Projective and polynomial superflows. I

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 $\mathbb{C}$.