Real Bers embedding on the line: Fisher-Rao linearization, Schwarzian curvature, and scattering coordinates

We develop a real-analytic counterpart of the Bers embedding for the Fréchet Lie group ${\text{Diff}}^{-\mathrm{\infty }}(\mathbb{R})$ of decay-controlled diffeomorphisms of the line, and establish its connection to $L^p$ Fisher-Rao geometry on densities. For $p\in [1,\mathrm{\infty })$, the $p$-root map $\phi \mapsto p({\phi }^{\mathrm{\prime }1\mathrm{/}p}-1)$ isometrically linearizes the homogeneous ${\dot{W}}^{1,p}$ Finsler metric on ${\text{Diff}}^{-\mathrm{\infty }}(\mathbb{R})$, yielding explicit geodesics and a canonical flat connection whose Eulerian geodesic equation is the generalized Hunter-Saxton equation; for $p=\mathrm{\infty }$, logarithmic coordinates $\phi \mapsto \log {\phi }^{\mathrm{\prime }}$ provide a global isometry and the Schwarzian derivative emerges as the projective curvature. We construct a real Bers map ${\beta }^{-\mathrm{\infty }}\text{\hspace{0.17em}⁣}:{\text{Diff}}^{-\mathrm{\infty }}(\mathbb{R})\mathrm{/}\text{Aff}(\mathbb{R})\to W^{\mathrm{\infty },1}(\mathbb{R})$ via this Schwarzian, prove it is a Fréchet-smooth injective immersion whose linearization admits a tame right inverse given by an explicit Volterra operator, and characterize its image through Sturm-Liouville spectral theory. We introduce an $L^p$-Schwarzian family $S_p$ that interpolates between affine and projective cocycles, establish full asymptotic expansions as $p\to \mathrm{\infty }$ in Fréchet and Orlicz-Sobolev scales, and extend the Bers embedding to Orlicz diffeomorphism groups. Through the Jacobian correspondence, these structures transfer to a manifold of densities asymptotic to Lebesgue measure, where the nonlinear Eulerian transport reduces to a pointwise Riccati law and the Schwarzian becomes the score curvature governing Fisher information. The compact-manifold $L^p$ Fisher-Rao linearization of Bauer, Bruveris, Harms, and Michor is recalled as a guiding framework.