Gauss–Kuzmin–Wirsing Operator
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Gauss–Kuzmin–Wirsing operator is the
transfer operator In mathematics, the transfer operator encodes information about an iterated map and is frequently used to study the behavior of dynamical systems, statistical mechanics, quantum chaos and fractals. In all usual cases, the largest eigenvalue is 1 ...
of the Gauss map that takes a positive number to the fractional part of its reciprocal. (This is not the same as the Gauss map in
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
.) It is named after Carl Gauss, Rodion Kuzmin, and Eduard Wirsing. It occurs in the study of continued fractions; it is also related to the Riemann zeta function.


Relationship to the maps and continued fractions


The Gauss map

The Gauss function (map) ''h'' is : :h(x)=1/x-\lfloor 1/x \rfloor. where \lfloor 1/x \rfloor denotes the
floor function In mathematics, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least integer greater than or eq ...
. It has an infinite number of
jump discontinuities Continuous functions are of utmost importance in mathematics, functions and applications. However, not all functions are continuous. If a function is not continuous at a limit point (also called "accumulation point" or "cluster point") of its do ...
at ''x'' = 1/''n'', for positive integers ''n''. It is hard to approximate it by a single smooth polynomial.


Operator on the maps

The Gauss–Kuzmin–Wirsing operator G acts on functions f as : fx) = \int_0^1 \delta(x-h(y)) f(y) \, dy = \sum_^\infty \frac f \left(\frac \right). it has the fixed point \rho(x) = \frac, unique up to scaling, which is the density of the measure invariant under the Gauss map.


Eigenvalues of the operator

The first
eigenfunction In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
of this operator is :\frac 1\ \frac 1 which corresponds to an
eigenvalue In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
of ''λ''1 = 1. This eigenfunction gives the probability of the occurrence of a given integer in a continued fraction expansion, and is known as the Gauss–Kuzmin distribution. This follows in part because the Gauss map acts as a truncating shift operator for the
continued fraction A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, ...
s: if : x= ;a_1,a_2,a_3,\dots/math> is the continued fraction representation of a number 0 < ''x'' < 1, then : h(x)= ;a_2,a_3,\dots Because h is conjugate to a Bernoulli shift, the eigenvalue \lambda_1=1 is simple, and since the operator leaves invariant the Gauss–Kuzmin measure, the operator is ergodic with respect to the measure. This fact allows a short proof of the existence of Khinchin's constant. Additional eigenvalues can be computed numerically; the next eigenvalue is ''λ''2 = −0.3036630029... and its absolute value is known as the Gauss–Kuzmin–Wirsing constant. Analytic forms for additional eigenfunctions are not known. It is not known if the eigenvalues are
irrational Irrationality is cognition, thinking, talking, or acting without rationality. Irrationality often has a negative connotation, as thinking and actions that are less useful or more illogical than other more rational alternatives. The concept of ...
. Let us arrange the eigenvalues of the Gauss–Kuzmin–Wirsing operator according to an absolute value: :1=, \lambda_1, > , \lambda_2, \geq, \lambda_3, \geq\cdots. It was conjectured in 1995 by Philippe Flajolet and Brigitte Vallée that : \lim_ \frac = -\varphi^2, \text \varphi=\frac 2. In 2018, Giedrius Alkauskas gave a convincing argument that this conjecture can be refined to a much stronger statement: : \begin & (-1)^\lambda_n=\varphi^ + C\cdot\frac+d(n)\cdot\frac, \\ pt& \text C=\frac=1.1019785625880999_; \end here the function d(n) is bounded, and \zeta(\star) is the Riemann zeta function.


