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%
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%**************** MAC.TEX ***********************************
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% origine: harvmac + modifications J. Zinn-Justin
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%****************************
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%  End definition of Euler Fraktur font.

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% *******************************************************************
%       Dictionnaires francais et anglais
\def\dicof{
\gdef\Resume{RESUME}
\gdef\Toc{Table des mati\`eres}
\gdef\soumisa{Soumis \`a:}
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\def\dicoa{
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\gdef\Toc{Table of Contents}
\gdef\soumisa{Submitted to}
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% ****** extrait de definit.tex (obsolete ?)
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% ********* A few math symbols
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\def\Re{\mathop{\rm Re}\nolimits}
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\def\tr{\mathop{\rm tr}\nolimits}
\def\frac#1#2{{\textstyle{#1\over#2}}}
\def\today{\number\day/\number\month/\number\year}
\def\leaderfill{\leaders\hbox to 1em{\hss.\hss}\hfill}
% ******************** LOGOS **********************************************
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\def\spht{
\centerline{CEA, Service de Physique Th\'eorique, CE-Saclay}
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\centerline{CEA/DSM/DRECAM/Service de Physique de l'Etat Condens\'e}
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% ************** double alignment in eqalignno style **********************
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%***************************************************************************
%********* titlepage, headline, section, subsection, sub, appendix *********
%***************************************************************************
%********* introduce equation number file: for non-causal quotation
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%
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% ******************* titlepage **********************************
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% **************** beginning
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% ***************** input table of contents
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% **************************** \subsection *************************
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\edef\ecrire{\write\toc{\string\par\string\itemitem
{\prefix\the\nosection.\the\nosubsection} {#1}
\string\leaderfill{\noexpand\number\pageno}}}\ecrire
}
%
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% ?????
\def\sub#1{\medskip\vskip\parskip
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%*********
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\global\newcount\refno \global\refno=1
\newwrite\rfile
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\def\ref{[\the\refno]\nref}
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%
\def\semi{;\hfil\break}
\def\addref#1{\immediate\write\rfile{\noexpand\item{}#1}} %now unnecessary
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\vskip0pt plus.1\vsize\penalty-100\vskip0pt plus-.1
\vsize\bigskip\vskip\parskip\centerline{{\bf References}}\bigskip%
{\frenchspacing%
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\nonfrenchspacing
\fi}
%
\def\startrefs#1{\immediate\openout\rfile=\jobname.ref\refno=#1}
%
\def\xref{\expandafter\xr@f}\def\xr@f[#1]{#1}
\def\refs#1{[\r@fs #1{\hbox{}}]}
\def\r@fs#1{\ifx\und@fined#1\message{reflabel \string#1 is undefined.}%
\xdef#1{(?.?)}\fi \edef\next{#1}\ifx\next\em@rk\def\next{}%
\else\ifx\next#1\xref#1\else#1\fi\let\next=\r@fs\fi\next}
%************************
%
\newwrite\lfile
{\escapechar-1\xdef\pctsign{\string\%}\xdef\leftbracket{\string\{}
\xdef\rightbracket{\string\}}\xdef\numbersign{\string\#}}
\def\writedefs{\immediate\openout\lfile=labeldef.tmp \def\writedef##1{%
\immediate\write\lfile{\string\def\string##1\rightbracket}}}
%
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\the\pageno\string\startrefs\leftbracket\the\refno\rightbracket%
\string\def\string\secsym\leftbracket\secsym\rightbracket%
\string\secno\the\secno\string\meqno\the\meqno}\immediate\closeout\lfile}}
%
\def\writestoppt{}\def\writedef#1{}
%*************************************************************************
%Macro de numerotation automatique
%*************************************************************************
% numbering without naming
\def\eqnn{\global\advance\neqno by 1 \ifinner\relax\else%
\eqno\fi(\eqprefix\the\neqno)}
%
% numbering and attaching a name: \eqnd{\ename}
\def\eqnd#1{\global\advance\neqno by 1 \ifinner\relax\else%
\eqno\fi(\eqprefix\the\neqno)\eqlabel#1
{\xdef#1{($\eqprefix\the\neqno$)}}
\edef\ewrite{\write\sym{\string\def\string#1{($\eqprefix%
\the\neqno$)}}%
}\ewrite%
}
%
% for eqalignno, allows (1a) (1b)...
\def\eqna#1{\wrlabel#1\global\advance\neqno by1
{\xdef #1##1{\hbox{$(\eqprefix\the\neqno##1)$}}}
\edef\ewrite{\write\sym{\string\def\string#1{($\eqprefix%
\the\neqno$)}}%
}\ewrite%
}
%
\def\em@rk{\hbox{}}
\def\xeqn{\expandafter\xe@n}\def\xe@n(#1){#1}
\def\xeqna#1{\expandafter\xe@na#1}\def\xe@na\hbox#1{\xe@nap #1}
\def\xe@nap$(#1)${\hbox{$#1$}}
% \eqns allows to quote several equations, suppressing unnecessary ()
\def\eqns#1{(\e@ns #1{\hbox{}})}
\def\e@ns#1{\ifx\und@fined#1\message{eqnlabel \string#1 is undefined.}%
\xdef#1{(?.?)}\fi \edef\next{#1}\ifx\next\em@rk\def\next{}%
\else\ifx\next#1\xeqn#1\else\def\n@xt{#1}\ifx\n@xt\next#1\else\xeqna#1\fi
\fi\let\next=\e@ns\fi\next}
%*************************** figure macros ****************************
\def\fig{fig.~\the\figno\nfig}
\def\nfig#1{\xdef#1{\the\figno}%
\immediate\write\sym{\string\def\string#1{\the\figno}}%
\global\advance\figno by1}%
\def\xfig{\expandafter\xf@g}\def\xf@g fig.\penalty\@M\ {}%
\def\figs#1{figs.~\f@gs #1{\hbox{}}}%
\def\f@gs#1{\edef\next{#1}\ifx\next\em@rk\def\next{}\else%
\ifx\next#1\xfig #1\else#1\fi\let\next=\f@gs\fi\next}%
%
\long\def\figure#1#2#3{\midinsert
#2\par
{\elevenpoint
\setbox1=\hbox{#3}
\ifdim\wd1=0pt\centerline{{\bf Figure\ #1}\hskip7.5mm}%
\else\setbox0=\hbox{{\bf Figure #1}\quad#3\hskip7mm}
\ifdim\wd0>\hsize{\narrower\noindent\unhbox0\par}\else\centerline{\box0}\fi
\fi}
\wrlabel#1\par
\endinsert}
%*************************** table macros ****************************
\def\tab{table~\uppercase\expandafter{\romannumeral\the\tabno}\ntab}
\def\ntab#1{\xdef#1{\the\tabno}
\immediate\write\sym{\string\def\string#1{\the\tabno}}
\global\advance\tabno by1}
\long\def\table#1#2#3{\topinsert
#2\par
{\elevenpoint
\setbox1=\hbox{#3}
\ifdim\wd1=0pt\centerline{{\bf Table
\uppercase\expandafter{\romannumeral#1}}\hskip7.5mm}%
\else\setbox0=\hbox{{\bf Table
\uppercase\expandafter{\romannumeral#1}}\quad#3\hskip7mm}
\ifdim\wd0>\hsize{\narrower\noindent\unhbox0\par}\else\centerline{\box0}\fi
\fi}
\wrlabel#1\par
\endinsert}
%***********************************************************************
\catcode`@=12
\def\draftend{\immediate\closeout\sym\immediate\closeout\toc
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\smalltrue
\draftstart
\preprint{T95/028}
\title{The Brjuno functions and their regularity properties}
\authorname{S. Marmi}
\address{\centerline{Dipartimento di Matematica ``U. Dini'', Universit\`a di
Firenze}
\centerline{50134 Firenze, ITALY}
}
\authorname{P. Moussa}
\address{\saclay}
\authorname{J.-C. Yoccoz}
\address{\centerline{Universit\'e de Paris-Sud, Math\'ematiques,}
\centerline{B\^at.425, 91405 Orsay, FRANCE}
}
\abstract
We show that various possible versions of the Brjuno function,
based on different kinds of continued fraction developments, are all
equivalent and we study their regularity ($L^p$, BMO and H\"older)
properties, through a systematic analysis of the functional
equation which they fulfill.
\endabstract
\vfill
\submitted{Communications in Mathematical Physics}
\eject
\input amssym.def
\input amssym.tex
\magnification 1200
\pageno=1

\catcode`\@=11

\hsize=125 mm   \vsize =187mm
\hoffset=4mm    \voffset=10mm
\pretolerance=500 \tolerance=1000 \brokenpenalty=5000

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\auteurcourant{Marmi, Moussa, Yoccoz}
\titrecourant{The Brjuno functions and their regularity
properties}
\def\mois{\ifcase\month\or January\or February\or March\or April\or
May\or June\or July\or August\or September\or October\or November\or
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%\Date
\medskip
\centerline{\tit The Brjuno functions and their regularity properties}
\bigskip
\centerline{S. Marmi\footnote{$^1$}{Dipartimento di Matematica ``U. Dini'',
Universit\`a di Firenze, Viale Morgagni 67$/$A, 50134 Firenze, Italy},
P. Moussa\footnote{$^2$}{Service de Physique Th\'eorique, CEA, C.E.
Saclay, 91191 Gif-Sur-Yvette, France}
and J.-C. Yoccoz\footnote{$^2$}{Universit\'e de Paris-Sud,
Math\'ematiques. B\^at. 425, 91405-Orsay, France}}
\vskip 2. truecm
\centerline{\bf Abstract}
We show that various possible versions of the Brjuno function,
based on different kinds of continued fraction developments, are all
equivalent and we study their regularity ($L^p$, BMO and H\"older)
properties, through a systematic analysis of the functional
equation which they fulfill.
\vskip 1. truecm
\beginsection{\bf 0. Introduction}\par
When an irrational rotation is analytically perturbed,
it is a natural question to ask whether or not there exists a neighborhood of
the fixed point where the dynamics looks like the unperturbed case.
More precisely, does there exist a local  holomorphic coordinate for which
the perturbed transformation is expressed as an ordinary rotation?
When the rotation number satisfies the Brjuno condition [Br],
such a coordinate exists in a domain called a Siegel disk.
The Brjuno function tells more:
it gives an estimate of minus the logarithm of the size of the Siegel disks
as a function of the rotation number [Yo]. A similar result also holds
in the simpler case of certain complex area--preserving maps
[Ma, Da].
In this work, we first analyze the relation between the Brjuno function
and the various kind of continued fractions. We establish the functional
equation fulfilled by the Brjuno function, and show that its solution
requires the inversion of an operator $T$. We estimate the value of
the spectral radius of the operator $T$
for  all $L^p$ norms, and also for the BMO
(Bounded Mean Oscillation) space.
The Brjuno function is obtained as the action of $(1-T)^{-1}$ on
a logarithmic function which belongs to  the BMO--space. Since
the spectral
radius of $T$ is smaller than one,
the Brjuno function is also in this space.
Noticing that the adjoint of $T$ is nothing else than the
Ruelle-Frobenius-Perron operator associated to the dynamical system which
generates the continued fraction, the identification of the space
adapted to $T$ seems to us promising for the study of  dynamical
properties.  Finally, the action of $T$ on continuous
functions is described with respct to H\"older-continuity properties.
We show that regular perturbations (at least $C^{1/2}$) of the
logarithmic term do modify the solution only by a $C^{1/2}$
contribution, so that the most singular part remains unchanged.
We anticipate that  this result might be much  more general:
it might happen that
the geometric renormalisation for holomorphic dynamical systems
produces only $C^1$ perturbations to the renormalisation
equation, and the most singular part of minus the logarithm of
the size of the stability domains as function of the rotation number
could be universally (that is modulo $C^{1/2}$) described by the Brjuno
function. This is in agreement with the numerical
results of [Ma]. A further motivation  for the present study,
and particularly the BMO-space results, is the problem of building
a complex analytic extension of the Brjuno function.
This will be the suject of a subsequent paper.\par
One of us (S. M.) wishes to thank the Italian CNR for financial
support, and A. Beretti and S. Isola for useful discussions.
Part of this work was made during a visit of the
second author (P. M.), who  thanks  the Department
of Mathematics `U. Dini' of the University of Florence and the
INFN for hospitality and  financial support.\par
\vskip 1. truecm
\beginsection{\bf 1. On a family of continued fraction
transformations}\par
Let $\alpha \in [1/2,1]$ and let $x \in \Bbb R$. We define
$$
[x]_\alpha = \min \{ p \in \Bbb Z \mid x < \alpha + p\}\; , \eqno(1.1)
$$
that is
$$
[x]_\alpha = p \hbox{ iff } \;\;\;\alpha - 1 + p \le x < \alpha + p \; .
$$
Note that
$$
 [x]_\alpha = [x-\alpha +1]\; ,
$$
where $[\;]=[\;]_1$ denotes the usual integer part of a real number.
