# Perron-Frobenius operators and the Klein-Gordon equation

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

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## 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 [12], 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 [12], 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 [13] 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.

## How to cite

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Canto-Martín, Francisco, Hedenmalm, Håkan, and Montes-Rodríguez, Alfonso. "Perron-Frobenius operators and the Klein-Gordon equation." Journal of the European Mathematical Society 016.1 (2014): 31-66. <http://eudml.org/doc/277383>.

@article{Canto2014,
abstract = {For a smooth curve $\Gamma$ and a set $\Lambda$ in the plane $\mathbb \{R\}^2$, let $AC(\Gamma ;\Lambda )$ 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 [12], we say that $(\Gamma ,\Lambda )$ is a Heisenberg uniqueness pair if $AC(\Gamma ;\Lambda )=\lbrace 0\rbrace$. 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 $AC(\Gamma ;\Lambda )$ when it is nonzero. 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 $\Lambda _\{\alpha ,\beta \}=(\alpha \mathbb \{Z\} \times \lbrace 0\rbrace )\cup (\lbrace 0\rbrace \times \beta \mathbb \{Z\})$, where $\alpha , \beta$ are positive reals. We will also consider $\Gamma _+$, the branch of $x_1x_2=1$ where $x_1>0$. In [12], it is shown that $AC(\Gamma ; \Lambda _\{\alpha ,\beta \})=\lbrace 0\rbrace$ if and only if $\alpha \beta \le 1$. Here, we show that for $\alpha \beta > 1$, we get a rather drastic “phase transition”: $AC(\Gamma ; \Lambda _\{\alpha , \beta \})$ is infinite-dimensional whenever $\alpha \beta >1$. It is shown in [13] that $AC(\Gamma _+;\Lambda _\{\alpha ,\beta \})=\lbrace 0\rbrace$ if and only if $\alpha \beta <4$. Moreover, at the edge $\alpha \beta =4$, the behavior is more exotic: the space $AC(\Gamma _+;\Lambda _\{\alpha ,\beta \})$ is one-dimensional. Here, we show that the dimension of $AC(\Gamma _+;\Lambda _\{\alpha ,\beta \})$ is infinite whenever $\alpha \beta >4$. Dynamical systems, and more specifically Perron-Frobenius operators, will play a prominent role in the presentation.},
author = {Canto-Martín, Francisco, Hedenmalm, Håkan, Montes-Rodríguez, Alfonso},
journal = {Journal of the European Mathematical Society},
keywords = {trigonometric system; inversion; Perron-Frobenius operator; Koopman operator; invariant measure; Klein–Gordon equation; ergodic theory; Heisenberg uniqueness; Fourier transform; trigonometric systems; Perron-Frobenius operators; Klein-Gordon equation},
language = {eng},
number = {1},
pages = {31-66},
publisher = {European Mathematical Society Publishing House},
title = {Perron-Frobenius operators and the Klein-Gordon equation},
url = {http://eudml.org/doc/277383},
volume = {016},
year = {2014},
}

TY - JOUR
AU - Canto-Martín, Francisco
AU - Hedenmalm, Håkan
AU - Montes-Rodríguez, Alfonso
TI - Perron-Frobenius operators and the Klein-Gordon equation
JO - Journal of the European Mathematical Society
PY - 2014
PB - European Mathematical Society Publishing House
VL - 016
IS - 1
SP - 31
EP - 66
AB - For a smooth curve $\Gamma$ and a set $\Lambda$ in the plane $\mathbb {R}^2$, let $AC(\Gamma ;\Lambda )$ 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 [12], we say that $(\Gamma ,\Lambda )$ is a Heisenberg uniqueness pair if $AC(\Gamma ;\Lambda )=\lbrace 0\rbrace$. 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 $AC(\Gamma ;\Lambda )$ when it is nonzero. 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 $\Lambda _{\alpha ,\beta }=(\alpha \mathbb {Z} \times \lbrace 0\rbrace )\cup (\lbrace 0\rbrace \times \beta \mathbb {Z})$, where $\alpha , \beta$ are positive reals. We will also consider $\Gamma _+$, the branch of $x_1x_2=1$ where $x_1>0$. In [12], it is shown that $AC(\Gamma ; \Lambda _{\alpha ,\beta })=\lbrace 0\rbrace$ if and only if $\alpha \beta \le 1$. Here, we show that for $\alpha \beta > 1$, we get a rather drastic “phase transition”: $AC(\Gamma ; \Lambda _{\alpha , \beta })$ is infinite-dimensional whenever $\alpha \beta >1$. It is shown in [13] that $AC(\Gamma _+;\Lambda _{\alpha ,\beta })=\lbrace 0\rbrace$ if and only if $\alpha \beta <4$. Moreover, at the edge $\alpha \beta =4$, the behavior is more exotic: the space $AC(\Gamma _+;\Lambda _{\alpha ,\beta })$ is one-dimensional. Here, we show that the dimension of $AC(\Gamma _+;\Lambda _{\alpha ,\beta })$ is infinite whenever $\alpha \beta >4$. Dynamical systems, and more specifically Perron-Frobenius operators, will play a prominent role in the presentation.
LA - eng
KW - trigonometric system; inversion; Perron-Frobenius operator; Koopman operator; invariant measure; Klein–Gordon equation; ergodic theory; Heisenberg uniqueness; Fourier transform; trigonometric systems; Perron-Frobenius operators; Klein-Gordon equation
UR - http://eudml.org/doc/277383
ER -

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