Perron-Frobenius operators and the Klein-Gordon equation

Abstract

For a smooth curve $\Gamma$ and a set $\Lambda$ in the plane ${\mathbb{R}}^2$, let $\mathrm{AC}(\Gamma ;\Lambda )$ be the space of finite Borel measures in the plane supported on $\Gamma$, absolutely continuous with respect to the arc length and whose Fourier transform vanishes on $\Lambda$. Following \cite{hh}, we say that $(\Gamma ,\Lambda )$ is a Heisenberg uniqueness pair if $\mathrm{AC}(\Gamma ;\Lambda )=\{0\}$. In the context of a hyperbola $\Gamma$, the study of Heisenberg uniqueness pairs is the same as looking for uniqueness sets $\Lambda$ of a collection of solutions to the Klein-Gordon equation. In this work, we mainly address the issue of finding the dimension of $\mathrm{AC}(\Gamma ;\Lambda )$ when it is non\-zero. We will fix the curve $\Gamma$ to be the hyperbola $x_1{x}_2=1$, and the set $\Lambda ={\Lambda }_{\alpha ,\beta }$ to be the lattice-cross

where $\alpha ,\beta$ are positive reals. We will also consider ${\Gamma }_{+}$, the branch of $x_1{x}_2=1$ where $x_1>0$. In \cite{hh}, it is shown that $\mathrm{AC}(\Gamma ;{\Lambda }_{\alpha ,\beta })=\{0\}$ if and only if $\alpha \beta \le 1$. Here, we show that for $\alpha \beta >1$, we get a rather drastic ``phase transition'': $\mathrm{AC}(\Gamma ;{\Lambda }_{\alpha ,\beta })$ is infinite-dimensional whenever $\alpha \beta >1$. It is shown in \cite{HM2} that $\mathrm{AC}({\Gamma }_{+};{\Lambda }_{\alpha ,\beta })=\{0\}$ if and only if $\alpha \beta <4$. Moreover, at the edge $\alpha \beta =4$, the behavior is more exotic: the space $\mathrm{AC}({\Gamma }_{+};{\Lambda }_{\alpha ,\beta })$ is one-dimensional. Here, we show that the dimension of $\mathrm{AC}({\Gamma }_{+};{\Lambda }_{\alpha ,\beta })$ is infinite whenever $\alpha \beta >4$. Dynamical systems, and more specifically Perron-Frobenius operators, will play a prominent role in the presentation.