Projective 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 (over \(\mathbb{R}\) and \(\mathbb{C}\)): 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 \(\phi_{\mathbb{D}_{4d+2}}\) whose group of symmetries is the dihedral group \(\mathbb{D}_{4d+2}\). In the current paper we explore the superflow \(\phi_{\mathbb{D}_{10}}\), which leads to investigation of abelian functions over curve of genus \(6\). The three dimensional theory of projective flows is more involved. We investigate two different \(3\)-dimensional superflows, whose group of symmetries are, respectively, the symmetric group \(S_{4}\) (the ring of invariants is polynomial), and the octahedral group \(\mathbb{O}\) (the ring of invariants is not polynomial), both of order 24 and isomorphic, though contragradient to one another 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 classify all \(3\)-dimensional superflows, and investigate superflows over \(\mathbb{C}\).