%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% In order to print the figures, search for: %%% FIGURES: BEGINNING %%% and create from the part of file below two new ps-files calling them %%% f1.ps and f2.ps (each file has to begin after the word %%% %BEGINSFIGURE and has to end before the word %ENDSFIGURE. %%% Then return to the main file and search for: %%% FIGPRINT %%% and write a % symbol before each line of %%% the three-line command: \def\epsffile#1 . %%% %%% If no one of the above operations is performed, no figure %%% will be printed! %%% %%% The file is in Plain Tex. Tex twice. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcount\mgnf\newcount\tipi\newcount\tipoformule\newcount\driver \newcount\indice \driver=1 %dvips \mgnf=0 %ingrandimento \tipi=2 %uso caratteri: 2=cmcompleti, 1=cmparziali, 0=amparziali \tipoformule=0 %=0 da numeroparagrafo.numeroformula; se no numero %assoluto \indice=1 %=1 prepara ma non compila un nuovo indice; altro %usa il vecchio %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% INCIPIT \ifnum\mgnf=0 \magnification=\magstep0 \hsize=14truecm\vsize=24.truecm \parindent=0.3cm\baselineskip=0.45cm\fi \ifnum\mgnf=1 \magnification=\magstep1\hoffset=0.truecm \hsize=14truecm\vsize=24.truecm \baselineskip=18truept plus0.1pt minus0.1pt \parindent=0.9truecm \lineskip=0.5truecm\lineskiplimit=0.1pt \parskip=0.1pt plus1pt\fi %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%% GRECO %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \let\a=\alpha \let\b=\beta \let\g=\gamma \let\d=\delta \let\e=\varepsilon \let\z=\zeta \let\h=\eta \let\th=\vartheta \let\k=\kappa \let\l=\lambda \let\m=\mu \let\n=\nu \let\x=\xi \let\p=\pi \let\r=\rho \let\s=\sigma \let\t=\tau \let\f=\varphi \let\ph=\varphi \let\c=\chi \let\ps=\psi \let\y=\upsilon \let\o=\omega \let\si=\varsigma \let\G=\Gamma \let\D=\Delta \let\Th=\Theta \let\L=\Lambda \let\X=\Xi \let\P=\Pi \let\Si=\Sigma \let\F=\Phi \let\Ps=\Psi \let\O=\Omega \let\Y=\Upsilon %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% EQUAZIONI CON NOMI SIMBOLICI %%% %%% per assegnare un nome simbolico ad una equazione basta %%% scrivere \Eq(...) o, in \eqalignno, \eq(...) o, %%% nelle appendici, \Eqa(...) o \eqa(...): %%% dentro le parentesi e al posto dei ... %%% si puo' scrivere qualsiasi commento; %%% per assegnare un nome simbolico ad una figura, basta scrivere %%% \geq(...); per avere i nomi %%% simbolici segnati a sinistra delle formule e delle figure si deve %%% dichiarare il documento come bozza, iniziando il testo con %%% \BOZZA. Sinonimi \Eq,\EQ,\EQS; \eq,\eqs; \Eqa,\Eqas;\eqa,\eqas. %%% All' inizio di ogni paragrafo si devono definire il %%% numero del paragrafo e della prima formula dichiarando %%% \numsec=... \numfor=... (brevetto Eckmannn); all'inizio del lavoro %%% bisogna porre \numfig=1 (il numero delle figure non contiene la sezione. %%% Si possono citare formule o figure seguenti; le corrispondenze fra nomi %%% simbolici e numeri effettivi sono memorizzate nel file \jobname.aux, che %%% viene letto all'inizio, se gia' presente. E' possibile citare anche %%% formule o figure che appaiono in altri file, purche' sia presente il %%% corrispondente file .aux; basta includere all'inizio l'istruzione %%% \include{nomefile} %%%%%%%%% NOTA BENE %%% Per il buon funzionamento dei riferimenti occorre togliere il %%% commento della riga \openout15=\jobname.aux ORA COMMENTATA, %%% e compilare DUE volte. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \global\newcount\numsec\global\newcount\numfor \global\newcount\numapp\global\newcount\numcap \global\newcount\numfig\global\newcount\numpag \global\newcount\numnf \def\SIA #1,#2,#3 {\senondefinito{#1#2}% \expandafter\xdef\csname #1#2\endcsname{#3}\else \write16{???? ma #1,#2 e' gia' stato definito !!!!} \fi} \def \FU(#1)#2{\SIA fu,#1,#2 } \def\etichetta(#1){(\veroparagrafo.\veraformula)% \SIA e,#1,(\veroparagrafo.\veraformula) % \global\advance\numfor by 1% \write15{\string\FU (#1){\equ(#1)}}% \write16{ EQ #1 ==> \equ(#1) }} \def\etichettaa(#1){(A\veraappendice.\veraformula) \SIA e,#1,(A\veraappendice.\veraformula) \global\advance\numfor by 1 \write15{\string\FU (#1){\equ(#1)}} \write16{ EQ #1 ==> \equ(#1) }} \def\getichetta(#1){Fig. \verafigura \SIA g,#1,{\verafigura} \global\advance\numfig by 1 \write15{\string\FU (#1){\graf(#1)}} \write16{ Fig. #1 ==> \graf(#1) }} \def\retichetta(#1){\numpag=\pgn\SIA r,#1,{\verapagina} \write15{\string\FU (#1){\rif(#1)}} \write16{\rif(#1) ha simbolo #1 }} \def\etichettan(#1){(n\verocapitolo.\veranformula) \SIA e,#1,(n\verocapitolo.\veranformula) \global\advance\numnf by 1 \write16{\equ(#1) <= #1 }} \newdimen\gwidth \gdef\profonditastruttura{\dp\strutbox} \def\senondefinito#1{\expandafter\ifx\csname#1\endcsname\relax} \def\BOZZA{ \def\alato(##1){ {\vtop to \profonditastruttura{\baselineskip \profonditastruttura\vss \rlap{\kern-\hsize\kern-1.2truecm{$\scriptstyle##1$}}}}} \def\galato(##1){ \gwidth=\hsize \divide\gwidth by 2 {\vtop to \profonditastruttura{\baselineskip \profonditastruttura\vss \rlap{\kern-\gwidth\kern-1.2truecm{$\scriptstyle##1$}}}}} \def\verapagina{ {\romannumeral\number\numcap}.\number\numsec.\number\numpag}} \def\alato(#1){} \def\galato(#1){} \def\veroparagrafo{\number\numsec}\def\veraformula{\number\numfor} \def\veraappendice{\number\numapp} \def\verapagina{\number\pageno}\def\veranformula{\number\numnf} \def\verafigura{{\romannumeral\number\numcap}.\number\numfig} \def\verocapitolo{\number\numcap}\def\veranformula{\number\numnf} \def\Eqn(#1){\eqno{\etichettan(#1)\alato(#1)}} \def\eqn(#1){\etichettan(#1)\alato(#1)} \def\Eq(#1){\eqno{\etichetta(#1)\alato(#1)}} \def\eq(#1){\etichetta(#1)\alato(#1)} \def\Eqa(#1){\eqno{\etichettaa(#1)\alato(#1)}} \def\eqa(#1){\etichettaa(#1)\alato(#1)} \def\dgraf(#1){\getichetta(#1)\galato(#1)} \def\drif(#1){\retichetta(#1)} \def\eqv(#1){\senondefinito{fu#1}$\clubsuit$#1\else\csname fu#1\endcsname\fi} \def\equ(#1){\senondefinito{e#1}\eqv(#1)\else\csname e#1\endcsname\fi} \def\graf(#1){\senondefinito{g#1}\eqv(#1)\else\csname g#1\endcsname\fi} \def\rif(#1){\senondefinito{r#1}\eqv(#1)\else\csname r#1\endcsname\fi} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% Numerazione verso il futuro ed eventuali paragrafi %%%%%%%%%%%%%%%%%% precedenti non inseriti nella scheda da compilare %%%%%%%%%%%%%%%%%% e elenco referenze bibliografiche creato in %%%%%%%%%%%%%%%%%% \jobname.bib \openin14=\jobname.aux \ifeof14 \relax \else \input \jobname.aux \closein14 \fi %14 e' libero !! %%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \let\EQS=\Eq\let\EQ=\Eq \let\eqs=\eq \let\Eqas=\Eqa \let\eqas=\eqa %\newcount\tipoformule %\tipoformule=1 %=0 da numeroparagrafo.numeroformula; se no numero % %assegnato \ifnum\tipoformule=1\let\Eq=\eqno\def\eq{}\let\Eqa=\eqno\def\eqa{} \def\equ{}\fi %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%% Numerazione pagine {\count255=\time\divide\count255 by 60 \xdef\hourmin{\number\count255} \multiply\count255 by-60\advance\count255 by\time \xdef\hourmin{\hourmin:\ifnum\count255<10 0\fi\the\count255}} \def\oramin{\hourmin } \def\data{\number\day/\ifcase\month\or gennaio \or febbraio \or marzo \or aprile \or maggio \or giugno \or luglio \or agosto \or settembre \or ottobre \or novembre \or dicembre \fi/\number\year;\ \oramin} %\setbox200\hbox{$\scriptscriptstyle \data $} \newcount\pgn \pgn=1 \def\foglio{\number\numsec:\number\pgn \global\advance\pgn by 1} \def\foglioa{A\number\numsec:\number\pgn \global\advance\pgn by 1} %\footline={\rlap{\hbox{\copy200}\ $\st[\number\pageno]$}\hss\tenrm %\foglio\hss} %\footline={\rlap{\hbox{\copy200}\ $\st[\number\pageno]$}\hss\tenrm %\foglioa\hss} % %\footline={\rlap{\hbox{\copy200}\ $\st[\number\pageno]$}\hss\tenrm\foglio\hss} %\footline={\rlap{\hbox{\copy200}}\hss\tenrm\folio\hss} \footline={\rlap{\hbox{\copy200}}} %%%%%%%%%%%%POSTSCRIPT % % Inizializza le macro postscript e il tipo di driver di stampa. % Attualmente le istruzioni postscript vengono utilizzate solo se il driver % e' DVILASER ( \driver=0 ), DVIPS ( \driver=1) o PSPRINT ( \driver=2); % qualunque altro valore di \driver produce un output in cui le figure % contengono solo i caratteri inseriti con istruzioni TEX (vedi avanti). % \newdimen\xshift \newdimen\xwidth % % inserisce una scatola contenente #3 in modo che l'angolo superiore sinistro % occupi la posizione (#1,#2) % \def\ins#1#2#3{\vbox to0pt{\kern-#2 \hbox{\kern#1 #3}\vss}\nointerlineskip} % % Crea una scatola di dimensioni #1x#2 contenente il disegno descritto in % #4.ps; in questo disegno si possono introdurre delle stringhe usando \ins % e mettendo le istruzioni relative al #3 (che puo' anche mancare); % al disotto del disegno, al centro, e' inserito il numero della figura % calcolato tramite \geq(#4). % Il file #4.ps contiene le istruzioni postscript, che devono essere scritte % presupponendo che l'origine sia nell'angolo inferiore sinistro della % scatola, mentre per il resto l'ambiente grafico e' quello standard. % Se \driver=2, e' necessario dilatare la figura in accordo al valore di % \magnification, correggendo i parametri P1 e P2 nell'istruzione % \special{#4.pst P1 P2 scale} % \def\insertplot#1#2#3#4{ \par \xwidth=#1 \xshift=\hsize \advance\xshift by-\xwidth \divide\xshift by 2 \vbox{ \line{} \hbox{ \hskip\xshift \vbox to #2{\vfil \ifnum\driver=0 #3 \special{ps::[local,begin] gsave currentpoint translate} \special{ps: plotfile #4.ps} \special{ps::[end]grestore} \fi \ifnum\driver=1 #3 \special{psfile=#4.ps} \fi \ifnum\driver=2 #3 \ifnum\mgnf=0 \special{#4.ps 1. 