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 eiα 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[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 B(x) is defined for irrational x and satisfies

for all irrational x between 0 and 1.

References[edit]


See also[edit]