%%%%%%%%%%Paper Marmi, Moussa, Yoccoz %%%%%%%% \def\service{T} % ******************** output macros ******************************** \catcode`\@=11 %%% saclay A4 paper: \def\unredoffs{\voffset=11mm \hoffset=0.5mm} \def\redoffs{\voffset=-12.5truemm\hoffset=-6truemm} \def\speclscape{\special{landscape}} % %---------------------------------------------------------------------% \newbox\leftpage \newdimen\fullhsize \newdimen\hstitle \newdimen\hsbody \newdimen\hdim \tolerance=400\pretolerance=800 %\tolerance=1000\hfuzz=2pt %\def\fontflag{cm} % % \newif\ifsmall \smallfalse \newif\ifdraft \draftfalse \newif\iffrench \frenchfalse \newif\ifeqnumerosimple \eqnumerosimplefalse % % \nopagenumbers \headline={\ifnum\pageno=1\hfill\else\hfil{\headrm\folio}\hfil\fi} \def\draftstart{ \ifsmall \message{(Reduced size)} \let\l@r=L \magnification=1000 \vsize=190truemm \redoffs% \hstitle=115truemm\hsbody=115truemm\fullhsize=10truein\hsize=\hsbody % \output={\ifnum\pageno=0 %%% This is the HUTP version \shipout\vbox{\speclscape{\hsize\fullhsize\makeheadline} \hbox to \fullhsize{\hfill\pagebody\hfill}}\advancepageno \else \almostshipout{\leftline{\vbox{\pagebody\makefootline}}}\advancepageno \fi} \headline={\hfil\oddpage\hfil\hfil\headrm\folio\hfil} \gdef\oddpage{} \def\almostshipout##1{\if L\l@r \count1=1 \message{[\the\count0.\the\count1]} \global\setbox\leftpage=##1 \global\let\l@r=R \xdef\oddpage{\ifnum\count0=1\else\headrm\the\count0\fi} \else \count1=2 \shipout\vbox{\speclscape{\hsize\fullhsize\makeheadline} \hbox to\fullhsize{\box\leftpage\hfil##1}} \global\let\l@r=L\fi} \else \message{(Normal size)} \magnification=1200 \unredoffs\hsize=130mm\vsize=190mm \hsbody=\hsize \hstitle=\hsize %take default values for unreduced format \fi \ifdraft \special{! userdict begin /bop-hook {gsave 100 160 translate 50 rotate 0 0 moveto /Times-Roman findfont 50 scalefont setfont 0.95 setgray (PRELIMINARY VERSION) show grestore} def end} \footline={{\bf\hfil Version \today}} \writelabels \else \nolabels \overfullrule=0pt \fi \iffrench % \fhyph \dicof \else \dicoa \fi } %**************** MAC.TEX *********************************** %************************************************************ % origine: harvmac + modifications J. Zinn-Justin % + modifications J.-M. Drouffe % fonts, Dirac slash \font\elevrm=cmr9 \font\elevit=cmti9 \font\subrm=cmr7 \newdimen\chapskip \font\twbf=cmssbx10 scaled 1200 \font\ssbx=cmssbx10 \font\twbi=cmmib10 scaled 1200 \font\caprm=cmr9 \font\capit=cmti9 \font\capbf=cmbx9 \font\capsl=cmsl9 \font\capmi=cmmi9 \font\capex=cmex9 \font\capsy=cmsy9 \chapskip=17.5mm \def\makeheadline{\vbox to 0pt{\vskip-22.5pt \line{\vbox to8.5pt{}\the\headline}\vss}\nointerlineskip} %*************************************************** \font\tbfi=cmmib10 \font\tenbi=cmmib7 \font\fivebi=cmmib5 \textfont4=\tbfi \scriptfont4=\tenbi \scriptscriptfont4=\fivebi \font\headrm=cmr10 \font\headit=cmti10 \font\twmi=cmmi10 scaled 1200 %**************************** \font\eightrm=cmr6 \font\sixrm=cmr5 \font\eightmi=cmmi6 \font\sixmi=cmmi5 \font\eightsy=cmsy6 \font\sixsy=cmsy5 \font\eightbf=cmbx6 \font\sixbf=cmbx5 \skewchar\capmi='177 \skewchar\eightmi='177 \skewchar\sixmi='177 \skewchar\capsy='60 \skewchar\eightsy='60 \skewchar\sixsy='60 \def\elevenpoint{ \textfont0=\caprm \scriptfont0=\eightrm \scriptscriptfont0=\sixrm \def\rm{\fam0\caprm} \textfont1=\capmi \scriptfont1=\eightmi \scriptscriptfont1=\sixmi \textfont2=\capsy \scriptfont2=\eightsy \scriptscriptfont2=\sixsy \textfont3=\capex \scriptfont3=\capex \scriptscriptfont3=\capex \textfont\itfam=\capit \def\it{\fam\itfam\capit} % \it is family 4 \textfont\slfam=\capsl \def\sl{\fam\slfam\capsl} % \sl is family 5 \textfont\bffam=\capbf \scriptfont\bffam=\eightbf \scriptscriptfont\bffam=\sixbf \def\bf{\fam\bffam\capbf} % \bf is family 6 \textfont4=\tbfi \scriptfont4=\tenbi \scriptscriptfont4=\tenbi \normalbaselineskip=13pt \setbox\strutbox=\hbox{\vrule height9.5pt depth3.9pt width0pt} \let\big=\elevenbig \normalbaselines \rm} \catcode`\@=11 \font\tenmsa=msam10 \font\sevenmsa=msam7 \font\fivemsa=msam5 \font\tenmsb=msbm10 \font\sevenmsb=msbm7 \font\fivemsb=msbm5 \newfam\msafam \newfam\msbfam \textfont\msafam=\tenmsa \scriptfont\msafam=\sevenmsa \scriptscriptfont\msafam=\fivemsa \textfont\msbfam=\tenmsb \scriptfont\msbfam=\sevenmsb \scriptscriptfont\msbfam=\fivemsb \def\hexnumber@#1{\ifcase#1 0\or1\or2\or3\or4\or5\or6\or7\or8\or9\or A\or B\or C\or D\or E\or F\fi } % The following 13 lines establish the use of the Euler Fraktur font. % To use this font, remove % from beginning of these lines. \font\teneuf=eufm10 \font\seveneuf=eufm7 \font\fiveeuf=eufm5 \newfam\euffam \textfont\euffam=\teneuf \scriptfont\euffam=\seveneuf \scriptscriptfont\euffam=\fiveeuf \def\frak{\ifmmode\let\next\frak@\else \def\next{\Err@{Use \string\frak\space only in math mode}}\fi\next} \def\goth{\ifmmode\let\next\frak@\else \def\next{\Err@{Use \string\goth\space only in math