We will consider the iteration of
$$
A_\alpha : (0,\alpha) \mapsto [0,\alpha] \eqno(1.2)
$$
defined by
$$
A_\alpha (x) = \left|\ {\ 1\ \over x} -
\left[\ {\ 1\ \over x}\ \right]_\alpha\ \right|\;, \eqno(1.3)$$
with branches
$$\eqalign{
A_{\al}(x)=&{1\over x}-k \quad \hbox{for}\quad{1\over
k+\al}<x\le{1\over k}\cr
A_{\al}(x)=&k- {1\over x} \quad \hbox{for}\quad{1\over
k}<x\le{1\over k+\al -1}\;\; .\cr }$$
Let
$$
G = {\sqrt{5}+1\over 2}\; , \;\;\; \Gamma = \sqrt{2}+1
$$
and
$$
g = G^{-1} = {\sqrt{5}-1\over 2} \; , \;\;\; \gamma =\Gamma^{-1}=
\sqrt{2}-1\; .
$$
\Proc{Theorem 1.1.}{The dynamical system defined by the iteration of
(1.3) preserves an absolutely continuous (w.r.t. Lebesgue) probability
measure $m_\alpha$ with density $c_\alpha \rho_\alpha (x)$:
$m_\alpha (dx) = c_\alpha \rho_\alpha (x) dx$. The density is  given by :
\item{(i)} if $g\le \alpha \le 1$, then
$$0 \le 1-\alpha \le A_{\alpha}(\alpha)=\alpha^{-1}-1\le\alpha\ ,$$ and
$$
\eqalign{ c_\alpha &= {1\over \log (1 + \alpha)} \; , \cr
            \rho_\alpha (x) & = {1 \over 1+x}
            \chi_{(\alpha^{-1}-1 ,\alpha)}(x) +
                                {1 \over 2+x}
            \chi_{(1-\alpha, \alpha^{-1}-1]} (x) \cr
                            & + \left( {1 \over 2+x} + {1 \over 2-x}\right)
            \chi_{(0,1 - \alpha]} (x) \; , \cr}
            \eqno(1.4)
$$
\item{(ii)} if $1-\gamma \le \alpha <g$, then
$$0 \le A_{\al}(\al)=2-\alpha^{-1} < 1 -\al\le A_{\al}(\al-1)=
(1-\al)^{-1}-2<\al\ ,$$ and
$$
\eqalign{ c_\alpha &= {1\over \log G} \cr
            \rho_\alpha (x) & = {1 \over G+x}
            \chi_{((1-\alpha)^{-1}-2,\alpha)}(x) +
                                {1 \over 2+x}
            \chi_{(1-\alpha , (1-\alpha)^{-1}-2]} (x) \cr
                    & + \left( {1 \over x+2} + {1 \over G+1-x} \right)
            \chi_{(2-\alpha^{-1},1 - \alpha]} (x) \cr
                    & + \left( {1 \over x+2} + {1 \over 2-x} \right)
            \chi_{(0,2 -\alpha^{-1}]} (x) \; , \cr}
            \eqno(1.5)
$$
\item{(iii)} if ${1 \over 2} \le \alpha < 1-\gamma $, then
$$0 \le A_{\al}(\al)=2-\alpha^{-1} \le A_{\al}(\al-1)=
(1-\al)^{-1}-2<1-\al\le \al\ ,$$ and
$$
\eqalignno{ c_\alpha =& {1\over \log G} \; , \cr
            \rho_\alpha (x)  = &{1 \over G+x} \chi_{(1-\alpha,\alpha)}(x) +
            \left( {1 \over G+x} + {1 \over G+1-x}\right)
            \chi_{((1-\alpha)^{-1}-2 , 1-\alpha]} (x) \cr
                     +& \left( {1 \over x+2}\right. + \left.{1 \over G+1-x}
            \right) \chi_{( 2-\alpha^{-1}, (1-\alpha)^{-1}-2 ]} (x)\cr
	    +& \left( {1 \over x+2} + {1 \over 2-x}
            \right) \chi_{(0,2- \alpha^{-1}]} (x) \; ,\cr
            &&(1.6)\cr}
$$
where $\chi_{(a,b)} (x)$ denotes the
characteristic function of the interval $(a,b)$.}
\par
\medskip
The theorem can be proven by direct calculations, which are very tedious
for general values of $\alpha$. It is  much easier to derive it from
similar results
of Nakada [Na], who considers the map $\tilde{A}_\alpha = \left| {1\over x}
\right| - \left[\left|{1\over x}\right|\right]_\alpha$, instead of
$A_\alpha = |\tilde{A}_\alpha |$.
Note that if $\alpha =1$, one recovers Gauss' result:
$c_1\rho_1(x) = {\dst1\over\dst (1+x) \log 2 }$, and if
$\al=1/2$, one finds
$c_{1/2}\rho_{1/2}(x) = {\dst1\over\dst \log G }\left({\dst1\over\dst
G+x}+{\dst 1\over\dst G+1-x} \right)$.
\par\medskip
\remark{1.2.} From the above theorem, it is obvious
that the density of the measure $m_{\al}$
is bounded above and below,
so that $m_{\al}$ and the  Lebesgue measure are equivalent.
More precisely, bounding each term of the type
$(b\pm x)^{-1}$ by its values at $0$ or $1$, one easily
shows that there exists a constant $c_0$ such that
for any $\al\in [1/2,1]$,
$$c_0^{-1}\le c_{\al} \rho_{\al}\le c_0\; .\eqno(1.7)$$
One finds
$$c_0=2/\log G=4.156...\; ,\eqno(1.8)$$
which can be reduced to $1/\log G$ when $\al=1/2$.
\par
\medskip
\remark{1.3.} It is well known that the Kolmogorov-Sinai entropy
of these maps is given by
$$
h(\alpha ) = -2\int_0^\alpha c_\alpha \rho_\alpha (x)\log x\,dx \; .
$$
If $\alpha =1$ one obtains
$$
h(1) = {\pi^2\over 6\log 2}\; ,
$$
for $\alpha =1/2$ one obtains,
$$
h(1/2) = {\pi^2\over 6\log G}\; ,
$$
a result already given by Rieger [Ri].
For general $\alpha$, we find that
$c_\alpha h(\alpha )$ does not depend on $\alpha$,
thus
$$
h(\alpha ) = \cases{ {\dst\pi^2\over\dst 6\log (1+\alpha )} &if
$g\le \alpha \le 1\;$, \cr
{\dst\pi^2\over\dst 6\log G} & if $1/2\le \alpha\le g\;$,
\cr}
$$
as already shown by  Nakada [Na]. Using the terminology of the thermodynamical
formalism for dynamical systems, these results show the existence of a
phase transition at $\alpha = g$.
\par
\medskip
To each $x \in \Bbb R \setminus \Bbb Q$ we associate a continued fraction
expansion by iterating $A_\alpha$ as follows. Let
$$
\eqalign{x_0 & = | x - [x]_\alpha| \cr
            a_0 & = [x]_\alpha \cr} \eqno(1.9)
$$
then one obviously has
$$
x_0 = a_0 + \varepsilon_0 x_0\quad\hbox{ where }\quad
\varepsilon_0 = \cases{ +1 \hbox{ iff } x \ge [x]_\alpha \cr
                          -1 \hbox{ otherwise }\; . \cr} \eqno(1.10)
$$
We now define inductively for all $n \ge 0$
$$
\eqalign{x_{n+1} & = A_\alpha(x_n) \cr
            a_{n+1} & = \left[ {1 \over x_n} \right]_\alpha  \ge 1\cr}
            \eqno(1.11)
$$
thus
$$
x_{n}^{-1} = a_{n+1} + \varepsilon_{n+1} x_{n+1}\quad\hbox{ where}
\quad\varepsilon_{n+1} =
\cases{ +1 \hbox{ iff }  x_n^{-1} \ge a_{n+1}\cr
        -1 \hbox{ otherwise }\; . \cr} \eqno(1.12)
$$
Therefore we have
$$
x=a_0 + \varepsilon_0 x_0=a_0+{\varepsilon_0 \over a_1 + \varepsilon_1
     x_1}= \ldots =a_0 + \displaystyle{\varepsilon_0 \over a_1
     + \displaystyle{\varepsilon_1  \over a_2 + \ddots +
     \displaystyle{\varepsilon_{n-1} \over a_n + \varepsilon_n x_n}}}
     \eqno(1.13)
$$
and we will write
$$
x=[(a_0,\varepsilon_0),(a_1,\varepsilon_1),\ldots ,(a_n,\varepsilon_n),
     \ldots] \;. \eqno(1.14)
$$
Note that for $\alpha = 1$ we recover the standard continued fraction
expansion defined through the iteration of Gauss' map $x \mapsto x^{-1}
\hbox{ mod } 1$, and all $\varepsilon_n = +1$. When $\alpha = 1/2$ one
has the
so-called nearest integer continued fraction, and $a_n \ge 2$ for all
$n \ge 1$.
\par
The nth-convergent is defined by
$$
{p_n \over q_n} = [(a_0,\varepsilon_0),(a_1,\varepsilon_1),\ldots ,
                     (a_n,\varepsilon_n)] =
                     a_0 + \displaystyle{\varepsilon_0 \over a_1
     + \displaystyle{\varepsilon_1  \over a_2 + \ddots +
     \displaystyle{\varepsilon_{n-1} \over a_n }}}
     \;. \eqno(1.15)
$$
and it is immediate to check that the numerators $p_n$ and denominators
$q_n$ are recursively determined by
$$
p_{-1}=q_{-2}=1 \;\;,\;\;\;p_{-2}=q_{-1}=0 \;\;,\eqno(1.16)
$$
and for all $n \ge 0$
$$
\eqalign{ p_n &= a_n p_{n-1} + \varepsilon_{n-1} p_{n-2} \; , \cr
            q_n &= a_n q_{n-1} + \varepsilon_{n-1} q_{n-2} \; . \cr}
           \eqno(1.17)
$$
Moreover
$$
\eqalignno{x &= {p_n + p_{n-1} \varepsilon_n x_n \over q_n + q_{n-1}
       \varepsilon_n x_n } &(1.18) \cr
              x_n &= - \varepsilon_n {q_n x -p_n \over q_{n-1} x - p_{n-1}}
              &(1.19) \cr
              q_n p_{n-1} - p_n q_{n-1} &= (-1)^n \varepsilon_0 \ldots
              \varepsilon_{n-1}\;\; .  &(1.20) \cr}
$$
Let
$$
\beta_n = \Pi_{i=0}^n x_i = (-1)^n \varepsilon_0 \ldots
  \varepsilon_n (q_n x - p_n)\quad\hbox{for\ }n\ge 0,\quad
  \hbox{and\ }\be_{-1}=1\;\; .\eqno(1.21)
$$
Then
$$
\eqalign{x_n &= {\beta_n \over \beta_{n-1}} \cr
            \beta_{n-2} &= a_n \beta_{n-1} + \varepsilon_n \beta_n \cr}
            \eqno(1.22)
$$
{}From the definitions given one easily proves by induction
the following proposition \par
\medskip
\Proc{Proposition 1.4.} {Given $\alpha \in [1/2,1]$, for all
$x \in \Bbb R \setminus \Bbb Q$ and for all $n \ge 1$ one has
\item{(i)}\qquad $q_{n+1} > q_n > 0$;
\item{(ii)}\qquad $p_n > 0$ when $x>0$ and $p_n< 0$ when $x<0$;
\item{(iii)}\qquad $ \left|q_n x - p_n\right|
={\dst 1\over\dst q_{n+1}+\varepsilon_{n+1}q_nx_{n+1}}$,
so that ${\dst 1\over\dst 1+\alpha}<\beta_nq_{n+1}<{\dst 1\over\dst\alpha}$\ ;
\item{(iv)}\qquad if $\alpha>g\ ,\ \beta_n\le\alpha g^n$;
\item{(v)}\qquad
if $\alpha\le g\ ,\ \beta_n\le\alpha \gamma^n$.}
\par
\medskip
\proof
One gets parts (i) and (ii)
by recursion using (1.17), in fact it is obvious only when $\al=1$.
When $\al\neq1$, one could alternatively use Lemma 1.8 below.
Part (iii) is easily obtained from (1.18).
\par
The proof of (iv) is also easy: either $x_k \le g$ for all $k=0,\ldots ,n$,
or $x_k> g$ for some $k$. Then $x_{k+1}=x_k^{-1}-1$ and
$x_{k+1}<g^{-1}-1=g$, thus $x_kx_{k+1}=1-x_k<1-g=g^2$.
In the sequence $\beta_n=x_0\cdots x_n$ one then isolates the pairs
$x_k, x_{k+1}$ such that $x_k>g$ (since for each pair $x_kx_{k+1}<g^2$).
The other terms in $\beta_n$ are all smaller or equal to $g$, except
for $x_n<\alpha$.
\par
The proof of (v) is more complicated: the result is obvious if among
$x_0,x_1,\ldots,x_n$, there are less than two of them taking values greater
than $\ga$. Otherwise, let $x_k$ and $x_{k+p+1}$  be
two successive values greater
than $\ga$. Note that $x_k>\ga$ implies $x_{k+1}=|2-x_k^{-1}|<\ga$,
therefore we must have $p\ge1$. Now statement (v) is an immediate consequence
of the following assertion which we will then prove:\par
{\it if $x_k>\gamma$, $p\ge1$,  $x_{k+1},\ldots ,x_{k+p} <\gamma$ and
$x_{k+p+1}>\gamma$ then $\Pi_{i=k}^{k+p} x_i <\gamma^{p+1}$.}\par
We divide the proof into some different cases.\par
\item{(1)} If $\ga<x_k\le 1/2$, then $x_{k+1}=x_{k}^{-1}-2$ and
$x_kx_{k+1}=1-2x_k<1-2\gamma =\gamma^2$, and the assertion holds.
By the way this closes the proof in the case $\al=1/2$.
