2016-09-202016-09-202014Canto Martín, F.M., Hedenmalm, H. y Montes Rodríguez, A. (2014). Perron-Frobenius operators and the Klein-Gordon equation. Journal of the European Mathematical Society, 16 (1), 31-66.1435-98551435-9863http://hdl.handle.net/11441/45142For a smooth curve Γ and a set Λ in the plane R2, let AC(Γ; Λ) be the space of finite Borel measures in the plane supported on Γ, absolutely continuous with respect to the arc length and whose Fourier transform vanishes on Λ. Following [12], we say that (Γ, Λ) is a Heisenberg uniqueness pair if AC(Γ; Λ) = {0}. In the context of a hyperbola Γ, the study of Heisenberg uniqueness pairs is the same as looking for uniqueness sets Λ of a collection of solutions to the Klein-Gordon equation. In this work, we mainly address the issue of finding the dimension of AC(Γ; Λ) when it is nonzero. We will fix the curve Γ to be the hyperbola x1x2 = 1, and the set Λ = Λα,β to be the lattice-cross Λα,β = (αZ × {0}) ∪ ({0} × βZ), where α, β are positive reals. We will also consider Γ+, the branch of x1x2 = 1 where x1 > 0. In [12], it is shown that AC(Γ; Λα,β) = {0} if and only if αβ ≤ 1. Here, we show that for αβ > 1, we get a rather drastic “phase transition”: AC(Γ; Λα,β) is infinite-dimensional whenever αβ > 1. It is shown in [13] that AC(Γ+; Λα,β) = {0} if and only if αβ < 4. Moreover, at the edge αβ = 4, the behavior is more exotic: the space AC(Γ+; Λα,β) is one-dimensional. Here, we show that the dimension of AC(Γ+; Λα,β) is infinite whenever αβ > 4. Dynamical systems, and more specifically Perron-Frobenius operators, will play a prominent role in the presentation.application/pdfengAttribution-NonCommercial-NoDerivatives 4.0 Internacionalhttp://creativecommons.org/licenses/by-nc-nd/4.0/Trigonometric systemInversionComposition operatorKlein-Gordon equationErgodic theoryinfo:eu-repo/semantics/articleinfo:eu-repo/semantics/openAccesshttps://doi.org/10.4171/JEMS/427