Abstract. Let x = (x, y). A plane flow is a function F (x, t) : R 2 × R → 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 (x, t) = φ(xt)/t; here φ 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 R all rational solutions of the two dimensional translation equationWe 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.

Abstract. The Minkowski question mark function ?(x) arises as a real distribution of rationals in the Farey tree. We examine the generating function of moments of ?(x). It appears that the generating function is a direct dyadic analogue of period functions for Maass wave forms and it is defined in the cut plane ‫ރ‬ \ (1, ∞). The exponential generating function satisfies an integral equation with kernel being the Bessel function. The solution of this integral equation leads to the definition of dyadic eigenfunctions, arising from a certain Hilbert-Schmidt operator. Finally, we describe p-adic distribution of rationals in the Stern-Brocot tree. Surprisingly, the Eisenstein series G 2 (z) does manifest in both real and p-adic cases.

Abstract. Let x = (x, y). Previously we have found all rational solutions of the 2-dimensional projective translation equation, or PrTE, (1 − z)φ(x) = φ(φ(xz)(1 − z)/z); here φ(x) = (u(x, y), v(x, y)) is a pair of two (real or complex) functions. Solutions of this functional equation are called projective flows. A vector field of a rational flow is a pair of 2-homogenic rational functions. On the other hand, only special pairs of 2-homogenic rational functions give rise to rational flows. In this paper we are interested in all non-singular (satisfying the boundary condition) and unramified (without branching points, i.e. single-valued functions in C 2 \ {union of curves}) projective flows whose vector field is still rational. If an orbit of the flow is given by homogeneous rational function of degree N , then N is called the level of the flow. We prove that, up to conjugation with 1-homogenic birational plane transformation, these are of 6 types: 1) the identity flow; 2) one flow for each non-negative integer N -these flows are rational of level N ; 3) the level 1 exponential flow, which is also conjugate to the level 1 tangent flow; 4) the level 3 flow expressable in terms of Dixonian (equianharmonic) elliptic functions; 5) the level 4 flow expressable in terms of lemniscatic elliptic functions; 6) the level 6 flow expressable in terms of Dixonian elliptic functions again. This reveals another aspect of the PrTE: in the latter four cases this equation is equivalent and provides a uniform framework to addition formulas for exponential, tangent, or special elliptic functions (also addition formulas for polynomials and the logarithm, though the latter appears only in branched flows). Moreover, the PrTE turns out to have a connection with Pólya-Eggenberger urn models. Another purpose of this study is expository, and we provide the list of open problems and directions in the theory of PrTE; for example, we define the notion of quasi-rational projective flows which includes curves of arbitrary genus. This study, though seemingly analytic, is in fact algebraic and our method reduces to algebraic transformations in various quotient rings of rational function fields.

Abstract. Previously, several natural integral transforms of the Minkowski question mark function F (x) were introduced by the author. Each of them is uniquely characterized by certain regularity conditions and the functional equation, thus encoding intrinsic information about F (x). One of them, the dyadic period function G(z), was defined as a Stieltjes transform. In this paper we introduce a family of "distributions" F p (x) for p ≥ 1, such that F 1 (x) is the question mark function and F 2 (x) is a discrete distribution with support on x = 1. We prove that the generating function of moments of F p (x) satisfies the three-term functional equation. This has an independent interest, though our main concern is the information it provides about F (x). This approach yields the following main result: we prove that the dyadic period function is a sum of infinite series of rational functions with rational coefficients.