\item{(2)} Let $x_k>1/2$, thus $x_{k+1}=2-x_k^{-1}<g^2$. Now observe
that the image of $[1/3,g^2]$ is $[0,g^2]$. If
$x_{k+1}\le1/3$ we let $m=1$ , and  if $x_{k+1}>1/3$ we let $ m\ge 2$
such that
$$
x_{k+1},\ldots ,x_{k+m-1}\in (1/3,g^2]\; , \;\;\;
x_{k+m}\in [0,1/3]\; .
$$
Note that $p\ge m\ge 1$.
\item{(2.1)} If $m\ge 4$, then $x_kx_{k+1}\cdots x_{k+m}\le
{1\over 3}g^{2m-1}$, since $x_k<g$, $x_{k+1},\ldots ,x_{k+m-1}\le g^2$ and
$x_{k+m}\le1/3$. A numerical exercise shows that for $m\ge4$,
${1\over3}g^{2m-1}<\ga^{m+1}$, and the assertion follows.
We must now consider the three cases left: $m=1$, $m=2$ and $m=3$.
\item{(2.2)} $m=1$. If $x_k<1-\gamma$, then $x_kx_{k+1}=2x_k-1<\gamma^2$;
otherwise $x_k\ge 1-\gamma$, and $x_{k+1}\ge 1-\sqrt{2}/2$
so that $2/7<x_{k+1}\le1/3$ and
$x_{k+2}=x_{k+1}^{-1}-3$,  $x_{k+1}x_{k+2}=1-3x_{k+1}$ and finally
$x_kx_{k+1}x_{k+2}=3-5x_k<5\gamma-2=\gamma^3$, which shows the
assertion.
\item{(2.3)} $m=2$, then $x_{k+2}=3-x_{k+1}^{-1}$ and
$x_kx_{k+1}x_{k+2}=5x_k-3$. If this is
smaller than $\gamma^3$, then the assertion follows. If not,
which is equivalent to assume
$x_k>\sqrt{2}-4/5$, then $x_{k+1}>(48-25\sqrt{2})/34$
and $x_{k+2}>0.3111\ldots$, so that  $x_{k+2}\in [2/7,1/3]$.
Thus $x_{k+3}=x_{k+2}^{-1}-3$ and
$x_kx_{k+1}x_{k+2}x_{k+3}=8-13x_k<8-13(\sqrt{2}-4/5)$. A numerical
exercise shows that this last number is smaller than
$\gamma^4$, which completes the
assertion in this case.
\item{(2.4)} $m=3$, then assume $x_kx_{k+1}x_{k+2}x_{k+3}=13x_k-8>\gamma^4$.
This would be  equivalent to $x_k>(25-12\sqrt{2})/13$.
However, from the definition of $m$, one gets
$x_{k+3}\le1/3$  which implies  $x_k\le21/34$, and the two inequalities
on $x_k$ are contradictory.
Thus $x_kx_{k+1}x_{k+2}x_{k+3}\le \gamma^4$, and the  proof of the
assertion is completed.
\qed
\par
\medskip
\remark{1.5.} From {\it (iii)} and {\it (iv)} one gets
if $\alpha>g\ ,\ q_n\ge{\dst1
\over\dst\alpha(1+\alpha)}G^{n-1}$, and similarly, from {\it (iii)}
and {\it (v)}  if $\alpha\le g\ ,\ q_n
\ge{\dst1\over\dst\alpha(1+\alpha)}\Gamma^{n-1}$.
\par
\medskip
\remark{1.6.} From {\it (iii)} one gets
$${1\over 2q_nq_{n+1}}< {1 \over q_n (q_n + q_{n+1})}\le
 {1 \over q_n (\alpha q_n + q_{n+1})}< \left| x - {p_n \over q_n} \right| <
   {1 \over q_n q_{n+1}} \eqno(1.23)$$
if $\varepsilon_{n+1} = +1$, whereas
$$
 {1 \over q_n q_{n+1}} < \left| x - {p_n \over q_n} \right| <
   {1 \over q_n ( q_{n+1} - (1-\alpha)q_n )}<{1\over\alpha q_n^2} \eqno(1.24)
$$
if $\varepsilon_{n+1} = -1$. Note also that assertions {\it (iv)} and
{\it (v)} remain valid for $x\in \Bbb Q$, with  the convention that
$\be_n=0$ as soon as one of the $x_k\ ,k\le n$, vanish (in which case
the $x_k$ with larger order are undefined).
\par
\medskip
\remark{1.7.} Using the estimates of the Remark 1.5,
$q_k\ge \max(1,G^{k-1}/2)$, and  the elementary inequality
$\log q_k\le (2/e)q_k^{1/2}$, there exists two positive constants
$c_1$ and $c_2$ such that
$$
\eqalignno{
\sum_{k=0}^\infty {\log q_{k}\over q_{k}} & \le c_1=
{2\over e} \left( 3+{\sqrt{2}\over G-\sqrt{G}}\right)=5.214...
\; ,&(1.25) \cr
\sum_{k=0}^\infty {\log 2\over q_{k}} & \le c_2=5\log 2=3.465...
\; ,&(1.26) \cr} $$
for all $\alpha\in [1/2,1]$ and for all $x\in (0,\alpha )$.
\par
Let $P_n/Q_n$ denote the n-th convergent to $x$ according to the
standard continued fraction expansions (i.e. obtained by the iteration of
the Gauss map $A_1$). Following the method  of [Bo],  we shall now
relate the n-th convergents of the $\alpha$--continued fractions
to $P_n/Q_n$.  In fact the following result is obtained through a
repeated use of the identity:
$$A-{1\over B+x}=A-1+{1\over1+{\dst 1\over\dst B-1+x}}\ .$$
\par
\medskip
\Proc{Lemma 1.8.} {For fixed $x\in  \Bbb R\setminus\Bbb Q$, let
$k^\alpha\, : \Bbb N \to \Bbb N$ be the
arithmetic function inductively defined by $k^\alpha (-1)=-1$ and
$$
k^\alpha (n+1) = \cases{ k^\alpha (n)+1 &if $\varepsilon_{n+1}=+1$,\cr
                 k^\alpha (n)+2 &if $\varepsilon_{n+1}=-1$,\cr}
$$
where $\varepsilon_{n+1}$ is defined as in (1.10) and (1.12).
Then $k^\alpha$ is strictly increasing and for all $n\in \Bbb N$
$$
{p_n\over q_n}={P_{k^\alpha (n)}\over Q_{k^\alpha (n)}}\; .
$$
Moreover, when $k^\alpha (n+1)=k^\alpha (n)+2$, we have for the denominators of
the convergent of Gauss'continued fraction
$Q_{k^\alpha (n+1)} = Q_{k^\alpha (n)+2}=Q_{k^\alpha (n)+1}+
Q_{k^\alpha (n)}$.}
\par
\bigskip
In the remainder of this Section,
we collect a few technical facts concerning the nearest
integer continued fraction which will be systematically used in
the various proofs of Sections 3 and 4. The reader
who is mainly interested in  the results
can skip the following Lemmas.
\par\medskip
Let $A=A_{1/2}$. We say that $x$ and $x'$ belong to the same
branch of $A^n$, when  $x_{k}$ and $x'_{k}$ belong to the
same branch of $A$ for $0\le k\le n-1$.  Then the coefficients
$a_k, \varepsilon_k$,
of the expansion of $x$ and the coefficients $a'_k, \varepsilon'_k$
of the expansion of $x'$ do coincide for $0\le k\le n$.
\medskip
We now define an integer $n(x,x')$ which represents the number
of iteration steps needed to separate the orbits of $x$ $x'$.
\Proc{Definition 1.9.}{Let $x,x'$ be two distinct irrationals in $(0,1/2)$.
The {\rm splitting order} $n(x,x')$ is the greatest integer $m$ such that
$x,x'$ belong to the same branch or to two adjacent branches of $A^m$.
We define also the integer $\de(x,x')$ such that
$m=n(x,x')-\de(x,x')$ is the greatest integer such that $x,x'$
belong to the same branch of $A^m$.}
\par
\smallskip
We shall see in the sequel that $\de(x,x')$ is equal to
$0$, $1$, or $2$. Indeed there are four possible situations,
provided we also include the cases where $x$ and $x'$ are
permuted (for brevity we will write $n$ and $\de$ for
$n(x,x')$ and $\de(x,x') $):
\item{(A)} $x$ and $x'$ belong to the same branch of $A^n$. Then
$\de=0$.
\item{(B)} $x$ and $x'$ belong to the same branch of $A^{n-1}$, and
there exists $k\ge 3$ such that
$$
{2\over 2k+1} < x_{n-1} < {1\over k} < x_{n-1}' < {2\over 2k-1} \; .
\eqno(1.27)
$$
Since $x$ and $x'$ belong to the same branch of $A^{n-1}$ and to
adjacent branches of $A^n$, in this case $\de=1$.
\item{(C)} $x$ and $x'$ belong to the same branch of $A^{n-2}$ and there
exists $k\ge 3$ such that
$$
{5\over 5k-2} < x_{n-2} < {2\over 2k-1} < x_{n-2}' < {5\over 5k-3}\; .
\eqno(1.28)
$$
In this case, both $x_n$ and $x'_n$ belong to $[2/5,1/2]$.
$x$ and $x'$ belong to adjacent branches of $A^{n-1}$ as well
as of $A^n$, and $\de=2$.
\item{(D)} $x$ and $x'$ belong to the same branch of $A^{n-1}$ and there
exists $k\ge 3$ such that
$$
{1\over k} < x_{n-1} < {2\over 2k-1} < x_{n-1}' < {1\over k-1}\; .
\eqno(1.29)
$$
In this case $\de=1$, as in case (B) above, but  one must add
the condition that one of the numbers
$x_n,x_n'$ (at least) does not belong to $[2/5,1/2]$, otherwise,
one is in case (C).
\par\smallskip\noindent
{}From the above definitions, for all $l\le n-\delta$ one has
$$
\eqalign{
a_l=a_l' \; ,\;\; \varepsilon_l&=\varepsilon_l'\; ,\;\;
p_l =p_l'\; , \;\; q_l=q_l'\; ,\cr
|\beta_l(x)-\beta_l(x')| &=q_l|x-x'|\; , \cr
|x-x'| &= |x_l-x_l'|\beta_{l-1}(x)\beta_{l-1}(x')\; . \cr}
\eqno(1.30)
$$
{ In the case (B)} one has
$$ a_n=a'_n=k\; ,\;
p_n=p_n'\; , \;\; q_n=q_n'\; , \;\;\varepsilon_n=+1\; , \;
\varepsilon_n'=-1\; .
\eqno(1.31)
$$
Let
$$
x''={p_n\over q_n}\in (x,x')\; .
\eqno(1.32)
$$
Then one has ${x''}_{n-1}=k^{-1}$, $\beta_{n-1}(x'')=q_n^{-1}$,
${x''}_n=0$ and
$$
\eqalign{
|x-x''| &=q_n^{-1}\beta_{n-1}(x)x_n=q_n^{-1}\beta_n(x)\cr
|x'-x''| &= q_n^{-1}\beta_{n-1}(x')x_n'=q_n^{-1}\beta_n(x')\; .\cr}
\eqno(1.33)
$$
{ In the case (C)} one has
$$
\eqalign{
a_{n-1} &= k\; ,\;\; \varepsilon_{n-1}=-1\; , \;\; a_n=2\; , \;\;
\varepsilon_n=+1\cr
a_{n-1}' &= k-1\; ,\;\; \varepsilon_{n-1}'=+1\; , \;\; a_n'=2\; , \;\;
\varepsilon_n'=+1\cr
q_{n-1} &=q_{n-1}'+q_{n-2}\; , \;\; q_n=q_n'=q_{n-1}+q_{n-1}'\cr
p_{n-1} &=p_{n-1}'+p_{n-2}\; , \;\; p_n=p_n'=p_{n-1}+p_{n-1}'\;
.\cr} \eqno(1.34)
$$
Let
$$
x'' = {p_n\over q_n}={p_{n-1}+p_{n-1}'\over
q_{n-1}+q_{n-1}'}\in (x,x')\; .
\eqno(1.35)
$$
Then one has ${x''}_{n-2}={2/(2k-1)}$, $\beta_{n-2}(x'')=2q_n^{-1}$,
${x''}_{n-1} ={1/2}$, $\beta_{n-1}(x'')=q_n^{-1}$, ${x''}_n=0$ and
$$
\eqalign{
|x-x''| &=q_n^{-1}\beta_{n-1}(x)x_n=q_n^{-1}\beta_n(x)\cr
|x'-x''| &= q_n^{-1}\beta_{n-1}(x')x_n'=q_n^{-1}\beta_n(x')\; .\cr}
\eqno(1.36)
$$
{ In the case (D)} one has
$$
\cases{ a_n=k\; , \;\varepsilon_n=-1\cr
        a_n'=k-1\; , \;\varepsilon_n'=+1\cr}
\;\;\;
\cases{ q_n=q_n'+q_{n-1}\cr
        p_n=p_n'+p_{n-1}\; .\cr}
\eqno(1.37)
$$
Let
$$
x'' = {p_n+p_n'\over q_n+q_n'}\in (x,x')\; .
\eqno(1.38)
$$
Then one has ${x''}_{n-1}=2/(2k-1)$,
${x''}_{n}=1/2$, $\beta_{n-1}(x'')=2(q_n+q_n')^{-1}$ and
$$
\eqalign{
|x-x''| &=2(q_n+q_n')^{-1}\beta_{n-1}(x)\left({1\over 2}-x_n\right)\cr
|x'-x''| &=2(q_n+q_n')^{-1}\beta_{n-1}(x')\left({1\over 2}-x_n'\right)\cr}
\eqno(1.39)
$$
We recall that in the case (D) one also has
$$
\max \left({1\over 2}-x_n,{1\over 2}-x'_n\right)\ge {1\over 10}\; .