1. scale}\fi \ifnum\mgnf=1 \special{#4.ps 1.2 1.2 scale}\fi\fi \ifnum\driver=5 #3 \fi} \hfil}}} \newdimen\xshift \newdimen\xwidth \newdimen\yshift % \def\eqfig#1#2#3#4#5{ \par\xwidth=#1 \xshift=\hsize \advance\xshift by-\xwidth \divide\xshift by 2 \yshift=#2 \divide\yshift by 2 \line{\hglue\xshift \vbox to #2{\vfil \ifnum\driver=0 #3 \special{ps::[local,begin] gsave currentpoint translate}% \special{ps: plotfile #4.ps} \special{ps::[end]grestore}\fi \ifnum\driver=1 #3 \special{psfile=#4.ps}\fi \ifnum\driver=2 #3 \special{ \ifnum\mgnf=0 #4.ps 1. 1. scale \fi \ifnum\mgnf=1 #4.ps 1.2 1.2 scale\fi} \fi}\hfill\raise\yshift\hbox{#5}}} \def\figini#1{ %\csname write13 \endcsname \def\8{\write13} \message{************************************************************} \message{Sto disegnando la fig. #1} \message{************************************************************} \catcode`\%=12\catcode`\{=12\catcode`\}=12 \catcode`\<=1\catcode`\>=2 \openout13=#1.ps} \def\figfin{ \closeout13 \catcode`\%=14\catcode`\{=1 \catcode`\}=2\catcode`\<=12\catcode`\>=12 \message{l' ho disegnata!} \message{************************************************************} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CARATTERI %%%%%%%%%%%%%% \newskip\ttglue %% am \def\TIPIO{ \font\setterm=amr7 %\font\settei=ammi7 \font\settesy=amsy7 \font\settebf=ambx7 %\font\setteit=amit7 %%%%% cambiamenti di formato %%% \def \settepunti{\def\rm{\fam0\setterm}% passaggio a tipi da 7-punti \textfont0=\setterm %\textfont1=\settei \textfont2=\settesy %\textfont3=\setteit %\textfont\itfam=\setteit \def\it{\fam\itfam\setteit} \textfont\bffam=\settebf \def\bf{\fam\bffam\settebf} \normalbaselineskip=9pt\normalbaselines\rm }\let\nota=\settepunti} \def\annota#1#2{\footnote{${}^#1$}{\nota#2}} %%%%%%% %%cm completo \def\TIPITOT{ \font\twelverm=cmr12 \font\twelvei=cmmi12 \font\twelvesy=cmsy10 scaled\magstep1 \font\twelveex=cmex10 scaled\magstep1 \font\dodici=cmbx10 scaled\magstep1 \font\twelveit=cmti12 \font\twelvett=cmtt12 \font\twelvebf=cmbx12 \font\twelvesl=cmsl12 \font\ninerm=cmr9 \font\ninesy=cmsy9 \font\eightrm=cmr8 \font\eighti=cmmi8 \font\eightsy=cmsy8 \font\eightbf=cmbx8 \font\eighttt=cmtt8 \font\eightsl=cmsl8 \font\eightit=cmti8 \font\sixrm=cmr6 \font\sixbf=cmbx6 \font\sixi=cmmi6 \font\sixsy=cmsy6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \font\twelvetruecmr=cmr10 scaled\magstep1 \font\twelvetruecmsy=cmsy10 scaled\magstep1 \font\tentruecmr=cmr10 \font\tentruecmsy=cmsy10 \font\eighttruecmr=cmr8 \font\eighttruecmsy=cmsy8 \font\seventruecmr=cmr7 \font\seventruecmsy=cmsy7 \font\sixtruecmr=cmr6 \font\sixtruecmsy=cmsy6 \font\fivetruecmr=cmr5 \font\fivetruecmsy=cmsy5 %%%% definizioni per 10pt %%%%%%%% \textfont\truecmr=\tentruecmr \scriptfont\truecmr=\seventruecmr \scriptscriptfont\truecmr=\fivetruecmr \textfont\truecmsy=\tentruecmsy \scriptfont\truecmsy=\seventruecmsy \scriptscriptfont\truecmr=\fivetruecmr \scriptscriptfont\truecmsy=\fivetruecmsy %%%%% cambio grandezza %%%%%% \def \eightpoint{\def\rm{\fam0\eightrm}% switch to 8-point type \textfont0=\eightrm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm \textfont1=\eighti \scriptfont1=\sixi \scriptscriptfont1=\fivei \textfont2=\eightsy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex \textfont\itfam=\eightit \def\it{\fam\itfam\eightit}% \textfont\slfam=\eightsl \def\sl{\fam\slfam\eightsl}% \textfont\ttfam=\eighttt \def\tt{\fam\ttfam\eighttt}% \textfont\bffam=\eightbf \scriptfont\bffam=\sixbf \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\eightbf}% \tt \ttglue=.5em plus.25em minus.15em \setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt}% \normalbaselineskip=9pt \let\sc=\sixrm \normalbaselines\rm \textfont\truecmr=\eighttruecmr \scriptfont\truecmr=\sixtruecmr \scriptscriptfont\truecmr=\fivetruecmr \textfont\truecmsy=\eighttruecmsy \scriptfont\truecmsy=\sixtruecmsy }\let\nota=\eightpoint} \newfam\msbfam %per uso in \TIPITOT \newfam\truecmr %per uso in \TIPITOT \newfam\truecmsy %per uso in \TIPITOT %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%Scelta dei caratteri %\newcount\tipi \tipi=0 %e' definito all'inizio \newskip\ttglue \ifnum\tipi=0\TIPIO \else\ifnum\tipi=1 \TIPI\else \TIPITOT\fi\fi %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% DEFINIZIONI LOCALI %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\aps{{\it a posteriori}} \let\0=\noindent\let\\=\noindent \def\pagina{{\vfill\eject}} \def\bra#1{{\langle#1|}}\def\ket#1{{|#1\rangle}} \def\media#1{{\langle#1\rangle}} \def\ie{\hbox{\it i.e.\ }}\def\eg{\hbox{\it e.g.\ }} \let\dpr=\partial %\def\\{\hfill\break} \let\circa=\cong \def\*{\vglue0.3truecm}\let\0=\noindent \let\==\equiv \let\txt=\textstyle\let\dis=\displaystyle %\def\Idea{{\it Idea:\ }}\def\mbe{{\\*\hfill\hbox{\it %mbe\kern0.5truecm}}\vskip3.truept} \def\1{{-1}} \let\io=\infty \def\V#1{\,\vec#1} \def\Dpr{\V\dpr\,} \let\I=\int \let\ig=\int \def\tende#1{\,\vtop{\ialign{##\crcr\rightarrowfill\crcr \noalign{\kern-1pt\nointerlineskip} \hskip3.pt${\scriptstyle #1}$\hskip3.pt\crcr}}\,} \def\otto{\,{\kern-1.truept\leftarrow\kern-5.truept\to\kern-1.truept}\,} \def\fra#1#2{{#1\over#2}} \global\newcount\numpunt \def\XWPR{{\it a priori}} \def\ap#1{\def\9{#1}{\if\9.\global\numpunt=1\else\if\9,\global\numpunt=2\else \if\9;\global\numpunt=3\else\if\9:\global\numpunt=4\else \if\9)\global\numpunt=5\else\if\9!\global\numpunt=6\else \if\9?\global\numpunt=7\else\global\numpunt=8\fi\fi\fi\fi\fi\fi \fi}\ifcase\numpunt\or{\XWPR.}\or{\XWPR,}\or {\XWPR;}\or{\XWPR:}\or{\XWPR)}\or {\XWPR!}\or{\XWPR?}\or{\XWPR\ \9}\else\fi} %%%%%%%%%%%%%%%%%%%%%%%% \def\fiat{{}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%DEFINIZIONI LOCALI \def\rot{{\hbox{$\,$rot$\,$}}}\def\DO{{\dpr\O}} \def\V#1{{\underline#1}}\def\A{\V A} \def\2{{1\over2}} \def\PP{{\cal P}}\def\EE{{\cal E}}\def\MM{{\cal M}} \def\CC{{\cal C}}\def\FF{{\cal F}}\def\HH{{\cal H}} \def\TT{{\cal T}}\def\NN{{\cal N}}\def\BB{{\cal B}} \def\AA{{\cal A}}\def\VV{{\cal V}}\def\SSS{{\cal S}} \def\RRR{{\cal R}}\def\LL{{\cal L}}\def\JJ{{\cal J}} \def\T#1{{#1_{\kern-3pt\lower7pt\hbox{$\widetilde{}$}}\kern3pt}} \def\VVV#1{{\underline #1}_{\kern-3pt \lower7pt\hbox{$\widetilde{}$}}\kern3pt\,} \def\W#1{#1_{\kern-3pt\lower7.5pt\hbox{$\widetilde{}$}}\kern2pt\,} \def\Re{{\rm Re}\,}\def\Im{{\rm Im}\,} \def\lis{\overline}\def\tto{\Rightarrow} \def\etc{{\it etc}} \def\acapo{\hfill\break} \def\mod{{\rm mod}\,} \def\per{{\rm per}\,} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\indica{\leaders \hbox to 0.5cm{\hss.\hss}\hfill} \openout15=\jobname.aux %\write15 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Introduzione \def\guida{\leaders\hbox to 1em{\hss.\hss}\hfill} %\overfullrule=0pt \def\pp{{\bf p}}\def\qq{{\bf q}}\def\ii{{\bf i}}\def\xx{{\bf x}} \let\h=\eta\let\x=\xi\def\ie{{\it i.e. }} \def\Overline#1{{\bar#1}} \newtoks\footline \footline={\hss\tenrm\folio\hss} %\headline{\hss\tenrm \it DRAFT 4, not for circulation} \footline={\rlap{\hbox{\copy200}}\tenrm\hss \number\pageno\hss} \let\ciao=\bye %\BOZZA \def\annota#1#2{\footnote{${}^#1$}{\nota#2}} \def\qed{\raise1pt\hbox{\vrule height5pt width5pt depth0pt}} \def\mm{{\bf m}} \def\nn{{\bf n}} \def\aa{{\bf a}} \def\bb{{\bf b}} \font\titolo=cmbx12 \font\titolone=cmbx10 scaled\magstep 2 \font\cs=cmcsc10 \font\ss=cmss10 \font\sss=cmss8 \font\crs=cmbx8 \ifnum\mgnf=0 \def\ZZ{\hbox{{\ss Z}\kern-3.3pt{\ss Z}}} \def\zz{\hbox{{\sss Z}\kern-3.3pt{\sss Z}}} \fi \ifnum\mgnf=1 \def\ZZ{\hbox{{\ss Z}\kern-3.6pt{\ss Z}}} \def\zz{\hbox{{\sss Z}\kern-3.6pt{\sss Z}}} \fi \ifnum\mgnf=0 \def\RR{\hbox{I\kern-1.4pt{\ss R}}} \def\rr{\hbox{{\crs I}\kern-1.3pt{\sss R}}} \fi \ifnum\mgnf=1 \def\RR{\hbox{I\kern-1.7pt{\ss R}}} \def\rr{\hbox{{\crs I}\kern-1.6pt{\sss R}}} \fi \ifnum\mgnf=0 \def\openone{\leavevmode\hbox{\ninerm 1\kern-3.3pt\tenrm1}}% \fi \ifnum\mgnf=1 \def\openone{\leavevmode\hbox{\ninerm 1\kern-363pt\tenrm1}}% \fi % EPSF.TEX macro file: % Written by Tomas Rokicki of Radical Eye Software, 29 Mar 1989. % Revised by Don Knuth, 3 Jan 1990. % Revised by Tomas Rokicki to accept bounding boxes with no % space after the colon, 18 Jul 1990. % % TeX macros to include an Encapsulated PostScript graphic. % Works by finding the bounding box comment, % calculating the correct scale values, and inserting a vbox % of the appropriate size at the current position in the TeX document. % % To use with the center environment of LaTeX, preface the \epsffile % call with a \leavevmode. (LaTeX should probably supply this itself % for the center environment.) % % To use, simply say % \input epsf % somewhere early on in your TeX file % \epsfbox{filename.ps} % where you want to insert a vbox for a figure % % Alternatively, you can type % % \epsfbox[0 0 30 50]{filename.ps} % to supply your own BB % % which will not read in the file, and will instead use the bounding % box you specify. % % The effect will be to typeset the figure as a TeX box, at the % point of your \epsfbox command. By default, the graphic will have its % `natural' width (namely the width of its bounding box, as described % in filename.ps). The TeX box will have depth zero. % % You can enlarge or reduce the figure by saying % \epsfxsize= \epsfbox{filename.ps} % (or % \epsfysize= \epsfbox{filename.ps}) % instead. Then the width of the TeX box will be \epsfxsize and its % height will be scaled proportionately (or the height will be % \epsfysize and its width will be scaled