HOME

TheInfoList



OR:

In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a Brjuno number (sometimes spelled Bruno or Bryuno) is a special type of
irrational number In mathematics, the irrational numbers are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, ...
named for Russian mathematician Alexander Bruno, who introduced them in .


Formal definition

An
irrational number In mathematics, the irrational numbers are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, ...
\alpha is called a Brjuno number when the infinite sum :B(\alpha) = \sum_^\infty \frac converges to a finite number. Here: * q_n is the denominator of the th convergent \tfrac of the continued fraction expansion of \alpha. * B is a Brjuno function


Examples

Consider the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \fr ...
: :\phi = \frac = 1+\cfrac. Then the ''n''th convergent \frac can be found via the
recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
: :\begin p_n = p_ + p_ & \text p_0=1,p_1=2, \\ q_n = q_ + q_ & \text q_0=q_1=1. \end It is easy to see that q_ for n \ge 2, as a result :\frac < \frac \text n \ge 2 and since it can be proven that \sum_^\infty \frac < \infty 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 = _0,a_1,a_2,\ldots/math> with (a_n) defined as :a_n = \begin 10 & \text n = 0,1, \\ q_n^ & \text n \ge 2. \end Then q_>q_n^\frac, so we have by the
ratio test In mathematics, the ratio test is a convergence tests, test (or "criterion") for the convergent series, convergence of a series (mathematics), series :\sum_^\infty a_n, where each term is a real number, real or complex number and is nonzero wh ...
that \sum_^\infty \frac 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 In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deriv ...
with linear part e^ 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 ()th convergent is exponentially larger than that of the th convergent. Thus, in contrast to the
Liouville number In number theory, a Liouville number is a real number x with the property that, for every positive integer n, there exists a pair of integers (p,q) with q>1 such that :0<\left, x-\frac\<\frac. The inequality implies that Liouville numbers po ...
s, they do not have unusually accurate
diophantine approximation In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria. The first problem was to know how well a real number can be approximated ...
s by
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...
s.


Brjuno function


Brjuno sum

The Brjuno sum or Brjuno function B is :B(\alpha) = \sum_^\infty \frac where: * q_n is the denominator of the th convergent \tfrac of the
continued fraction A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, ...
expansion of \alpha.


Real variant

The real Brjuno function B(\alpha) is defined for irrational numbers \alpha : B : \R \setminus \Q \to \R \cup \ and satisfies :\begin B(\alpha) &= B(\alpha+1) \\ B(\alpha) &= - \log \alpha + \alpha B(1/\alpha) \end for all irrational \alpha between 0 and 1.


Yoccoz's variant

Yoccoz's variant of the Brjuno sum defined as follows:scholarpedia: Quadratic Siegel disks
/ref> :Y(\alpha)=\sum_^ \alpha_0\cdots \alpha_ \log \frac, where: * \alpha is irrational real number: \alpha\in \R \setminus \Q * \alpha_0 is the fractional part of \alpha * \alpha_ is the fractional part of \frac This sum converges if and only if the Brjuno sum does, and in fact their difference is bounded by a universal constant.


See also

*
Irrationality measure In mathematics, an irrationality measure of a real number x is a measure of how "closely" it can be Diophantine approximation, approximated by Rational number, rationals. If a Function (mathematics), function f(t,\lambda) , defined for t,\lambd ...
*
Markov constant In number theory, specifically in Diophantine approximation theory, the Markov constant M(\alpha) of an irrational number \alpha is the factor for which Dirichlet's approximation theorem can be improved for \alpha. History and motivation Cert ...


References

* * * *


Notes

{{reflist Dynamical systems Number theory