\eqno(1.40)
$$
\bigskip
We give now to lemmas which relate  the separation between two
numbers and their splitting orders.\par\bigskip
\Proc{Lemma 1.10}{There exists a positive constant $c_3$ independent
on  $x$ and $x'$, such that for all $l<n =n(x,x')$, we have
$$
c_3^{-1}\beta_l(x')<\beta_l(x)<c_3\beta_l(x')\; . \eqno(1.41)
$$
Indeed one can take  $c_3=9/2$.}\par
\proof For $l<n-\de$, this is just a consequence of (1.30) and
Proposition 1.4 {\it (iii)}, and the constant obtained in this
case  is 3. When $\de=1$, one gets from
(1.27) or (1.29) $2/3\le x_{n-1}/x'_{n-1}\le 3/2$, which leads
to a constant 9/2.  When $\de=2$, one gets from (1.28),
$12/13\le x_{n-2}/x'_{n-2}\le 13/12$, and $4/5\le
x_{n-1}/x'_{n-1}\le 5/4$, so that we also get for the constant
$65/16<9/2$.
\qed
\bigskip
\Proc{Lemma 1.11}{There exists a positive constant $c_4>0$ such that
for all $x,x'\in (0,1/2)$, and $n\ge n(x,x')$, one has
$$
\max( \beta_n(x),\beta_n(x'))\le c_4|x-x'|^{1/2}\; .
\eqno(1.42)
$$
Indeed one can take $c_4=9\sqrt{15}/2=17.42...$.}
\par
\proof In the case (D) one has $ |x-x'|=|x-x''|+|x''-x'|$, so
that
$$|x-x'|\ge {1\over 5}(q_n+q_n')^{-1}\inf
(\beta_{n-1}(x), \beta_{n-1}(x'))\ge {1\over 15}q_n^{-2}\ge
{1\over 60}\be_{n-1}^2(x)\; , \eqno(1.43)
$$
since $q_n=q'_n+q_{n-1}>q'_n$, and $(2/3)q_n^{-1}\le\be_{n-1}(x)\le
2q_n^{-1}$. The  previous Lemma then  shows  that
$|x-x'|\ge  (c_3^2/60)\be_{n-1}^2(x')$.
Since $\be_n\le (1/2)\be_{n-1}$, the constant $c_4$ is at most
equal to $c_3\sqrt{15}=9\sqrt{15}/2$.
\par
In the cases (B) and (C) one has
$$
|x-x'|=q_n^{-1}(\beta_n(x)+\beta_n(x'))\ge q_n^{-1}\max
(\beta_n(x),\beta_n(x'))\; .
\eqno(1.44)
$$
However, $q_n^{-1}\ge (1/2)\be_{n-1}(x)\ge
(1/2c_3)\be_{n-1}(x')= (1/9)\be_{n-1}(x')$,
and therefore we get $c_4\ge 3/\sqrt{2}$.\par
Finally in the case (A) one has $|x_n^{-1}-{x'}_n^{-1}|\ge 1$
since $x,x'$ do not belong to two adjacent branches of $A^{n+1}$,
from which follows that
$ |x_n-x_n'|\ge |x_n||x'_n| $.
Suppose $x_n>x'_n$ (the other case resulting by symmetry), then
if $x'_n \ge x_n/2$, we get $ |x_n-x_n'|\ge |x_n|^2/2> $,
and if $x'_n<x_n/2$, we get $|x_n-x'_n|=x_n(1-x'_n/x_n)
>x_n/2>x^2_n/2$. Therefore
$$ |x_n-x_n'|\ge {1\over 2}[\max (x^2_n,x'^2_n)] \eqno(1.45) $$
and
$$
|x-x'|= \beta_{n-1}(x)\beta_{n-1}(x')|x_n-x'_n|
\ge{ \beta_{n-1}(x)\beta_{n-1}(x') \over 2}[\max (x^2_n,x'^2_n)]
$$
and using the previous Lemma,
$$ |x-x'|  \ge{ \max (\beta^2_{n-1}(x),\beta^2_{n-1}(x'))
\over 2c_3}[\max (x^2_n,x'^2_n)]\eqno(1.46)$$
so that one gets $c_4\ge 3$ in this case.
\qed
\par
\medskip
\Proc{Lemma 1.12}{ Let $J$ the interval of definition of one
single branch of $A^m$, and $|J|$ its length. One of its
end-points is equal  to $p_m/q_m$ . We have
$${1\over3q_m^2}\le  |J|\le {1\over q_m^2}\ ,\qquad \hbox{\sl
and for $x\in J$\ ,}\qquad {1\over4}q_m^2\le {\left|
dA^m(x)\over dx\right|} \le {9\over4}q_m^2 \; ,\eqno(1.47)$$
so that
$$ {1\over12}\le |J|{dA^m(x)\over dx}\le {9\over4}
\; .\eqno(1.48)$$}\par
\proof The end-points of $J$ are obtained in setting $x_m=0$ or
$x_m=\pm1/2$ in (1.18), that  is $p_m/q_m$ and $(2p_m\pm p_{m-1})
/(2q_m\pm q_{m-1})$ repectively. Therefore we have
$|J|^{-1}=q_m(2q_m\pm q_{m-1})$. On the other hand, one gets
$dx/dx_m$ from (1.18), so that its inverse $|dA^m/dx|=(q_m\pm
q_{m-1}x_m)^2$, and the Lemma follows easily.
\qed\par\medskip
\Proc{Lemma 1.13}{ Let $J$and $J'$ be the intervals of definition
of two adjacent branches of $A^m$, with respective lengths
$|J|$ and $|J'|$, then there exists a constant $c_5$
(which can be taken $c_5=12$), such that
$$ c_5^{-1}\le {|J|\over |J'|}\le c_5 \; .\eqno(1.49)$$ }\par
\proof Let $x$ be the common  end-point. If $A^m(x)=0$,
then one has $x=p_m/q_m$, and the other end-points are
$(2p_m\pm p_{m-1})/(2q_m\pm q_{m-1})$. Therefore $q_m=q'_m$,
and $|J|/|J'|= (2q_m\pm q_{m-1})/(2q_m\pm q_{m-1})\le 3$.
If $A^m(x)=1/2$, then  one of the two intervals has the form
$[p_m/q_m,(2p_m\pm p_{m-1})/(2q_m\pm q_{m-1})]$
and the same  holds for the other, but with $p_m$ and $q_m$ replaced
by $p'_m$ and $q'_m$ respectively.
For the same reasons as in case (C) above (see Equation
(1.34)), we have $p_m=p'_m\pm p_{m-1}$, $q_m=q'_m\pm q_{m-1}$,
$p_{m-1}=p'_{m-1}$, $q_{m-1}=q'_{m-1}$, so that $q_m/q'_m\le3$,
and the length ratio
$|J|/|J'|=q_n(2q_n \pm q_{n-1})/q'_n(2q'_n\pm q'_{n-1})\le
3q_n^2/{q'}_n^2 \le 12$. Note that in both cases we have
$${1\over 3}\le {q_m\over q'_m}\le 3\; .\eqno(1.50)$$
\qed
\vfill\eject
\beginsection{\bf 2. The Brjuno functions}\par
Following Yoccoz [Yo] we define a (generalized) Brjuno function:
\par
\medskip
\Proc{Definition 2.1}{The {\it $\alpha$-Brjuno function}
$B_\alpha \,: \Bbb R \setminus \Bbb Q \to \overline\Bbb R$ is defined by
the formula
$$
B_\alpha (x) = - \sum_{i=0}^\infty \beta_{i-1}(x) \log x_i
\eqno(2.1)
$$
where the  $x_n$ follow  $x_0=x$ by repeated iterations of
$A_{\al}$, as defined in (1.10) and (1.11), and the $\be_n$'
are given by (1.21). We have posed $\beta_{-1} = 1$.}
\par
\medskip
\remark{2.2.} It is useful to extend the above definition
$x\in \Bbb Q$, by setting $B_{\al}(x)=+\infty$, or $\exp(-B_{\al}(x))=0$.
The Brjuno function defined in [Yo] corresponds to
$B_{1/2}$, the one defined by the nearest integer continued fraction
map $A_{1/2}$.
\par
\medskip
\Proc{Proposition 2.3.} {Given $\alpha \in [1/2,1]$ one has
\item{(i)} $B_\alpha (x) = B_\alpha (x+1) $  for all $x \in \Bbb R $;
\item{(ii)} For all $x \in (0,\alpha) $
$$
B_\alpha (x) = - \log x + x B_\alpha \left( {1 \over x} \right)\;
;
\eqno(2.2)
$$
\item{(iii)} if $x \in [\alpha - 1, 0) $
then $B_\alpha (-x) = B_\alpha (x)$;
\item{(iv)} there exists a constant $C_1 >0$ (independent of $\alpha$)
such that for all
$x \in \Bbb R \setminus \Bbb Q$ one has
$$
\left| B_\alpha (x) - \sum_{j=0}^\infty {\log q_{j+1} \over q_j}
\right| \le C_1
\eqno(2.3)
$$
where $\{q_j\}_{j \ge 0}$ denotes the sequence of the denominators
of the convergents to $x$ of the $\alpha$-continued
fraction expansion.}
\par
\medskip
\proof Given $x\in \Bbb R\setminus \Bbb Q$, the sequences
$(x_i)_{i\ge 0}$ and $(\beta_i)_{i\ge 0}$ associated to $x$ and
$x+1$ are the same, which proves (i). The same is true for $x$
and $-x$ if $x\in (\alpha -1,0)$, which proves (iii).
\par
If $x\in (0,\alpha )$, let $y=1/x$ and denote by $y_i$, $a_i(y)$,
$\beta_i(y)$, and $x_i$, $a_i(x)$, $\beta_i (x)$ the sequences
(1.11) and (1.21) associated to $y$ and to $x$ respectively.
{}From (1.9) and (1.10) it follows that $x_0=x$, $a_0(y)=a_1(x)$,
$y_0=x_1$ and by induction for all $n\ge 0$
$y_n=x_{n+1}$ and $\beta_n(y)=(\beta_{n+1} (x))/x$. Thus
$$
\eqalign{
B_\alpha (y)
&= -\sum_{i=0}^\infty\beta_{i-1}(y)\log y_i
=-\log y_0-\sum_{i=1}^\infty {1\over x}\beta_i(x)\log x_{i+1} \cr
& = -{1\over x}\sum_{i=1}^\infty \beta_{i-1}(x)\log x_i
= {1\over x}[B_\alpha (x)+\log x]\; ,\cr}
$$
which proves (ii).
\par
To prove (iv) we first remark that (1.21) implies
$$
q_i\beta_{i-1}+\varepsilon_iq_{i-1}\beta_i=1
$$
for all $i\ge 0$.  Then
$$
\eqalign{
-B_\alpha (x) &+\sum_{i=0}^\infty {\log q_{i+1}\over q_i}
= \sum_{i=0}^\infty \beta_{i-1}\log {\beta_i\over\beta_{i-1}}
+\sum_{i=0}^\infty \left(\beta_{i-1}+\varepsilon_i{q_{i-1}\over q_i}
\beta_i\right)\log q_{i+1} \cr
&= \sum_{i=0}^\infty \beta_{i-1}\log\beta_iq_{i+1} -
\sum_{i=0}^\infty \beta_{i-1}\log\beta_{i-1} +
\sum_{i=0}^\infty \varepsilon_i{q_{i-1}\over q_i}\beta_i\log
q_{i+1}\; , \cr}
$$
but by (1.21), Proposition 1.4  {\it (iii)}, and the estimates
of Remark 1.7.,
$$
\eqalign{
\left|\sum_{i=0}^\infty \beta_{i-1}\log\beta_iq_{i+1}\right| &\le
2\sum_{i=0}^\infty {\log 2\over q_i} \le 2c_2 \; , \cr
\left|\sum_{i=0}^\infty \beta_{i-1}\log\beta_{i-1} \right| &\le
2\sum_{i=0}^\infty {\log 2+\log q_i\over q_i}\le 2(c_1+c_2)\; , \cr
\left|\sum_{i=0}^\infty \varepsilon_i{q_{i-1}\over q_i}\beta_i\log
q_{i+1}\right| &\le
2\sum_{i=0}^\infty {\log q_{i+1}\over q_{i+1}}\le 2 c_1\; , \cr}
$$
from which it follows that
$$
\left|B_\alpha (x)-\sum_{i=0}^\infty {\log q_{i+1}\over q_i} \right|
\le C_1=4(c_1+c_2)\; . \;\;\;
$$\qed
\par
\medskip
By means of Lemma 1.8 one can prove the following
\par
\Proc{Proposition 2.4.}{There exists a positive constant $C_2>0$ such that
for all $\alpha \in [1/2,1]$ and for all $x \in \Bbb R \setminus \Bbb Q$
one has
$$
\left| B_\alpha (x) - \sum_{j=0}^\infty {\log Q_{j+1} \over Q_j} \right|
   \le C_2 \eqno(2.4)
$$}
\par
\medskip
\proof Thanks to {\it (iv)}, Proposition 2.3, it suffices to compare
$\sum_{j=0}^\infty (1/q_j)\log q_{j+1}$ with
$\sum_{j=0}^\infty (1/Q_j)\log Q_{j+1} $. By Lemma 2.3, one has
$q_j=Q_{k(j)}$ for all $j$, where for brevity we write
$k(j)$ for $k^\alpha (j)$. Thus
$$
\sum_{j=0}^\infty {\log q_{j+1} \over q_j} =
\sum_{k(j+1)=k(j)+1} {\log Q_{k(j+1)} \over Q_{k(j)}}
+ \sum_{k(j+1)=k(j)+2} {\log Q_{k(j+1)} \over Q_{k(j)}}\; .
$$
Using the fact that
$Q_{k(j+1)} = Q_{k(j)+2}=Q_{k(j)+1}+Q_{k(j)}$
we have
$$
{\log  Q_{k+2}\over Q_{k}} = {\log  (Q_{k+1}+Q_k)\over Q_{k}}
={\log Q_{k+1}\over Q_{k}}+{\log \left(1+Q_k/Q_{k+1}\right)\over Q_k}
$$
but
$$
0 \le {\log \left(1+Q_k/ Q_{k+1}\right)\over Q_k}
\le {\log 2\over Q_k}
\; ,
$$
By applying the estimates of Remark 1.7 one gets the result:
$$
\left| \sum_{j=0}^\infty {\log q_{j+1} \over q_j} -
\sum_{j=0}^\infty {\log Q_{j+1} \over Q_j} \right| \le 2c_2 + c_1 \; .