proportiontally). The % width (and height) is restored to zero after each use. % % A more general facility for sizing is available by defining the % \epsfsize macro. Normally you can redefine this macro % to do almost anything. The first parameter is the natural x size of % the PostScript graphic, the second parameter is the natural y size % of the PostScript graphic. It must return the xsize to use, or 0 if % natural scaling is to be used. Common uses include: % % \epsfxsize % just leave the old value alone % 0pt % use the natural sizes % #1 % use the natural sizes % \hsize % scale to full width % 0.5#1 % scale to 50% of natural size % \ifnum#1>\hsize\hsize\else#1\fi % smaller of natural, hsize % % If you want TeX to report the size of the figure (as a message % on your terminal when it processes each figure), say `\epsfverbosetrue'. % \newread\epsffilein % file to \read \newif\ifepsffileok % continue looking for the bounding box? \newif\ifepsfbbfound % success? \newif\ifepsfverbose % report what you're making? \newdimen\epsfxsize % horizontal size after scaling \newdimen\epsfysize % vertical size after scaling \newdimen\epsftsize % horizontal size before scaling \newdimen\epsfrsize % vertical size before scaling \newdimen\epsftmp % register for arithmetic manipulation \newdimen\pspoints % conversion factor % \pspoints=1bp % Adobe points are `big' \epsfxsize=0pt % Default value, means `use natural size' \epsfysize=0pt % ditto % \def\epsfbox#1{\global\def\epsfllx{72}\global\def\epsflly{72}% \global\def\epsfurx{540}\global\def\epsfury{720}% \def\lbracket{[}\def\testit{#1}\ifx\testit\lbracket \let\next=\epsfgetlitbb\else\let\next=\epsfnormal\fi\next{#1}}% % \def\epsfgetlitbb#1#2 #3 #4 #5]#6{\epsfgrab #2 #3 #4 #5 .\\% \epsfsetgraph{#6}}% % \def\epsfnormal#1{\epsfgetbb{#1}\epsfsetgraph{#1}}% % \def\epsfgetbb#1{% % % The first thing we need to do is to open the % PostScript file, if possible. % \openin\epsffilein=#1 \ifeof\epsffilein\errmessage{I couldn't open #1, will ignore it}\else % % Okay, we got it. 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If so, \def\epsfclipon{\def\epsfclipstring{ clip}}% \def\epsfclipoff{\def\epsfclipstring{}}% % \def\epsfsetgraph#1{% \epsfrsize=\epsfury\pspoints \advance\epsfrsize by-\epsflly\pspoints \epsftsize=\epsfurx\pspoints \advance\epsftsize by-\epsfllx\pspoints % % If `epsfxsize' is 0, we default to the natural size of the picture. % Otherwise we scale the graph to be \epsfxsize wide. % \epsfxsize\epsfsize\epsftsize\epsfrsize \ifnum\epsfxsize=0 \ifnum\epsfysize=0 \epsfxsize=\epsftsize \epsfysize=\epsfrsize \epsfrsize=0pt % % We have a sticky problem here: TeX doesn't do floating point arithmetic! % Our goal is to compute y = rx/t. The following loop does this reasonably % fast, with an error of at most about 16 sp (about 1/4000 pt). % \else\epsftmp=\epsftsize \divide\epsftmp\epsfrsize \epsfxsize=\epsfysize \multiply\epsfxsize\epsftmp \multiply\epsftmp\epsfrsize \advance\epsftsize-\epsftmp \epsftmp=\epsfysize \loop \advance\epsftsize\epsftsize \divide\epsftmp 2 \ifnum\epsftmp>0 \ifnum\epsftsize<\epsfrsize\else \advance\epsftsize-\epsfrsize \advance\epsfxsize\epsftmp \fi \repeat \epsfrsize=0pt \fi \else \ifnum\epsfysize=0 \epsftmp=\epsfrsize \divide\epsftmp\epsftsize \epsfysize=\epsfxsize \multiply\epsfysize\epsftmp \multiply\epsftmp\epsftsize \advance\epsfrsize-\epsftmp \epsftmp=\epsfxsize \loop \advance\epsfrsize\epsfrsize \divide\epsftmp 2 \ifnum\epsftmp>0 \ifnum\epsfrsize<\epsftsize\else \advance\epsfrsize-\epsftsize \advance\epsfysize\epsftmp \fi \repeat \epsfrsize=0pt \else \epsfrsize=\epsfysize \fi \fi % % Finally, we make the vbox and stick in a \special that dvips can parse. % \ifepsfverbose\message{#1: width=\the\epsfxsize, height=\the\epsfysize}\fi \epsftmp=10\epsfxsize \divide\epsftmp\pspoints \vbox to\epsfysize{\vfil\hbox to\epsfxsize{% \toks0={#1 }% \ifnum\epsfrsize=0\relax \special{PSfile=\the\toks0 llx=\epsfllx\space lly=\epsflly\space urx=\epsfurx\space ury=\epsfury\space rwi=\number\epsftmp \epsfclipstring}% \else \epsfrsize=10\epsfysize \divide\epsfrsize\pspoints \special{PSfile=\the\toks0 llx=\epsfllx\space lly=\epsflly\space urx=\epsfurx\space ury=\epsfury\space rwi=\number\epsftmp\space rhi=\number\epsfrsize \epsfclipstring}% \fi \hfil}}% \global\epsfxsize=0pt\global\epsfysize=0pt}% % % We still need to define the tricky \epsfaux macro. This requires % a couple of magic constants for comparison purposes. % {\catcode`\%=12 \global\let\epsfpercent=%\global\def\epsfbblit{%BoundingBox}}% % % So we're ready to check for `%BoundingBox:' and to grab the % values if they are found. % \long\def\epsfaux#1#2:#3\\{\ifx#1\epsfpercent \def\testit{#2}\ifx\testit\epsfbblit \epsfgrab #3 . . . \\% \epsffileokfalse \global\epsfbbfoundtrue \fi\else\ifx#1\par\else\epsffileokfalse\fi\fi}% % % Here we grab the values and stuff them in the appropriate definitions. % \def\epsfempty{}% \def\epsfgrab #1 #2 #3 #4 #5\\{% \global\def\epsfllx{#1}\ifx\epsfllx\epsfempty \epsfgrab #2 #3 #4 #5 .\\\else \global\def\epsflly{#2}% \global\def\epsfurx{#3}\global\def\epsfury{#4}\fi}% % % We default the epsfsize macro. % \def\epsfsize#1#2{\epsfxsize} % % Finally, another definition for compatibility with older macros. % \let\epsffile=\epsfbox %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % FIGPRINT % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%% IF YOU WANT TO PRINT THE FIGURES %%%%%%%%%%%%%% %%%%%%%%%%%%%%% WRITE A SYMBOL % BEFORE EACH OF THE %%%%%%%%%%%%%% %%%%%%%%%%%%%%% THREE LINES OF THE COMMAND BELOW %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % \def\epsffile#1{\vbox to \epsfysize{\vfil \hbox to 2 truecm{\hss {\bf #1} \hss} \vfil} } % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \null \vskip3.truecm \centerline{\titolone Large deviation rule for Anosov flows} \vskip1.truecm \centerline{{\titolo Guido Gentile}$^\dagger$} \vskip.2truecm \centerline{$\dagger$ IHES, 35 Route de Chartres, Bures sur Yvette, France.} \vskip1.truecm \line{\vtop{ \line{\hskip1.5truecm\vbox{\advance \hsize by -3.1 truecm \\{\cs Abstract.} {\it The volume contraction in dissipative reversible transitive Anosov flows obeys a large deviation rule (fluctuation theorem).}} \hfill} }} \vskip2.truecm \centerline{\titolo 1. Introduction and formalism} \*\numsec=1\numfor=1 \\In this paper the results in [G3] are extended to the case of Anosov flows. The interest and the physical motivation are explained in [CG1] and [G2]: we briefly review them. For dissipative systems the existence and properties of a non equilibrium stationary state are not known in general. Nevertheless there are systems in which such a state exists and has been extensively studied: Anosov systems and, more generally, Axiom $A$ systems. The content of the {\sl chaotic hypothesis} proposed in [CG1] generalizing the Ruelle's principle for turbulence, [R5], is that, as far as only macroscopical quantities have to be computed, a many particle system in a stationary state out of equilibrium can be regarded as if it was an Axiom $A$ system. In [CG1] a theorem of large deviations is heuristically proven for dissipative reversible transitive Anosov systems, by using the properties of the stationary state (SRB measure), and it is shown to agree with the results of the numerical experiments in [ECM]. A rigorous proof for Anosov diffeomorphisms is performed in [G3]. As the physical systems which one wants to study through mathematical models evolve in a continuous way, it can be interesting to check if the theorem still holds if one consider Anosov flows instead of diffeomorphisms. This program is achieved in the present paper: it will be shown that the study of Anosov flows can be reduced to the study of Anosov diffeomorphisms (more rigorously of maps which have all the ``good'' properties of Anosov diffeomorphisms, in a sense which will be explained below, after Proposition 1.9), so that the large deviation rule for dissipative reversible transitive Anosov diffeomorphisms is extended to the case of flows. If the systems is Axiom $A$ but not Anosov, something can still be said: see comments after Theorem 3.6. \* In this section we simply review the basic notions and results on Axiom $A$ flows, Markov partitions and symbolic dynamics, essentially taken from [B2] and [BR], and in \S 2 we introduce the SRB measures for Axiom $A$ flows. Even if in the end we will confine ourselves on Anosov flows, it can be worthwile to start with more general definitions (also in view of possible future extensions to Axiom $A$ flows of the results holding for Anosov flows), as the discussion in this introductory part can be carried out with no relevant change both for Axiom $A$ and Anosov flows. In \S 3 we consider dissipative reversible Axiom $A$ flows, study conditions under which they reduce to Anosov flows, and state the fundamental result of the paper (a large deviation rule for the volume contraction in dissipative reversible transitive Anosov flows), which will be proven in \S 4. \* Let $M$ be a differentiable ($C^{\io}$) compact Riemannian manifold and $f^t\!\!: M \to M$ a differentiable flow. \* \\{\bf 1.1.} {\cs Definition.} {\it A closed $f^t$-invariant set $X\subset M$ containing no fixed points is {\sl hyperbolic} if the tangent bundle restricted to $X$ can be written as the Whitney sum\annota{1}{\nota That is for each $x\in X$, the decomposition \equ(1.1) becomes $T_xM=E_x\oplus E_x^s \oplus E_x^u$.