Continuous spectrum

The eigenvalues form a discrete spectrum, when the operator is limited to act on functions on the unit interval of the real number line. More broadly, since the Gauss map is the shift operator on
Baire space In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior. According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are ...
\mathbb^\omega, the GKW operator can also be viewed as an operator on the function space \mathbb^\omega\to\mathbb (considered as a
Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
, with basis functions taken to be the
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , then the indicator functio ...
s on the cylinders of the
product topology In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...
). In the later case, it has a continuous spectrum, with eigenvalues in the unit disk , \lambda, <1 of the complex plane. That is, given the cylinder C_n \, the operator G shifts it to the left: GC_n = C_ /math>. Taking r_(x) to be the indicator function which is 1 on the cylinder (when x\in C_n /math>), and zero otherwise, one has that Gr_=r_. The series :f(x)=\sum_^\infty \lambda^ r_(x) then is an eigenfunction with eigenvalue \lambda. That is, one has fx)=\lambda f(x) whenever the summation converges: that is, when , \lambda, <1. A special case arises when one wishes to consider the
Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This Measure (mathematics), measure was introduced by Alfrà ...
of the shift operator, that is, a function that is invariant under shifts. This is given by the Minkowski measure ?^\prime. That is, one has that G?^\prime = ?^\prime.


Ergodicity

The Gauss map is in fact much more than ergodic: it is exponentially mixing, but the proof is not elementary.


Entropy

The Gauss map, over the Gauss measure, has entropy \frac . This can be proved by the Rokhlin formula for entropy. Then using the Shannon–McMillan–Breiman theorem, with its equipartition property, we obtain Lochs' theorem.


Measure-theoretic preliminaries

A covering family \mathcal C is a set of measurable sets, such that any open set is a ''disjoint'' union of sets in it. Compare this with base in topology, which is less restrictive as it allows non-disjoint unions. Knopp's lemma. Let B \subset outer regular, we can take an open set B' that is close to B^c, meaning the symmetric difference has arbitrarily small measure \mu(B' \Delta B^c) < \epsilon. At the limit, \mu(B' \cap B) \geq \gamma \mu(B') becomes have 0 \geq \gamma \mu(B^c).


The Gauss map is ergodic

Fix a sequence a_1, \dots, a_n of positive integers. Let \frac = [0;a_1, \dots, a_n]. Let the interval \Delta_n be the open interval with end-points ;a_1, \dots, a_n [0;a_1, \dots, a_n+1]. Lemma. For any open interval (a, b) \subset (0, 1), we have\mu(T^(a,b) \cap \Delta_n) = \mu((a,b)) \mu(\Delta_n) \underbrace_ Proof. For any t \in (0, 1) we have ;a_1, \dots, a_n + t= \frac by standard continued fraction theory. By expanding the definition, T^(a,b) \cap \Delta_n is an interval with end points ;a_1, \dots, a_n + a ;a_1, \dots, a_n+ b/math>. Now compute directly. To show the fraction is \geq 1/2, use the fact that q_n \geq q_. Theorem. The Gauss map is ergodic. Proof. Consider the set of all open intervals in the form ( ;a_1, \dots, a_n ;a_1, \dots, a_n+1. Collect them into a single family \mathcal C. This \mathcal C is a covering family, because any open interval (a, b)\setminus \Q where a, b are rational, is a disjoint union of finitely many sets in \mathcal C. Suppose a set B is T-invariant and has positive measure. Pick any \Delta_n \in \mathcal C. Since Lebesgue measure is outer regular, there exists an open set B_0 which differs from B by only \mu(B_0 \Delta B) < \epsilon. Since B is T-invariant, we also have \mu(T^B_0 \Delta B) = \mu(B_0 \Delta B) < \epsilon. Therefore, \mu(T^B_0 \cap \Delta_n) \in \mu(B\cap \Delta_n) \pm \epsilonBy the previous lemma, we have\mu(T^B_0 \cap \Delta_n) \geq \frac 12 \mu(B_0) \mu(\Delta_n) \in \frac 12 (\mu(B) \pm \epsilon) \mu(\Delta_n) Take the \epsilon \to 0 limit, we have \mu(B \cap \Delta_n) \geq \frac 12 \mu(B) \mu(\Delta_n). By Knopp's lemma, it has full measure.


Relationship to the Riemann zeta function

The GKW operator is related to the Riemann zeta function. Note that the zeta function can be written as :\zeta(s)=\frac-s\int_0^1 h(x) x^ \; dx which implies that :\zeta(s)=\frac-s\int_0^1 x \left x^ \right, dx by change-of-variable.


