Brjuno number

In mathematics, a Brjuno number is a special type of irrational number.

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 $n$ th convergent ${\frac {p_{n}}{q_{n}}}$ of the continued fraction expansion of $\alpha$ .
$B$ is a Brjuno function

Name

The Brjuno numbers are named after Alexander Bruno, who introduced them in Brjuno (1971); they are also occasionally spelled Bruno numbers or Bryuno numbers.

Importance

The Brjuno numbers are important in the one–dimensional analytic small divisors problems. Bruno improved the diophantine condition in Siegel's Theorem, showed that germs of holomorphic functions with linear part $e^{2\pi i\alpha }$ are linearizable if $\alpha$ is a Brjuno number. Jean-Christophe Yoccoz (1995) showed in 1987 that this condition is also necessary, and for quadratic polynomials is necessary and sufficient.

Properties

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

Brjuno function

The real Brjuno function $B(x)$ is defined for irrational x and satisfies

B(x)=B(x+1)

B(x)=-\log x+xB(1/x)

for all irrational x between 0 and 1.

gollark: Also also, "convention over configuration" being stupid. Yes, the choice of four spaces vs two isn't too significant, but being able to choose means you'll have code you can possibly read a bit more easily, and also public/privateness via *capitalization* just (in my opinion) looks ugly and is annoying if you want to change privacy.

gollark: i.e. generic slices/maps/channels but not actual generics, == being ***maaaaagic*** (admittedly like in most languages, I think), and `make`/`new`.

gollark: Also, as well as that, how it just special-cases stuff instead of implementing reusable solutions.

gollark: e.g. no map function existing or even being possible means that you have *readable* code with a for loop, but it's harder to understand *why that's there* and *what it's for*.

gollark: The main problem I have with it is that it conflates readability (you can see what the code is doing at a low level) with comprehensibility (you know what and why it's doing at a higher one).

References

Brjuno, Alexander D. (1971), "Analytic form of differential equations. I, II", Trudy Moskovskogo Matematičeskogo Obščestva, 25: 119–262, ISSN 0134-8663, MR 0377192
Lee, Eileen F. (Spring 1999), "The structure and topology of the Brjuno numbers" (PDF), Proceedings of the 1999 Topology and Dynamics Conference (Salt Lake City, UT), Topology Proceedings, 24, pp. 189–201, MR 1802686
Marmi, Stefano; Moussa, Pierre; Yoccoz, Jean-Christophe (2001), "Complex Brjuno functions", Journal of the American Mathematical Society, 14 (4): 783–841, doi:10.1090/S0894-0347-01-00371-X, ISSN 0894-0347, MR 1839917
Yoccoz, Jean-Christophe (1995), "Théorème de Siegel, nombres de Bruno et polynômes quadratiques", Petits diviseurs en dimension 1, Astérisque, 231, pp. 3–88, MR 1367353

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