Brjuno number

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

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 ${p_{n}}/{q_{n}}$ of the continued fraction expansion of $\alpha$ .

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 showed that germs of holomorphic functions with linear part e^2πiα are linearizable if α is a Brjuno number. Yoccoz (1995) showed in 1987 that this condition is also necessary for quadratic polynomials. For other germs the question is still open.

Properities

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.

References

Brjuno, A. 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