mode}}\fi\next} \def\frak@#1{{\frak@@{#1}}} \def\frak@@#1{\fam\euffam#1} % End definition of Euler Fraktur font. \edef\msa@{\hexnumber@\msafam} \edef\msb@{\hexnumber@\msbfam} \def\msb{\tenmsb\fam\msbfam} \def\Bbb{\ifmmode\let\next\Bbb@\else \def\next{\errmessage{Use \string\Bbb\space only in math mode}}\fi\next} \def\Bbb@#1{{\Bbb@@{#1}}} \def\Bbb@@#1{\fam\msbfam#1} \font\sacfont=eufm10 scaled 1440 \catcode`\@=12 % \def\sla#1{\mkern-1.5mu\raise0.4pt\hbox{$\not$}\mkern1.2mu #1\mkern 0.7mu} \def\Dbar{\mkern-1.5mu\raise0.4pt\hbox{$\not$}\mkern-.1mu {\rm D}\mkern.1mu} \def\Abar{\mkern1.mu\raise0.4pt\hbox{$\not$}\mkern-1.3mu A\mkern.1mu} % ******************************************************************* % Dictionnaires francais et anglais \def\dicof{ \gdef\Resume{RESUME} \gdef\Toc{Table des mati\`eres} \gdef\soumisa{Soumis \`a:} } \def\dicoa{ \gdef\Resume{ABSTRACT} \gdef\Toc{Table of Contents} \gdef\soumisa{Submitted to} } % ****** extrait de definit.tex (obsolete ?) \def\fileth{\noalign{\hrule}} \def\saut{\noalign{\smallskip}} \def\alignement{\offinterlineskip\halign} \def\filetv{\vrule} \def\colgauche{\strut\ } \def\coldroite{\ } \def\filetdroit{\cr} \def\filetvide{height2pt} \def\colvide{\omit} \def\fintableau{} \def\uniset{\rlap{\elevrm 1}\kern.15em 1} \def\bkR{{\rm I\kern-.17em R}} \def\bkC{{\rm \kern.24em \vrule width.05em height1.4ex depth-.05ex \kern-.26em C}} % ********* A few math symbols \def\e{\mathop{\rm e}\nolimits} \def\sgn{\mathop{\rm sgn}\nolimits} \def\Im{\mathop{\rm Im}\nolimits} \def\Re{\mathop{\rm Re}\nolimits} \def\d{{\rm d}} \def\ud{{\textstyle{1\over 2}}} \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 ********************************************** \def\saclay{\if S\service \spec \else \spht \fi} \def\spht{ \centerline{CEA, Service de Physique Th\'eorique, CE-Saclay} \centerline{F-91191 Gif-sur-Yvette Cedex, FRANCE}} \def\spec{ \centerline{CEA/DSM/DRECAM/Service de Physique de l'Etat Condens\'e} \centerline{CE Saclay, F-91191 Gif-sur-Yvette Cedex, FRANCE}} % \def\logo{ \if S\service % Logo SPEC \font\sstw=cmss10 scaled 1200 \font\ssx=cmss8 \vtop{\hsize 9cm {\sstw {\twbf P}hysique de l'{\twbf E}tat {\twbf C}ondens\'e \par} \ssx SPEC -- DRECAM -- DSM\par \vskip 0.5mm \sstw CEA -- Saclay \par } \else % Logo SPHT \vtop{\hsize 9cm \special{" /Helvetica-Bold findfont 9 scalefont setfont 0 -80 translate 2 73 moveto (PHYSIQUE\ \ THEORIQUE) show 35 38 moveto (CEA-DSM) show 0.7 setgray /Helvetica-Bold findfont 26.5 scalefont setfont 0 50 moveto (SACLAY) show 0 setgray 1.5 setlinewidth 0 41 moveto 32 41 lineto stroke 80 41 moveto 110 41 lineto stroke}} \fi } % ************************************************************************* \catcode`\@=11 % ************** double alignment in eqalignno style ********************** \def\deqalignno#1{\displ@y\tabskip\centering \halign to \displaywidth{\hfil$\displaystyle{##}$\tabskip0pt&$\displaystyle{{}##}$ \hfil\tabskip0pt &\quad \hfil$\displaystyle{##}$\tabskip0pt&$\displaystyle{{}##}$ \hfil\tabskip\centering& \llap{$##$}\tabskip0pt \crcr #1 \crcr}} % ************** double eqalign ****************************************** \def\deqalign#1{\null\,\vcenter{\openup\jot\m@th\ialign{ \strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil &&\quad\strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$ \hfil\crcr#1\crcr}}\,} %*************************************************************************** %********* titlepage, headline, section, subsection, sub, appendix ********* %*************************************************************************** %********* introduce equation number file: for non-causal quotation \openin 1=\jobname.sym \ifeof 1\closein1\message{<< (\jobname.sym DOES NOT EXIST) >>}\else% \input\jobname.sym\closein 1\fi % \newcount\nosection \newcount\nosubsection \newcount\neqno \newcount\notenumber \newcount\figno \newcount\tabno \def\content{\jobname.toc} \def\symbols{\jobname.sym} %\def\Figures{\jobname.fig} %\def\Tables{\jobname.tab} \newwrite\toc \newwrite\sym %\newwrite\Fig %\newwrite\Tab % ******************* titlepage ********************************** %\def\authorname#1{\maketitle{\bf #1}\smallskip} \def\authorname#1{\centerline{\bf #1}\smallskip} \def\address#1{ #1\medskip} % \newdimen\hulp \def\maketitle#1{ \edef\oneliner##1{\centerline{##1}} \edef\twoliner##1{\vbox{\parindent=0pt\leftskip=0pt plus 1fill\rightskip=0pt plus 1fill \parfillskip=0pt\relax##1}} \setbox0=\vbox{#1}\hulp=0.5\hsize \ifdim\wd0<\hulp\oneliner{#1}\else \twoliner{#1}\fi} \def\pacs#1{{\bf PACS numbers:} #1\par} \def\submitted#1{{\it {\soumisa} #1}\par} % **************** beginning \def\title#1{\gdef\titlename{#1} \maketitle{ %\ssbx\uppercase\expandafter \twbf {\titlename}} \vskip3truemm\vfill \nosection=0 \neqno=0 \notenumber=0 \figno=1 \tabno=1 \def\prefix{} \def\eqprefix{} \mark{\the\nosection} \message{#1} \immediate\openout\sym=\symbols } \def\preprint#1{\vglue-10mm \line{ \logo \hfill {#1} }\vglue 20mm\vfill} \def\abstract{\vfill\centerline{\Resume} \smallskip \begingroup\narrower \elevenpoint\baselineskip10pt} \def\endabstract{\par\endgroup \bigskip} % ***************** input table of contents \def\mktoc{\centerline{\bf \Toc} \medskip\caprm \parindent=2em \openin 1=\jobname.toc \ifeof 1\closein1\message{<< (\jobname.toc DOES NOT EXIST. TeX again)>>}% \else\input\jobname.toc\closein 1\fi \bigskip} %******************************* section *********************************** \def\section#1\par{\vskip0pt plus.1\vsize\penalty-100\vskip0pt plus-.1 \vsize\bigskip\vskip\parskip \message{ #1} \ifnum\nosection=0\immediate\openout\toc=\content% \edef\ecrire{\write\toc{\par\noindent{\ssbx\ \titlename} \string\leaderfill{\noexpand\number\pageno}}}\ecrire\fi% ajout \advance\nosection by 1\nosubsection=0 \ifeqnumerosimple \else \xdef\eqprefix{\prefix\the\nosection.}\neqno=0\fi \vbox{\noindent\bf\prefix\the\nosection\ #1} \mark{\the\nosection}\bigskip\noindent \xdef\ecrire{\write\toc{\string\par\string\item{\prefix\the\nosection} #1 \string\leaderfill {\noexpand\number\pageno}}}\ecrire} % appendix \def\appendix#1#2\par{\bigbreak\nosection=0 \notenumber=0 \neqno=0 \def\prefix{A} \mark{\the\nosection} \message{\appendixname} \leftline{\ssbx APPENDIX} \leftline{\ssbx\uppercase\expandafter{#1}} \leftline{\ssbx\uppercase\expandafter{#2}} \bigskip\noindent\nonfrenchspacing \edef\ecrire{\write\toc{\par\noindent{{\ssbx A}\ {\ssbx#1\ #2}}\string\leaderfill{\noexpand\number\pageno}}}\ecrire}% % **************************** \subsection ************************* \def\subsection#1\par {\vskip0pt plus.05\vsize\penalty-100\vskip0pt plus-.05\vsize\bigskip\vskip\parskip\advance\nosubsection by 1 \vbox{\noindent\it\prefix\the\nosection.\the\nosubsection\ \it #1}\smallskip\noindent \edef\ecrire{\write\toc{\string\par\string\itemitem {\prefix\the\nosection.\the\nosubsection} {#1} \string\leaderfill{\noexpand\number\pageno}}}\ecrire } % \def\note #1{\advance\notenumber by 1 \footnote{$^{\the\notenumber}$}{\sevenrm #1}} % ????? \def\sub#1{\medskip\vskip\parskip {\indent{\it #1}.}} %\parindent=1em %\newinsert\margin %\dimen\margin=\maxdimen %\count\margin=0 \skip\margin=0pt % ********************* references harvmac style \def\nolabels{\def\wrlabel##1{}\def\eqlabel##1{}\def\reflabel##1{}} \def\writelabels{\def\wrlabel##1{\leavevmode\vadjust{\rlap{\smash% {\line{{\escapechar=` \hfill\rlap{\sevenrm\hskip.03in\string##1}}}}}}}% \def\eqlabel##1{{\escapechar-1\rlap{\sevenrm\hskip.05in\string##1}}}% \def\reflabel##1{\noexpand\llap{\noexpand\sevenrm\string\string\string##1}}} %********* %\catcode`\@=11 \global\newcount\refno \global\refno=1 \newwrite\rfile % \def\ref{[\the\refno]\nref} \def\nref#1{\xdef#1{[\the\refno]}\writedef{#1\leftbracket#1}% \ifnum\refno=1\immediate\openout\rfile=\jobname.ref\fi \global\advance\refno by1\chardef\wfile=\rfile\immediate \write\rfile{\noexpand\item{#1\ }\reflabel{#1\hskip.31in}\pctsign}\findarg} % horrible hack to sidestep tex \write limitation \def\findarg#1#{\begingroup\obeylines\newlinechar=`\^^M\pass@rg} {\obeylines\gdef\pass@rg#1{\writ@line\relax #1^^M\hbox{}^^M}% \gdef\writ@line#1^^M{\expandafter\toks0\expandafter{\striprel@x #1}% \edef\next{\the\toks0}\ifx\next\em@rk\let\next=\endgroup\else\ifx\next\empty% \else\immediate\write\wfile{\the\toks0}\fi\let\next=\writ@line\fi\next\relax}} \def\striprel@x#1{} \def\em@rk{\hbox{}} % \def\semi{;\hfil\break} \def\addref#1{\immediate\write\rfile{\noexpand\item{}#1}} %now unnecessary % \def\listrefs{ \ifnum\refno=1 \else \immediate\closeout\rfile\writestoppt\baselineskip=14pt% \vskip0pt plus.1\vsize\penalty-100\vskip0pt plus-.1 \vsize\bigskip\vskip\parskip\centerline{{\bf References}}\bigskip% {\frenchspacing% \parindent=20pt\escapechar=` \input \jobname.ref\vfill\eject}% \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}}} % \def\writestop{\def\writestoppt{\immediate\write\lfile{\string\pageno% \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 } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \draftstart \preprint{T95/016} \title{Continued fraction transformations, Brjuno functions and BMO spaces} \authorname{S. Marmi} \address{\centerline{Dipartimento di Matematica \lq\lq U. Dini\rq\rq , Universit\`a di} \centerline{Firenze, } \centerline{Viale Morgagni 67/A, 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 The small divisors problem which is raised by stability questions in classical mechanics has an analog in holomorphic dynamical systems, namely the existence of Siegel disks. The size of these disks is well represented by the Brjuno functions. We analyse the relation between these functions and the various continued fraction transformations, and display the functional equation which is fulfilled by these highly singular functions. The analysis of this functional equation shows that the Brjuno function belong to the BMO space, and that a regular perturbation of this equation leads to a modification of the singular function which is 1/2-H\"older continuous. This leads us to believe that the most singular part of the size of the stability domains as function of the rotation number, is `universal' up to a 1/2-H\"older continuous function. \endabstract \vfill \submitted{Note CEA} \eject %%%% Marmi, Moussa, Yoccoz %%%%% %%%%%%%% macros %%%%% \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 \catcode`\;=\active \def;{\relax\ifhmode\ifdim\lastskip>\z@ \unskip\fi\kern.2em\fi\string;} \catcode`\:=\active \def:{\relax\ifhmode\ifdim\lastskip>\z@\unskip\fi \penalty\@M\ \fi\string:} \catcode`\!