Peano curves with smooth footprints

Abstract

We construct Peano curves $\gamma :[0,\infty )\to {\mathbb{R}}^2$ whose "footprints" $\gamma ([0,t])$, $t>0$, have $C^{\infty }$ boundaries and are tangent to a common continuous line field on the punctured plane ${\mathbb{R}}^2\setminus \{\gamma (0)\}$. Moreover, these boundaries can be taken $C^{\infty }$-close to any prescribed smooth family of nested smooth Jordan curves contracting to a point.

17 pages, 4 figures