The Gauss-kuzmin-wirsing Operator

This paper presents a review of the Gauss-Kuzmin-Wirsing (GKW) operator. The GKW operator is the transfer operator of the Gauss map, and thus has connections to the theory of continued fractions – specifically, it is the shift operator for continued fractions. The operator appears to have a reasonably smooth, well-behaved structure, however, no closed-form analytic solutions are known, and these are not easy to obtain. Eigenvalues and eigenfunctions can be obtained numerically, but little else is known in the mathematical literature. While this paper does attempt to be a review, it is incomplete; it is more of a diary of research results. Connections to the Minkowski Question Mark Function are probed. In particular, the Question Mark is used to define a transfer operator which is conjugate to the GKW. This conjugate operator is solvable, and can be shown to have fractal eigenfunctions. However, the spectrum of this operator is not at all the same as that of the GKW. This is because the Jacobian of the transformation relating the two is given by (?′◦?−1)(x) , which is wellknown as the prototypical “multi-fractal measure”. Nonetheless, conjugacy allows the eigenfunctions of the one to be used to construct eigenfunctions of the other; in this sense, a “solution” of the GKW operator is undertaken. The presentation given here assumes little math background beyond basic linear algebra and analytic function theory. This paper is part of a set of chapters that explore the relationship between the real numbers, the modular group, and fractals.x 1. THE GAUSS-KUZMIN-WIRSING OPERATOR This text is a diary of ongoing research results. As such, it not always coherent, and is somewhat disorganized. It is only sporadically updated. It starts with a review of basic ideas, and moves on to a series of original results. The overall layout is as follows: • Present the Gauss-Kuzmin-Wirsing (GKW) operator, including basic facts, theorems, relationships, numerical studies. Explore an asymptotic expansion for the GKW operator. Introduce the Ruelle-Mayer (transfer) operator. • Show that the Minkowski Question Mark converts the GKW into a sawtooth. • Discuss how the Cantor set is a model for the unit interval, and is the appropriate setting for discussing the question mark, fractals, dyadic monoid self similarity, and also for solving the GKW. • Solve the two sawtooth transfer operators (these are exactly solvable). Provide a discrete spectrum of polynomial solutions. For dyadic sawtooth, provide a complete set of fractal eigenfunctions, possessing a continuous spectrum. • Show how to get GKW eigenfunctions from the dyadic sawtooth eigenfunctions; present a complete(?) set of fractal eigenfunctions for the GKW operator. • Show how the continuous solutions arise as the kernel of an operator; discuss differentiability. • Review the Farey Map Date: 2 January 2004 (revised 2008, 31 Nov 2010) . 1 THE GAUSS-KUZMIN-WIRSING OPERATOR 2 • Appendixes providing details for various results. 2. THE GAUSS-KUZMIN-WIRSING OPERATOR The map that that acts as the shift operator for continued fractions is h(x) = 1 x − ⌊ 1 x ⌋ That is, if one writes out the continued-fraction expansion for x ∈ [0,1]: (2.1) x = 1 a1 + 1 a2+ 1 a3+··· ≡ [a1,a2,a3, · · · ] then one has that h(x) = [a2,a3, · · · ] whence the name “shift operator”. This map is often called the Gauss Map. Note that shift operators, as linear operators, are studied as a subtopic of Banach Space theory, and often appear in applied mathematics texts devoted to the engineering topics of control theory, stability theory and filter design[21]; they are studied in Operator Theory as a topic in pure mathematics[22]. However, in these texts, shift operators are typically applied to sequences of functions defined on Hardy spaces, or more generally on Hilbert spaces. It appears that the shift as applied to continued fractions is very nearly unstudied – and no wonder – it appears nearly intractable when approached with standard analytic tools. The usual tools and techniques seem unapplicable; thus, much of this paper is devoted to finding tools and techniques that are relevant. The Ruelle-Frobenius-Perron or transfer operator associated with the Gauss map is known as the Gauss-Kuzmin-Wirsing (GKW)[17, 32] operator Lh. It is the pushback of h, and as such, is a linear map between spaces of functions on the unit interval (topological vector spaces)[30]. That is, given the vector space of functions from the closed unit interval to the real numbers F = { f | f : [0,1]→ R} then Lh is a linear operator mapping F to F . Given f ∈F , it is represented by (2.2) [Lh f ] (x) = ∞ ∑ n=1 1 (n+ x)2 f ( 1 n+ x ) This operator is bounded; its largest eigenvalue is 1. The GKW operator Lh is a special case of what is sometimes called the Ruelle-Mayer operator[13] [Gs f ] (z) = ∞ ∑ n=1 1 (n+ z)s f ( 1 n+ z ) for general complex s and z. This operator, with a value of s = 4, occurs in the study of the Gaussian reduction algorithm applied to modular lattices[13]. Neither the GKW nor the Mayer-Ruelle operators have been “solved”, in the sense that there is no known closed-form analytic solution expressing its all of its eigenfunctions and eigenvectors. The GKW operator has one classically known eigenvector, f (x) = 1/(1+x), which corresponds to the unit eigenvalue; this solution was given by Gauss. THE GAUSS-KUZMIN-WIRSING OPERATOR 3 Kuzmin considers iterating this operator, and shows that given any continuous, differentiable function g(x) with bounded derivative on the unit interval, that the iterate converges uniformly to f (x) = C/(1+ x). That is, by defining gk+1(x) = [Lhgk] (x) as the k’th iterate of g(x), then gk(x)→C/(1+x) uniformly, for all bounded, differentiable g. Thus, as a corollary, this eigenvector is unique[15, section 15]. An alternative way of understanding this result is via the Frobenius-Perron theorem, which asserts that the eigenfunction associated with the maximal eigenvalue is unique. The operator is not normal (i.e. LhL T h is not equal to L T h Lh); this is typically the case for transfer operators. Thus, the left and right eigenvectors are distinct, although they share common eigenvalues. To use proper matrix algebra language, these should be called “singular values”, although we will persist in calling them eigenvalues below; and likewise diagonalization should properly be called “singular value decomposition”, and the left and right eigenvectors are properly called the left and right singular vectors. Alternately, if one considers the operator as acting on a Banach space, then right singular vectors form a basis for for the Banach space, and the left vectors are the dual. When the domain of the Mayer-Ruelle operator is restricted to certain Banach spaces, then the operator is a nuclear operator – that is, it has a discrete spectrum, and its eigenvectors form a basis for Banach space. By considering the operator restricted to a Hardy space, Daudé etal show that the spectrum is real when s is real[13]. The first eigenvalue below 1.0 is approximately 0.3036, and is known as the GKW constant [6, 12, 13][xxx need Babenko ref for original discussion]. As is typically the case for transfer operators, when the right eigenvectors are smooth functions, then the left eigenvectors are linear combinations of derivatives of Dirac delta functions, located at 0 and 1. All of this is explored in greater detail, in the rest of this paper. Aside from the analytic solutions, there is also large class of fractal, discontinuouseverywhere functions associated with eigenvalue 1. The prototypical such solution is the derivative of the Minkowski Question Mark function ?(x). That is, [ Lh? ] (x) =?′(x) A proper construction for the everywhere-discontinuous function ?′, and the derivation of the above identity, is given in [30], together with a construction of a class of other similar solutions. 2.1. Relation to the Riemann Zeta Function. The Gauss map is connected to the Riemann zeta function by a Mellin Transform: ζ (s) = 1 s−1 − s ∫ 1