# Perron-Frobenius operators and the Klein-Gordon equation

• Volume: 016, Issue: 1, page 31-66
• ISSN: 1435-9855

## Abstract

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For a smooth curve $\Gamma$ and a set $\Lambda$ in the plane ${ℝ}^{2}$, let $AC\left(\Gamma ;\Lambda \right)$ 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 , we say that $\left(\Gamma ,\Lambda \right)$ is a Heisenberg uniqueness pair if $AC\left(\Gamma ;\Lambda \right)=\left\{0\right\}$. 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 $AC\left(\Gamma ;\Lambda \right)$ when it is nonzero. 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 ${\Lambda }_{\alpha ,\beta }=\left(\alpha ℤ×\left\{0\right\}\right)\cup \left(\left\{0\right\}×\beta ℤ\right)$, where $\alpha ,\beta$ are positive reals. We will also consider ${\Gamma }_{+}$, the branch of ${x}_{1}{x}_{2}=1$ where ${x}_{1}>0$. In , it is shown that $AC\left(\Gamma ;{\Lambda }_{\alpha ,\beta }\right)=\left\{0\right\}$ if and only if $\alpha \beta \le 1$. Here, we show that for $\alpha \beta >1$, we get a rather drastic “phase transition”: $AC\left(\Gamma ;{\Lambda }_{\alpha ,\beta }\right)$ is infinite-dimensional whenever $\alpha \beta >1$. It is shown in  that $AC\left({\Gamma }_{+};{\Lambda }_{\alpha ,\beta }\right)=\left\{0\right\}$ if and only if $\alpha \beta <4$. Moreover, at the edge $\alpha \beta =4$, the behavior is more exotic: the space $AC\left({\Gamma }_{+};{\Lambda }_{\alpha ,\beta }\right)$ is one-dimensional. Here, we show that the dimension of $AC\left({\Gamma }_{+};{\Lambda }_{\alpha ,\beta }\right)$ is infinite whenever $\alpha \beta >4$. Dynamical systems, and more specifically Perron-Frobenius operators, will play a prominent role in the presentation.

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