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A pair $(\Gamma,\Lambda)$, where $\Gamma\subset\mathbb{R}^2$ is a locallyrectifiable curve and $\Lambda\subset\mathbb{R}^2$ is a {\em Heisenberguniqueness pair} if an absolutely continuous (with respect to arc length)finite complex-valued Borel measure supported on $\Gamma$ whose Fouriertransform vanishes on $\Lambda$ necessarily is the zero measure. Recently, itwas shown by Hedenmalm and Montes that if $\Gamma$ is the hyperbola$x_1x_2=M^2/(4\pi^2)$, where $M>0$ is the mass, and $\Lambda$ is thelattice-cross $(\alpha\mathbb{Z}\times\{0\}) \cup(\{0\}\times\beta\mathbb{Z})$, where $\alpha,\beta$ are positive reals, then$(\Gamma,\Lambda)$ is a Heisenberg uniqueness pair if and only if $\alpha\betaM^2\le4\pi^2$. The Fourier transform of a measure supported on a hyperbolasolves the one-dimensional Klein-Gordon equation, so the theorem supplies verythin uniqueness sets for a class of solutions to this equation. The case of thesemi-axis $\mathbb{R}_+$ as well as the holomorphic counterpart remained open.In this work, we completely solve these two problems. As for the semi-axis, weshow that the restriction to $\mathbb{R}_+$ of the above exponential systemspans a weak-star dense subspace of $L^\infty(\mathbb{R}_+)$ if and only if$0<\alpha\beta<4$, based on dynamics of Gauss-type maps. This has aninterpretation in terms of dynamical unique continuation. As for theholomorphic counterpart, we show that the above exponential system with$m,n\ge0$ spans a weak-star dense subspace of $H^\infty_+(\mathbb{R})$ if andonly if $0<\alpha\beta\le1$. To obtain this result, we need to develop newharmonic analysis tools for the dynamics of Gauss-type maps, related to theHilbert transform. Some details are deferred to a separate publication.

The Klein-Gordon equation, the Hilbert transform, and dynamics of Gauss-type maps