=\active \def!{\relax\ifhmode\ifdim\lastskip>\z@ \unskip\fi\kern.2em\fi\string!} \catcode`\?=\active \def?{\relax\ifhmode\ifdim\lastskip>\z@ \unskip\fi\kern.2em\fi\string?} \def\^#1{\if#1i{\accent"5E\i}\else{\accent"5E #1}\fi} \def\"#1{\if#1i{\accent"7F\i}\else{\accent"7F #1}\fi} %\frenchspacing \catcode`\@=12 \newif\ifpagetitre \pagetitretrue \newtoks\hautpagetitre \hautpagetitre={\hfil} \newtoks\baspagetitre \baspagetitre={\hfil} \newtoks\auteurcourant \auteurcourant={\hfil} \newtoks\titrecourant \titrecourant={\hfil} \newtoks\hautpagegauche \newtoks\hautpagedroite \hautpagegauche={\hfil\the\auteurcourant\hfil} \hautpagedroite={\hfil\the\titrecourant\hfil} \newtoks\baspagegauche \baspagegauche={\hfil\tenrm\folio\hfil} \newtoks\baspagedroite \baspagedroite={\hfil\tenrm\folio\hfil} \headline={\ifpagetitre\the\hautpagetitre \else\ifodd\pageno\the\hautpagedroite \else\the\hautpagegauche\fi\fi} \footline={\ifpagetitre\the\baspagetitre \global\pagetitrefalse \else\ifodd\pageno\the\baspagedroite \else\the\baspagegauche\fi\fi} \def\nopagenumbers{\def\folio{\hfil}} \hautpagetitre={\hfill\tenrm \hfill} \hautpagegauche={\tenrm\folio\hfill\tenrm\the\auteurcourant} \hautpagedroite={\tenrm\the\titrecourant\hfill\tenrm\folio} \baspagegauche={\hfil} \baspagedroite={\hfil} \auteurcourant{Marmi, Moussa, Yoccoz} \titrecourant{Continued fractions, Brjuno functions and BMO spaces} \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 December\fi} \def\Date{\rightline{\mois\ /\ \the\day\ /\/ \the\year}} \hfuzz=0.3pt \font\tit=cmb10 scaled \magstep1 \def\dst{\displaystyle} \def\sst{\scriptstyle} \def\hfb{\hfill\break\indent} \def\ie{{\it i.e.\ }} \def\R{\Bbb R} \def\T{\Bbb T} \def\Z{\Bbb Z} \def\Q{\Bbb Q} \def\C{\Bbb C} \def\al{\alpha} \def\be{\beta} \def\ga{\gamma} \def\de{\delta} \def\la{\lambda} \def\Lloc{L^1_{\rm loc}(\R)} \def\mean#1#2{{1\over |#1|}\int_{#1}#2\,dx} \def\Mean#1#2{{1\over |#1|}\int_{#1}|#2|\,dx} \def\BM#1{\hbox{${\rm BMO}(#1)$}} \def\norm#1#2{||#1||_{*,#2}} \def\Norm#1#2#3{||#1||_{*,#3,#2}} \def\lnorm#1#2#3{||#1||_{#3,#2}} \def\Dnorm#1#2{\sup_{#1}{1\over |I|}\int_{I}|#2-#2_I|\,dx} \def\hnorm#1#2{\hbox{$\vert#1\vert_{#2}$}} \def\Hnorm#1#2{\hbox{$||#1||_{#2}$}} \def\remark#1{\noindent{\it Remark\ }#1\ } \def\proof{\noindent{\it Proof.\ }} \def\qed{\hfill$\square$\par\smallbreak} \def\Proc#1#2\par{\medbreak \noindent {\bf #1\enspace }{\sl #2}% \par\ifdim \lastskip <\medskipamount \removelastskip% \penalty 55\medskip \fi} %%% end macros %%%%% %%%%% Section 1 %%%% %\Date \vglue 2.5 truecm \centerline{\tit Continued fraction transformations, Brjuno functions and BMO spaces} \vskip 1truecm \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, C.E. Saclay, 91191 Gif-Sur-Yvette, France}, and J.-C. Yoccoz \footnote{$^3$}{Universit\'e de Paris-Sud, Math\'ematiques. B\^at. 425, 91405-Orsay, France}} \vskip 1.5 truecm \beginsection{\bf Abstract}\par The small divisors problem which is raised by stability questions in classical mechanics has an analog in holomorphic dynamical systems, namely the existence of Siegel disks. The size of these disks is well represented by the Brjuno functions. We analyse the relation between these functions and the various continued fraction transformations, and display the functional equation which is fulfilled by these highly singular functions. The analysis of this functional equation shows that the Brjuno function belong to the BMO space, and that a regular perturbation of this equation leads to a modification of the singular function which is 1/2-H\"older continuous. This leads us to believe that the most singular part of the size of the stability domains as function of the rotation number, is `universal' up to a 1/2-H\"older continuous function.\par \rightline{\hphantom{November 23 1994}} \vfill\eject \vglue 2.5 truecm \beginsection{\bf 0. Introduction}\par The Brjuno condition tells which invariant disks (called Siegel disks) persist when an irrational rotation is analytically perturbed. 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]. In this work, we first analyse 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 show that $T$ is a contracting operator 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. Therefore 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 generate the continued fraction, the identification of the space adapted to $T$, seems to us promising for the dynamical properties. Finally, the action of $T$ on continuous function is decribed according to H\"older's 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 remain unchanged. We anticipate that this result might be much more general: the geometric renormalisation for holomorphic dynamical systems will likely produce 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.