Matrix elements

Consider the
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
expansions at ''x'' = 1 for a function ''f''(''x'') and g(x)= fx). That is, let :f(1-x)=\sum_^\infty (-x)^n \frac and write likewise for ''g''(''x''). The expansion is made about ''x'' = 1 because the GKW operator is poorly behaved at ''x'' = 0. The expansion is made about 1 âˆ’ ''x'' so that we can keep ''x'' a positive number, 0 ≤ ''x'' ≤ 1. Then the GKW operator acts on the Taylor coefficients as :(-1)^m \frac = \sum_^\infty G_ (-1)^n \frac, where the matrix elements of the GKW operator are given by :G_=\sum_^n (-1)^k \left \zeta (k+m+2)- 1\right This operator is extremely well formed, and thus very numerically tractable. The Gauss–Kuzmin constant is easily computed to high precision by numerically diagonalizing the upper-left ''n'' by ''n'' portion. There is no known closed-form expression that diagonalizes this operator; that is, there are no closed-form expressions known for the eigenvectors.


Riemann zeta

The Riemann zeta can be written as :\zeta(s)=\frac-s \sum_^\infty (-1)^n t_n where the t_n are given by the matrix elements above: :t_n=\sum_^\infty \frac . Performing the summations, one gets: :t_n=1-\gamma + \sum_^n (-1)^k \left \frac - \frac \right/math> where \gamma is the
Euler–Mascheroni constant Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (), defined as the limiting difference between the harmonic series and the natural logarith ...
. These t_n play the analog of the Stieltjes constants, but for the
falling factorial In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial \begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) . \end ...
expansion. By writing :a_n=t_n - \frac one gets: ''a''0 = −0.0772156... and ''a''1 = −0.00474863... and so on. The values get small quickly but are oscillatory. Some explicit sums on these values can be performed. They can be explicitly related to the Stieltjes constants by re-expressing the falling factorial as a polynomial with
Stirling number In mathematics, Stirling numbers arise in a variety of Analysis (mathematics), analytic and combinatorics, combinatorial problems. They are named after James Stirling (mathematician), James Stirling, who introduced them in a purely algebraic setti ...
coefficients, and then solving. More generally, the Riemann zeta can be re-expressed as an expansion in terms of Sheffer sequences of polynomials. This expansion of the Riemann zeta is investigated in the following references. The coefficients are decreasing as :\left(\frac\right)^e^ \cos\left(\sqrt-\frac\right) + \mathcal \left(\frac\right).


References


General references

* A. Ya. Khinchin, ''Continued Fractions'', 1935, English translation University of Chicago Press, 1961 ''(See section 15).'' * K. I. Babenko, ''On a Problem of Gauss'', Soviet Mathematical Doklady 19:136–140 (1978) * K. I. Babenko and S. P. Jur'ev, ''On the Discretization of a Problem of Gauss'', Soviet Mathematical Doklady 19:731–735 (1978). * A. Durner, ''On a Theorem of Gauss–Kuzmin–Lévy.'' Arch. Math. 58, 251–256, (1992). * A. J. MacLeod, ''High-Accuracy Numerical Values of the Gauss–Kuzmin Continued Fraction Problem.'' Computers Math. Appl. 26, 37–44, (1993). * E. Wirsing, ''On the Theorem of Gauss–Kuzmin–Lévy and a Frobenius-Type Theorem for Function Spaces.'' Acta Arith. 24, 507–528, (1974).


Further reading

* Keith Briggs,
A precise computation of the Gauss–Kuzmin–Wirsing constant
' (2003) ''(Contains a very extensive collection of references.)'' * Phillipe Flajolet and Brigitte Vallée,
On the Gauss–Kuzmin–Wirsing Constant
'' (1995). * Linas Vepsta
The Bernoulli Operator, the Gauss–Kuzmin–Wirsing Operator, and the Riemann Zeta
(2004) (PDF)


External links

* * {{DEFAULTSORT:Gauss-Kuzmin-Wirsing operator Continued fractions Dynamical systems