In
mathematics, a Brjuno number (sometimes spelled Bruno or Bryuno) is a special type of
irrational number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
named for Russian mathematician
Alexander Bruno, who introduced them in .
Formal definition
An
irrational number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
is called a Brjuno number when the infinite sum
:
converges to a finite number.
Here:
*
is the denominator of the th
convergent of the
continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...
expansion of
.
*
is a
Brjuno function
Importance
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
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
are
linearizable if
is a Brjuno number. showed in 1987 that this condition is also necessary, and for quadratic polynomials is necessary and sufficient.
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 1 + \log_2(d) ~) no pair of integers ~(\,p,\,q\,)~ exists ...
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 by r ...
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 (e.g. ). The set of all rat ...
s.
Brjuno function
Brjuno sum
The Brjuno sum or Brjuno function
is
:
where:
*
is the denominator of the th
convergent of the
continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...
expansion of
.
Real variant
![Brjuno function](https://upload.wikimedia.org/wikipedia/commons/2/22/Brjuno_function.png)
The real Brjuno function
is defined for irrational numbers
:
and satisfies
:
for all irrational
between 0 and 1.
Yoccoz's variant
Yoccoz's variant of the Brjuno sum defined as follows:
scholarpedia: Quadratic Siegel disks
/ref>
:
where:
* is irrational real number:
* is the fractional part of
* is the fractional part of
This sum converges if and only if the Brjuno sum does, and in fact their difference is bounded by a universal constant.
See also
* Transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and .
Though only a few classes ...
References
*
*
*
*
Notes
{{reflist
Dynamical systems
Number theory