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 pn/qn 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 (n + 1)st convergent is exponentially larger than 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 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.

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.

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