Peano curves with smooth footprints (Q303842)

The main result of the paper is the following theorem: There exist a Peano curve \(\gamma:[0,\infty)\to \mathbb{R}^2\) and a continuous line field \(\Lambda\) on the punctured plane \(\mathbb{R}^2\setminus\{\gamma(0)\}\) such that for all \(t > 0\), the boundary of the set \(\gamma([0,t])\) is a \(C^\infty\)-curve \(C_t\) containing the point \(\{\gamma (t)\}\) and tangent to the line field \(\Lambda\) at each point. Moreover, it is possible to choose the Peano curve \(\gamma\) in such a way that each curve \(C_t\) is \(C^\infty\)-close to the circle \(x^2+y^2= t^2\). The proof is established via several lemmas and technical propositions.