} of three $Tf^t$-invariant continuous subbundles % $$ T_X M = E + E^s + E^u \; , \Eq(1.1) $$ % where $E$ is the one-dimensional bundle tangent to the flow, and % $$ \eqalign{ (a) \kern.5truecm & \| T f^t w \| \le c\,e^{-\l t}\|w\| \; , \quad \hbox{for } w\in E^s \; , \quad t\ge 0 \; , \cr (b) \kern.5truecm & \| T f^{-t} w \| \le c\,e^{-\l t}\|w\| \; , \quad \hbox{for } w\in E^u \; , \quad t\ge 0 \; , \cr} \Eq(1.2) $$ % for some positive constants $c,\l$; $\|\cdot\|$ denotes the norm induced by the Riemann metric.} \* More generally, if $Y$ is the union of a hyperbolic set as above and a finite number of hyperbolic fixed points, we also say that $Y$ is hyperbolic, [ER], \S F.2. %One can choose $t_0>0$ and change $\l$, so that (a) and (b) hold %with $t\ge t_0$ and $c=1$, [BR], %(the metric is then said to be {\sl adapted} to $f^{t_0}$, %as in the case of diffeomorphisms, [Ma,HP]). \* \\{\bf 1.2.} {\cs Definition.} {\it A closed $f^t$-invariant set $\L$ is a {\sl basic hyperbolic set} if \acapo (1) $\L$ contains no fixed points and is hyperbolic; \acapo (2) the periodic orbits of $f^t|\L$ are dense in $\L$; \acapo (3) $f^t|\L$ is topologically transitive;\annota{2}{\nota A flow $f^t\!:\L \to \L$ is topologically transitive if, for all $U$, $V\subset \L$ open nonempty, $U \cap f^t V \neq \emptyset$ for some $t>0$.} \acapo (4) there is an open set $U \supset \L$ with $\L = \cap_{t\in{\rr}} f^tU$.} \* Definition 1.2 is taken from [BR], \S 1. Usually one defines a basic hyperbolic set as a set which either satisfies Definition 1.2 or is a hyperbolic fixed point. In the following we will be interested in basic hyperbolic sets which are not a single point: this motivates Definition 1.2. \* \\{\bf 1.3.} {\cs Definition.} {\it A basic hyperbolic set $\L$ for which the set $U$ in item (4) can be chosen satisfying $f^tU\subset U$ for all $t\ge t_0$, for fixed $t_0$, is defined to be an {\sl attractor}.} \* A point $x\in M$ is called {\sl nonwandering} if, for every neighbourhood $V$ of $x$ and every $t_0\in\RR$, there is a $t>t_0$ such that % $$ f^tV \cap V \neq \emptyset \; . $$ % The {\sl nonwandering set} is defined as % $$ \O = \{ x \in M : \hbox{$x$ is nonwandering} \} \; . $$ % \* \\{\bf 1.4.} {\cs Definition.} {\it A flow $f^t\!\!:M\to M$ is said to satisfy {\sl Axiom $A$} if the nonwandering set $\O$ is the disjoint union of a set satisfying (1) and (2) of Definition 1.2 and a finite number of hyperbolic fixed points.} \* Smale's {\sl spectral decomposition theorem} ([Sm], Theorem 5.2; see also [PS], Theorem 2.1) states that, if the flow $f^t\!\!:M\to M$ satisfies Axiom $A$, and if we denote by $\FF$ the set of hyperbolic fixed points in $\O$, then $\O\setminus\FF$ is the disjoint union of a finite number of basic hyperbolic sets. \* \\{\bf 1.5.} {\cs Definition.} {\it A flow $f^t\!\!:M\to M$ is an {\sl Anosov flow} if $M$ is hyperbolic.} \* An Anosov flow satisfies Axiom A (Anosov's {\sl closing lemma}, [A]; see also [B3], \S 3.8). By Smale's spectral decomposition theorem, given an Anosov flow $f^t\!\!:M\to M$, then one can decompose $\O=\cup_{j=1}^N \L_j$, where $\L_1,\ldots,\L_N$ are basic hyperbolic sets, and one can consider the restriction $f^t|\L_j$, $\forall j=1,\ldots,N$, which is topologically transitive. If one has $\O=M$, then each $f^t|\L_j$ is a {\sl transitive Anosov flow}.\annota{3}{\nota Note that there are Anosov flows for which $\O\neq M$, [FW]: such flows are obviously non transitive. In the case of maps, the identity $\O=M$ is conjectured to hold for all Anosov diffeomorphisms, [Sm], Problem 3.4.} Standard examples of Axiom $A$ flows are the suspension of an Axiom $A$ diffeomorphism, \eg the solenoid, [Sm], and the geodesic flow on a compact manifold with negative curvature, [A], which is an Anosov flow, (see also [B4]). \* Let $\L$ be a basic hyperbolic set. For any $x\in\L$, the stable and unstable manifolds are defined as % $$ \eqalign{ W_x^s & = \{ y\in M \; : \; \lim_{t\to\io} d(f^tx,f^ty) = 0 \} \; , \cr W_x^u & = \{ y\in M \; : \; \lim_{t\to\io} d(f^{-t}x,f^{-t}y) = 0 \} \; , \cr} \Eq(1.3) $$ % where $d$ is the distance induced by the Riemann metric, and % $$ W_\L^s = \bigcup_{x\in\L}W_x^s \; , \quad W_\L^u = \bigcup_{x\in\L}W_x^u \; . \Eq(1.4) $$ % If $\L$ is an attractor, $W_\L^s$ is its {\sl basin}: $\lim_{t\to\io} d(f^tx,\L) = 0$ $\forall x\in W_{\L}^s$. %The stable and unstable manifolds are fibers (or leaves) of the %foliations of $M$ determined by the subbundles $E^s$ and $E^u$. %The fibers are $C^{\io}$, (as they are as smooth as the flow %itself, [AS], and the flow is $C^{\io}$), but the foliations %are only H\"older continuous in general. %Let $B_x(\e,T)$ be a closed $\e$-neighbourhood of $x$ for the %distance % %$$ \d\,(x,y) = \sup_{0\le t \le T} d(f^tx,f^ty) \; ; $$ % %then $W_{x,\e}^s=W_x^s\cap B_x(\e,\io)$ and %$W_{x,\e}^u=W_x^u\cap B_x(\e,\io)$. \* For $x\in\L$, we set % $$ \eqalign{ W_{x,\e}^s & = \{ y \in W_x^s \; : \; d(f^tx,f^ty) \le \e \; \forall t\ge 0 \} \; , \cr W_{x,\e}^u & = \{ y \in W_x^u \; : \; d(f^{-t}x,f^{-t}y) \le \e \; \forall t\ge 0 \} \; . \cr} $$ \* Let $D$ be a differentiable closed disk (\ie a closed element of a $C^{\io}$ manifold of dimension $\hbox{dim}(M)-1$, if $\hbox{dim}(M)$ is the dimension of $M$), containing a point $x\in\L$ and trasverse to the flow. For any closed subset $T\subset D$ containing $x$ and for any $y\in T$ such that $d(x,y) \le \a_1$ for a suitable $\a_1$, let us define $\langle x,y \rangle$ as the intersection $W^s_{f^vx,\e}\cap W^u_{y,\e}$, for a suitable $v$, (by choosing $\a_1$ small enough, one can always take $|v|\le \e$): such an intersection is well defined (\ie it is a single point) and lies in $\L$, [Sm]. Then let us introduce the {\sl canonical coordinate} $\langle x,y \rangle_D$ as the {\sl projection} of $\langle x,y \rangle$ on $D$, [B2], \S 1: this means that there exists a constant $\x\ge0$ such that, for $|r|\le \x$, $f^r\langle x, y \rangle_D$ $=$ $\langle x,y \rangle$. The subset $T$ is called a {\sl rectangle} if $\langle x,y \rangle_D \in T$ for any $x,y\in T$, and in this case we can define $\langle x,y \rangle_T\=\langle x,y \rangle_D$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% FIGURA 1.1 (F1.PS) %%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % \midinsert $$ \vcenter{\epsfysize=6.cm \epsffile{f1.ps} } $$ \centerline{{\cs Fig.1.} {\it Canonical coordinates.}} \endinsert % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Then, for $x\in T$, we set $W^s_x(T)=\{\langle x,y\rangle \; : \; y \in T \}$ and $W^u_x(T)=\{\langle y,x\rangle \; : \; y \in T \}$: we can say that $W^s_x(T)$ and $W_x^u(T)$ are the projection of the stable manifold and, respectively, of the unstable manifold of $x$ on the rectangle $T$ containing $x$ (the projection is meant along the flow), and $T$ becomes the direct product of $W_x^s(T)$ and $W_x^u(T)$. \* \\{\bf 1.6.} {\cs Definition.} {\it Choose a basic hyperbolic set $\L$. We call $\TT=\{T_1,\ldots,T_\NN \}$ a {\sl proper family of rectangles}, if there are positive constants $\a$ and $\a_1$ such that \acapo (1) $T_j\subset\L$ is a closed rectangle; \acapo (2) if $\G(\TT)=\bigcup_{j=1}^{\NN} T_j$, then $\L \subset \bigcup_{0\le t\le\a}f^{-t}\G(\TT)$; \acapo (3)} $T_j\subset \hbox{int}\,D_j$, {\it where $D_j$ is a $C^{\io}$ closed disk transverse to the flow, such that: (3.1)} $\hbox{diam}\,(D_j)\le \a_1$ {\it $\forall j$, (3.2)} $\hbox{dim}\,(D_j) = \hbox{dim}\,(M)-1$ {\it $\forall j$, (3.3)} $T_j = \overline{\hbox{int}\,T_j}$ {\it $\forall j=1,\ldots,\NN$, (where} $\hbox{int}\,T_j$ {\it is the interior of $T_j$ as a subset of $\L\cap D_j$), and (3.4) for $i\neq j$, at least one of the sets $D_i\cap \bigcup_{o\le t\le\a}f^tD_j $ and $D_j\cap \bigcup_{o\le t\le\a}f^tD_i$ is empty (in particular $D_i\cap D_j=\emptyset$).} \* The above definition is taken from [B2], Definition 2.1. [In [B2], $\a_1=\a$ and $\TT$ is called a proper family of rectangles ``of size $\a$''; in general it can be useful to consider $\a$ and $\a_1$ as independent parameters, so that one can change one of them without affecting the other one.] Note that, if the flow $f^t\!\!:M\to M$ is a topologically mixing Anosov flow,\annota{4}{\nota A transitive Anosov flow is said to be topologically mixing if the stable and unstable manifolds $W_x^s$ and $W_x^u$ are dense in $M$ for some (and then for each) $x\in M$.} %then its stable and unstable foliations then $E^s$ and $E^u$ are not jointly integrable (\ie $E^s\oplus E^u$ is not integrable, [Pl], Proposition 1.6), [Pl], Lemma 1.4, Lemma 1.5, Theorem 1.8. This means that, if $\e'$ is so chosen that for any $x\in\L$, $y\in W^u_{x,\e}$ and $\x\in W^s_{x,\e'}$ one has $W^u_{\x,\e}\cap\bigcup_{-\e\le t\le\e}f^t W^s_{y,\e}\neq\emptyset$, then $W^u_{\x,\e}\cap W^s_{y,\e}=\emptyset$. Therefore, in such a case, the disks $D_j$'s can {\it not} be constructed so that %$W_{x,\e}^s\cap W_{x,\e}^u %\subset D_j$, if $D_j$ is the disk containing $x$, (equivalently the conditions $W_{x,\e}^s(T)=W_{x,\e}^s\cap T$ and $W_{x,\e}^u(T)=W_{x,\e}^u\cap T$ are simultaneously possible. Given $x\in\G(\TT)$, let $t'(x)$ be the first positive time required for $f^tx$ to cross $\G(\TT)$. If $x\in T_i$, for some $i=1,\ldots,\NN$, then $f^{t'(x)}x\in T_j$, for some $j\neq i$; set $t(x)=t'(x)$ for $x\in \hbox{int}\,T_i$ and extend it by continuity to the boundaries of $T_i$. Then define $\HH_\TT x = f^{t(x)}x$, for any $x\in\G(\TT)$, [B2], \S 2: $t(x)$ is called {\sl ceiling function} and $\HH_\TT$ {\sl Poincar\'e map}. There exists a $t_0\in(0,\a)$ such that $t(x)>t_0$ $\forall x\in \G(\TT)$. The function $\HH_{\TT}$ is continuous on % $$ \G'(\TT) = \{ x \in \G(\TT) \; : \; \HH_{\TT}^kx \in \bigcup_{j=1}^{\NN} \hbox{int}\,T_j \; \forall k \in \ZZ \} \; , \Eq(1.5) $$ % and $\G'(\TT)$ is dense in $\G(\TT)$, being a countable intersection of dense open subsets ({\sl Baire's theorem}, [Bb], Ch. IX, \S 5.3). \* \\{\bf 1.7.