In mathematics, a Brjuno number is a special type of irrational number.
An irrational number is called a Brjuno number when the infinite sum
converges to a finite number
- is the denominator of the nth convergent of the continued fraction expansion of .
- is a Brjuno function
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 are linearizable if 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.
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 real Brjuno function is defined for irrational x and satisfies
- for all irrational x between 0 and 1.
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- 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