\par \vfill \eject \beginsection \vbox{\bf\noindent 1. On a family of continued fraction transformations and the related Brjuno functions}\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 $$ A_\alpha (x) = \left|\ {\ 1\ \over x} - \left[\ {\ 1\ \over x}\ \right]_\alpha\ \right| \;. \eqno(1.3) $$ \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 ${\sqrt{5} -1 \over 2} \le \alpha \le 1$ $$ \eqalign{ c_\alpha &= {1\over \log (1 + \alpha)} \; , \cr \rho_\alpha (x) & = {1 \over 1+x} \chi_{({1-\alpha \over \alpha},\alpha)}(x) + {1 \over 2+x} \chi_{(1-\alpha, {1-\alpha \over \alpha}]} (x) \cr & + \left( {1 \over x+2} + {1 \over 2-x}\right) \chi_{(0,1 - \alpha]} (x) \; , \cr} \eqno(1.4) $$ \item{(ii)} if $2 - \sqrt{2} \le \alpha < {\sqrt{5} -1 \over 2}$ $$ \eqalign{ c_\alpha &= {1\over \log G} \cr \rho_\alpha (x) & = {1 \over G+x} \chi_{({2 \alpha - 1 \over 1-\alpha},\alpha)}(x) + {1 \over 2+x} \chi_{(1-\alpha , {2 \alpha -1 \over 1-\alpha}]} (x) \cr & + \left( {1 \over x+2} + {1 \over G+1-x} \right) \chi_{({2 \alpha -1 \over \alpha},1 - \alpha]} (x) \cr & + \left( {1 \over x+2} + {1 \over 2-x} \right) \chi_{(0,{2 \alpha -1 \over \alpha}]} (x) \; , \cr} \eqno(1.5) $$ \item{(iii)} if ${1 \over 2} \le \alpha < 2 - \sqrt{2} $ $$ \eqalignno{ c_\alpha =& {1\over \log G} \; , \cr \rho_\alpha (x) = &{1 \over G+x} \chi_{(1-\alpha,\alpha)}(x) + \left( {1 \over G+x} + {1 \over G+1-x}\right) \chi_{({2 \alpha -1 \over 1-\alpha}, 1-\alpha]} (x) \cr + \left( {1 \over x+2}\right. &+ \left.{1 \over G+1-x} \right) \chi_{({2 \alpha -1 \over \alpha},{2 \alpha -1 \over 1 - \alpha}]} (x) + \left( {1 \over x+2} + {1 \over 2-x} \right) \chi_{(0,{2 \alpha -1 \over \alpha}]} (x) \; ,\cr &&(1.6)\cr} $$ where $G={\sqrt{5}+1 \over 2}$ and $\chi_{(a,b)} (x)$ denotes the characteristic function of the interval $(a,b)$.} \par \medskip \remark{1.2.} Note that if $\alpha =1$ one finds Gauss' result: $c_1\rho_1(x) = {1\over (1+x) \log 2 }$. \par \medskip \proof By Theorem 4.1.1 of [LM] $m_\alpha$ will be an invariant probability measure for $A_\alpha$ if and only if its density $\rho_\alpha$ is a fixed point of the Perron-Frobenius operator $P_\alpha$ associated to $A_\alpha$: if $B$ denotes any measurable subset of $(0,\alpha )$, and $f$ any summable function on $(0,\alpha )$, $P_\alpha$ is defined by $$ \int_B (P_\alpha f)(x) dx = \int_{A_\alpha^{-1}(B)} f(x) dx \; . $$ The Perron-Frobenius operator for all these maps has the following form: $$ (P_\alpha f)(x) = \sum_{m\ge m_+}^{+\infty} {1\over (m+x)^2} f\left({1\over m+x}\right) + \sum_{m\ge m_-}^{+\infty} {1\over (m-x)^2} f\left({1\over m-x}\right) \; , $$ where $m_+$ and $m_-$ depend both on $\alpha$ and on $x$. \par If $\alpha =1$ one has $m_+=1$ and $m_- = +\infty$ and it is immediate to check that ${1\over 1+x}$ is a fixed point of $P_1$. \par If ${\sqrt{5} -1 \over 2} < \alpha < 1$ one has three possible cases: \item{(a)} $x\in (0,1-\alpha ]$ then $m_+ = 2$, $m_- =2$ and $\rho_\alpha (x)= {1\over x+2}+{1\over 2-x}$; \item{(b)} $x\in (1-\alpha , {1-\alpha\over\alpha}]$ then $m_+=2$, $m_-=+\infty$ and $\rho_\alpha (x)={1\over x+2}$; \item{(c)} $x \in ({1-\alpha\over\alpha}, \alpha )$ then $m_+=1$, $m_-=+\infty$ and $\rho_\alpha (x) = {1\over 1+x}$. \par Let us consider case (c). By (1.4) it follows that the two sums in the definition of ${\cal P}_\alpha$ must be splitted into three parts according to which interval contains $1/(m\pm x)$: $$ \eqalign{ (P_\alpha \rho_\alpha )(x) &= \sum_{m=2}^{m_1} {1\over (m+x)^2} \displaystyle{1\over 1+ \displaystyle{1\over m+x}} + \sum_{m=2}^{m_2} {1\over (m-x)^2} \displaystyle{1\over 1+ \displaystyle{1\over m-x}} \cr & + {1\over m_1+x+1}-{1\over m_1+x+{3\over 2}} + {1\over m_2-x+1}-{1\over m_2-x+{3\over 2}}\cr & + \sum_{m=m_1+2}^\infty {1\over (m+x)^2}\left( \displaystyle{1\over 2+ \displaystyle{1\over m+x}} + \displaystyle{1\over 2- \displaystyle{1\over m+x}}\right) \cr & + \sum_{m=m_2+2}^\infty {1\over (m-x)^2}\left( \displaystyle{1\over 2+ \displaystyle{1\over m-x}} + \displaystyle{1\over 2- \displaystyle{1\over m-x}}\right)\cr}\; . $$ Using $$ \eqalign{ {1\over (m\pm x)^2} \displaystyle{1\over 1+ \displaystyle{1\over m\pm x}} &= {1\over m\pm x} - {1\over m+1\pm x} \cr {1\over (m+x)^2} \left( \displaystyle{1\over 2+ \displaystyle{1\over m+x}} + \displaystyle{1\over 2- \displaystyle{1\over m+x}}\right) & = {1\over m+x-{1\over 2}}-{1\over m+x+{1\over 2}} \cr {1\over (m-x)^2} \left( \displaystyle{1\over 2+ \displaystyle{1\over m-x}} + \displaystyle{1\over 2- \displaystyle{1\over m-x}}\right) & = {1\over m-x-{1\over 2}}-{1\over m-x+{1\over 2}} \cr} $$ it is easy to show that all