} {\cs Definition.} {\it A proper family of rectangles $\TT$ is called a {\sl Markov partition} (or {\sl Markov family}, or {\sl Markov pavement}), if \acapo (1) for $x\in T_i$, $\HH_\TT x \in T_j$, one has $\HH_\TT y \in T_j$ $\forall y \in W^s_x(T_i)$; \acapo (2) for $x\in T_i$, $\HH_{\TT}^{-1} x \in T_j$, one has $\HH_{\TT}^{-1} y \in T_j$ $\forall y \in W^u_x(T_i)$.} \* The above definition is taken from [B2], Definition 2.3. Define % $$ \eqalign{ & \dpr^s\TT = \bigcup_{j=1}^{\NN} \dpr^s T_j \; , \quad \dpr^u\TT = \bigcup_{j=1}^{\NN} \dpr^u T_j \; , \quad \dpr\TT = \dpr^s\TT \cup \dpr^u\TT \; , \cr & \D^s\TT = \bigcup_{0\le t\le\a} f^t \dpr^s\TT \; , \quad \D^u\TT = \bigcup_{0\le t\le\a} f^{-t} \dpr^u\TT \; ; \cr} $$ % where % $$ \eqalign{ \dpr^s T & = \{ \langle y_1, y_2 \rangle_T \; : \; y_1\in \dpr W_{x,\e}^u(T) \hbox{ and } y_2\in W_{x,\e}^s(T) \}\; , \cr % \dpr^u T & = \{ \langle y_1, y_2 \rangle_T \; : \; y_1\in W_{x,\e}^u(T)\hbox{ and } y_2\in\dpr W_{x,\e}^s(T) \}\;,\cr} $$ % if $\dpr W_{x,\e}^u(T)$ and $\dpr W_{x,\e}^s(T)$ denote the boundaries of $W_{x,\e}^u(T)$ and $W_{x,\e}^s(T)$ as subsets, respectively, of $W_{x}^u(T)\cap\L$ and $W_{x}^s(T)\cap\L$. \* \\{\bf 1.8.} {\cs Proposition.} {\it If $\TT$ is a Markov partition, one has $f^t\D^s\TT \subset\D^s\TT$ and $f^{-t}\D^u\TT \subset \D^u\TT$ $\forall t\ge0$.} \* The proof is in [B2], Proposition 2.6. Then the following fundamental result is proven in [B2], Theorem 2.5. \* \\{\bf 1.9.} {\cs Proposition.} {\it Any basic hyperbolic set $\L$ admits a proper family of rectangles %of size arbitrarily small which is a Markov partition.} \* By construction, the discontinuity set of $\HH_\TT$, \ie $\G(\TT)\setminus\G'(\TT)$, is covered by the evolution of some stable and unstable manifolds, so that $\HH_{\TT}\!\!:\G'(\TT)\to\G'(\TT)$ can be studied as it was an Axiom $A$ diffeomorphism, (see [B3] for a review). \* \\{\bf 1.10.} {\it Symbolic dynamics.} Let us introduce a $\NN\times\NN$ matrix $A$ such that % $$ A_{ij} = \cases{ 1 & if there exists $x\in \hbox{int}\,T_i$ such that $\HH_\TT x \in \hbox{int}\,T_j$, \cr 0 & otherwise, \cr} $$ % ({\sl transition matrix}), and let us define the {\sl space of the compatible strings} % $$ \MM = \{ \mm \= \{m_i\}_{i\in{\zz}} \; : \; m_i\in\{1,\ldots,\NN\} \, , \; A_{m_im_{i+1}}=1 \; \forall i\in{\ZZ} \} , \Eq(1.6) $$ % and the map $\s \! : \MM \to \MM$ by $\s\mm = \{m_i'\}_{i\in{\zz}}$, where $m_i'=m_{i+1}$. If $\{1,\ldots,\NN\}$ is given the discrete topology and $\{1,\ldots,\NN\}^{\zz}$ the product topology, $\MM$ becomes a compact metrizable space and $\s$ a topologically transitive homeomorphism ({\sl subshift of finite type}); furthermore, because of the transitivity of $f^t$, $\s$ can be supposed to be topologically mixing,\annota{5}{\nota A homeomorphism $f\! : X\to X$ is topologically mixing if, for all $U$, $V \subset X$ open nonempty, $U\cap f^n V \neq \emptyset$ for all sufficiently large $n$.} [BR], Lemma 2.1. A metric on $\MM$ can be $d(\mm,\nn)=d_1e^{-d_2N}$, with $d_1,d_2>0$, if $m_i=n_i$, $\forall |i|\le N$, [B2], \S 1. For $\psi \!: \MM \to {\RR}$ a positive continuous function, \ie $\psi\in C(\MM)$, and % $$ Y = \{(\mm,s)\; :\; s\in[0,\psi(\mm)],\;\mm\in\MM\} \; , \Eq(1.7) $$ % identify the points $(\mm,\psi(\mm))$ and $(\s\mm,0)$ for all $\mm\in\MM$, so obtaining a new compact metric space $\L(A,\psi)$, [BW]. If $q: \; Y \to \L(A,\psi)$ is the quotient map, then the {\sl suspension flow} (or {\sl special flow}) $g^t\! :\L(A,\psi) \to \L(A,\psi)$ is defined as % $$ g^tq(\mm,s) = q(\s^k\mm,v) \; , $$ % where $k$ is chosen so that % $$ v = t + s - \sum_{j=0}^{k-1} \psi(\s^j\mm) \; \in [0,\psi(\s^k\mm)] \; . $$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% FIGURA 1.1 (F2.PS) %%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % \midinsert $$ \vcenter{\epsfysize=6.cm \epsffile{f2.ps} } $$ \centerline{{\cs Fig.2.} {\it Suspension flow.}} \endinsert % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% For $\psi\in C(\MM)$, let % $$ \hbox{var}_N \psi = \sup \{ |\psi(\mm) - \psi(\nn)|\;:\; \mm,\nn\in\MM , \; m_i=n_i \; \forall |i|\le N \} \; $$ % and let % $$ \FF = \{ \psi \in C(\MM)\; : \; \exists \; c_1,c_2>0 \hbox{ so that var}_N\psi\le c_1\,e^{-c_2N} \; \forall N\ge0\}. \Eq(1.8) $$ A flow $g^t$ with $\psi\in\FF$ is called a {\sl hyperbolic symbolic flow}, [B2], Definition 1.3, and its class is the same as the class of all one-dimensional basic sets for flows, [B1]. \* \\{\bf 1.11.} {\cs Proposition.} {\it If $\L$ is a basic hyperbolic set, there exists a positive $\psi\in\FF$ and a continuous surjection $\r\!:\;\L(A,\psi)\to\L$ ({\sl symbolic code}) such that $\r \circ g^t = f^t \circ \r$.} \* The proof is in [B1], \S 2. If $N=\bigcup_{t=-\io}^{\io} f^t\dpr\TT$, and we set $\L_0=\L\setminus N$ and $\MM_0=\L(A,\psi)\setminus\r^{-1}(N)$, then $\r$ is a continuous bijection between $\L_0$ and $\MM_0$. Note that, if $x\in\G(\TT)$, then $x=\r(\mm,0)$, $\mm\in\MM$, and one has $t(x)=\psi(\mm)$. \vskip1.truecm \centerline{\titolo 2. Equilibrium states and SRB measures} \*\numsec=2\numfor=1 \\Given a homeomorphism $f$, $M(f)$ denotes the set of $f$-invariant Borel probability measures; if $F=\{f^t\}_{t\in{\rr}}$, then $M(F)=\bigcap_{t\in{\rr}}M(f^t)$. If $g^t$ is the suspension flow, we set $G=\{g^t\}$. Let $\L$ be a basic hyperbolic set. For $x\in\L$, if $E_x^s$ and $E_x^u$ denote, respectively, the subbundles tangent to $W_x^s$ and $W_x^u$ in $x$, let $\l_{0,t}(x)$, $\l_{u,t}(x)$ and $\l_{s,t}(x)$ be the jacobians of the linear maps, respectively, $Df^t\!: E_x\to E_{f^tx}$, $Df^t\!: E_x^u\to E_{f^tx}^u$ and $Df^t\!: E_x^s\to E_{f^tx}^s$, and % $\l_t(x)$ $=$ $\l_{0,t}(x)$ $\l_{s,t}(x)$ $\l_{u,t}(x)$ $\chi_{t}^1(x)$ $\chi_{t}^2(x)$, % where $\chi_t^1(x)= \sin[\psi^1(f^tx)]/\sin[\psi^1(x)]$ and $\chi_t^2(x)=\sin[\psi^2(f^tx)]/\sin[\psi^2(x)]$, being $\psi^1(x)$ the angle between $E^s_x$ and $E^u_x$ in $x$, and $\psi^2(x)$ the angle between $E^s_x \oplus E^u_x$ and the flow direction in $x$. Note that $\l_{u,t+t'}(x)=\l_{u,t}(f^{t'}x)\,\l_{u,t'}(x)$ and analogous relations hold for $\l_{s,t}$, $\l_{0,t}$, $\chi_t^1$ and $\chi_t^2$: so that all such quantities are {\sl cocycles}, in the sense of [R4], Definition B2. By the transversality properties and the absence of fixed points of the basic hyperbolic sets (see Definitions 1.1 and 1.2), there exists a positive constant $B_1$ such that $B^{-1}_1<\chi_t^1(x),\chi_t^2(x),\l_{0,t}(x)0,\l>0$. In particular $h_\pm$ (and $h_0$) enjoy the above property ({\sl short range}), by the properties of the Markov partition introduced in \S 1. \* \\{\bf 2.4.} {\cs Proposition.} {\it For any smooth function} $g\!:M\to\RR$, {\it if $\L$ is an attractor for the Axiom $A$ flow $f^t\!:M\to M$, and $W_\L^s$ is its basin, one has % $$ \lim_{T\to\io} {1\over T} \int_0^T dt \, g(f^{t}x) = \int_{\L} \m_{+} (dy) \, g(y) $$ % for $\m_0$-almost all $x\in W_\L^s$. Analogously, if $\L'$ is an attractor for the opposite flow $f^{-t}\!\!:M\to M$, one has % $$ \lim_{T\to\io} {1\over T} \int_0^T dt \, g(f^{-t}x) = \int_{\L'} \m_{-} (dy) \, g(y) $$ % for $\m_0$-almost all $x\in W_{\L'}^u$, where $W_{\L'}^u$ is the basin of $\L'$: $\lim_{t\to\io}d\,(f^{-t}x,\L')=0$ $\forall x\in W_{\L'}^u$.} \* The proof is in [BR], Theorem 5.1. \* Consider transitive Anosov flows. Let us construct the Markov partition $\TT_L = \bigvee_{j=-L}^{L} \HH_{\TT}^{-j} \TT$, (this means that, if $\mm_{[-L,L]}$ $\=$ $(m_{-L},\ldots,m_L)$ and $T_{\mm_{[-L,L]}}$ $=$ $\bigcap_{j=-L}^{L} \HH_{\TT}^{-j} T_{m_j}$, for $T_{m_j}\in\TT$ $\forall j$ $=$ $-L,\ldots,L$, then $T_{\mm_{[-L,L]}}\in\TT_L$), and let $x_{\mm_{[-L,L]}^0}$ be a suitable point in $T_{\mm_{[-L,L]}}$, where $\mm_{[-L,L]}^0\in\MM$, (\ie $\mm_{[-L,L]}^0$ is a compatible string), with $(\mm_{[-L,L]}^0)_i$ $=$ $(\mm_{[-L,L]})_i$, $\forall |i|\le L$; for instance we can choose the symbols corresponding to the sites $|j|\ge\pm(L+1)$ such that $A_{m_jm_{j+1}}=1$, so that the dependence on $\mm_{[-L,L]}$ is only via the symbols $m_{\pm L}$. We can define % $$ \int_{\L} \m_{L,k}(dx) \, g(x) = {\sum_{\mm_{[-L,L]}} \JJ_{u,k}^{-1} (x_{\mm_{[-L,L]}^0}) \, \int_0^{\psi(\mm_{[-L,L]}^0)} dt \, g(f^tx_{\mm_{[-L,L]}^0}) \over \sum_{\mm_{[-L,L]}} \JJ_{u,k}^{-1} (x_{\mm_{[-L,L]}^0}) \, \psi(\mm_{[-L,L]}^0) } \, \; , \Eq(2.5) $$ % which is called {\sl approximating distribution for $\m_+$.} In fact the following result holds. \* \\{\bf 2.5.} {\cs Proposition (Approximation theorem).} {\it Let $f^t\!\!:M\to M$ be a transitive Anosov flow. If $\m_{L,k}$ is defined as in \equ(2.5), then, for any smooth function} $g\!: M \to \RR$, {\it one has % $$ \lim_{k\to\io \atop L\ge k/2} \int \m_{L,k}(dx) \, g(x) = \int \m_+(dx) \, g(x) \; , $$ % where $\m_+$ is the forward SRB measure.