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 $x_1x_2=1$, 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 $x_1x_2=1$ where $x_1>0$. In \cite{hh}, it is shown that $\mathrm{AC}(\Gamma;\Lambda_{\alpha,\beta})=\{0\}$ if and only if $\alpha\beta\le1$. 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.