\; \;
$$
and one gets $C_2=6c_2+5c_1$.\qed
\par
\medskip
\remark{2.5.} The {\it Brjuno numbers} [Br] are usually
defined by the {\it Brjuno condition}
$$
\sum_{i=0}^\infty {\log Q_{i+1}\over Q_i} < +\infty\; .
$$
Proposition 2.4 shows that the $\alpha$-Brjuno functions $B_\alpha$
are finite at $x$ if and only if $x$ is a Brjuno number and that
all the generalized Brjuno functions differ one from the other
for a $L^\infty$ function.
\par
On the other hand, the advantage of the functions $B_\alpha$ with respect
to the Brjuno condition is that they verify a nice functional equation (2.2)
under the action of the modular group $\hbox{SL}\,(2,\Bbb Z)$.
\par
\medskip
Another important characterization of the generalized Brjuno functions
comes from their ``uniqueness'', as it is an immediate consequence of Theorem
2.6 below.
\par
\medskip
Let us consider the operator
$$
(T_\nu f)(x)=x^\nu f\left({1\over x}\right)\; ,
\eqno(2.5)
$$
if $x\in (0,\alpha )$, where $\nu \ge 0$,
defined for the moment on measurable functions
of $\Bbb R$ which verify
$$
f(x)=f(x+1) \;  \hbox{for almost every}\, x\in \Bbb R\; ,
\;\;\;f(-x)=f(x)\;\hbox{ for a.e.}\, x\in (0,1-\alpha)\; .
\eqno(2.6)
$$
It is understood that the function $T_\nu f$ is completed outside $(0,\alpha)$
by imposing on $T_\nu f$ the same parity and periodicity conditions which
are expressed for $f$ in (2.6).
\par
The functional equation for the $\alpha$-Brjuno function can be written
in the form
$$
[(1-T_1)B_\alpha ](x)=-\log x\; ,
\eqno(2.7)
$$
for all $x\in (0,\alpha )$, complemented with the periodicity and
symmetry conditions (2.6). This suggest to study the operator
$T_\nu$ on the Banach spaces
$$
X_{\alpha ,p}= \left\{ f : \Bbb R \to \Bbb R \mid f \,\hbox{verifies (2.6)}
\; , \;\; f\in L^p((0,\alpha),dm_\alpha (x)),\right\}
\eqno(2.8)
$$
endowed with the norm of $L^p((0,\alpha ),dm_\alpha (x))$,
where $$dm_\al(x)=c_\al \rho_\al(x)\, dx\eqno(2.9)$$
is the invariant measure defined in Section 1, so that
$$
||f ||_{\alpha ,p}=\left(\int_0^\alpha |f(x)|^p dm_\alpha (x)
\right)^{1/p}\; ,
\eqno(2.10)
$$
as $\alpha$ varies in
$(1/2,1)$ and $p\in [1,\infty ]$. Note that if $p<p'$
one has the obvious inclusion $X_{\alpha ,p'}\subset X_{\alpha ,p}$
and let
$$
X_\alpha = \cap_{p\ge 1}X_{\alpha ,p}\; .
\eqno(2.11)
$$
If $(1-T_1)$ is invertible in the considered space, then (2.7)
has a unique solution for $B_{\alpha}$, provided that the
right hand side also belongs to the space, which is easy to check.
The invertibility  property  is given by the following theorem, which
states in particular that the spectral radius of $T_1$ is strictly
smaller than 1.
\medskip
\Proc{Theorem 2.6.}{$T_\nu$ is a linear bounded operator from
$X_{\alpha ,p}$ into itself for all $\nu >0$, for all $\alpha\in [{1\over
2},1]$
and for all $p\in [1,\infty ]$. Indeed its spectral radius on
$X_{\alpha ,p}$ satisfies}
$$
r(T_\nu )\le \cases{ g^{\nu }\; , & if $\alpha >g$\cr
                     \gamma^{\nu }\; , & if $\alpha\le g$.\cr}
                     \eqno(2.12)
$$
\par
\medskip
\proof
It is a simple calculation. We observe first that
$$
(T_{\nu}^nf)(x) = (\be_{n-1}(x))^{\nu} f(x_n)=
(\be_{n-1}(x))^{\nu} (f\circ A^n_{\al})(x)\; ,\eqno(2.13)
$$
therefore
$$
\eqalign{
\int_0^\alpha |T_\nu^nf(x)|^p m_\alpha (x)dx &=
\int_0^\alpha (\beta_{n-1}(x))^{\nu p}|f(A_\alpha^n(x))|^p dm_\alpha (x)\cr
& \le
[\alpha\gamma_\alpha^{n-1}]^{\nu p}\int_0^\alpha |f(x)|^p dm_\alpha (x)\cr}
\eqno(2.14)
$$
where we have used Proposition 1.4 (iv) and (v) (therefore $\gamma_\alpha = g$
if $\alpha > g$, $\gamma_\alpha = \gamma$ if $\alpha\le g$) and
the invariance of the measure $dm_\alpha(x) $ w.r.t. $A_\alpha$.
{}From (2.14) it immediately follows that
$$
||T_\nu^nf||_{\alpha ,p}\le [\alpha \gamma_\alpha^{n-1}]^{\nu}
||f||_{\alpha ,p}
\eqno(2.15)
$$
and one gets (2.12) by taking the $1/n$--th root of both sides.
\qed
The use of the invariant measure in (2.8) makes the evaluation
of the spectral radius remarkably simple. We of course
get the same result if we replace the measure in (2.8) by the
Lebesgue measure which is equivalent (see Remark 1.2).
In particular, the above Theorem implies that the spectral
radius is also given by (2.12) for the operator $T_{\nu}$ in the
spaces $L^p({\Bbb T})$, naturally introduced using the periodicity
property.
It is more difficult to tell whether $T_1$ is itself contracting
(see [MMY] for some results in this direction), we only mention here
that in the case $\al=1/2$, $T_1$ is a contraction for all Lebesgue
$L^p$-norms on $[0,1/2]$.
\par
\vskip 1. truecm
\beginsection{\bf 3. The Brjuno function and  the BMO space}\par
In the previous Section, it has been shown that  the Brjuno
functions $B_{\al}$ belong to $L^p({\Bbb T})$ and therefore to the
intersection $\bigcap_{p=1}^{\infty} L^p({\Bbb T})$. The purpose
of this Section is to show a stronger result: the Brjuno functions
$B_{\al}$ belong to ${\rm BMO}({\Bbb T})$. We recall the
definition and the main properties of BMO spaces in Appendix
A.\par
In fact,  we already know that all Brjuno functions $B_{\al}$
differ by  $L^\infty$ functions, and  since $L^\infty\subset\,$BMO,
it will
be enough to prove that $B_\alpha$ is in ${\rm BMO}({\Bbb T})$
for a fixed value of $\alpha$.
In this section we fix $\alpha =1/2$ and denote $A_{1/2}$ and
$B_{1/2}$ simply by $A$ and $B$ respectively. We will also
write
$$dm(x)=dm_{1/2}(x)={1\over\log G}\left({1\over G+x}+
{1\over G+1-x}\right)\, dx\ .\eqno(3.1)$$
\par
\medskip\noindent
On the interval $I$, we define now the mean value $f_I$ of $f$
$$f_I={1\over m_I}\int_I f(x)\,dm(x)
\quad,\quad m_I=\int_I dm(x)\; ,\eqno(3.2)$$
and its quadratic oscillation ${\cal O}_I(f)$
$$\eqalignno{ {\cal O}_I(f)&=\left({1\over m_I}
\int_I \big(f(x)-f_I\big)^2\, dm(x)\right)^{1/2}\cr &=
\left({1\over 2m_I^2}\int_{I\times I}\big(f(s)-f(t)\big)^2
\,dm(s)\,dm(t) \right)^{1\over2}\; &(3.3)\cr}
$$
We now consider the space
$X_*\subset X_{1/2}=\cap_{p=1}^{\infty}X_{1/2,p}$
defined as:
$$
X_* = \{f\in\hbox{BMO}(\Bbb R)\, \mid f(x+1)=f(x)\;
\forall x\in \Bbb R\; ,
\; f(-x)=f(x) \;\forall x\in [0,1/2]\}\; , \eqno(3.4)
$$
with the norm
$$
|| f||_* = |f|_* +\Hnorm{f}{2}  \; ,
\eqno(3.5)
$$
where  $\Hnorm{f}{2}=
|| f||_{L^2((0,1/2), dm)}$, and
$$
|f|_* = \sup_{I\subset [0,1/2]} {\cal O}_I(f)\; .\eqno(3.6)
$$
Therefore we have
$$\eqalignno{||f||_*&=\sup_{I\subset[0,1/2]}\left({1\over 2m_I^2}
\int_{I\times I} \big(f(s)-f(t)\big)^2\,dm(s)\,
dm(t)\right)^{1\over 2}\cr&+ \left(\int_0^{1/2}(f(x))^2\,
dm(x)\right)^{1\over 2}\ .&(3.7)\cr}$$
\par
\medskip
\remark{3.1.}
Due to the equivalence between the measure $m$ and the Lebesgue
measure, one gets an equivalent norm in replacing $m(I)$ by the
length $|I|$ in (3.7)
We explain in the appendix why the norm we use here is equivalent
to the usual BMO norm: it differs with the usual one [Gr,GCRF] in
two respects, first, we use here the invariant measure $dm$ instead of
the Lebesgue measure, second, we use here a $L^2$-norm definition
of BMO instead of the usual $L^1$-norm definition. The equivalence
between the  $L^1$ and $L^2$ definitions is a corollary of the
John-Nirenberg Theorem which is far from obvious. Furthermore,
the BMO-norm on $[0,1/2]$ is equivalent to the BMO-norm on
${\Bbb T}$ only for even functions, which is the case for
$B_{1/2}$.
\par
\medskip
We shall now prove the following theorem,
\Proc{Theorem 3.2.}{The Brjuno function $B=B_{1/2}$ belongs to
$X_*$, and therefore to
BMO(${\Bbb T}$). For $1/2\le\al\le1$, the functions $B_{\al}$
also belong to BMO(${\Bbb T}$). }\par
The proof follows immediately from the following Theorem 3.3,
which states in particular that for $\al=1/2$,
$1-T_1$ is invertible in $X_*$.
And it is easy to show by direct computations that, for
$\al=1/2$, the right hand side of (2.7), namely the even and periodic
function equal to $\log x$ on $(0,1/2]$, is  in $X_*$.
It results that $B=B_{1/2}$ is also in $X_*$, as well as all $B_{\al}$
for $1/2\le\al\le1$. Note that $B\not\subset L^{\infty}$ because
the logarithmic function is unbounded.
\Proc{Theorem 3.3.}{ In the case $\al=1/2$, and for all $\nu>0$,
$T_{\nu}$ is a bounded linear operator from $X_*$ to $X_*$.
Indeed, its spectral radius in $X_*$ is at most equal
to $\ga^{\nu}=(\sqrt{2}-1)^{\nu}$\ .}\par
\proof
In order to prove the theorem, we must estimate ${\cal O}_I(T_\nu^mf)$
for $m\ge 0$, $I\subset [0,1/2]$. Let $I=[x,x']$, $n=n(x,x')$ the
splitting order of $x$ and $x'$.
We divide the proof into three cases.
\par\medskip
{\it First case:} one has $m>n$.
\par
Let $\widehat{I}$ denote the union of the domains of the branches of
$A^m$ which meet the interval $I$. Then one has
$I\subset\widehat{I}$, and from (1.49), $1\le |\widehat {I}|/|I|
\le (1+2c_5)$, since in this case $I$ contains the full
interval of at least one branch of $A^m$.
Setting $g=T_\nu^mf$, one gets
$$
{\cal O}_I^2(g)={1\over m_I}\int_I(g-g_I)^2\, dm\le{1\over m_I}
\int_Ig^2\, dm\le c_0(1+2c_5) {1\over |\widehat{I}|}\int_{\widehat{I}}
g^2\, dm\; , \eqno(3.8)
$$
where we have used (1.7) and (1.49). For the domain $J$ of any
complete branch of $A^m$, we take $A^m(t)$ instead of $t$ as the
integration variable. Then, it follows from (1.48) that
$$
\int_J (T_\nu^mf)^2\, dm\le ||\beta_{m-1}||^{2\nu}_{{\cal C}^0}
\int_J(f\circ A^m)^2\, dm \le 12|J|c_0^2
||\beta_{m-1}||^{2\nu}_{{\cal C}^0} ||f||^2_2
\eqno(3.9)
$$
where $||\ ||_{\cal C}^0$ denotes the sup-norm on  $[0,1/2]$.
