Peano curves with smooth footprints

TL;DR

This paper constructs Peano curves with smooth, tangent footprints that can approximate any given smooth nested curve family, revealing new possibilities for smooth space-filling curves with controlled boundary properties.

Contribution

It introduces a method to create Peano curves with smooth footprints tangent to a continuous line field, generalizing previous space-filling curve constructions.

Findings

Footprints have $C^ abla$ boundaries tangent to a continuous line field.

Footprints can approximate any smooth nested Jordan curve family.

Constructs smooth space-filling curves with prescribed boundary behavior.

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.