Brjuno number

In mathematics, a Brjuno number (sometimes spelled Bruno or Bryuno) is a special type of irrational number named for Russian mathematician Alexander Bruno, who introduced them in Brjuno (1971).

Formal definition[edit]

An irrational number $\alpha$ is called a Brjuno number when the infinite sum

B(\alpha )=\sum _{n=0}^{\infty }{\frac {\log q_{n+1}}{q_{n}}}

converges to a finite number.

Here:

$q_{n}$ is the denominator of the $n$ th convergent ${\tfrac {p_{n}}{q_{n}}}$ of the continued fraction expansion of $\alpha$ .
$B$ is a Brjuno function

Importance[edit]

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 $e^{2\pi i\alpha }$ are linearizable if $\alpha$ 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.

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 $n$ th convergent. Thus, in contrast to the Liouville numbers, they do not have unusually accurate diophantine approximations by rational numbers.

Brjuno function[edit]

Brjuno sum[edit]

The Brjuno sum or Brjuno function $B$ is

B(\alpha )=\sum _{n=0}^{\infty }{\frac {\log q_{n+1}}{q_{n}}}

where:

$q_{n}$ is the denominator of the $n$ th convergent ${\tfrac {p_{n}}{q_{n}}}$ of the continued fraction expansion of $\alpha$ .

Real variant[edit]

The real Brjuno function $B(\alpha )$ is defined for irrational numbers $\alpha$ ^[1]

B:\mathbb {R} \setminus \mathbb {Q} \to \mathbb {R} \cup \{+\infty \}

and satisfies

{\begin{aligned}B(\alpha )&=B(\alpha +1)\\B(\alpha )&=-\log \alpha +\alpha B(1/\alpha )\end{aligned}}

for all irrational $\alpha$ between 0 and 1.

Yoccoz's variant[edit]

Yoccoz's variant of the Brjuno sum defined as follows:^[2]

Y(\alpha )=\sum _{n=0}^{\infty }\alpha _{0}\cdots \alpha _{n-1}\log {\frac {1}{\alpha _{n}}},