Brjuno number

In mathematics, a Brjuno number (sometimes spelled Bruno or Bryuno) is a special type of irrational number named for Russian mathematician Alexander Bruno, who introduced them in {{harvtxt|Brjuno|1971}}.

Formal definition

An irrational number $\alpha$ is called a Brjuno number when the infinite sum

: $B(\alpha) = \sum_{n=0}^\infty \frac{\log q_{n+1}}{q_n}$

converges to a finite number.

Here:

$q_n$ is the denominator of the {{mvar|n}}th convergent $\tfrac{p_n}{q_n}$ of the continued fraction expansion of $\alpha$ .
$B$ is a Brjuno function

Examples

Consider the golden ratio {{phi}}:

: $\phi = \frac{1+\sqrt{5}}{2} = 1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{\ddots}}}}.$

Then the nth convergent $\frac{p_n}{q_n}$ can be found via the recurrence relation:{{sfn|Lee|1999|p=192}}

: $\begin{cases}
p_n = p_{n-1} + p_{n-2} & \text{ with } p_0=1,p_1=2, \\
q_n = q_{n-1} + q_{n-2} & \text{ with } q_0=q_1=1.
\end{cases}$

It is easy to see that $q_{n+1} for n \ge 2, as a result$

: $\frac{\log{q_{n+1}}}{q_n} < \frac{2\log{q_{n}}}{q_n} \text{ for } n \ge 2$

and since it can be proven that $\sum_{n=0}^\infty \frac{\log q_n}{q_n} < \infty$ for any irrational number, {{phi}} is a Brjuno number. Moreover, a similar method can be used to prove that any irrational number whose continued fraction expansion ends with a string of 1's is a Brjuno number.{{sfn|Lee|1999|p=193–194}}

By contrast, consider the constant $\alpha = [a_0,a_1,a_2,\ldots]$ with $(a_n)$ defined as

: $a_n = \begin{cases}
10 & \text{ if } n = 0,1, \\
q_n^{q_{n-1}} & \text{ if } n \ge 2.
\end{cases}$

Then $q_{n+1}>q_n^\frac{2q_n}{q_{n-1}}$ , so we have by the ratio test that $\sum_{n=0}^\infty \frac{\log q_{n+1}}{q_n}$ diverges. $\alpha$ is therefore not a Brjuno number.{{sfn|Lee|1999|p=193}}

Importance

The Brjuno numbers are important in the one-dimensional analytic small divisors problems. Bruno improved the diophantine condition in Siegel's Theorem by showing that germs of holomorphic functions with linear part $e^{2\pi i \alpha}$ are linearizable if $\alpha$ is a Brjuno number. {{harvs|txt|first=Jean-Christophe |last=Yoccoz|authorlink=Jean-Christophe Yoccoz|year=1995}} showed in 1987 that Brjuno's condition is sharp; more precisely, he proved that for quadratic polynomials, this condition is not only sufficient but also necessary for linearizability.

Properties

Intuitively, these numbers do not have many large "jumps" in the sequence of convergents, in which the denominator of the ({{math|n + 1}})th convergent is exponentially larger than that of the {{mvar|n}}th convergent. Thus, in contrast to the Liouville numbers, they do not have unusually accurate diophantine approximations by rational numbers.

Brjuno function

=Brjuno sum=

The Brjuno sum or Brjuno function $B$ is

: $B(\alpha) = \sum_{n=0}^\infty \frac{\log q_{n+1}}{q_n}$

where:

$q_n$ is the denominator of the {{mvar|n}}th convergent $\tfrac{p_n}{q_n}$ of the continued fraction expansion of $\alpha$ .

=Real variant=

File:Brjuno function.png

The real Brjuno function $B(\alpha)$ is defined for irrational numbers $\alpha$ [https://arxiv.org/abs/math/9912018v1 Complex Brjuno functions by S. Marmi, P. Moussa, J.-C. Yoccoz ]

: $B : \R \setminus \Q \to \R \cup \{ +\infty \}$

and satisfies

: $\begin{align}
B(\alpha) &= B(\alpha+1) \\
B(\alpha) &= - \log \alpha + \alpha B(1/\alpha)
\end{align}$

for all irrational $\alpha$ between 0 and 1.

=Yoccoz's variant=

Yoccoz's variant of the Brjuno sum defined as follows:[http://www.scholarpedia.org/article/Siegel%20disks/Quadratic%20Siegel%20disks scholarpedia: Quadratic Siegel disks]

: $Y(\alpha)=\sum_{n=0}^{\infty} \alpha_0\cdots \alpha_{n-1} \log \frac{1}{\alpha_n},$

where:

$\alpha$ is irrational real number: $\alpha\in \R \setminus \Q$
$\alpha_0$ is the fractional part of $\alpha$
$\alpha_{n+1}$ is the fractional part of $\frac{1}{\alpha_n}$

This sum converges if and only if the Brjuno sum does, and in fact their difference is bounded by a universal constant.

References

{{Citation | last1=Brjuno | first1=Alexander D. | authorlink = Alexander Bruno | title=Analytic form of differential equations. I, II | mr=0377192 | year=1971 | journal=Trudy Moskovskogo Matematičeskogo Obščestva | issn=0134-8663 | volume=25 | pages=119–262}}
{{citation

| last = Lee | first = Eileen F.

| contribution = The structure and topology of the Brjuno numbers

| contribution-url = http://topology.nipissingu.ca/tp/reprints/v24/tp24114.pdf

| mr = 1802686

| pages = 189–201

| series = Topology Proceedings

| title = Proceedings of the 1999 Topology and Dynamics Conference (Salt Lake City, UT)

| volume = 24

| date = Spring 1999}}

{{Citation | last1=Marmi | first1=Stefano | last2=Moussa | first2=Pierre | last3=Yoccoz | first3=Jean-Christophe | author3-link = Jean-Christophe Yoccoz | title=Complex Brjuno functions | doi=10.1090/S0894-0347-01-00371-X | mr=1839917 | year=2001 | journal=Journal of the American Mathematical Society | issn=0894-0347 | volume=14 | issue=4 | pages=783–841| doi-access=free }}
{{citation

| last = Yoccoz | first = Jean-Christophe | authorlink = Jean-Christophe Yoccoz

| contribution = Théorème de Siegel, nombres de Bruno et polynômes quadratiques

| mr = 1367353

| pages = 3–88

| series = Astérisque

| title = Petits diviseurs en dimension 1

| volume = 231

| year = 1995}}

Notes

Category:Dynamical systems

Category:Number theory