Therefore one gets
$$
\int_{\widehat{I}}(T_\nu^mf)^2\, dm\le 12c_0^2
||\beta_{m-1}||^{2\nu}_{{\cal C}^0} ||f||^2_2|\widehat{I}|\; ,
\eqno(3.10)
$$
from which it follows that
$$
{\cal O}_I(T_\nu^mf)\le 2c_0\sqrt{3c_0(1+2c_5)}
||\beta_{m-1}||^{\nu}_{{\cal C}^0} ||f||_2\; .
\eqno(3.11)
$$
\par\medskip
{\it Second case:} one has $m\le n-\delta$.
\par
Then $I$ is contained in the domain $J$ of a single branch of $A^m$.
Let $I_1=A^m(I)\subset [0,1/2]$.
We have
$$
2m_I^2{\cal O}_I^2(T^mf)=
\int_{I\times I}(T_\nu^mf(s)-T_\nu^mf(t))^2\,dm(s)\,dm(t)\le
2(M_1+M_2)\; , \eqno(3.12)
$$
where
$$
\eqalignno{
M_1 &= \int_{I\times I}\beta_{m-1}^{2\nu}(s)(f(A^ms)-f(A^mt))^2
\, dm(s)\, dm(t)\; , &(3.13)\cr
M_2 &= \int_{I\times I}(f(A^mt))^2(\beta_{m-1}^\nu (s)-
\beta_{m-1}^\nu (t))^2\, dm(s)\, dm(t)\; . &(3.14)\cr}
$$
Now, from the bound (1.47) on $dA^m/dx$, we deduce
upper and lower bounds on the ratio $|I_1|/|I|$
$${q_m^2\over4}\le {|I_1|\over |I|} \le {9q_m^2\over4}\ ,\quad
\hbox{and,}\quad
\left|{dA^m(x)\over dx}\right|^{-1}\le {4\over q_m^2}
\le9{|I|\over|I_1|} \le 9c_0^2{m_{I}\over m_{I_1}}\; .\eqno(3.15)
$$
Taking $A^m(s)$ and $A^m(t)$ as new integration variable, one
gets
$$
\eqalign{
M_1 &\le c_0^4 ||\beta_{m-1}||^{2\nu}_{{\cal C}^0}
\max_{x\in I_1}\left( \left|{dA^m\over dx}\right|^{-2} \right)
\int_{I_1\times I_1}(f(s')-f(t'))^2\, dm(s')\, dm(t')\cr
& \le 81c_0^8||\beta_{m-1}||^{2\nu}_{{\cal C}^0}2m_I^2
{\cal O}^2_{I_1}(f) =c_6^2 ||\beta_{m-1}||^{2\nu}_{{\cal C}^0}
m_I^2 {\cal O}^2_{I_1}(f)
\; ,\cr} \eqno(3.16)
$$
with $c_6=9\sqrt{2}c_0^4$.  On the other hand, from (1.30) one gets
$$
|\beta_{m-1}(s)-\beta_{m-1}(t)|\le q_{m-1}|I|\le |I|^{1/2}\; ,
\eqno(3.17)
$$
since from (1.47), one has $|I|\le|J|\le q_m^{-2}\le q_{m-1}^{-2}$.
Then ,using the obvious inequality $|x^{\nu}-y^{\nu}|\le
\nu \max(x^{\nu-1},y^{\nu-1})|x-y|$, and Lemma 1.10,  we get
$$
\eqalignno{
M_2 &\le \nu^2\int_{I\times I}(f(A^mt))^2
\max( \beta_{m-1}^{2\nu -2}(s), \beta_{m-1}^{2\nu -2}(t))
(\beta_{m-1}(s)-\beta_{m-1}(t))^2\, dm(s)dm(t)\cr
&\le \nu^2|I|c_3^2\int_{I\times I}(f(A^mt))^2
\beta_{m-1}^{2\nu -2}(t)\, dm(s)dm(t)&(3.18)\cr}
$$
$$M_2 \le \nu^2 |I|m_Ic_0c_3^2||\beta_{m-1}||^{2\nu}_{{\cal C}^0}
\int_I(f(A^m(t)))^2 \beta_{m-1}^{-2}(t)dt\le
c_7^2||\beta_{m-1}||^{2\nu}_{{\cal C}^0}m_I^2||f||^2_2\; .
\eqno(3.19)
$$
We have bounded $\be_{m-1}$ in the integral using Proposition
1.4 {\it(iii)} and taken $A^m(t)$ as new integration variable
using (1.47), so that setting $c_7=3\nu c_0^{3/2}c_3$,
one obtains
$$\eqalignno{
{\cal O}_I(T^m_\nu f)&\le ||\beta_{m-1}||^{\nu}_{{\cal C}^0}
\left(c_6^2{\cal O}_{I_1}^2(f)+ c_7^2||f||_2^2\right)^{1/2}\cr
&\le ||\beta_{m-1}||^{\nu}_{{\cal C}^0}
\left(c_6{\cal O}_{I_1}(f)+ c_7||f||_2\right) \; .
&(3.20)\cr}
$$
\par\medskip
{\it Third case:} one has $n-\delta <m\le n$.
\par
Thus one is in one of the cases (B), (C) or (D), discussed
above  after Definition 1.9 : the interval $I$ is contained
in the union of two adjacent branches of $A^m$ and the point $x''$
is the common point of the two branches.
Let $I^-=[x,x'']$, $I^+=[x'',x']$ and we still define $M_1$
and $M_2$ as in Equations (3.13) and (3.14), so that we get
(3.12) as a  bound of the oscillation on $I$.
We first bound  $M_2$ as in the previous case, but now
from (1.30),
$$\eqalignno{ |\beta_{m-1}(s)-\beta_{m-1}(t)|&\le
|\beta_{m-1}(s)-\beta_{m-1}(x'')|
+|\beta_{m-1}(x'')-\beta_{m-1}(t)|\cr&\le
q_{m-1}|I_+|+q_{m-1}'|I_-|\cr
&\le |I_+|^{1/2}+|I_-|^{1/2}\le\sqrt{2}|I|^{1/2}\; &(3.21)\cr}
$$
where the beforelast inequality is obtained as (3.17) above.
Then
$$
\int_I f(A^m(t))^2\beta_{m-1}^{-2}(t)dt\le
\left(\int_{I^-}+\int_{I^+}\right)f(A^m(t))^2\beta_{m-1}^{-2}(t)dt\le
2c_{7}^2||f||^2_2\; ,
\eqno(3.22)
$$
from which it follows that
$$
M_2\le 4c_{7}^2||\beta_{m-1}||^{2\nu}_{{\cal C}^0}m_I^2||f||^2_2\; .
\eqno(3.23)
$$
On the other hand
$$
M_1\le c_0^2||\beta_{m-1}||^{2\nu}_{{\cal C}^0}\int_{I\times I}
(f(A^m(s))-f(A^m(t)))^2\,dsdt\; .
\eqno(3.24)
$$
Let $I_1^+=A^m(I^+)$, $I_1^-=A^m(I^-)$; one has, using (3.15),
$$
\eqalign{
\int_{I^\varepsilon\times I^\varepsilon} (f(A^m(s))-f(A^m(t)))^2dsdt
&\le 81{|I^\varepsilon |^2\over |I_1^\varepsilon |^2}
\int_{I_1^\varepsilon\times I_1^\varepsilon} (f(s')-f(t'))^2ds'dt'\cr
&\le 81 c_0^4 2m_I^2{\cal O}_{I_1^\varepsilon}^2(f)\; .\cr}
\eqno(3.25)
$$
Finally one gets
$$
\int_{I^+\times I^-} (f(A^m(s))-f(A^m(t)))^2dsdt
\le 81{|I^+||I^-|\over |I_1^+||I_1^-|}\int_{I_1^+\times I_1^-}
(f(s')-f(t'))^2ds'dt'\; . \eqno(3.26)
$$
Let $I_1=I_1^+\cup I_1^-$. From the discussion of cases (B), (C)
and (D) made after Definition 1.9, it follows that
$I_1^+$ and $I_1^-$ have a common end-point which is either 0 or
1/2. Therefore, either $I_1^+\subset I_1^-$ or
$I_1^-\subset I_1^+$, then
$$
\int_{I_1^+\times I_1^-}(f(s')-f(t'))^2ds'dt'\le c_0^2 2m_{I_1}^2
{\cal O}_{I_1}^2(f)\; .
\eqno(3.27)
$$
However, as read from (3.15),
$ {\dst|I^+|\over \dst|I_1^+|}q_m^2$, and
${\dst|I^-|\over\dst |I_1^-|}{q'}^2_m$
both belong to the interval $[4/9,4]$. Then, from (1.50), one gets
$$
{|I^+||I^-|\over |I_1^+||I_1^-|}\le
c_8^2\left({|I|\over |I_1|}\right)^2\; , \eqno(3.28)
$$
with $c_8=2^23^4$, from which it follows that
$$
\int_{I^+\times I^-}(f(A^m(s))-f(A^m(t)))^2dsdt\le
c^2_{9}2m_I^2{\cal O}_{I_1}^2(f)\; .
\eqno(3.29)
$$
with $c_9=81c_8^2c_0^4$. Putting all constants together, one
therefore finds $c_{10}$ and $c_{11}$ such that
$$
\eqalign{
M_1 &\le c_{10} ||\beta_{m-1}||^{2\nu}_{{\cal C}^0}m_I^2
({\cal O}_{I_1^+}^2(f)+{\cal O}_{I_1^-}^2(f)+
{\cal O}_{I_1}^2(f))\; , \cr
{\cal O}_{I}(T_\nu^mf)&\le c_{11}||\beta_{m-1}||^{\nu}_{{\cal C}^0}
({\cal O}_{I_1^+}(f)+{\cal O}_{I_1^-}(f)+
{\cal O}_{I_1}(f)+||f||_2)\; . \cr}
\eqno(3.30)
$$
In the three cases we have considered, we get a bound for the
oscillation of $T_{\nu}^m(f)$ as a linear combination with
constant coefficients of the sup of the
oscillation of $f$, the norm $||f||_2$, and
{\it an overall factor
$||\beta_{m-1}||^{\nu}_{{\cal C}^0}$ which contains the only
dependance  on $m$}. Since (12.5) gives the same result for
the $L^2$-norm, we easily deduce that, for the norm (3.5),
there exist a constant $c_{12}$ independant of $m$ and $f$,
such that
$$
||T_\nu^mf||_*\le c_{12} ||\beta_{m-1}||^{\nu}_{{\cal C}^0} ||f||_*
\le \left(\ga^{\nu}\right)^m(c_{12}/2\ga)||f||_*\eqno(3.31)
$$
The first consequence is that $T_{\nu}$ is a bounded linear
operator from $X_*$ to $X_*$, the second is that its
spectral radius $r(T_{\nu})=\lim ||(T_{\nu})^m||^{1/m}$
is bounded by $\gamma^\nu$.\qed
\par\medskip
The spectral radius computation is rather general,
and does not require a tight adjustment for the constants:
more specific results on the norm of $T_{\nu}$ itself depend on
the peculiar norm taken. See [MMY] for some results of this
kind.
We observe that the computation of the spectral radius
for the $L^p$-norm and the BMO-norm both come from the
leading behaviour of  $\be_{m-1}$, as given by Theorem 1.4.
The embedding of the BMO space in between the $L^p$ spaces
and $L^{\infty}$ makes this result very natural. We expect
now interesting consequences for the complex extension of the
Brjuno functions.
\par
\vskip 1. truecm
\beginsection 4. The Brjuno function and its H\"older
stability properties\par
The functional equation (1.29) for the Brjuno function for $\alpha=1/2$ is
$$
[(1-T_1)B_{1/2}](x)=-\log x\; ,
\eqno(4.1)
$$
for all $x\in (0,1/2)$, complemented with the condition that $B=B_{1/2}$
is even
and periodic.  In this Section we will suppose that the right hand side of
this equation is pertubed, by an additional term $f$, which is
less singular than the
logarithmic function. We want to study the singular properties of the
perturbed solution. Since  the equation is linear, we only need to consider
the action on $f$ of $T_1$ and $(1-T_1)^{-1}$, which
will be conveniently called the
Brjuno operator ${\bf B}$. We will consider even and periodic functions $f$
which are {\it continuous}. It is sufficient to know the value of $f$ on
$[0,1/2]$, so we assume $f\in {\cal C}^0_{[0,1/2]}$. One can check  that
$T_1f$ (resp $T_{\nu}$ for $\nu>0$) is also continuous provided we
set $(T_1f)(0)=0$ (resp $(T_{\nu}f)(0)=0$). We need now the usual
H\"older's type semi-norms for continuous functions.
\Proc{Definition 4.1.}{Let $f\in {\cal C}^0_{[0,1/2]}$, Then we define the
H\"older's $\eta$-norm as
$$\hnorm{f}{\eta}=
\sup_{0\le x<y\le1/2} {\dst|f(x)-f(y)|\over\dst|x-y|^{\eta}}\ ,\eqno(4.2)$$
with $0<\eta\le 1$.
