2013/12/21 by Jaming, Philippe, Kellay, Karim
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.1312.6236
Let Λ be a set of lines in ℝ2 that intersect at the origin. For Γ⊂ℝ2 a smooth curve, we denote by AC(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\inAC(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.