---
tag: YDK77Q21X3
kind: definition
name: Brjuno number
msc: [37F50, 11J70]
defines: 5WR304YRST
uses: [0DZXAXEJQ9, 04P2RJ4WPS, 0D474C7ES1, KNDYA9T8ZK]
sources:
  - url: https://en.wikipedia.org/wiki/Brjuno_number
---

## Definition

Let $\alpha$ be an [[irrational number|0DZXAXEJQ9]].
Let $p_n / q_n$, for $n \ge 0$, be the [[convergents|0D474C7ES1]] of the [[continued fraction|04P2RJ4WPS]] expansion of $\alpha$, with $q_n \ge 1$.
The number $\alpha$ is a *Brjuno number* if
$$\sum_{n = 0}^{\infty} \frac{\log q_{n+1}}{q_n} < \infty.$$

## Remarks

Example: for the golden mean $\alpha = (\sqrt{5} - 1)/2$, the $q_n$ are Fibonacci numbers.
They satisfy $q_{n+1} \le 2 q_n$ and $q_n \ge (3/2)^{n-1}$ for $n \ge 1$.
So the terms are at most $\log(2 q_n) / q_n$, which decay geometrically, and the series converges.
Non-example: choose partial quotients $a_n \ge 1$ inductively with $a_{n+1} \ge e^{q_n}$.
Then $q_{n+1} \ge a_{n+1} q_n \ge e^{q_n}$, so each term is at least $1$ and the series diverges.
Every Diophantine number, and hence almost every real number, is a Brjuno number.
Brjuno proved that if $\alpha$ is a Brjuno number, every germ $f(z) = e^{2\pi i \alpha} z + O(z^2)$ is linearizable at $0$.
Yoccoz proved that for the quadratic germ $e^{2\pi i \alpha} z + z^2$ the condition is also necessary.
So $e^{2\pi i \alpha} z + z^2$ has a [[Siegel disc|KNDYA9T8ZK]] around $0$ exactly when $\alpha$ is a Brjuno number.