} \* The measure $\m_{L,k}$ can be written as $\m_{L,k}\=\m_{\n_{L,k}}$, where $\n_{L,k}$ is the approximating distribution for $\n_+ \in M(\s)$ defined on $\MM$. In the following we shall use the notation % $$ \int \n_+(d\mm) \, \lis g(\mm) = \int_{\L} \m_+(dx)\,g(x) \; ,\qquad \int \n_{L,k}(d\mm) \, \lis g(\mm) = \int_{\L} \m_{L,k}(dx)\,g(x)\;, $$ % where % $$ \lis g(\mm) = \int_0^{\psi(\mm)} dt \, g(\r(\mm,t)) \; , $$ % with $\psi(\mm)\in(t_0,\a)$ $\forall \mm\in\MM$. For any subset $A\subset \MM$, we denote by $\n_+(A)$ the $\n_+$-measure of $A$: $\n_+(A)=\int_A\n_+(d\mm)$. We conclude this section with a comment inherited from [CG2]. \* \\{\bf 2.6.} {\it Remark.} In Proposition 2.5, we could define the approximating distribution with $\JJ_{u,k}(x)\to\JJ_{u,k}(x)\,\d_k(x)$, where $\d_k(x)$ $=$ $\sin(\HH_{\TT}^{k/2}x)/\sin(\HH_{\TT}^{-k/2}x)$, which corresponds to considering a Gibbs state with a different boundary condition (with the difference becoming irrelevant in the limit as $k\to\io$, because of the absence of phase transitions for one-dimensional Gibbs states with short range interactions, [R2,GL]). Note that the factors $\d_k(x)$ are {\sl cocycles}, according to [R4], Definition B2. \vskip1.truecm \centerline{\titolo 3. Reversible dissipative systems and results.} \*\numsec=3\numfor=1 \\Let us consider flows $f^t\!\!:M\to M$ verifying the following conditions (A) and (B). \* \\{\bf 3.1.} {\cs Definition.} {\it The flow $f^t\!\!:M\to M$ is (A) {\sl dissipative} if % $$ \s_\pm = - \int_\L \m_{\pm}(dx) \, \ln J^{\pm1}(x) > 0 \; , $$ % and (B) {\sl reversible} if there is an isometric involution $i\!:M \to M$, $i^2=\openone$, such that: $if^t=f^{-t}i$.} \* If $f^t\!\!:M\to M$ is transitive and reversible, then, for any $x\in M$, the stable and the unstable manifolds have the same dimension, so that the the dimension of $M$ is odd. By reversibility, one has $\s_+=\s_-$, $J(x)=J^{-1}(ix)$, $iW_x^u=W_{ix}^s$, [G3], \S 2, and $\JJ_{u,k}$ $=$ $\JJ_{s,k}^{-1}(ix)$, [G3], \S 4. Moreover, if $\L$ is an attractor for the flow $f^t\!\!: M\to M$, then $\L'=i\L$ is an attractor for the opposite flow $f^{-t}\!\!:M \to M$, so that $W_{\L}^s=iW_{\L'}^u$, with the notations in Proposition 2.4. \* \\{\bf 3.2.} {\cs Lemma.} {\it Let $\L$ be an attractor for the Axiom A flow $ f^t\!\!: M\to M $. If \acapo (a) the flow is transitive on $M$, or \acapo (b) the flow is reversible and $i\L=\L$, \acapo then $\L$ is a connected component of $M$ and $f^t|\L$ is an Anosov flow.} \* \\{\bf 3.3.} {\it Proof of Lemma 3.2.} If $f^t\!\!: M\to M$ is transitive, $\L$ is dense in $M$ (because of the $f^t$-invariance of $\L$), and, as $\L$ is closed, then $\L=M$; this proves (a).\annota{6}{\nota It is not necessary to assume the existence of an attractor in order to deduce from transitivity that $f^t\!\!:M\to M$ is an Anosov flow: in fact transitivity implies trivially $\O=M$.} If $\L$ is an attractor, one has $m(W_{\L}^s)>0$, where $m$ is the measure on $M$ derived from the Riemann metric, [BR], Theorem 5.6. If one sets $\L'=i\L$, one has $iW_{\L'}^u=W_{\L}^s$, by reversibility. If $i\L=\L$, then $m(W_{\L}^u)=m(W_{\L}^s)$. But $\L=W^u_{\L}$, hence $m(\L)>0$, so that $\L$ is a connected component of $M$ and $f^t|\L$ is an Anosov flow, [R5], [BR], Corollary 5.7. Then (b) follows. \qed \* \\{\bf 3.4.} {\cs Remark.} In hypothesis (b) of Lemma 3.2, one can assume $iW_{\L}^s=W_{\L}^s$ instead of $i\L=\L$: in fact $i\L=\L$ yields $\L=W_{\L}^s$, and, if $iW_{\L}^s=W_{\L}^s$, one has $i\L\subset iW_{\L}^s= W_{\L}^s$, hence $i\L=\L$, (because $i\O=\O$). \* Note that a stretched exponential bound on the correaltion functions is obtained, [Ch], for three-dimensional topologically mixing Anosov flows satisfying an extra assumption (``uniform nonintegrability'' of the ``foliations'' $E^s$ and $E^u$, [Ch], \S 13, Assumption A5), while it is known that topologically mixing Axiom $A$ flows can have correlations function decaying arbitrarily slowly, [R6,Po]. \* \\{\bf 3.5.} {\cs Definition.} {\it We define the {\sl dimensionless volume contraction rate} at $x\in\G(\TT)$ and over a time $k$ as % $$ \e_k(x) ={1\over\s_+k} \sum_{j=-k/2}^{k/2-1} \ln J^{-1}(\HH_\TT^jx) = {1\over\s_+k} \ln \JJ^{-1}_k(x) \; , $$ % where $\JJ^{-1}_k(x) \= \prod_{j=-k/2}^{k/2-1} J^{-1}(\HH_\TT^jx)$, and we set $\lis\e_k(\mm)=\int_0^{\psi(\mm)}dt\,\e_k(\r(\mm,t))$.} \* Then the following result holds, which can be interpreted as a large deviation rule, (see [La,CG1]). \* \\{\bf 3.6.} {\cs Theorem (Fluctuation theorem).} {\it Let $\L$ be an attractor for the dissipative reversible transitive Anosov flow $f^t\!\!:M\to M$. There exists $p^*>0$ such that the SRB distribution $\m_+=\m_{\n_+}$ on $\L$ verifies % $$ p-\d\le \lim_{k\to\io}{1\over\s_+k}\ln{\n_+(\{\mm : \; \lis\e_k(\mm)\in[p-\d,p +\d]\}) \over \n_+(\{\mm : \; \lis\e_k(\mm)\in-[p-\d,p+\d]\})} \le p+\d \; , $$ % for all $p$ and $\d$ such that $|p|+\d0$ and for $p\in(-p^*,p^*)$, $|p|+\dp-\d-\h'(k)\cr} $$ % where $\h(k),\h'(k)>0$ and $\h(k),\h'(k)\to 0$ for $k\to\io$.} \* For $q\in\ZZ$ and $n$ odd, set $X=\{q,{q+1},\ldots,{q+n-1}\} \=[q,q+n-1]$ and define $\mm_X$ $=$ $(m_q,$ $m_{q+1},$ $\ldots,$ $m_{q+n-1})$ and $\lis X$ $=$ $q+(n-1)/2$ (the {\sl center} of $X$). If $\mm\in\MM$, let $\mm_X^0$ be an arbitrary configuration $\{(m_X^0)_i\}_{i\in\zz}$ such that $(m^0_X)_i$ $=$ $(m_X)_i$, $\forall i=q, \ldots,q+n-1$. One can write % $$ {1\over\s_+}\ln J^{-1}(\r(\mm,0)) = \sum_{\lis X=0} E_X(\mm_X) \; , \qquad h_+(\mm) = \sum_{\lis X=0} H_X(\mm_X) \; , \Eq(4.2) $$ % where $E_X(\mm_X)$ and $H_X(\mm_X)$ are translation invariant and exponentially decaying functions, \ie, if $\th$ denotes translation to the right, % $$ \eqalign{ E_{\th X}(\mm_X) & \= E_X(\mm_X) \; , \qquad |E_X(\mm_X)| \le b_1^{(E)}\, e^{-b_2^{(E)} n} \; , \cr H_{\th X}(\mm_X) & \= H_X(\mm_X) \; , \qquad |H_X(\mm_X)| \le b_1^{(H)}\, e^{-b_2^{(H)} n} \; , \cr} $$ % for suitable positive constant $b_1^{(E)}$, $b_2^{(E)}$, $b_1^{(H)}$ and $b_2^{(H)}$. Then $k\lis\e_k(\mm)$ can be written as % $$ k\lis\e_k(\mm) \; = \sum_{\lis X \in[-k/2,k/2-1]} E_X(\mm_X) \; , $$ % and, if $k\lis\e_k^N(\mm)=\sum^{(N)}E_X(\mm_X)$, with $\sum^{(N)}$ denoting summation over the sets $X\subseteq[-k/2-N,k/2+N]$, $N\ge 0$, while $\lis X \in [-k/2,k/2-1]$, one has the approximation formula % $$ |k\lis\e^N_k(\mm)-k\lis\e_k(\mm)|\le b_1 e^{-b_2 N} \; , $$ % where $b_1=(e+1)[(e-1)\,(1-\exp(-b_2^{(E)}))]^{-1}b_1^{(E)}$, $b_2=b_2^{(E)}$, and $N$ can be chosen $N=0$. Then % $$ \eqalign{ \n_+ & (\{\mm \; : \;\lis\e^0_k(\mm)\in I_{p,\d-b_1/k}\}) \; \le \; \n_+(\{\mm \; : \;\lis\e_k(\mm)\in I_{p,\d}\}) \cr & \le \; \n_+(\{\mm \; : \;\lis\e^0_k(\mm)\in I_{p,\d+b_1/k}\}) \; . \cr} $$ % From the general theory of one-dimensional Gibbs states, [R1,R3], (see Proposition 2.5 in \S 2), one has that the $\n_+$-probability of a configuration $\mm^0_{[-L/2,L/2]}$, $L\ge k/2$, is % $$ { e^{-{\sum}^* H_X (\mm_X^0) } \; B\left(\mm_{[-L/2,L/2]}\right) \over \Big[ \sum_{\mm_{[-L/2,L/2]}} e^{-{\sum}^* H_X (\mm_X^0) } \Big] } \; , $$ % where $\sum^*$ denotes summation over all the $X\subseteq [-L/2,L/2]$, with $\lis X \in [-k/2,k/2-1]$, and $B(\mm_{[-L/2,L/2]})$ depends on $\mm_{[-L/2,L/2]}$, but verifies the bound $|\ln B(\mm_{[-L/2,L/2]})|\le \ln B_2$, for a suitable $B_2>0$ and uniformly in $L$. As $\psi(\mm)\in(t_0,\a)$, we can define $\tilde B_2 = \max\{\a,t_0^{-1}\}$, so that $|\ln \psi(\mm)| \le \ln \tilde B_2$. Then, for any $L\ge k/2$, one has, for $B_3=B_2\tilde B_2$, % $$ \eqalign{ \n_+ (\{\mm & : \;\lis\e_k(\mm) \in I_{p,\d}\}) \; \le \; \n_+(\{\mm : \;\lis \e^0_k(\mm)\in I_{p,\d+b_1/k}\}) \cr \; \le \; & B_3\,\n_{L,k}(\{\mm : \;\lis\e^0_k(\mm)\in I_{p,\d+b_1/k}\}) \; \le \; B_3\, \n_{L,k}(\{\mm : \;\lis\e_k(\mm)\in I_{p,\d+2b_1/k}\}) \; , \cr} $$ % and likewise a lower bound is obtained by replacing $B_3$ by $B_3^{-1}$ and $b_1$ by $-b_1$. Then, if $I_{p,\d}\subset (-p^*,p^*)$ the set of the rectangles $T_{\mm_{[-L,L]}}\in\TT_L$ with center $x$ such that $\e_k(x)\in I_{p,\d}$ is not empty, and we have obtained the following rewriting of Lemma 4.3. \* \\{\bf 4.4.} {\cs Lemma.} {\it The distributions $\n_+$ and $\n_{L,k}$, $L\ge k/2$, verify the inequalities % $$ {1\over k\s_+}\ln { \n_+(\{\mm : \;\lis\e_k(\mm)\in I_{p,\d\mp 2b_3/k}\}) \over \n_+(\{\mm : \;\lis\e_k(\mm)\in- I_{p,\d\pm 2b_3/k}\}) } \,\cases{ < {1\over k\s_+} \ln B_3^2 + {1\over k\s_+} \ln {\n_{L,k}(\{\mm : \; \lis\e_k(\mm)\in I_{p,\d}\}) \over \n_{L,k}(\{\mm : \;\lis\e_k(\mm)\in- I_{p,\d}\}) } \cr > - {1\over k\s_+} \ln B_3^2 + {1\over k\s_+} \ln {\n_{L,k}(\{\mm : \; \lis\e_k(\mm)\in I_{p,\d}\}) \over \n_{L,k}(\{\mm : \; \lis\e_k(\mm)\in- I_{p,\d}\}) } \cr} $$ % for $I_{p,\d}\subset (-p^*,p^*)$ and for $k$ so large that $p+\d+2b_3/k< p^*$.} \* Hence Lemma 4.3 follows if the following result can be proven. \* \\{\bf 4.5.} {\cs Lemma.} {\it There is a constant $\lis b$ such that the approximate distribution $\m_{L,k}$ verifies the inequalities % $${ 1 \over \s_+k } \ln { \n_{L,k}(\{\mm : \;\lis\e_k(\mm)\in I_{p,\d} \}) \over \n_{L,k}(\{\mm : \;\lis\e_k(\mm)\in-I_{p,\d}\}) } \, \cases{\le p+\d+ \lis b/k\cr \ge p-\d -\lis b/k \cr} $$ % for $k$ large enough (so that $|p|+\d+\lis b/k\min_{\mm_{[-L,L]}} \JJ^{-1}_{u,k}\left(x_{\mm_{[-L,L]}}\right) \JJ^{-1}_{s,k}\left(x_{\mm_{[-L,L]}}\right) \cr} $$ % where the maxima are evaluated as $\mm_{[-L,L]}$ varies with $\e_k(x_{\mm_{-L,L}})\in I_{p,\d}$. We can replace $\JJ^{-1}_{u,k}(x)\,\JJ^{-1}_{s,k}(x)$ with $\JJ_k^{-1}(x)B^{\pm1}_4$, $B_4=B_1^3$, and $B_1$ is defined at the beginning of \S 2. By definition of the set of $\mm_{[-L,L]}$'s in the maximum operation in the last inequalities one has $[\s_+k]^{-1}\ln \JJ^{-1}_k(x_{\mm_{[-L,L]}}) \in I_{p,\d}$: then Lemma 4.5 follows with $\lis b =\s_+^{-1}\ln B_4$. From the chain of implications 4.5 $\to$ 4.4 $\to$ 4.3 $\to$ 3.6, Theorem 3.6 follows and a bound $O(k^{-1})$ is found on the speed at which the limits are approached: in fact the limit in Lemma 4.1 is reached at speed $O(k^{-1})$, and the regularity of $\z(s)$, the size of $\h(k)$ and $\h'(k)$ and the error term in Lemma 4.5 have all order $O(k^{-1})$. \vskip1.truecm \0{\bf Acknowledgements.