamma$ generated by $\Lambda$. The investigation of this dynamical system allows us to establish that $(\Gamma,\Lambda)$ 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 $(\Gamma,\Lambda)$ is a Heisenberg Uniqueness Pair.