# Perron-Frobenius operators and the Klein-Gordon equation

### Francisco Canto-Martín

Universidad de Sevilla, Spain### Haakan Hedenmalm

Royal Institute of Technology, Stockholm, Sweden### Alfonso Montes-Rodríguez

Universidad de Sevilla, Spain

## Abstract

For a smooth curve $Γ$ and a set $Λ$ in the plane $R_{2}$, 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 non-zero. We will fix the curve $Γ$ to be the hyperbola $x_{1}x_{2}=1$, and the set $Λ=Λ_{α,β}$ to be the lattice-cross

where $α,β$ are positive reals. We will also consider $Γ_{+}$, the branch of $x_{1}x_{2}=1$ where $x_{1}>0$. In [12], it is shown that $AC(Γ;Λ_{α,β})={0}$ if and only if $αβ≤1$. Here, we show that for $αβ>1$, we get a rather drastic “phase transition”: $AC(Γ;Λ_{α,β})$ is infinite-dimensional whenever $αβ>1$. It is shown in [13] that $AC(Γ_{+};Λ_{α,β})={0}$ if and only if $αβ<4$. Moreover, at the edge $αβ=4$, the behavior is more exotic: the space $AC(Γ_{+};Λ_{α,β})$ is one-dimensional. Here, we show that the dimension of $AC(Γ_{+};Λ_{α,β})$ is infinite whenever $αβ>4$. Dynamical systems, and more specifically Perron–Frobenius operators, will play a prominent role in the presentation.

## Cite this article

Francisco Canto-Martín, Haakan Hedenmalm, Alfonso Montes-Rodríguez, Perron-Frobenius operators and the Klein-Gordon equation. J. Eur. Math. Soc. 16 (2014), no. 1, pp. 31–66

DOI 10.4171/JEMS/427