The projective translation equation and rational plane flows. I

Abstract

Let x = (x, y). A plane flow is a function ${F(\mathbf{x}, t) : \mathbb{R}^2 \times \mathbb{R} \mapsto \mathbb{R}^{2}}$ such that F(F(x, s), t) = F(x, s + t) for (almost) all real numbers x, y, s, t (the function F might not be well-defined for certain x, t). In this paper we investigate rational plane flows which are of the form ${F(\mathbf{x}, t) = \phi(\mathbf{x}t)/t;}$ here ${\phi}$ is a pair of rational functions in 2 real variables. These may be called projective flows, and for a description of such flows only the knowledge of Cremona group in dimension 1 is needed. Thus, the aim of this work is to completely describe over ${\mathbb{R}}$ all rational solutions of the two dimensional translation equation ${(1 - z)\phi(\mathbf{x}) = \phi(\phi(\mathbf{x}z)(1 - z)/z)}$. We show that, up to conjugation with a 1-homogenic birational plane transformation (1-BIR), all solutions are as follows: a zero flow, two singular flows, an identity flow, and one non-singular flow for each non-negative integer N, called the level of the flow. The case N = 0 stands apart, while the case N = 1 has special features as well. Conjugation of these canonical solutions with 1-BIR produce a variety of flows with different properties and invariants, depending on the level and on the conjugation itself. We explore many more features of these flows; for example, there are 1, 4, and 2 essentially different symmetric flows in cases N = 0, N = 1, and N ≥ 2, respectively. Many more questions related to rational flows will be treated in the second part of this work.