Brjuno number

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

Formal definition[edit]

An irrational number is called a Brjuno number when the infinite sum

converges to a finite number. Here is the denominator of the nth convergent of the continued fraction expansion of .

Name[edit]

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[edit]

The Brjuno numbers are important in the one–dimensional analytic small divisors problems. Bruno showed that germs of holomorphic functions with linear part are linearizable if is a Brjuno number. Jean-Christophe Yoccoz (1995) showed in 1987 that this condition is also necessary for quadratic polynomials. For other germs the question is still open.

Properties[edit]

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[edit]

The real Brjuno function is defined for irrational x and satisfies

for all irrational x between 0 and 1.

See also[edit]

References[edit]

  • 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