} I would thank G. Gallavotti for having proposed the argument and for continuous encouragement and suggestions. I'm also indebted to M. Cassandro for discussions about the paper [CO], to F. Bonetto for comments and remarks, and in particurar to D. Ruelle for hospitality at IHES, and for enlightening discussions about the theory of Axiom $A$ systems and critical comments on the manuscript. \vskip1.truecm \centerline{\titolo Appendix A1. Proof of Proposition 2.2} \*\numapp=1\numfor=1 \\The proof of the statements in Proposition 2.2 can be adapted from [G3]. In fact we can study the map $S=\HH_T\!\!: \G'(\TT)\to\G'(\TT)$ as it was an Anosov diffeomorphism. For any $x\in\G(\TT)$, $W_x^s(T)$ and $W_x^u(T)$ are the stable and the unstable manifolds of $x$, if $T$ is the rectangle in $\G(\TT)$ containing $x$; the angle $\a(x)$ between them is bounded by two constants $K_0^{-1}0$, let $p_0$ be such that $\b p-\z(p) > \l(\b)-c_1\d$ for any $p\in[p_0-\d,p_0+\d]$, for a suitable constant $c_1$. Then for $k$ large enough (so that $D_10$, one can write, for $k$ large enough (so that $D_10$, then there exists a domain $\o$ of the complex plain centered in the origin, such that, for $\F\in\BB$, the limit % $$ q(\b) = \lim_{k\to\io} {1\over k} \ln \int \n_{k/2,k}(d\mm)\, e^{\b U^{(\F)}(\mm) } $$ % exists and is analytic in $\b\in\o$.} \* Once the existence of the limit $q(\b)$ is proven, the convexity can be proven with standard methods, [R1], and the linearity in $\b$ for $\b\to\io$ follows from the definition of $p^*$ in \S 4. %is an easy application of the {\sl law of large numbers}, ([Fe], Ch. X, \S 2). \* \\{\bf A3.2.} {\it Decimation and first cluster expansion.} Let $\L_p$ be the interval centered at the origin of lenght $|\L_p|=2pM+(2p+1)L$, and decompose $\L_p$ into consecutive blocks $A_{-p}$,$B_{-p}$,$A_{-p+1}$,$\ldots$,$A_{p-1}$,$B_{p-1}$,$A_{p}$, such that $|B_i|=M$ $\forall i=-p,\ldots,p-1$, and $|A_i|=L$ $\forall i=-p,\ldots,p$, with $L\le M$. Set $\ZZ=\lim_{p\to\io} \L_p$, and define $\G^A_p = \{A_i\}_{i=-p}^{p}$ and $\G^B_p= \{B_i\}_{i=-p}^{p-1}$. Consider the function % $$ Z_p(H+\b E) = \sum_{\mm_{\L_p}}e^{-U^{(H+\b E)}(\mm_{\L_p})} \; , $$ % where the potentials $H=\{H_X(\mm_X)\}_{X\subset \zz}$ and $E=\{E_X(\mm_X)\}_{X\subset \zz}$ are in $\BB$. Note that $U^{(H+\b E)}=U^{(H)}+U^{(\b E)}$, so that the real analyticity in $\b$ of % $$ \lim_{p\to\io}|\L_p|^{-1} Z_p(H+\b E) $$ % yields Lemma A3.1 (the sign of $\b$ being irrelevant). If one chooses $H$ and $B$ as defined in \S 4, after Lemma 4.3, then from Lemma A3.1 Proposition 4.2 follows, for $A_{ij}\=1$ ($A$ is the matrix introduced in \S 1.8). The extension to the general case is trivial. \* Call $\aa_i=\mm_{A_i}$ and $\bb_i=\mm_{B_i}$ the configurations in the blocks $A_i$ and $B_i$, and $\aa_S$ [$\bb_S$] the configuration in $S$, if $S$ is the union of sets in $\G^A_p$ [$\G^B_p$]. We have % $$ \eqalign{ U^{(H+\b E)}(\mm_{\L_p}) & = \sum_{i=-p}^{p} \a^{(H+\b E)}(\aa_i) + \sum_{i=-p}^{p-1} J^{(H)}_i(\aa_i, \bb_i, \aa_{i+1}) \cr & + \sum_{D \in \G^D_p} W^{(H+\b E)}_D(\aa_D) + \sum_{i=-p}^{p-1} J^{(\b E)}_i(\aa_i, \bb_i,\aa_{i+1}) + \sum_{C \in \bar\G_p} W^{(H+\b H)}_C (\mm_C) \; , \cr} $$ % where the sets $\G^D_p$ and $\bar \G_p$ are defined as follows: % $$ \G^D_p = \{ D = A_{i_1} \cup \ldots \cup A_{i_k} \; : \; 2 \le k \le p \, , \hbox{ and } D \neq A_i\cup A_{i+1} \, , \quad \forall i\in\ZZ \} \; , $$ % $$ \eqalign{ \bar\G_p = \{ & C = A_{i_1} \cup \ldots \cup A_{i_k} \cup B_{i_1'} \cup \ldots \cup B_{i_{k'}'} \; : \; 0\le k \le p \, , \; 1 \le k' \le p+1 \, , \cr & \hbox{and } C\neq A_i\cup B_i \cup A_{i+1} \, , \; C\neq A_i B_i \, ,\; C\neq B_i A_{i+1} \, , \; C \neq B_i \, , \forall i\in\ZZ \} \; , \cr} $$ % and % $$ \eqalign{ \a^{(\F)}(\aa_i) & = \sum_{X\subset A_i} \F_X(\aa_X) \; , \cr % J^{(\F)}_i(\aa_i, \bb_i, \aa_{i+1}) & = \sum_{X \subset A_i\cup B_i \cup A_{i+1} \atop X \cap B_i \neq \emptyset} \F_X(\mm_X) + \sum_{X \subset A_i \cup A_{i+1} \atop X \cap A_i \neq \emptyset \, , \; X \cap A_{i+1} \neq \emptyset} \F_X(\aa_X) \; , \cr % W^{(\F)}_D (\aa_D) & = \sum_{X\subset D \atop X \cap A_{i_h} \neq \emptyset \; \forall A_{i_h} \subset D} \F_X(\aa_D) \; , \cr % W^{(\F)}_C (\mm_C) & = \sum_{X \subset C \atop X \cap A_{i_h} \neq \emptyset \; \forall A_{i_h} \subset C \, , \; X \cap B_{i_h'} \neq \emptyset \; \forall B_{i_h'}\subset C} \F_X(\mm_C) \; . \cr} $$ % Note that $\G_p^D=\emptyset$ if only connected subsets $X$ are allowed for interaction $\F$, as it is the case when $H$ and $E$ are given as in \S 4. If we define % $$ Z_{B_i}^{(\F)}(\aa_i,\aa_{i+1}) = \sum_{\bb_i} e^{J_i^{(\F)}(\aa_i,\bb_i,\aa_{i+1})} \; , \Eqa(A3.1)$$ % and % $$ \eqalign{ \exp[-\tilde U^{(H+\b E)}(\aa_{\G^A_p})] & = \sum_{\bb_{\G^B_p}} \Big[ \prod_{i=-p}^{p-1} { e^{J^{(H)}_i(\aa_i, \bb_i, \aa_{i+1})} \over \sum_{\bb_i} e^{ J^{(H)}_i(\aa_i, \bb_i, \aa_{i+1}) } } \Big] \cdot \cr & \cdot \Big[ \prod_{C\in \bar\G_p} e^{W^{(H+\b E)}_C (\mm_C) }\Big] \cdot \Big[ \prod_{i=-p}^{p-1} e^{J^{(\b E)}_i (\aa_i, \bb_i, \aa_{i+1})} \Big] \; , \cr} $$ % then we can average over the variables associated to $\G^B_p$ ({\sl decimation procedure}, see [KH1, KH2]) % $$ \eqalign{ \sum_{\mm_{\L_p}} U^{(H+\b E)}(\mm_{\L_p}) & = \sum_{\aa_{\G^A_p}} \Big\{ \sum_{i=-p}^{p} \a^{(H+\b E)}(\aa_i) + \sum_{i=-p}^{p-1} \ln Z_{B_i}^{(\F)}(\aa_i,\aa_{i+1}) \cr & + \sum_{D \in \G^D_p} W^{(H+\b E)}_D(\aa_D) + \tilde U^{(H+\b E)}(\aa_{\G^A_p}) \Big\} \; . \cr} $$ % To each $C \in \bar\G_p$ we associate a bond $\p(C)$, and to each $B\in\G^B_p$ a bond $\p(B)$, such that the lenght of a bond $\p(B)$ is $|\p(B)|=1$ and the lenght of a bond $\p(C)$, denoted as $|\p(C)|$, is given by the number of blocks $A$'s and $B$'s contained in $C$. Consider % $$ \RRR = \{ C_1 , \ldots , C_k , B_1 , \ldots , B_h \Big\} \; , \qquad C_{s} \subset \bar\G_p \; \forall s=1,\ldots,k \; , $$ % and set $\tilde C = C \cap \G^B_p$ and $\tilde \RRR = \{ \tilde C_1 , \ldots , \tilde C_k , B_1 , \ldots , B_h \Big\}$: $|\p(\tilde C)|$ is given by the number of sets $B$'s contained in $C$. We set $\tilde B=B$. The set of bonds corresponding to $\RRR$, \ie % $$ R = \Big\{ \p(C_1), \ldots, \p(C_k), \p(B_1) , \ldots , \p(B_h) \Big\} \; , \qquad C_{s} \subset \bar\G \; \forall s=1,\ldots,k \; , \Eqa(A3.2) $$ % is a {\sl polymer} (see [GMM]) if, for any choise of bonds $\p(X_l)$ and $\p(X_j)$, with $X_l, X_j\in\RRR$, there exist $X_{i_1}, \ldots,X_{i_r}\in\RRR$, $r\le k+h$, such that $X_{i_1}=X_l$, $X_{i_r}=X_j$ and $\tilde X_{i_h} \cap \tilde X_{i_h+1} \neq \emptyset$ $\forall h=1,\ldots,r-1$. Then one has % $$ \exp[-\tilde U^{(H+\b E)}(\aa_{\G^A_p})] = 1 + \sum_{n=1}^{\io} \sum_{R_1,\ldots,R_n \atop \tilde R_i \cap \tilde R_j = \emptyset} \prod_{i=1}^n \z (R_i) \; , $$ % where $\tilde R$ is defined as $R$ in \equ(A3.2), but with $C_s$ replaced with $\tilde C_s$ $\forall s=1,\ldots,k$, and % $$ \eqalign{ \z(R) & = \sum_{\bb_{\tilde \RRR}} \Big[ \prod_{B_i \subset \tilde \RRR} {e^{J^{(H)}_i(\aa_i, \bb_i, \aa_{i+1})} \over \sum_{\bb_i} e^{ J^{(H)}_i(\aa_i, \bb_i, \aa_{i+1}) } } \Big] \cdot \cr & \cdot \Big[ \prod_{s=1}^{k} \Big( e^{W^{(H+\b E)}_{C_s} (\mm_{C_s}) } - 1 \Big) \Big] \cdot \Big[ \prod_{s'=1}^{h} \Big( e^{J^{\b E}_{i_{s'}} (\aa_{i_{s'}}, \bb_{i_{s'}}, \aa_{i_{s'}+1})} - 1 \Big) \Big] \; , \cr} $$ % is the {\sl activity} of the polymer $R$. Here $B_i\subset \tilde\RRR$ means $B_i\in\RRR$ or $B_i\subset \tilde C_j\in\tilde\RRR$ for some $j=1,\ldots,k$. We set $|\tilde R|$ $=$ $\sum_{s=1}^k|\p(\tilde C)|$ $+h$ (as $|\p(B)|=1$). \* \\{\bf A3.3.} {\cs Lemma.} {\it Given a potential $\Phi\in\BB$, and considered a polymer $R$, the activity $\z(R)$ satisfies the inequality % $$ |\z(R)| \le \r^{|\tilde R|} \prod_{s=1}^k w_{C_s} \prod_{s'=1}^h j_{B_{s'}} \; , $$ % where $\r=e^{-r}$, $w_C = 2\,e^{r|\p(\tilde C)|} \| W_C^{(H+\b E)}\|_{\io}$ and $j_{B_i}=2\,e^r$ $\|J_i^{(\b E)}\|_{\io}$, being $\|f_X\|_{\io}$ the supremum norm for the continuous function $f_X$, and $w_C$ and $j_B$ are positive constants such that $\max\{w_C,j_B\} \le \ln [\sqrt{\r}(2-\sqrt{\r})]^{-1}$ for $\b$ small enough and $L$ sufficiently large.} \* \\{\bf A3.4.} {\it Proof of Lemma A3.3.} For complex $z$ such that $|z|<1/2$, one has $|e^z-1|\le 2|z|$. Since % $$ \lim_{L\to\io} e^{r'L} \, \| W_C^{(H+\b E)} \|_{\io} = 0 \; , \qquad \forall r'0 \; , $$ % we can apply the above inequality, and obtain % $$ |\z(R)| \le \sum_{\bb_{\tilde R}} \Big[ \prod_{B_i\subset \tilde R} {e^{J^{(H)}_i(\aa_i, \bb_i, \aa_{i+1})} \over \sum_{\bb_i} e^{J^{(H)}_i(\aa_i, \bb_i, \aa_{i+1})} } \Big] \cdot \Big( \prod_{s=1}^k 2 \| W_C^{(H+\b E)} \| \Big) \cdot \Big( \prod_{s'=1}^h 2 \|J_{j_{s'}}^{(\b E)}\| \Big) \; ; $$ % then we can \acapo \\(1) extract a factor $e^{-r|\p(\tilde C)|}$ from $\| W_C^{(H+\b E)} \|_{\io}$ and a factor $e^{-r}$ from $\|J_{j_{s'}}^{(\b E)}\|_{\io}$, because of the exponential decay of the interaction, and\\ \acapo \\(2) make $w_C$ and $j_B$ arbitrarily small by taking $L$ sufficiently large, and % $$ \eqalign{ & L > {1\over r} \ln \Big[ 4 (1-\r)^{-1} \|H+\b E\|_{\BB} \max\{1, \ln [\sqrt{\r} (2-\sqrt{\r})] \} \Big] \; , \cr % & |\b| < {\r \over 4M\|E\|_{\BB} } \min\{ 1,\ln[\sqrt{\r} (2-\sqrt{\r})]^{-1}\} \; , \cr} $$ % so obtaining Lemma A3.3. \qed \* \\{\bf A3.5.