Brjuno number

In mathematics, a Brjuno number is an irrational number α such that

$\sum_{n=0}^\infty \frac{\log q_{n+1}}{q_n} <\infty$

where p_n/q_n are the convergents of the continued fraction expansion of α. 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.

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

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