terms cancel except for ${1\over x+2} + {1\over 2-x}$. \par The remaining cases (a) and (b) are simpler, and the proof of (ii) and (iii) follows the same kinds of ideas (note that $G^2=G+1$, $G-1=1/G$ and $2-G=1/G^2$). \qed \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 $$ h(\alpha ) = \cases{ {\pi^2\over 6\log (1+\alpha )} &if ${\sqrt{5}-1\over 2}\le \alpha \le 1\;$, \cr {\pi^2\over 6\log G} & if ${1\over 2}\le \alpha\le {\sqrt{5}-1\over 2}\;$, \cr} $$ as already shown by Nakada [Na]. These results show the existence of a phase transition at $\alpha = {\sqrt{5}-1\over 2}$. \par \medskip To each $x \in \Bbb R \setminus \Bbb Q$ we associate a continued fraction expansion by iterating $A_\alpha$ as follows. Let $$ \eqalign{x_0 & = | x - [x]_\alpha| \cr a_0 & = [x]_\alpha \cr} \eqno(1.7) $$ then one obviously has $$ x_0 = a_0 + \varepsilon_0 x_0 \eqno(1.8) $$ where $$ \varepsilon_0 = \cases{ +1 \hbox{ iff } x \ge [x]_\alpha \cr -1 \hbox{ otherwise } \cr} \eqno(1.9) $$ We now define inductively for all $n \ge 0$ $$ \eqalign{x_{n+1} & = A_\alpha(x_n) \cr a_{n+1} & = \left[ {1 \over x_n} \right]_\alpha \ge 1\cr} \eqno(1.10) $$ thus $$ x_{n}^{-1} = a_{n+1} + \varepsilon_{n+1} x_{n+1} \eqno(1.11) $$ where $$ \varepsilon_{n+1} = \cases{ +1 \hbox{ iff } {1 \over x_n} \ge a_{n+1}\cr -1 \hbox{ otherwise } \cr} \eqno(1.12) $$ 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 $$ \beta_n = \Pi_{i=0}^n x_i = (-1)^n \varepsilon_0 \ldots \varepsilon_n (q_n x - p_n) \eqno(1.21) $$ 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 $\alpha>{\dst\sqrt{5}-1\over\dst2}\ ,\ \beta_n\le\alpha \left({\dst\sqrt{5}-1\over\dst2}\right)^n$; \item{(v)}\qquad if $\alpha>{\dst\sqrt{5}-1\over\dst2}\ ,\ q_n\ge{\dst1 \over\dst\alpha(1+\alpha)}\left({\dst\sqrt{5}+1\over\dst2}\right)^{(n-1)}$; \item{(vi)}\qquad if $\alpha\le{\dst\sqrt{5}-1\over\dst2}\ ,\ \beta_n\le\alpha \left(\sqrt{2}-1)\right)^n$; \item{(vii)}\qquad if $\alpha\le{\dst\sqrt{5}-1\over\dst2}\ ,\ q_n \ge{\dst1\over\dst\alpha(1+\alpha)}\left(\sqrt{2}+1)\right)^{(n-1)}$.} \par \medskip \remark{1.5.} We only quote here some aspects of the proof of the above theorem: one gets parts (i) and (ii) by recursion using (1.17). Part (iii) is easily obtained from (1.18), and is used to deduce part (v) from (iv) and part (vii) from (vi). Part iv) and vi) are easy to prove when $\alpha$ is close to one or one half, respectively. However the proof is much more intricate around $\alpha=(\sqrt{5}-1)/2$ (see [MMY]). \par \medskip \remark{1.6.} From (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 $\varepsilon_{n+1} = -1$. \par \medskip \remark{1.7.} By (v) and (vii), Proposition 1.4, there exists two positive constants $c_1$ and $c_2$ such that $$ \eqalign{ \sum_{k=0}^\infty {\log q_{k}\over q_{k}} & \le c_1 \; , \cr \sum_{k=0}^\infty {\log 2\over q_{k}} & \le c_2\; , \cr} $$ for all $\alpha\in [1/2,1]$ and for all $x\in (0,\alpha )$. \par \medskip Following Yoccoz [Yo] we define a (generalized) Brjuno function: \par \medskip \Proc{Definition 1.8.}{The {\it $\alpha$-Brjuno function} $B_\alpha \,: \Bbb R \setminus \Bbb Q \to \bar\Bbb R$ is defined by the formula $$ B_\alpha (x) = - \sum_{i=0}^\infty \beta_{i-1} \log x_i \eqno(1.25) $$ where we have posed $\beta_{-1} = 1$.} \par \medskip \remark{1.9.} 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 1.10.} {Given $\alpha \in [1/2,1]$ one has \item{(i)} $B_\alpha (x) = B_\alpha (x+1) $ for all $x \in \Bbb R \setminus \Bbb Q$; \item{(ii)} For all $x \in (0,\alpha) \cap \Bbb R \setminus \Bbb Q$ $$ B_\alpha (x) = - \log x + x B_\alpha \left( {1 \over x} \right) $$ \item{(iii)} if $x \in [\alpha - 1, 0) \cap \Bbb R \setminus \Bbb Q$ then $B_\alpha (-x) = B_\alpha (x)$. \item{(iv)} There exists a constant $C_\alpha >0$ such that for all $x \in \Bbb R \setminus \Bbb Q$ one has $$ \left| B_\alpha (x) - \sum_{j=0}^\infty {\log q_{j+1} \over q_j} \right| \le C_{\alpha} $$ 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 (1.10) and (1.21) associated to $y$ and to $x$ respectively. {}From (1.7) and (1.8) 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 $\beta_n(y)={\beta_{n+1} (x)\over x}$. 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 $$ q_i\beta_{i-1}+\varepsilon_iq_{i-1}\beta_i=1 $$ 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), (1.23) and (1.24) one has $$ \eqalign{ \left|\sum_{i=0}^\infty \beta_{i-1}\log\beta_iq_{i+1}\right| &\le 2\sum_{i=0}^\infty {\log 2\over q_i} \le 2c_2 \; , \cr \left|\sum_{i=0}^\infty \beta_{i-1}\log\beta_{i-1} \right| &\le 2\sum_{i=0}^\infty {\log 2+\log q_i\over q_i}\le 2(c_1+c_2)\; , \cr \left|\sum_{i=0}^\infty \varepsilon_i{q_{i-1}\over q_i}\beta_i\log q_{i+1}\right| &\le 2\sum_{i=0}^\infty {\log q_{i+1}\over q_{i+1}}\le 2 c_1\; , \cr} $$ from which it follows that $$ \left|B_\alpha (x)-\sum_{i=0}^\infty {\log q_{i+1}\over q_i} \right| \le 4(c_1+c_2)\; . \;\;\; $$\qed \par \medskip 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 $A_1$). Using the results of [Bo] one relates the n-th convergents of the $\alpha$--continued fractions to $P_n/Q_n$: \par \medskip \Proc{Lemma 1.11.