This is a seminorm since it vanishes on constant functions, so that we
introduce the norm:
$$\Hnorm{f}{\eta}=\hnorm{f}{\eta}+\Hnorm{f}{{\cal C}^0}\ ,\eqno(4.3)$$
where
$\Hnorm{f}{{\cal C}^0}=\max_{0\le x\le 1/2}|f(x)|$.
We say that $f\in {\cal C}^{\eta}_{[0,1/2]}$,
if $f\in {\cal C}^0_{[0,1/2]}$ and $\hnorm{f}{\eta}$ is finite.}\par
We now have:
\par
\medskip
\Proc{Theorem 4.2.}{Let $f\in {\cal C}^{\eta}_{[0,1/2]}$,
$0<\eta\le 1$. Then:
\item{(1)} $T^m_\nu f$ is H\"older continuous, of exponent
$\overline{\eta}=\min (\eta,\nu /2)$;
\item{(2)} If $\nu >2\eta$ (thus $\overline{\eta}=\eta$), $T_\nu$ is
a bounded linear operator
in ${\cal C}^\eta$, of spectral radius smaller or equal to
$\gamma^{\nu -2\eta}<1$. The operator ${\bf B}_{\nu}=
(1-T_{\nu})^{-1}$ is defined in this space and fulfils
${\bf B}_{\nu}=\sum_{m=0}^{\infty}T_{\nu}^m$.
\item{(3)} If $\nu =2\eta$ (thus $\overline{\eta}=\eta=\nu/2$),
there exists a
positive constant $c_{14}>0$ such that
$$
||T_\nu^m||_{{\cal C}^{\nu /2}(0,1/2)}\le c_{14}\; , \;\;
\hbox{for all}\; m\ge 0\; .
\eqno(4.4)
$$}
\par
\medskip
\proof
Let $x,x'\in [0,1/2]$, $m\ge 0$, $\eta\in (0,1]$, $\nu >0$.
We want to estimate
$$
|T_\nu^mf(x)-T_\nu^mf(x')|
\eqno(4.6)
$$
under the assumption that $f\in {\cal C}^\eta_{[0,1/2]}$. We
let $n=n(x,x')$.
\par\smallskip
{\it First case:} $m>n$. From (2.13), one has
$$
\eqalignno{
|T_\nu^mf(x)-T_\nu^mf(x')| &\le ||f||_{{\cal C}^0}(\beta_{m-1}^\nu (x)
+\beta_{m-1}^\nu (x'))\cr
&\le ||f||_{{\cal C}^0}(\beta_{n}^\nu (x)\beta_{m-n-2}^{\nu}(x_{n+1})
+\beta_{n-1}^\nu (x')\beta_{m-n-2}^{\nu}(x'_{n+1}))\cr
&\le 2c_4^{\nu}||f||_{{\cal C}^0}|x-x'|^{\nu /2}||
\beta_{m-n-2}||_{{\cal C}^0}^\nu\; ,
&(4.7)\cr}
$$
where we have used Lemma 1.11.\par\smallskip
{\it Second case:} $m\le n-\delta$. One has
$$
\eqalign{
|T_\nu^mf(x)-T_\nu^mf(x')| &\le (\beta_{m-1}(x))^\nu |f(A^m(x))-f(A^m(x'))|\cr
&+|f(A^m(x'))||\beta_{m-1}^\nu (x)-\beta_{m-1}^\nu (x')|\; .\cr}
\eqno(4.8)
$$
{}From (1.47) and Proposition 1.4 {\it (iii)}, one has
$$
|A^m(x)-A^m(x')| \le 9(\beta_{m-1}(x))^{-2}|x-x'|\; ,\eqno(4.9)
$$
and, using (1.30), and (1.41)
$$
\eqalignno{
|\beta_{m-1}^\nu (x)-\beta_{m-1}^\nu (x')|
&\le\nu \max( \beta_{m-1}^{\nu-1}(x),\beta_{m-1}^{\nu-1}(x'))
|\beta_{m-1}(x)-\beta_{m-1}(x')|\cr
&\le \nu c_3^{\nu-1}(\beta_{m-1} (x))^{\nu -1} q_{m-1}|x-x'|\cr
&\le \nu c_3^{\nu-1}(\beta_{m-1}(x))^{\nu -2}|x-x'|
\; .&(4.10)\cr}
$$
Therefore
$$
\eqalignno{&|x-x'|^{-\eta} |T_\nu^mf(x)-T_\nu^mf(x')|\cr
\le& (\beta_{m-1}(x))^{\nu -2\eta}
\left(9^{\eta}|f|_{\eta}+\nu c_3^{\nu-1} ||f||_{{\cal C}^0}
(\beta_{m-1}(x))^{2\eta-2} |x-x'|^{1-\eta} \right)\ ,\cr
\le& (\beta_{m-1}(x))^{\nu -2\eta} \left(9^{\eta}|f|_{\eta}+
\nu c_3^{\nu-1}||f||_{{\cal C}^0} (16/9)^{1-\eta} \right)=
K_f (\beta_{m-1}(x))^{\nu -2\eta}\; ,\qquad &(4.11)\cr}
$$
where we have used $|x-x'|\le |J|$, and (from 1.48)
$\be_{m-1}(x)\ge (3/4)q_m^{-1}\ge (3/4)|J|^{1/2}$, $J$ being
the domain of the branch of $A^m$ which contains $x$ and $x'$.
\par\smallskip
{\it Third case:} one has $n-\delta <m\le n$: one is then led
to consider the cases (B), (C) or (D) defined above, after
Definition 1.9. One then introduces
the intermediate point $x''$ and one gets the same estimates of the
second case. More precisely,
$$\eqalignno{|T_\nu^mf(x)-T_\nu^mf(x')|&\le
|T_\nu^mf(x)-T_\nu^mf(x'')|+ |T_\nu^mf(x')-T_\nu^mf(x'')|\cr
&\le 2K_f\max\left[ (\beta_{m-1}(x))^{\nu -2\eta} |x-x''|^\eta,
(\beta_{m-1}(x'))^{\nu -2\eta}) |x'-x''|^\eta\right]\cr
&\le 2K_fc_3^{\nu-2\eta} (\beta_{m-1}(x))^{\nu -2\eta} |x-x'|^\eta,
&(4.12)\cr}
$$
where we have used (1.41).\par\smallskip\noindent
One can summarize the possible cases, in view of  Theorem 4.3,
by stating that there exists always a constant $c_{13}$ such
that
$$
|T_\nu^mf(x)-T_\nu^mf(x')|\le c_{13}||f||_\eta
|\beta_{m-1}(x)|^{\nu -2\eta}
|x-x'|^\eta\; .
\eqno(4.13)
$$
The statements of the theorem result easily from the Equations
(4.3), (4.7), (4.11), (4.12), and from the obvious inequality for the
${\cal C}_0$ norm
$$
|T_\nu^mf(x)| \le ||f||_{{\cal C}^0}\sup[\beta_{m-1}^\nu (x)]
\le{1\over2}\ga^{(m-1)\nu} ||f||_{{\cal C}^0}\ .\eqno(4.14)
$$
\qed
\par\medskip
\remark{4.3.} It will be useful to notice here that an estimate
similar to (4.13) holds for
$|\varepsilon_m(x)T_\nu^mf(x)-\varepsilon_m(x')T_\nu^mf(x')|$
provided
$$f(0)=f(1/2)=0\ ,\eqno(4.15)$$
namely
$$
|\varepsilon_m(x)T_\nu^mf(x)-\varepsilon_m(x')
T_\nu^mf(x')|\le c_{13}||f||_\eta |\beta_{m-1}(x)|^{\nu -2\eta}
|x-x'|^\eta\; .  \eqno(4.16)
$$
In order to get (4.16), we follow the same argument as we used
in the previous proof, and distinguish three cases. The proof is
immediate in the first case where $m>n$, and the bound is
obtained in the same way as (4.7). In the second case where
$m\le n-\de$, we also get the result as in (4.8), since we now
have $\varepsilon_m(x)=\varepsilon_m(x')$. In the third case,
where $n-\de<m\le n$, one introduces once more the intermediate
point $x''$, and one observes that  either $x''_m=0$, or
$x''_m=1/2$, according to the discussion which follows Definition
1.9. It follows that $T_{\nu}^m(x'')=\be_{m-1}(x'')f(x''_m)=0$,
due to  condition (4.15). Therefore
$|\varepsilon_m(x)T_\nu^mf(x)-\varepsilon_m(x')T_\nu^mf(x')|\le
|T_\nu^mf(x)-T_\nu^mf(x'')|+ |T_\nu^mf(x')-T_\nu^mf(x'')| $,
and the required estimate is obtained as in (4.12). This
completes the proof of (4.16).\par\medskip
When $f\in {\cal C}^{\eta}_{[0,1/2]}$, with $\nu<2\eta$, we
easily deduce from the proof of the preceding theorem that $T_{\nu}f$
is in ${\cal C}^{\nu /2}$, and that ${\bf B}_{\nu}f$ is in
${\cal C}^{\theta}$, for any $\theta<\nu/2$. In fact,
one has a slightly stronger result for ${\bf B}_{\nu}f$
\Proc{Theorem 4.4.}{Let $f\in {\cal C}^{\eta}_{[0,1/2]}$,
\item{(1)} If $\nu <2\eta$, then the function
${\bf B_{\nu}}f=\sum_{m\ge 0}T_\nu^mf$ is in ${\cal C}^{\nu /2}$.
\item{(2)}If $\nu=2\eta$, then ${\bf B}_{\nu}f$ is in
${\cal C}^{\eta}$ for $0\le\eta <\nu/2$. Indeed it admits
$x^{\nu/2}|\log x|$ as continuity modulus.}\par
\proof We have, using (4.13) and setting $n=n(x,x')$,
$$
\eqalignno{
|{\bf B_{\nu}}&f(x)-{\bf B_{\nu}}f(x')| \le
\sum_0^n|T_\nu^mf(x)-T_\nu^mf(x')|+
\sum_{n+1}^\infty |T_\nu^mf(x)-T_\nu^mf(x')|\cr
&\le c_{13}||f||_\eta
|x-x'|^{\nu /2} \left( |x-x'|^{\eta-\nu/2}
\left(\sum_0^n(\beta_{m-1}(x))^{\nu -2\eta}\right)
+ \sum_{n+1}^\infty ||\beta_{m-n-2}||_{{\cal C}^0}^\nu\right)
\; .\cr& &(4.17)\cr} $$
{}From the bound on the $\be_n$, one sees immediately that
the second series in the previous inequality converges.
Furthermore all $x_k$ belong to $(0,1/2]$,  and one has
$$ \sum_0^n(\beta_{m-1}(x))^{\nu -2\eta}\le
(\beta_{n-1}(x))^{\nu -2\eta}\sum_{0}^n
\left({1\over 2^{n-m}}\right)^{2\eta-\nu}=
c_{15}(q_n(x))^{2\eta-\nu}\ ,$$
with  $c_{15}= (3/2)^{2\eta-\nu}\sum_{0}^n 2^{-(m(2\eta-\nu))}$,
which is finite.
Now, if $x$ and $x'$ belong to the same branch of $A^n$, let
$K$ be its interval of definition. If they belong to two
adjacent branches, let $K$ be the union of their intervals of
definition. In both cases one has $|x-x'|\le |K|$ and,
using Lemmas 1.12 and 1.13, one easily sees that $|K|q_n^2(x)$
is bounded, above and below,
so that finally, one can find $c_{16}$ such that
$$ \hbox{for}\quad \nu<2\eta\ ,\qquad
|{\bf B}_{\nu}f(x)- {\bf B}_{\nu}f(x')|
\le c_{16}||f||_{\eta} |x-x'|^{\nu /2}\; .\eqno(4.18)$$
When $\nu=2\eta$, we get
$$ |{\bf B}_{\nu}f(x)-{\bf B}_{\nu}f(x')| \le c_{17}(1+n(x,x'))
||f||_{\eta} |x-x'|^{\nu /2}\; .\eqno(4.19)$$
However, $x$ and $x'$ belong to the same branch $|J|$ of $A^{n-2}$,
so that, using (1.48), one gets $|x-x'|\le |J|\le q_{n-2}^{-2}$,
and following remark 1.5, $|x-x'|\le (3/4)\ga^{n-3}$. We
therefore have, up to some constant $c$, $\log |x-x'|\ge
n\log {\ga}+c$, which means that $n$ is bounded:
$n\le (\log (|x-x'|^{-1})-c)/\log\Gamma$. Therefore there
exists a constant $c_{18}$ such that
$$ \hbox{for}\quad \nu=2\eta\ ,\qquad
|{\bf B}_{\nu}f(x)-{\bf B}_{\nu}f(x')|
\le c_{18}||f||_{\eta} \log(|x-x'|^{-1})|x-x'|^{\nu /2}
\; .\eqno(4.20)$$
\qed
\par
As a consequence of the above theorem, we observe that if we
consider a perturbation of the functional equation (2.7) for
the Brjuno function (in the case $\al=1/2$), that is
$$\hbox{for\ \ }0<x\le{1\over2}\quad,\quad[(1-T_1)B_f](x)=
-\log x +f(x)\; ,$$
the function $B_f$ differs from $B$ by  {\it 1/2-H\"older
continuous function} when $f$ is analytic, or at least 1/2-
H\"older continuous. Therefore the `most singular' part of
$B_f$ does not depend on the perturbation when it is
sufficiently regular. The present result could provide an
explanation for the so-called `modular smoothing' of critical
functions observed in earlier works [BPV,MS].
