Brjuno number

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

Formal definition

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

${\displaystyle B(\alpha )=\sum _{n=0}^{\infty }{\frac {\log q_{n+1}}{q_{n}}}}$

converges to a finite number

Here:

• ${\displaystyle q_{n}}$ is the denominator of the nth convergent ${\displaystyle {\frac {p_{n}}{q_{n}}}}$ of the continued fraction expansion of ${\displaystyle \alpha }$.
• ${\displaystyle B}$ 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 ${\displaystyle e^{2\pi i\alpha }}$ are linearizable if ${\displaystyle \alpha }$ 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 ${\displaystyle B(x)}$ is defined for irrational x and satisfies

${\displaystyle B(x)=B(x+1)}$

${\displaystyle B(x)=-\log x+xB(1/x)}$ for all irrational x between 0 and 1.

See also

• Transcendental number

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