Editing Gauss–Kuzmin–Wirsing operator (section)

==Relationship to the maps and continued fractions==

===The Gauss map === 
[[File:Gauss function.svg|thumb|right|File:Gauss function]]
The Gauss function (map) ''h'' is :

:<math>h(x)=1/x-\lfloor 1/x \rfloor.</math>

where <math>\lfloor 1/x \rfloor</math> denotes the [[Floor and ceiling functions|floor function]].

It has an infinite number of [[Classification of discontinuities#Jump discontinuity|jump discontinuities]] at ''x'' = 1/''n'', for positive integers&nbsp;''n''. It is hard to approximate it by a single smooth polynomial.<ref>[https://www.springer.com/us/book/9781461484523 ''A Graduate Introduction to Numerical Methods From the Viewpoint of Backward Error Analysis'' by Corless, Robert, Fillion, Nicolas]</ref>

===Operator on the maps ===
The Gauss–Kuzmin–Wirsing [[transfer operator|operator]] <math> G</math> acts on functions <math>f</math> as

:<math>[Gf](x) = \int_0^1 \delta(x-h(y)) f(y) \, dy  = \sum_{n=1}^\infty \frac {1}{(x+n)^2} f \left(\frac {1}{x+n}\right).</math>
it has the fixed point <math>\rho(x) = \frac{1}{\ln 2 (1+x)}</math>, unique up to scaling, which is the density of the measure invariant under the Gauss map.

===Eigenvalues of the operator===

The first [[eigenfunction]] of this operator is

:<math>\frac 1{\ln 2}\ \frac 1{1+x}</math>

which corresponds to an [[eigenvalue]] of ''λ''<sub>1</sub>&nbsp;=&nbsp;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]]s: if

: <math>x=[0;a_1,a_2,a_3,\dots]</math>

is the continued fraction representation of a number 0&nbsp;<&nbsp;''x''&nbsp;<&nbsp;1, then

: <math>h(x)=[0;a_2,a_3,\dots].</math>

Because <math>h</math> is conjugate to a [[Bernoulli shift]], the eigenvalue <math>\lambda_1=1</math> 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''<sub>2</sub> = −0.3036630029... {{OEIS|A038517}}
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]].

Let us arrange the eigenvalues of the Gauss–Kuzmin–Wirsing operator according to an absolute value:

:<math>1=|\lambda_1|> |\lambda_2|\geq|\lambda_3|\geq\cdots.</math>

It was conjectured in 1995 by [[Philippe Flajolet]] and [[Brigitte Vallée]] that

:<math>
\lim_{n\to\infty} \frac{\lambda_n}{\lambda_{n+1}} = -\varphi^2, \text{ where } \varphi=\frac{1+\sqrt 5} 2.
</math>

In 2018, Giedrius Alkauskas gave a convincing argument that this conjecture can be refined to a much stronger statement:<ref>{{cite arXiv |last1=Alkauskas |first1=Giedrius |year=2018 |title=Transfer operator for the Gauss' continued fraction map. I. Structure of the eigenvalues and trace formulas |eprint=1210.4083 |class=math.NT}}</ref>

:<math>
\begin{align}
& (-1)^{n+1}\lambda_n=\varphi^{-2n} + C\cdot\frac{\varphi^{-2n}}{\sqrt{n}}+d(n)\cdot\frac{\varphi^{-2n}}{n}, \\[4pt]
& \text{where } C=\frac{\sqrt[4]{5}\cdot\zeta(3/2)}{2\sqrt{\pi}}=1.1019785625880999_{+};
\end{align}
</math>

here the function <math>d(n)</math> is bounded, and <math>\zeta(\star)</math> 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 (set theory)|Baire space]] <math>\mathbb{N}^\omega</math>, the GKW operator can also be viewed as an operator on the function space <math>\mathbb{N}^\omega\to\mathbb{C}</math> (considered as a [[Banach space]], with basis functions taken to be the [[indicator function]]s on the [[cylinder set|cylinders]] of the [[product topology]]).  In the later case, it has a continuous spectrum, with eigenvalues in the unit disk <math>|\lambda|<1</math> of the complex plane.  That is, given the cylinder <math>C_n[b]= \{(a_1,a_2,\cdots) \in \mathbb{N}^\omega : a_n = b \}</math>, the operator G shifts it to the left: <math>GC_n[b] = C_{n-1}[b]</math>. Taking <math>r_{n,b}(x)</math> to be the indicator function which is 1 on the cylinder (when <math>x\in C_n[b]</math>), and zero otherwise, one has that <math>Gr_{n,b}=r_{n-1,b}</math>.  The series

:<math>f(x)=\sum_{n=1}^\infty \lambda^{n-1} r_{n,b}(x)</math>

then is an eigenfunction with eigenvalue <math>\lambda</math>. That is, one has <math>[Gf](x)=\lambda f(x)</math> whenever the summation converges: that is, when <math>|\lambda|<1</math>.

A special case arises when one wishes to consider the [[Haar measure]] of the shift operator, that is, a function that is invariant under shifts.  This is given by the [[Minkowski's question mark function|Minkowski measure]] <math>?^\prime</math>. That is, one has that <math>G?^\prime = ?^\prime</math>.<ref>{{cite arXiv |last1=Vepstas |first1=Linas |year=2008 |title=On the Minkowski Measure |eprint=0810.1265 |class=math.DS}}</ref>