} {Let $k^\alpha\, : \Bbb N \to \Bbb N$ be the arithmetic function inductively defined by k(-1)=-1 and $$ k(n+1) = \cases{ k(n)+1 &if $\varepsilon_{n+1}=+1$,\cr k(n)+2 &if $\varepsilon_{n+1}=-1$,\cr} $$ where $\varepsilon_{k}$ is defined as in (1.10) and (1.11). Then $k$ is strictly increasing and for all $n\in \Bbb N$ $$ {p_n\over q_n}={P_{k(n)}\over Q_{k(n)}}\; . $$ Moreover, when $k(n+1)=k(n)+2$, we have for the denominators of the convergent of Gauss'continued fraction $Q_{k(n+1)} = Q_{k(n)+2}=Q_{k(n)+1}+Q_{k(n)}$} \par \medskip By means of Lemma 1.11 one can prove the following \par \Proc{Theorem 1.12.}{There exists a positive constant $C>0$ such that for all $\alpha \in [1/2,1]$ and for all $x \in \Bbb R \setminus \Bbb Q$ one has $$ \left| B_\alpha (x) - \sum_{j=0}^\infty {\log Q_{j+1} \over Q_j} \right| \le C \eqno(1.26) $$} \par \medskip \proof Thanks to (iv), Proposition 1.10, it suffices to compare $\sum_{j=0}^\infty {\log q_{j+1} \over q_j}$ with $\sum_{j=0}^\infty {\log Q_{j+1} \over Q_j}$. By Lemma 1.11, one has $q_j=Q_{k(j)}$ for all $j$ 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 $$ {\log Q_{k+2}\over Q_{k}} = {\log (Q_{k+1}+Q_k)\over Q_{k}} ={\log Q_{k+1}\over Q_{k}} + {\log \left(1+{Q_k\over Q_{k+1}}\right) \over Q_k} $$ but $$ 0 \le {\log \left(1+{Q_k\over Q_{k+1}}\right)\over Q_k} \le {\log 2\over Q_k} \; , $$ By applying the estimates of Remark 1.7 one gets the result: $$ \left| \sum_{j=0}^\infty {\log q_{j+1} \over q_j} - \sum_{j=0}^\infty {\log Q_{j+1} \over Q_j} \right| \le 2c_2 + c_1 \; . \; \; $$\qed \par \medskip \remark{1.13.} The {\it Brjuno numbers} [Br] are usually defined by the condition $$ \sum_{i=0}^\infty {\log Q_{i+1}\over Q_i} < +\infty\; . $$ Theorem 1.12 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 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 stated by Corollary 1.15 below. \par \medskip Let us consider the operator $$ (Tf)(x)=xf\left({1\over x}\right)\; , \eqno(1.27) $$ if $x\in (0,\alpha )$, defined for the moment on measurable functions of $\Bbb R$ which verify $$ f(x)=f(x+1) \; \hbox{for almost every}\, x\in \Bbb R\; , \;\;\;f(-x)=f(x)\;\hbox{ for a.e.}\, x\in (0,1-\alpha)\; . \eqno(1.28) $$ It is understood that the function $Tf$ is completed outside $(0,\alpha)$ by imposing on $Tf$ the same parity and periodicity conditions which are expressed for $f$ in (1.28). \par The functional equation for the $\alpha$-Brjuno function can be written in the form $$ [(1-T)B_\alpha ](x)=-\log x\; , \eqno(1.29) $$ for all $x\in (0,\alpha )$, complemented with the periodicity and symmetry conditions (1.28). This suggest to study the operator $T$ on the Banach spaces $$ X_{\alpha ,p}= \left\{ f : \Bbb R \to \Bbb R \mid f \,\hbox{verifies (1.28)} \; , \;\; f\in L^p(0,\alpha),\right\} \eqno(1.30) $$ endowed with the norm of $L^p(0,\alpha )$, as $\alpha$ varies in $(1/2,1)$ and $p\in [1,\infty ]$. Note that if $p
0$
$$(T_{(\nu)}f)(x)=x^{\nu}f\left({1\over x}\right)\; ,\eqno(1.42)$$
if $x\in (0,\alpha )$, and completed as in (1.28) outside $(0,\alpha)$.
The adjoint operators $T^*_{(\nu)}$ also
belong to the class of the Perron-Frobenius-Ruelle operators
associated to $A_\alpha$ (see remark 1.17 above).\par
Similarly as equations (1.33), (1.34), and (1.40),
we get the following results:\par
if\ \ ${1\over 2}<\alpha\le {\sqrt{5}-1\over 2}\ $, then
$$|| T_{(\nu)}||_{{\cal L}(X_{\alpha ,p})}\le
[\zeta (2+p\nu)-1+\zeta (2+p\nu,\alpha )-\alpha^{-(p\nu+2)}]^{1\over p}\ ,
\eqno(1.43)$$\par
if\ \ ${\sqrt{5}-1\over 2}<\alpha \le 1\ $, then
$$|| T_{(\nu)}||_{{\cal L}(X_{\alpha ,p})}\le
[\zeta (2+p\nu)-1+\zeta (2+p\nu,\alpha )-\alpha^{-(p\nu+2)}
+\alpha^{p\nu+2}]^{1\over p}\ ,\eqno(1.44)$$\par
and finally if $\alpha={1\over2}$
$$||T_{(\nu)}||_{{\cal L}(X_{1/2 ,p})}\le \left(2^{p\nu+2}(\zeta(p\nu+2)-1)-1
-\left({2\over3}\right)^{p\nu+2}\right)^{1\over p}\ .\eqno(1.45)$$
For $p=2$, this last estimate can be rewritten as
$$||T_{(\nu)}||_{{\cal L}(X_{1/2 ,2})}\le \left(2^{2\nu+2}(\zeta(2\nu+2)-1)-1
-\left({2\over3}\right)^{2\nu+2}\right)^{1\over 2}\ .\eqno(1.46)$$
Proceeding as in the proof of corollary 1.15, one easily checks
that $T_{(\nu)}$ is
a contraction on $X_{\alpha,p}$ for $p$ sufficiently large.\par
\vfill \eject
%%%%%%%% Section 2 %%%%%%
\beginsection 2. BMO and the Brjuno function\par
\vskip .5 truecm
In this section we want to prove that the Brjuno functions $B_\alpha$
belong to the space BMO (see [Gr], and also Appendix A for a short
account on some fundamental results concerning BMO). Since they all
differ by a $L^\infty$ function, and $L^\infty\subset\,$BMO, it will
be enough to prove that $B_\alpha$ is BMO for a fixed value of
$\alpha$.
\par
In this section we fix $\alpha =1/2$ and denote $B_{1/2}$ simply by $B$.
\par
\medskip
Let $I\subset [0,1/2]$ be any interval, and $f\in L^1([0,1/2])$.
