A dynamical system approach to Heisenberg uniqueness pairs

• Published in: Journal d'analyse mathématique (Jerusalem) 2018-02, Vol.134 (1), p.273-301
• Main Authors: Jaming, Philippe, Kellay, Karim
• Format: Article
• Language: English
• Subjects: Abstract Harmonic Analysis; Analysis; Classical Analysis and ODEs; Dynamical systems; Dynamical Systems and Ergodic Theory; Fourier transforms; Functional Analysis; Mathematics; Mathematics and Statistics; Partial Differential Equations; Set theory; Uniqueness; Wold theorem
• ISSN: 0021-7670; 1565-8538
• DOI: 10.1007/s11854-018-0010-6

Let Λ be a set of lines in R 2 that intersect at the origin. For a smooth curve Γ ⊂ R 2 , we denote by AC (Γ) the subset of finite measures on Γ that are absolutely continuous with respect to arc length on Γ. For μ ∈ AC (Γ), μ ^ denotes the Fourier transform of μ . Following Hedenmalm and Montes-Rodríguez, we say that (Γ,Λ) is a Heisenberg uniqueness pair if μ ∈ AC (Γ) is such that μ ^ = 0 on Λ implies μ = 0. The aim of this paper is to provide new tools to establish this property. To do so, we reformulate the fact that μ ^ 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. In many cases, 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 gives a better geometric intuition as to why (Γ,Λ) is a Heisenberg uniqueness pair. As a side result, we also give the first instance of a positive result in the classical Cramér-Wold theorem where finitely many projections suffice to characterize a measure (under strong support constraints).