Regularity of the limit set of embedded Poincaré Disks

Vincent Borelli

Let f be an embedding of a non compact manifold into an Euclidean space and p_n be a divergent sequence of points of M. If the image points f(p_n) converge, the limit is called a limit point of f. In this talk, we will build an embedding f of a closed disk into a 3-Euclidean ball whose restriction to the interior is a C^1 embedding of the Poincaré disk and which is, on the closed disk, \beta-Hölder for any 0<\beta<1. In particular, this embedding of the Poincaré disk has a closed embedded curve of Hausdorff dimension one as limit set.

https://www.ias.edu/video/regularity-limit-set-embedded-poincare-disks