Minimal Peano curve (Q735638)

This is an interesting article, studying systemically Peano curves. As is well-known, a continuous map \(p\) of an interval \(I\) to a square \(Q\) is called a Peano curve if \(p\) is a map of \(I\) onto \(Q\). We denote by \(P\) the set of all (unit) Peano curves \(p: [0,1]\to [0,1]\times [0,1]\). An important characteristic of a Peano curve \(p\) is given by its square-to-linear ratio of \(p\) for \(t\), \(t'\in [0,1]\) \((t\neq t')\), defined by \(\text{Sl}(p; t,t')= |p(t)- p(t')|^2/|t- t'|\). The square-to-linear ratio of \(p\), denoted by \(\text{Sl}(p)\), is defined by \(\text{Sl}(p)= \sup\{\text{Sl}(p; t,t'); t\neq t'\}\). \textit{E. V. Shchepin} [On fractal Peano curves. Proceedings of the Steklov Institute of Mathematics 247, 272--280 (2004); translation from Tr. Mat. Inst. Steklova 247, 294--303 (2004; Zbl 1124.28011)] proved that \(\text{Sl}(p)\geq 5\) for \(p\in P\). The authors assert that the exact calculation of \(\text{Sl}(p)\) is a rather difficult problem. Now, a \(p\in P\) is called regular fractal if the domain of definition \(I\) of \(p\) is decomposed into several equal segments \(I_i\) \((i= 1,\dots, q)\) (fractal periods) such that \(p|_{I_k}\) is similar to \(p\) for \(i= 1,\dots, q\). The (least possible) number \(q\) of fractal periods is called the fractal genus of \(p\). In this paper, the authors systematically analyse Peano curves of fractal genus 9. One of the main results shows that there exists a Peano curve \(p_0\) of fractal genus 9 with \(\text{Sl}(p_0)= 5{2\over 3}\), which is smaller than the square-to-linear ratio of the Peano-Hilbert curve depicted in Fig.1. The proof of \(\text{Sl}(p_0)= 5{2\over 3}\) is based on computer calculations. In fact, a theory is developed that allows one to find the exact value of \(\text{Sl}(p)\) for a regular Peano curve \(p\) on this basis of computer calculations. The authors prove as a byproduct that \(\text{Sl}(p)\) is a rational number for every regular Peano curve \(p\). Finally, the authors give an open problem for a continuous mapping of a unit interval onto a plane set of unit area concerning the square-to-linear ratio.