In mathematics, a Brjuno number is a special type of irrational numbers
Contents
Formal definition
An irrational number
converges to a finite number. Here
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 e2πiα 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