Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Brjuno function continuously estimates the size of quadratic Siegel
  disks   If alpha is an irrational number, we define Yoccoz's Brjuno function Phi by
Phi(alpha)=sum_{n geq 0} alpha_0*alpha_1*...*alpha_{n-1}*log(1/alpha_n), where
alpha_0 is the fractional part of alpha and alpha_{n+1} is the fractional part
of 1/alpha_n. The numbers alpha such that Phi(alpha)<infty are called the
Brjuno numbers. The quadratic polynomial P_alpha:z -> e^{2i pi alpha}z+z^2 has
an indifferent fixed point at the origin. If P_alpha is linearizable, we let
r(alpha) be the conformal radius of the Siegel disk and we set r(alpha)=0
otherwise. Yoccoz proved that Phi(alpha)=infty if and only if r(alpha)=0 and
that the restriction of alpha -> Phi(alpha)+log r(alpha) to the set of Brjuno
numbers is bounded from below by a universal constant. We proved that it is
also bounded from above by a universal constant. In fact, Marmi, Moussa and
Yoccoz conjecture that this function extends to $R$ as a H\"older function of
exponent 1/2. In this article, we prove that there is a continuous extension to
$R$.

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The categories of this paper are: chem-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: alg-geom
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The categories of this paper are: comp-gas