%%%%%%%% Papier 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 } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\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 \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{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 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} %\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 $$\eqalign{ A_{\al}(x)=&{1\over x}-k \quad \hbox{for}\quad{1\over k+\al} q_n > 0$; \item{(ii)}\qquad $p_n > 0$ when $x>0$ and $p_n< 0$ when $x<0$; \item{(iii)}\qquad $ \left|q_n x - p_n\right| ={\dst 1\over\dst q_{n+1}+\varepsilon_{n+1}q_nx_{n+1}}$, so that ${\dst 1\over\dst 1+\alpha}<\beta_nq_{n+1}<{\dst 1\over\dst\alpha}$\ ; \item{(iv)}\qquad if $\alpha>g\ ,\ \beta_n\le\alpha g^n$; \item{(v)}\qquad if $\alpha\le g\ ,\ \beta_n\le\alpha \gamma^n$.} \par \medskip \proof One gets parts (i) and (ii) by recursion using (1.17), in fact it is obvious only when $\al=1$. When $\al\neq1$, one could alternatively use Lemma 1.8 below. Part (iii) is easily obtained from (1.18). \par The proof of (iv) is also easy: either $x_k \le g$ for all $k=0,\ldots ,n$, or $x_k> g$ for some $k$. Then $x_{k+1}=x_k^{-1}-1$ and $x_{k+1}g$ (since for each pair $x_kx_{k+1}\ga$ implies $x_{k+1}=|2-x_k^{-1}|<\ga$, therefore we must have $p\ge1$. Now statement (v) is an immediate consequence of the following assertion which we will then prove:\par {\it if $x_k>\gamma$, $p\ge1$, $x_{k+1},\ldots ,x_{k+p} <\gamma$ and $x_{k+p+1}>\gamma$ then $\Pi_{i=k}^{k+p} x_i <\gamma^{p+1}$.}\par We divide the proof into some different cases.\par \item{(1)} If $\ga1/2$, thus $x_{k+1}=2-x_k^{-1}1/3$ we let $ m\ge 2$ such that $$ x_{k+1},\ldots ,x_{k+m-1}\in (1/3,g^2]\; , \;\;\; x_{k+m}\in [0,1/3]\; . $$ Note that $p\ge m\ge 1$. \item{(2.1)} If $m\ge 4$, then $x_kx_{k+1}\cdots x_{k+m}\le {1\over 3}g^{2m-1}$, since $x_k\sqrt{2}-4/5$, then $x_{k+1}>(48-25\sqrt{2})/34$ and $x_{k+2}>0.3111\ldots$, so that $x_{k+2}\in [2/7,1/3]$. Thus $x_{k+3}=x_{k+2}^{-1}-3$ and $x_kx_{k+1}x_{k+2}x_{k+3}=8-13x_k<8-13(\sqrt{2}-4/5)$. A numerical exercise shows that this last number is smaller than $\gamma^4$, which completes the assertion in this case. \item{(2.4)} $m=3$, then assume $x_kx_{k+1}x_{k+2}x_{k+3}=13x_k-8>\gamma^4$. This would be equivalent to $x_k>(25-12\sqrt{2})/13$. However, from the definition of $m$, one gets $x_{k+3}\le1/3$ which implies $x_k\le21/34$, and the two inequalities on $x_k$ are contradictory. Thus $x_kx_{k+1}x_{k+2}x_{k+3}\le \gamma^4$, and the proof of the assertion is completed. \qed \par \medskip \remark{1.5.} From {\it (iii)} and {\it (iv)} one gets if $\alpha>g\ ,\ q_n\ge{\dst1 \over\dst\alpha(1+\alpha)}G^{n-1}$, and similarly, from {\it (iii)} and {\it (v)} if $\alpha\le g\ ,\ q_n \ge{\dst1\over\dst\alpha(1+\alpha)}\Gamma^{n-1}$. \par \medskip \remark{1.6.} From {\it (iii)} one gets $${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) $$ \bigskip We give now to lemmas which relate the separation between two numbers and their splitting orders.\par\bigskip \Proc{Lemma 1.10}{There exists a positive constant $c_3$ independent on $x$ and $x'$, such that for all $l0$ such that for all $x,x'\in (0,1/2)$, and $n\ge n(x,x')$, one has $$ \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) $$ However, $q_n^{-1}\ge (1/2)\be_{n-1}(x)\ge (1/2c_3)\be_{n-1}(x')= (1/9)\be_{n-1}(x')$, and therefore we get $c_4\ge 3/\sqrt{2}$.\par Finally in the case (A) one has $|x_n^{-1}-{x'}_n^{-1}|\ge 1$ since $x,x'$ do not belong to two adjacent branches of $A^{n+1}$, from which follows that $ |x_n-x_n'|\ge |x_n||x'_n| $. Suppose $x_n>x'_n$ (the other case resulting by symmetry), then if $x'_n \ge x_n/2$, we get $ |x_n-x_n'|\ge |x_n|^2/2> $, and if $x'_nx_n/2>x^2_n/2$. Therefore $$ |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) $$ as $\alpha$ varies in $(1/2,1)$ and $p\in [1,\infty ]$. Note that if $p0$, for all $\alpha\in [{1\over 2},1]$ and for all $p\in [1,\infty ]$. Indeed its spectral radius on $X_{\alpha ,p}$ satisfies} $$ 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} $$ \par\medskip {\it Third case:} one has $n-\delta 0$) is also continuous provided we set $(T_1f)(0)=0$ (resp $(T_{\nu}f)(0)=0$). We need now the usual H\"older's type semi-norms for continuous functions. \Proc{Definition 4.1.}{Let $f\in {\cal C}^0_{[0,1/2]}$, Then we define the H\"older's $\eta$-norm as $$\hnorm{f}{\eta}= \sup_{0\le x2\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 $n-\delta n$, 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 $n-\de0$, 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 \beginsection References \par \item{[Bo]} W. Bosma ``Optimal continued fractions'', {\it Indag. Math.}, {\bf A90}, 1987, 353--379. \item{[Br]} A. D. Brjuno ``Analytical form of differential equations'' {\it Trans. Moscow Math. Soc.} {\bf 25} (1971), 131--288; {\bf 26} (1972), 199--239. \item{[BPV]} N. Buric, I. C. Percival, F. Vivaldi, `` Critical function and modular smoothing'' {\it Nonlinearity} {\bf 3} (1990) 21--37. \item{[Da]} A. M. Davie ``The critical function for the semistandard map'' {\it Nonlinearity} {\bf 7} (1994) 219--229. \item{[Ga]} E. F. 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