Brjuno number

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title: "Brjuno number" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["dynamical-systems", "number-theory"] description: "Special type of irrational number" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Brjuno_number" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Special type of irrational 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 .

Formal definition

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 nth convergent \tfrac{p_n}{q_n} of the continued fraction expansion of \alpha.
• B is a Brjuno function

Examples

Consider the golden ratio : :\phi = \frac{1+\sqrt{5}}{2} = 1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{\ddots}}}}. Then the nth convergent \frac{p_n}{q_n} can be found via the recurrence relation: :\begin{cases} p_n = p_{n-1} + p_{n-2} & \text{ with } p_0=1,p_1=2, \ q_n = q_{n-1} + q_{n-2} & \text{ with } q_0=q_1=1. \end{cases} It is easy to see that q_{n+1} for n \ge 2, as a result :\frac{\log{q_{n+1}}}{q_n} and since it can be proven that \sum_{n=0}^\infty \frac{\log q_n}{q_n} for any irrational number, is a Brjuno number. Moreover, a similar method can be used to prove that any irrational number whose continued fraction expansion ends with a string of 1's is a Brjuno number.

By contrast, consider the constant \alpha = [a_0,a_1,a_2,\ldots] with (a_n) defined as :a_n = \begin{cases} 10 & \text{ if } n = 0,1, \ q_n^{q_{n-1}} & \text{ if } n \ge 2. \end{cases} Then q_{n+1}q_n^\frac{2q_n}{q_{n-1}}, so we have by the ratio test that \sum_{n=0}^\infty \frac{\log q_{n+1}}{q_n} diverges. \alpha is therefore not a Brjuno number.

Importance

The Brjuno numbers are important in the one-dimensional analytic small divisors problems. Bruno improved the diophantine condition in Siegel's Theorem by showing that germs of holomorphic functions with linear part e^{2\pi i \alpha} are linearizable if \alpha is a Brjuno number. showed in 1987 that Brjuno's condition is sharp; more precisely, he proved that for quadratic polynomials, this condition is not only sufficient but also necessary for linearizability.

Properties

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

Brjuno sum

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 nth convergent \tfrac{p_n}{q_n} of the continued fraction expansion of \alpha.

Real variant

::figure[src="https://upload.wikimedia.org/wikipedia/commons/2/22/Brjuno_function.png" caption="Brjuno function"] ::

The real Brjuno function B(\alpha) is defined for irrational numbers \alpha : B : \R \setminus \Q \to \R \cup { +\infty }

and satisfies

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

for all irrational \alpha between 0 and 1.

Yoccoz's variant

Yoccoz's variant of the Brjuno sum defined as follows:

:Y(\alpha)=\sum_{n=0}^{\infty} \alpha_0\cdots \alpha_{n-1} \log \frac{1}{\alpha_n},

where:

• \alpha is irrational real number: \alpha\in \R \setminus \Q
• \alpha_0 is the fractional part of \alpha
• \alpha_{n+1} is the fractional part of \frac{1}{\alpha_n} This sum converges if and only if the Brjuno sum does, and in fact their difference is bounded by a universal constant.

References

• {{citation | last = Lee | first = Eileen F. | contribution = The structure and topology of the Brjuno numbers | contribution-url = http://topology.nipissingu.ca/tp/reprints/v24/tp24114.pdf | mr = 1802686 | pages = 189–201 | series = Topology Proceedings | title = Proceedings of the 1999 Topology and Dynamics Conference (Salt Lake City, UT) | volume = 24 | date = Spring 1999}}
• {{citation | last = Yoccoz | first = Jean-Christophe | authorlink = Jean-Christophe Yoccoz | contribution = Théorème de Siegel, nombres de Bruno et polynômes quadratiques | mr = 1367353 | pages = 3–88 | series = Astérisque | title = Petits diviseurs en dimension 1 | volume = 231 | year = 1995}}

Notes

References

1. [https://arxiv.org/abs/math/9912018v1 Complex Brjuno functions by S. Marmi, P. Moussa, J.-C. Yoccoz ]
2. [http://www.scholarpedia.org/article/Siegel%20disks/Quadratic%20Siegel%20disks scholarpedia: Quadratic Siegel disks]

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