The Hyperbolic Plane in $\mathbb{E}^3$

We build an explicit $C^1$ isometric embedding
$f_{\infty}:\mathbb{H}^2\to\mathbb{E}^3$ of the hyperbolic plane whose image is
relatively compact. Its limit set is a closed curve of Hausdorff dimension 1.
Given an initial embedding $f_0$, our construction generates iteratively a
sequence of maps by adding at each step $k$ a layer of $N_{k}$ corrugations. To
understand the behavior of $df_\infty$ we introduce a $formal$ $corrugation$
$process$ leading to a $formal$ $analogue$ $\Phi_{\infty}:\mathbb{H}^2\to
\mathcal{L}(\mathbb{R}^2,\mathbb{R}^3)$. We show a self-similarity structure
for $\Phi_{\infty}$. We next prove that $df_\infty$ is close to $\Phi_{\infty}$
up to a precision that depends on the sequence $N_*:= (N_{k})_k$. We then
introduce the $pattern$ $maps$ $\boldsymbol{\nu}_{\infty}^\Phi$ and
$\boldsymbol{\nu}_{\infty}$, of respectively $\Phi_{\infty}$ and $df_\infty$,
that together with $df_0$ entirely describe the geometry of the Gauss maps
associated to $\Phi_{\infty}$ and $df_\infty$. For well chosen sequences of
corrugation numbers, we finally show an asymptotic convergence of
$\boldsymbol{\nu}_{\infty}$ towards $\boldsymbol{\nu}_{\infty}^\Phi$ over
circles of rational radii.