We denote by
$$
f_I = {1\over |I|}\int_I f(x)dx
\eqno(2.1)
$$
the mean of $f$ over $I$. Using Lemma A.10, we also define the quadratic
oscillation ${\cal O}_I(f)$ as follows
$$
{\cal O}_I(f) = \left({1\over 2|I|^2}\int_I\int_I(f(s)-f(t))^2dsdt
\right)^{1\over2}\;.\eqno(2.2)$$
\par
As in the previous Section, we define for $1\le p\le\infty$
the space $X_p\equiv X_{1/2,p}$\ :
$$
X_p = \{f\in L^p([0,1/2]) \mid f(x+1)=f(x)\; \forall x\in \Bbb R\; ,
\; f(-x)=f(x) \;\forall x\in [0,1/2]\}\; , \eqno(2.3)
$$
with the usual $L^p$-norm on $[0,1/2]$. In the above definition, $f$
is extended from $[0,1/2]$ to $\R$, in such a way that it is even and
periodic with period 1.\par
We now consider the space
$$
X_* = \{f\in\hbox{BMO}(\Bbb R)\, \mid f(x+1)=f(x)\; \forall x\in \Bbb R\; ,
\; f(-x)=f(x) \;\forall x\in [0,1/2]\}\; , \eqno(2.4)
$$
with the norm
$$
|| f||_* =A |f|_* +B\lnorm{f}{[0,1/2]}{2} \; ,
\eqno(2.5)
$$
where $A>0$ and $B>0$ are fixed, $\lnorm{f}{[0,1/2]}{2}=
|| f||_{L^2(0,1/2)}$, and
$$
|f|_* = \sup_{I\subset [0,1/2]} {\cal O}_I(f)\; .\eqno(2.6)
$$
Therefore we have
$$||f||_*=A\sup_{I\subset[0,1/2]}\left({1\over 2|I|^2}\int_I
\int_I(f(s)-f(t))^2\,dsdt\right)^{1\over 2}+
B\left(\int_0^{1/2}(f(x))^2\,dx\right)^{1\over 2}\ .\eqno(2.7)$$
\par
\medskip
\remark{2.1.}
Using
John-Nirenberg's theorem and its corollary, the norm we use is equivalent
to the usual BMO norm (see Appendix A, Propositions A.4, A.6, A.8, and A.10).
\par
\medskip
\remark {2.2.}
One has the obvious inclusions $X_{\infty}\subsetneqq X_*
\subsetneqq X_{1}$. By elementary (but tedious) calculations, one checks
that the even and periodic function wich coincides with $\log x$ on
$(0,1/2]$ is in $X_*$, which can be written as
$\log |x-[x]_{1/2}|\in X_*$. \par\medskip
We now recall the definition of the operator $T$. For
$x\in\R$, let $[x]_{1/2}$ be the distance from $x$ to the nearest
integer. For $f\in L^1([0,1/2])$, we define $Tf$ as in (1.27 and 28)
$$(Tf)(x)=xf\left(\left|{1\over x}-\left[{1\over x}\right]_{1/2}\right|
\right)\quad\hbox{for}\quad x\in [0,1/2]\ .\eqno(2.8)$$
We continue $f$ , and $Tf$, to the full real axis so that they are
even and periodic, and (2.8) can be rewritten as $(Tf)(x)=xf(1/x)$,
for $0\le x\le 1/2$.
As already mentioned in Theorem 1.14, $Tf$ is a linear operator from
$X_{p}$ to $X_{p}$ for all $p\ge1$ including $\infty$. More precisely,
the norm of the operator $T$ in the various $X_p$ can be estimated.
We have
$$||T||_{{\cal L}(X_{\infty})}={1\over 2}\ .\eqno(2.9)$$
For finite $p\ge 1$, we use the bound described in Remark 1.18 above
when $\al=1/2$, namely
$$||T||_{{\cal L}(X_{p})}\le C_p\ ,\eqno(2.10)$$
with
$$C_p=\left(\sum_{n=2}^{\infty}\left({1\over n^{p+2}}+
{1\over (n+1/2)^{p+2}}\right)\right)^{{1\over p}}\ .\eqno(2.11)$$
By comparing the sums in (2.11) to an integral over
the interval $(3/2,\infty)$, which we cut at every integer and half-integer
points, we get an easy estimate
$$C_p\le\left(2\int_{3/2}^{\infty}{dx\over x^{p+2}}\right)^{1\over p}
=\left({2\over p+1}\right)^{1\over p}\left({2\over 3}\right)^{1+{1\over p}}
\le{2\over3}\ \ \ \hbox{for $p\ge 1$}\ .\eqno(2.12)$$
Therefore, as already stated in Theorem 2.14, the operator $T$
is a contraction on all $X_p$, for $1\le p\le\infty$.
We shall now consider the action of $T$ on $X_*$.
We have the following theorem.
\Proc{Theorem 2.3.}{$T$ is a
bounded linear operator $T:X_*\mapsto X_*$. Moreover it is a contraction
for the norm (2.7) with $A=1$ and $B\ge 7.405$ (as in (2.41) below), that is
$$||T||_{{\cal L}(X_{*})}\le c_*<1\ \ ,\ \ \hbox{\sl with}\ \ c_*=
{40\sqrt{3}\over 81}\sim 0.855 .\eqno(2.13)$$
In fact, if we allow to increase $B$ to a sufficiently large value,
one can reduce $c_*$ to be as close as one wants to
$(2/5)\sqrt{3}\sim 0.6928$.}\par
\proof\par
a) Let $f\in X_*$, we shall show that for the seminorm (2.6) we have
$|Tf|_*<\infty$, which will prove that $T$ is a linear operator
$T:X_*\mapsto X_*$.
For $I=[a,b]\subset[0,1/2]$, with $b>a$, we have from (2.6)
$$\eqalignno{({\cal O}_I(Tf))^2&={1\over2|I|^2}\int_{I\times I}
\left((Tf)(s)-(Tf)(t)\right)^2\,dsdt\cr=
&{1\over2|I|^2}\left[2|I|\int_{I}((Tf)(s))^2\,ds-2\left(\int_I
(Tf)(s)\,ds\right)^2\right]\le{1\over|I|}\int_I((Tf)(s))^2\,ds\cr
&\le{1\over|I|}\int_{J}(f(s))^2{ds\over s^4}\ ,&(2.14)}$$
where $J=[1/b,1/a]\subset [2,\infty]$. Let now $[m/2,n/2]$ be the smallest
interval with half integer edges, which contains $J$.
We have
$$1/b=\be+m/2\quad,\quad 1/a=n/2-\al\quad,\quad 0\le\al<1/2\quad,
\quad0\le\be<1/2\ .\eqno(2.15)$$
Since $b-a>0$, we have for the integers $m$ and $n$, $4\le m