\par
The above result seems to give a special role to the Brjuno
function $B=B_{1/2}$. However, the theorem below displays
a somewhat surprising result, namely that the difference
$B_{1}-B_{1/2}$ is not only bounded, as we already know, but
also 1/2-H\"older continuous. We first give a preparatory
statement.
\Proc{Proposition 4.5}{Let $B_1^+$ and $B_1^-$ be the even and
odd part of $B_1$. $B_1^+$ and $B_1^-$ are periodic, so that
they are determined by their values in $[0,1/2]$. We have
$$\eqalignno{\hbox{for}\quad x\in[0,1/2]
\quad ,\quad B^-_1(x)&= {x\over 2}\log\left({1-x\over x}
\right)&(4.21)\cr
B^+_1(x)&=xB^+_1\left({1\over x}\right) +g(x)-\log x\ ,\qquad\qquad
\quad&(4.22)\cr}$$
with, still for
$x\in[0,1/2]$,
$$g(x)=- {x\over 2}\log\left({1-x\over x}\right)
+xB_1^-\left({1\over x}\right)\; .\eqno(4.23)$$ }\par
\proof For $x\in[0,1/2]\cap [{\Bbb R}\setminus {\Bbb Q}]$, we
have $B_1(-x)=B_1(1-x)$, and $1<(1-x)^{-1}<2$, so that
$B_1(-x)=-\log(1-x)+(1-x)B_1((1-x)^{-1}-1)$.  But
$$B_1\left({1\over 1-x}-1\right)=B_1\left({x\over 1-x}\right)
=\log\left({1-x\over x}\right)+{x\over 1-x}
B_1\left({1\over x}\right) \; ,$$
since $0<x/(1-x)<1$. Therefore
$$B_1(-x)=-x\log(1-x)-(1-x)\log x+xB_1(x^{-1})\; .$$
Since we also have
$$ B_1(x)=-\log x+xB_1(x^{-1})\; ,$$
we easily get (4.21) by subtraction. By addition, we get
$$B_1^+(x)=xB_1\left({1\over x}\right)-\log x -{x\over 2}
\log\left({1-x\over x}\right)
= xB_1\left({1\over x}\right)-\log x -B_1^-(x) \; ,\eqno(4.24)$$
which leads to (4.22) since we already know $B_1^-$.
\qed
\medskip
The odd part of $B_1-B_{1/2}$ coincides with $B_1^-$, and (4.21)
shows that $B_1^-$ is in  ${\cal C}^{\eta}$ for $0\le\eta<1$.
The even part of $B_1-B_{1/2}$, namely
$$\Delta(x)=B_1^+(x)-B_{1/2}(x) \eqno(4.25)$$
satisfies $\Delta(x)=g(x)+x\Delta(x^{-1})$. From (4.23) and
Theorem 4.2, one deduces that $g\in{\cal C}^{\eta}$,
for $0\le\eta<1/2$. Then Theorem 4.4 tells that $\Delta\in
{\cal C}^{\eta}$ for $0\le\eta<1/2$, and the same holds for the
difference $B_1-B_{1/2}$. It is easy to see that the continuity
extends to $x=0$ and $x=1/2$, by setting
$\Delta(0)=\Delta(1/2)=0$.
In fact, the following theorem gives a
stronger result which includes the case $\eta=1/2$.\par
\Proc{Theorem 4.6.}{ The difference $B_1-B_{1/2}$ may be extended
from ${\Bbb R}\setminus {\Bbb Q}$ to ${\Bbb R}$ as an
$1/2$-H\"older continuous periodic function with period one.}\par
\proof
{}From  (4.24) and (4.25), we get for $0<x<1/2$,
$$\Delta(x)=x\Delta(x^{-1})+xB_1^-(x^{-1})-B_1^-(x)=
x_0\Delta(x_1)+\varepsilon_1 x_0B_1^-(x_1)-B_1^-(x_0)\; ,
\eqno(4.26)$$
where we have used the first step of the continued fraction
expansion of $x$. Solving (4.26) by iteration leads to
the even and periodic functions $\Delta$, $\Delta_1$, $\Delta_2$
defined for $0<x<1/2$ as:
$$\eqalignno{
\Delta(x)&=\Delta_1(x)+\Delta_2(x)\;,&(4.27)\cr
\Delta_1(x)&=-\sum_{n=0}^{\infty}\be_{n-1}(x)B_1^-(x_n)\; ,
&(4.28)\cr
\Delta_2(x)&=\sum_{n=1}^{\infty}\varepsilon_n(x)\be_{n-1}(x)
B_1^-(x_n) \; .&(4.29)\cr}$$
We deduce from Theorem 4.4 that $\Delta_1= -(1-T_1)^{-1}B_1^-$ is
an even  periodic function
in ${\cal C}^{1/2}$,  since $B_1^-$ is sufficiently regular.
We now observe that $B_1^-(0)=B_1^-(1/2)=0$, therefore
the evaluation of $|\Delta_2(x)-\Delta_2(x')|$ is made following
the proof of Theorem 4.4,   for $\nu=1$, using  (4.16)
instead of (4.13).
The conclusion is the same as Theorem 4.4, part (i), that is
that also $\Delta_2\in{\cal C}^{1/2}$.
Now, $\Delta$ is the even part of $B_1-B_{1/2}$,
and since the odd part of $B_1-B_{1/2}$ is nothing but $B_1^-$,
we deduce that   $B_1-B_{1/2}$ is $1/2$-H\"older-continuous.
\qed
\vskip 1 truecm
\beginsection Appendix:  B. M. O. Norms\par
Let $f\in\Lloc$. We define the mean value $f_I$ of $f$ on the interval
$I$ as:
$$ f_I=\mean{I}{f}\ ,\eqno({\rm A}.1)$$
where $|I|$ is the length of the interval $I$. Then we define
for any interval $U$
$$ \norm{f}{U}=\Dnorm{I\subset U}{f}\ .\eqno({\rm A}.2)$$
We then say $f$ belongs to the space \BM{U} if $\norm{f}{U}<\infty$,
\ie is finite. BMO is an abbreviation for `bounded mean oscillation'.
$\norm{f}{U}$ is a seminorm on \BM{U}, since for any constant $c$,
we have $\norm{f+c}{U}=\norm{f}{U}$. In particular $\norm{f}{U}=0$
if $f$ is constant on $U$. This applies to $U=\R$ and leads
to the space \BM{\R}, abbreviated as BMO
and the seminorm $\norm{f}{\R}$
on \BM{\R} will simply be written  $||f||_*$.
In fact this seminorm is a norm on the quotient space of function
in $\Lloc$ modulo the constant functions. With this norm, the quotient
space is complete.
We list now some  more or less classical results and lemmas [Gr,GRCF].
\Proc{Proposition A.1.}{The space $L^{\infty}(U)$ is a
subspace of \BM{U}, and
$\norm{f}{U}\le\inf_{\dst c}\lnorm{f-c}{U}{\infty}$}.\par
\Proc{Proposition A.2.}{Let $f$ in \BM{U} and let $I$ be
an interval. Then for any $\lambda>0$, the Lebesgue measure
of the set of points $t\in I$  such that $|f(t)-f_I|>\lambda$
is bounded by $K_1\exp(-K_2\lambda/\norm{f}{U})$.}\par
This is the  John-Nirenberg Theorem theorem, see Garnett [Gr].
The proof there is for $U=\R$,
but a careful reading show that it works for any $U$. The
constants $K_1$ and $K_2$ do not depend on $U$, $\lambda$, and
$f$. Roughly speaking, this theorem says that where $f$ is
unbounded, it behaves at most as  as a logarithmic function.
\Proc{Proposition A.3.}{We have the following `magic
reverse H\"older's inequality':
let $f\in\Lloc$ and suppose that for some interval $U\subset\R$, the
seminorm $\norm{f}{U}$ is finite, then for any bounded real $p\ge 1$,
there exists a constant $A_p$ such that
$$\sup_{I\subset U}\left(\mean{I}{|f-f_I|^p}\right)^{\sst1\over\sst p}
\le A_p\norm{f}{U}\ .\eqno({\rm A}.3)$$}\par
In fact it is an easy corollary of the John-Nirenberg
theorem, see Garnett [Gr].  The constant $A_p$ {\it does not
depend on }$U$, and may be shown to be smaller
than $pC$  with an explicit constant $C$. Note that the
inequality does not work in the limit
$p\to\infty$.\qed
The preceding proposition shows that replacing the $L^1$ norm in the
definition of the $BMO$ norm $\norm{f}{U}$, by the analogous $L^p$
norm (with $p$ finite), leads to the same $BMO$ space. More precisely
using the usual $L^p$ norm
$$\lnorm{f}{U}{p}=\left(\int_U |f|^p\,dx\right)^{\sst1\over\sst p}
\ ,\eqno({\rm A}.4)$$
we define
$$\Norm{f}{U}{p}=\sup_{I\subset U}
\left(\mean{I}{|f-f_I|^p}\right)^{\sst1\over\sst p}=\sup_{I\subset U}
|I|^{-{\sst1\over\sst p}}\lnorm{f-f_I}{I}{p}\ .\eqno({\rm A}.5)$$
We then have
\Proc{Proposition A.4.}{The space \BM{U}, is a subspace of $L^p(U)$
when $U$ is a bounded interval.}\par
Thus, \BM{U} is a subspace of $\cap_{p=1}^{\infty}L^{p}(U)$, but
{\it not} a subspace of $L^{\infty}(U)$.
In fact, on \BM{U}, we have a family of equivalent norms, as
shown in the following proposition.
\Proc{Proposition A.5.}{On \BM{U}, where $U$ is a bounded interval,
define for any real
$a>0$, and $b>0$, and for any integer $p\ge 1$ (finite), the
following family of norms
$$N(f,a,b,p)=a\Norm{f}{U}{p}+b \lnorm{f}{U}{p}\ ,
\eqno({\rm A}.6)$$ then
these norms are all equivalent for various $a$ and $b$
and $p$.}\par
In \BM{\R}, we will also define the seminorm $\norm{f}{\T}$
as follows
$$\norm{f}{\T}=\Dnorm{\dst|I|\le 1}{f}\ ,\eqno({\rm A}.7)$$
the supremum being taken over intervals $I\subset \R$ with length
less or equal to 1. The seminorm $\norm{f}{\T}$ is convenient
for periodic functions with period 1.
For any $f\in\BM{\R}$ we obviously have:
$\norm{f}{[-(1/2),+(1/2)]}\le\norm{f}{\T}\le\norm{f}{\R}\ .$
This observation will be useful if we now consider
functions $f\in\BM{\R}$ {\it which are even and periodic
with period 1}: for such  functions, we also have the
following result
\Proc{Proposition A.6.}{There exist constants $K_3>1$, $K_4>1$, $K_5>1$
such that for any
$f\in\BM{\R}$, which is even and periodic with period 1, we have
$$\matrix{\hbox{a)}\quad\hfill &\norm{f}{\R}\le K_3
\norm{f}{\T}\hfill&\cr
\hbox{b)}\quad\hfill &\norm{f}{\T}\le K_4\norm{f}{[0,1]}\hfill&\cr
\hbox{b)}\quad\hfill &\norm{f}{[0,1]}\le K_5\norm{f}{[0,1/2]}
\hfill&\ .\cr }$$}\par
See [MMY] for a detailed proof. Note that parts b) and c) are not
true if the periodic function $f$ is not even. A non trivial, but
immediate consequence of these results is the following corollary.
\Proc{Corollary A.7.}{Let $f$ be a function defined in $[0,1/2]$,
which belongs to \BM{[0,1/2]}. The function $g$ which is even and
periodic with period 1, and which coincides with $f$ on $[0,1/2]$
is in \BM{\R}.}\par
As indicated in Propositions A.5 and A.6, we have a wide choice
among  possible  equivalent norms. The $L^2$-norm is especially
useful, due to the following elementary identity:
$$
{1\over|I|}\int_I (f-f_I)^2 \,ds = {1\over 2|I|^2}
\int_{I\times I}(f(s)-f(t))^2 \,dsdt \; .  \eqno({\rm A}.8)
$$
Defining  the oscillation of $f$ in $I$ with the $L_2$-norm,
namely
$${\cal O}_I(f) = \left({1\over 2|I|^2}\int_{I\times I}
(f(s)-f(t))^2\, dsdt \right)^{1\over2}\ ,\eqno({\rm A}.9)$$
the norm $N(f,a,b,2)$ can be rewritten as
$$
N(f,a,b,2)=a\sup_{I\subset U}{\cal O}_I(f)+b \lnorm{f}{U}{2}\ ,
\eqno({\rm A}.10)$$
In the above expression, (A.9) and (A.10), replacing the
Lebesgue measure on $U$ by any equivalent mesure, leads to
an equivalent norm.\par
We now end this appendix by quoting
Fefferman's theorem, which makes the link between analysis
on the real line and harmonic or complex extension on the upper
halfplane.
\Proc{Proposition A.8}{The space \BM{\R} is the dual space of the Hardy
space $H^1$ on ${\Bbb R}$. If $f\in \Lloc$, $f\in\BM{\R}$ if and
only if there exist a constant $c$, and functions $\phi$ and
$\psi$ in $L^{\infty}$, such that
$f=c+\phi+H\psi$, where the Hilbert transform $H\psi$ is the
harmonic conjugate of $\psi$. Furthermore, $\phi$ and $\psi$
can be chosen such that $||\phi||_{\infty}\le C||f||_*$ and
$||\psi||_{\infty}\le C||f||_*$, with $C$ a constant.}
\par
\vskip 1 truecm
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\bye