Projective and polynomial superflows. I

TL;DR

This paper classifies and analyzes highly symmetric projective superflows in two and three dimensions, exploring their algebraic structures, symmetry groups, and associated elliptic and hyperelliptic functions.

Contribution

It provides a complete classification of 2D superflows with dihedral symmetry and investigates 3D superflows with tetrahedral and octahedral symmetries, linking them to elliptic and hyperelliptic functions.

Findings

Classified all 2D superflows with dihedral symmetry.

Described 3D superflows with tetrahedral and octahedral symmetry.

Connected superflows to elliptic and hyperelliptic functions.

Abstract

Let $x\in\mathbb{R}^{n}$. For $\phi:\mathbb{R}^{n}\mapsto\mathbb{R}^{n}$ and $t\in\mathbb{R}$, we put $\phi^{t}=t^{-1}\phi(xt)$. A projective flow is a solution to the projective translation equation $\phi^{t+s}=\phi^{t}\circ\phi^{s}$, $t,s\in\mathbb{R}$. Previously we have developed an arithmetic, topologic and analytic theory of $2$-dimensional projective flows: rational, algebraic, unramified, abelian flows, commuting flows. The current paper is devoted to highly symmetric flows - superflows. Within flows with a given symmetry, superflows are unique and optimal. Our first result classifies all $2$-dimensional superflows. For any positive integer $d$, there exists the superflow $\phi_{\mathbb{D}_{2d+1}}$ whose group of symmetries is the dihedral group $\mathbb{D}_{2d+1}$. In the current paper we explore the superflow $\phi_{\mathbb{D}_{5}}$, which leads to investigation of abelian functions over curve of genus $6$. The $3$-dimensional theory of projective flows is more involved. We investigate two different $3$-dimensional superflows, whose group of symmetries are, respectively, the full tetrahedral group $\widehat{\mathbb{T}}$ (all symmetries of a tetrahedron), and the octahedral group $\mathbb{O}$ (orientation preserving symmetries of an octahedron), both isomorphic, though non-equivalent as representations. The generic orbits of the first flow are space curves of genus $1$, and the flow itself can be analytically described in terms of Jacobi elliptic functions. The generic orbits of the second flow are curves of genus $9$, and the flow itself can be described in terms of Weierstrass elliptic functions (via reduction of hyper-elliptic functions to elliptic). In the second part of this work we will classify all $3$-dimensional superflows (including the icosahedral superflow), and in the third we investigate superflows over $\mathbb{C}$.