Eigenvalues of the Gauss-Kuzmin-Wirsing operator

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Numbers

$n$

$\lambda_n$

comment: Exactly $1$: the operator preserves the Gauss measure, whose density $\frac{1}{(1+x)\log2}$ is the eigenfunction (5).

-0.303663002898732658597448121901556233110877352253657895188245481467226995294246910984340812

comment: The Gauss-Kuzmin-Wirsing constant is $|\lambda_2|$, and the literature quotes it positive [7] [6]; the eigenvalue itself is negative.

0.100884509293104075305637642992486247444481593431675337003907259642759007927818416724732621

-0.0354961590216598454088916811346969995131873271635908886069884189025177241454069108539262751

0.0128437903624402648151602953020630217224895390188822097913127446079017504846546521591302192

-0.00471777751157103107387547499904633332232334208299968466282216479721293952102944895888555435

0.00174867512430551191331170967465212021561446528114536287371081591475124045243229450540354312

-0.000652020858320502900330267784809284370609589222144988279381535083960411175972415540357982374

0.000244131465524515812136850477916349318368626217322258416048183826704641834467701840910474626

10:

-0.0000916890837685933029776867371439100329386461532426983285894892748213937182577719803735739045

11:

0.0000345165461638542533132479591394783868805269215908520863117630191043309955921228709714913910

12:

-0.0000130176978770230305864019406353768610783385127922317862176727558675874534305648732322286639

13:

0.00000491678230246449125546106380230808090306692239457191153185557424163147038792237620822660517

14:

-0.00000185930735150904238198606749148557667472632299235730485975619416077422524636771973143272049

15:

7.03811343087039809246451587472823480368161218427568175195149546250300555976342762342643640e-7

16:

-2.66641343447956406171434243996502090069709651990384435705396875802753283024680673498656937e-7

17:

1.01090553221499246944625165929681012050455332075592349422077077137268900914736421391199221e-7

18:

-3.83496979502656472706259498561024363508693920422190869537232747304917784425754533069634418e-8

19:

1.45561383866802305370728397061356814939757306186135371796419623889550720869473912359324511e-8

20:

-5.52756793799760806048860801765727370277389461719887493165202632908765460713769487664184422e-9

21:

2.09991358297268764248798945279937160320784202256295409391211009086534591220064625998866891e-9

22:

-7.98045768272019623539353572600879018253270117404686860296829157480168698859351538825932902e-10

23:

3.03386294909857542193541454202487736870470765668925006162457520366331431874846919637266166e-10

24:

-1.15369541814466859546775197221467165861948370540033006837550596663625156405202132749243207e-10

25:

4.38834501139843608520604267866655039789399870102747706566941806170983515775079007497903582e-11

26:

-1.66960521511737675528195285901437786343672130522510803043702080325630760043197532688281990e-11

27:

6.35361242098269496304808082653766007307825296165876304634007694199837262724374901495877726e-12

28:

-2.41831684759530204590226838906611048727349887210551322193604526141292826933711968894945002e-12

29:

9.20627420057633985611927145650172454130177980238478959127433607455363351732147083790095307e-13

30:

-3.50530935459368636355375782080287322146907218017412372452860910543095991161703787932586712e-13

31:

1.33485714414356906348683864381185993764816853900456849368668398179254336349850681031124325e-13

32:

-5.08398310974063715121489784877710785214205896407068171576571754113655981912444302093175266e-14

33:

1.93655438830852674014027041215637952721988660348119832943314252006071540339587718062298435e-14

34:

-7.37747046440651957074551594324168679547724684391168289923353383899060889093699060162399942e-15

35:

2.81082457232607406706683008986516287868706718018837226436434320324363545276810927976509455e-15

36:

-1.07103865011607100414976336177078216128079455655501519698787821886019169615037692209695864e-15

37:

4.08148926363649029666100422050553819658792825656662261509857399849108265255640963778095879e-16

38:

-1.55550545323793104110551834494234423509377533754062206870711841694635960929666115979498895e-16

39:

5.92872462394695261241063063960700738665472182586903804893298823646537489471020057837518341e-17

40:

-2.25988112977161885347921002802101488821572498417754838754474519573764880056410176103254170e-17

41:

8.61474442975977795666028896113483779793953974574709399717127081121450777991036868197315351e-18

42:

-3.28420129321585877897586107292468009711516415997071810133988322368275302339223091590333955e-18

43:

1.25211997264208427204826448100541298064157957802307851971866793208946866904909698764809636e-18

44:

-4.77407569935796642106947555956382175640888826714606037367680077130851925209935827808935238e-19

45:

1.82036444567682802537364829730603455924592882371205966212571802885034660856559690994366677e-19

46:

-6.94147381297144046949252744601756059774841724610623320919389595626509922228323583973749762e-20

47:

2.64708602284749249025744224700289651853864717540854140248086197572844679693581519321179447e-20

48:

-1.00949983400539590567515339916852679556196523135003787466801047645682147847343166144568988e-20

49:

3.85003942249205119773984344615519437546062519741927390833764804210797393318228171528290521e-21

50:

-1.46839805816557558582431653623546773731212115186065047113246321246923770691369569173767646e-21

Definition

Let $\mathcal L$ be the Gauss-Kuzmin-Wirsing transfer operator for the Gauss continued-fraction map, defined by (1) [3] [1]. Its real eigenvalues $\lambda_n$ are ordered by decreasing absolute value and alternate in sign (6). This table gives $\lambda_n$, signed, for $n\geq1$; the leading eigenvalue is $\lambda_1=1$ (5).

Parameters

$n$

— eigenvalue number ($n\geq1$)

Formulas

(1)

$(\mathcal L f)(x)=\sum_{m=1}^{\infty} \frac{1}{(x+m)^2}f\left(\frac{1}{x+m}\right)$.

(2)

Briggs's Taylor-basis approximation has entries $M_{jk}=\frac{(-1)^j}{j!(-2)^k}\sum_{i=0}^k {k\choose i}(-2)^i (i+2)_j\left(\zeta(i+j+2)(2^{i+j+2}-1)-2^{i+j+2}\right)$, with $0\leq j,k\leq N$ [6] [4].

(3)

$\lim_{n\to\infty}\lambda_n/\lambda_{n+1}=-\varphi^2$, where $\varphi$ is the golden ratio [3].

Comments

(4)

For the certified range used here, Nisoli gives real simple eigenvalues with $|\lambda_1|>|\lambda_2|>\cdots>|\lambda_{50}|$ [1].

(5)

The leading eigenvalue is exactly $\lambda_1=1$, with the invariant density $\frac{1}{(1+x)\log2}$ of the Gauss map as its eigenfunction. It is a theorem rather than a computation, so the row is the integer.

(6)

The signs alternate: $(-1)^{n+1}\lambda_n>0$, and the source certifies the sign of each eigenvalue in a column of its own [2]. The Gauss-Kuzmin-Wirsing constant is $|\lambda_2|$, which is how the literature quotes it [7] [6]; that convention is about one row and the rest are given as they are.

Programs

(P1)

Sage

R = RealField(400)

def rising(a, j):
    out = R(1)
    for t in range(j):
        out *= a + t
    return out

def M_entry(j, k):
    total = R(0)
    for i in range(k + 1):
        total += binomial(k, i) * (-2)**i * rising(i + 2, j) * (
            zeta(i + j + 2) * (2**(i + j + 2) - 1) - 2**(i + j + 2))
    return (-1)**j * total / (factorial(j) * (-2)**k)

N = 40
M = matrix(R, N + 1, N + 1,
           lambda j, k: M_entry(j, k))
eigenvalues = sorted(M.eigenvalues(), key=lambda z: -abs(z))
# Later eigenvalues of this truncation include artifacts.
[(n, abs(eigenvalues[n - 1])) for n in range(2, 5)]

References

[1]

I. Nisoli, Certified spectral approximation of transfer operators and the Gauss map, arXiv preprint, 2026. (arXiv)

[2]

I. Nisoli, Eigenvalue, eigenvectors, projection coefficients and data for the GKW map, Harvard Dataverse, 2026. (doi)

[3]

G. Alkauskas, Transfer operator for the Gauss continued fraction map. I. Structure of the eigenvalues and trace formulas, arXiv preprint, 2012. (arXiv)

[4]

K. Briggs, A precise computation of the Gauss-Kuzmin-Wirsing constant, preliminary report, 2003.

Links

[5]

Wikipedia: Gauss-Kuzmin-Wirsing operator

[6]

MathWorld: Gauss-Kuzmin-Wirsing Constant

[7]

OEIS A038517: decimal expansion of the Gauss-Kuzmin-Wirsing constant

Similar tables

Khinchin's means — the invariant density for the leading eigenvalue gives the Gauss-Kuzmin distribution whose partial-quotient means are listed there

Lévy's constant — Lévy's constant gives the almost-sure denominator growth rate for the same regular continued fraction, while $\mathcal L$ is the transfer operator for the Gauss map

Lyapunov exponents of classical chaotic systems — the Gauss-map row there gives the Lyapunov exponent with respect to the same invariant measure

Values of the Riemann zeta function at rational numbers — zeta values occur in Briggs's matrix formula (2)

Golden ratio — the asymptotic ratio of consecutive eigenvalues is $-\varphi^2$

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds $\lambda_n$ for $1\leq n\leq50$, and the sequence continues for all $n\geq1$)

How they were obtained:

The values were read from Nisoli's certified spectral data set [2], file gkw_spectral_coefficients_K1024.tsv. That file gives the eigenvalue centers, enclosure radii, and resolvent certification data for a $K=1024$ Galerkin approximation computed in 2048-bit ball arithmetic. The source states that the first 50 nonzero eigenvalues are real and simple and that each certified enclosure carries at least 90 decimal digits [1].

The row $\lambda_1$ is the integer $1$, which is (5) and not a measurement; the source's own value for it, $0.999\ldots9$, is checked to enclose $1$ and is otherwise not used. The generator returns each other stored row as a real ball centered at the source eigenvalue, sign and all, and enlarged by the source column eval_encl. The $K=1024$ intervals were compared with the independent $K=512$ intervals in the same data set, and all stored rows overlap. The row $\lambda_2$ was compared, in absolute value, with OEIS A038517 [7]; the rows $2\leq n\leq24$ were compared with Alkauskas's 16-digit table [3]. The alternating sign pattern in (6) and the approach of $\lambda_n/\lambda_{n+1}$ toward $-\varphi^2$ were checked on the stored range.