} {\cs Remark.} Note that, if we had considered an interaction with exponential decay $e^{-r|X|}$ instead of $e^{-r\,{\rm diam}(X)}$, the same bound as in Lemma A2.3 would have followed. In fact the same cluster expansion can be still performed, and the only difference is that now $W_C^{(H+\b E)}$ decays as $e^{-r|\tilde C|}$: but this is sufficient in order to prove Lemma A3.3. \* \\{\bf A3.6.} {\it Second cluster expansion.} By Lemma A3.3, for suitable $\b$ and $L$, one has % $$ |\z(R)| \le \r^{|\tilde R|} \prod_{C\in\RRR} \CC_C \; , \qquad 0 < \CC_C \le K \; , \qquad K \le \ln[\sqrt{\r} (2-\sqrt{\r})]^{-1} \; ; $$ % then the conditions of [CO], Lemma 1, are satisfied, so that we can deduce (analogously to [CO], Lemma 2) % $$ \exp[ -\tilde U^{(H+\b E)}(\aa_{\G^A_p})] = \sum_{D\in \tilde\G^D_p} \tilde W^{(H+\b E)}_D(\aa_D) \; , $$ % where $\tilde\G^D_p$ is defined as $\G^D_p$, but with no restriction, and $\tilde W^{(H+\b E)}_D(\aa_D)$ is analytic in $\b\in\o_M$, if $\o_M$ is a circle around the origin of the complex plane whose radius tend to zero as $M\to\io$. One can write % $$ \tilde W^{(H+\b E)}_D(\aa_D) = \sum_{R_1,\ldots,R_n \atop \cup_{i=1}^n \RRR_i\setminus \tilde \RRR_i \subseteq D } \varphi_T(R_1,\ldots,R_n) \prod_{i=1}^n \z(R_i) \; , $$ % where the sum is over all the polymers $R_1,\ldots,R_n$ such that the product $\z(R_1)$ $\ldots$ $\z(R_n)$ depends only on the variables $\aa_D$, and $\varphi_T$ $(R_1,$ $\ldots,$ $R_n)$ is a suitable coefficient, (see [CO], Lemma 1; see also [GMM]). Then we can define (recall \equ(A3.1) and define ${\bf 1}$ as the configuration of a block $A$ with each element set equal to $1$) % $$ \bar\a^{(H)}(\aa_i) = \a^{(H)}(\aa_i) + \ln \Big[ { Z_{B_i}^{(H)}(\aa_i,{\bf 1}) \cdot Z_{B_{i-1}}^{(H)}({\bf 1},\aa_i) \over Z_{B_i}^{(H)}({\bf 1},{\bf 1}) \cdot Z_{B_{i-1}}^{(H)}({\bf 1},{\bf 1}) } \Big] \; , $$ % and % $$ \VV_D (\aa_D) = \cases{ \a_i^{(\b E)}(\aa_i) \; , & if $D=A_i \;$, \cr % \tilde W_{A_i\cup A_{i+1}}^{(H+\b E)}(\aa_i,\aa_{i+1}) + \ln \Big[ { Z_{B_i}^{(H)}(\aa_i,\aa_{i+1}) \cdot Z_{B_{i}}^{(H)}({\bf 1},{\bf 1}) \over Z_{B_i}^{(H)}(\aa_i,{\bf 1}) \cdot Z_{B_{i}}^{(H)}({\bf 1},\aa_{i+1}) } \Big] \; , & if $D=A_i\cup A_{i+1} \;$, \cr % W_D^{(H+\b E)}(\aa_D) + \tilde W^{(H+\b E)}_D(\aa_D) \; , & if $D \neq A_i, A_i\cup A_{i+1} \;$, \cr} $$ % and write % $$ \eqalign{ \sum_{\mm_{\L_p}} \exp[ -U^{(H+\b E)}(\mm_{\L_p}) ] & = \hbox{const.} \Big[ \prod_{i=-p}^{p} \sum_{\aa_i} e^{ \bar \a^{(H)}(\aa_i) } \Big] \cdot \cr & \cdot \sum_{\aa_{\G^A_p}} \Big[ \prod_{i=-p}^{p} { e^{ \bar \a^{(H)}(\aa_i) } \over \sum_{\aa_i} e^{ \bar \a^{(H)}(\aa_i) } } \Big] \cdot \Big[ \prod_{D\in\tilde\G^D_p} e^{\VV_D(\aa_D)} \Big] \; . \cr} $$ % We introduce a new cluster expansion by associating to each $D \in \tilde\G^D_p$ a bond $\p(D)$, and defining a polymer $S$ as % $$ S = \{ \p(D_1) , \ldots , \p(D_k) \} \; , \Eqa(A3.3) $$ % and $\SSS = \{D_1,\ldots,D_k\}$. Then % $$ \sum_{\mm_{\L_p}} \exp[-U^{(H+\b E)}(\mm_{\L_p}) ] = \Big[ \prod_{i=-p}^{p} \sum_{\aa_i} e^{ \bar \a^{(H)}(\aa_i) } \Big] \cdot \Big( 1 + \sum_{n=1}^{\io} \sum_{S_1,\ldots,S_n \atop S_i \cap S_j = \emptyset} \prod_{i=1}^n \Theta(S_i) \Big) \; , $$ % where % $$ \Theta(S) = \sum_{\aa_\SSS} \prod_{A_i \subset \SSS} { e^{ \bar \a^{(H)}(\aa_i) } \over \sum_{\aa_i} e^{ \bar \a^{(H)}(\aa_i) } } \prod_{D\in \SSS} \Big( e^{\VV_D(\aa_D)} - 1 \Big) $$ % is the activity of the polymer $S$. Here $A_i\subset\SSS$ means $A_i\subset D_j\in\SSS$ for some $j=1,\ldots,k$. \* \\{\bf A3.7.} {\cs Lemma.} {\it If $Z_{B_{i}}^{(H)}(\aa_i,\aa_{i+1})$ is defined as $Z_{B_{i}}^{(H)}(\aa_i,\aa_{i+1})=\sum_{\bb_{i}}\exp J_i^{(H)} (\aa_i,\bb_i,\aa_{i+1})$, then % $$ \lim_{M\to\io} \sup_{\aa_i,\aa_{i+1}} \Big\{ \ln \Big[ { Z_{B_i}^{(H)}(\aa_i,\aa_{i+1}) \cdot Z_{B_i}^{(H)}({\bf 1},{\bf 1}) \over Z_{B_i}^{(H)}(\aa_i,{\bf 1}) \cdot Z_{B_i}^{(H)}({\bf 1},\aa_{i+1}) } \Big] \Big\} = 0 \; , $$ % $\forall L<\io$.} \* \\{\bf A3.8.} {\it Proof of Lemma A3.7.} Let $\ZZ^+=\{i\in\ZZ\;:\;i\ge 0\}$ and $K_+=\{1,\ldots,\NN\}^{\zz^+}$. We denote by $C(K_+)$ the Banach space of the real continuous functions on $K_+$, and by $M^*(K_+)$ its dual, \ie the space of real measures on $K_+$. Given a configuration $\mm_N\in \{1,\ldots,\NN\}^{N}$, $N\ge1$, and a configuration $\mm_+\in K_+$, one can define the configuration $(\mm_N,\mm_+)\in K_+$ as % $$ (\mm_N,\mm_+)_i = \cases{ (\mm_N)_i \; , & for $i=0,\ldots,N \;$, \cr (\mm_+)_{i-N} \; , & for $i>N$. \cr} $$ Given $\F\in\BB$, an operator $\LL_{\F}\!\!:C(K_+)\to C(K_+)$ is defined by % $$ \LL_{\F}f\,(\mm_+) = \sum_{m_0=1}^{\NN} \exp\Big[ \sum_{X \subset \zz^+ \atop X \ni 0} \F_X(\mm_{X}) \Big] f(m_0,\mm_+) \; ; $$ % then there exist $\l_{\F}>0$, $h_{\F}\in C(K_+)$ and $\n_{\F}\in M^*(K_+)$ such that \acapo (a) $\LL_{\F}h_{\F}=\l_{\F}h_{\F}$, and \acapo (b) if $f\in C(K_+)$, $\lim_{k\to\io} \| \l_{\F}^{-k}\LL^k_{\F}f-\n_{\F}(f)\,h_{\F} \|_{\io} = 0$, uniformly for $\F$ in a bounded subset of a finite dimensional subspace of $\BB$. The proof of such a statement follow from [R1] and can be found in [G1], Ch. 18, Proposition XXXV and exercises, (it is essentially an adaptation from [R1], see also [GL]). Define the function in $C(K_+)$ % $$ f_{\aa_i}(\mm_+)=\exp\Big[ \sum_{X\subset A_i\cup A_{i+1}} \F_X(\mm_X) \Big] \; , $$ % where $\aa_i\in\{1,\ldots,\NN\}^L$. Note that $f_{\aa_i}(\mm_+)$ depends only on the first $L$ symbols of $\mm_+$, (so that the successive ones can be set equal to an arbitrary value, say 1). Then one has % $$ { Z_{B_i}^{(H)}(\aa_i,\aa_{i+1}) \cdot Z_{B_i}^{(H)}({\bf 1},{\bf 1}) \over Z_{B_i}^{(H)}(\aa_i,{\bf 1}) \cdot Z_{B_i}^{(H)}({\bf 1},\aa_{i+1}) } = { \LL_{\F}^M f_{\aa_i}(\aa_{i+1},{\bf 1}) \LL_{\F}^M f_{\bf 1}({\bf 1}) \over \LL_{\F}^M f_{\aa_i}({\bf 1}) \LL_{\F}^M f_{\bf 1}(\aa_{i+1},{\bf 1}) } \; , $$ % where ${\bf 1}$ appearing in $(\aa_{i+1},{\bf 1})$ is an element in $K_+$, while the subscript ${\bf 1}$ in $f_{\bf 1}$ is an element in $\{1,\ldots,\NN\}^L$ (\ie $\aa_i={\bf 1}$). Then from the property (b) above, Lemma A3.7 follows. \qed \* \\{\bf A3.9.} {\it Proof of Lemma A3.1.} We consider the cluster expansion envisaged in \S A3.5. From the interaction $\{\VV_D\}_{D\in\tilde\G^D_p}$, terms of the following form arise: % \acapo \\(a) $\sum_{R_1,\ldots,R_n}\varphi_T(R_1,\ldots,R_n) \, \z(R_1)\ldots\z(R_n)$, where the dependence on a configuration $\aa_i$ is only through the factors % $$ U_i={e^{J_i^{(H)}(\aa_i,\bb_i,\aa_{i+1})} \over \sum_{\bb_i} e^{J_i^{(H)}(\aa_i,\bb_i,\aa_{i+1})} } $$ % appearing in $\z(R)$; \acapo \\(b) for $D=A_i$, $\a_i^{(\b E)}(\aa_i)$; \acapo \\(c) $\sum_{R_1,\ldots,R_n}\varphi_T(R_1,\ldots,R_n) \, \z(R_1)\ldots\z(R_n)$, where the dependence on the configurations $\aa_i$ is (also) through terms $W_C^{(H+\b E)}(\mm_C)$ and $J_i^{(\b E)}(\aa_i,\bb_i,\aa_{i+1})$ in $\z(R)$; \acapo \\(d) for $D=A_i\cup A_{i+1}$, also % $$ \ln \Big[ { Z_{B_{i}}^{(H)}(\aa_i,\aa_{i+1}) \cdot Z_{B_{i}}^{(H)}({\bf 1},{\bf 1}) \over Z_{B_{i}}^{(H)}(\aa_i,{\bf 1}) \cdot Z_{B_{i}}^{(H)}({\bf 1},\aa_{i+1}) } \Big] \; . $$ % As a polymer $R$ contains at most $2|\tilde R|$ $A$-blocks through the factors $U_i$, we can extract also a factor $\r^{1/4}$ from each $A$-block appearing in terms of the form (a), by simply replacing $\z(R)$ with a new activity $\hat \z(R)=\r^{-|\tilde R|/2}\r(R)$. In (b), we can write $\a_i^{(\b E)}(\aa_i)=\r^{1/2}[\r^{-1/2} \a_i^{(\b E)}(\aa_i)]$, where $\r^{-1/2}\|\a_i^{(\b E)}\|_{\io}$ can be made arbitrarily small by taking $\b$ small enough. As far as the terms in (c) are concerned, we can extract a factor $e^{-r}=\r$ from each $A$-block, thanks to the exponential decay of the interaction, and the remaining factor can be made arbitrarily small by taking $L$ large and $\b$ small, (see the proof of Lemma A3.4 and the definition of the set $\G_p^D$ in \S A3.2). By Lemma A3.7, we can extract a factor $\r$ from each term in (d) by taking $M$ sufficiently large. \* Therefore we have a factor $\r^{1/4}$ $\forall A$ arising in (a) and a factor $\r^{1/2}$ $\forall A$ arising in (b), (c) and (d). Then we can bound % $$ |\Theta(S)| \le \tilde\r^{|S|}\prod_{D\in \SSS} \CC_D \; , $$ % where $\tilde\r =\r^{1/4}$, and % $$ \CC_D = \cases{ 2\r^{-1/2} \|\a^{(\b E)} \|_{\io} \; , & if $D=A_i \; $, \cr 2\r^{-1} \|\VV_D\|_{\io} \; , & if $D=A_i\cup A_{i+1} \; $, \cr 2 \big[ \|W_D\| + \|\hat W_D\|_{\io} \big] \; , & if $D\neq A_i\,,\,A_i\cup A_{i+1} \;$ , \cr} $$ % being $\hat W_D$ defined as $\tilde W_D$, but with $\hat \z(R)$ replacing $\z(R)$. If $\b\to 0$ and $M\to \io$, then % $$ \sum_X |\VV_X(\aa_X)|\,e^{|X|r'} $$ % can me made arbitrarily small, for some $r'0$, we have \acapo \\(1) for any $L$, $\exists M_1(L)$ such that for $\forall M\ge M_1(L)$ % $$ {2\over \r} \sup_{\aa_i,\aa_{i+1}} \Big\{ \ln \Big[ { Z_{B_{i}}^{(H)}(\aa_i,\aa_{i+1}) \cdot Z_{B_{i}}^{(H)}({\bf 1},{\bf 1}) \over Z_{B_{i}}^{(H)}(\aa_i,{\bf 1}) \cdot Z_{B_{i}}^{(H)}({\bf 1},\aa_{i+1}) } \Big] \Big\} \le {\bar\k \over 3} \; ; $$ % \acapo \\(2) $\exists L_0$ such that $\forall L\ge L_0$, $\forall M$ and $\forall \b\in\o_M$ (being $\o_M$ defined in \S A3.6) % $$ \sup_{A\in \G^A_p} \sum_{D\supset A} 2 \| \hat W_D(\aa_D) \|_{\io} \le {\bar\k \over 3} \; ; $$ \acapo % \\(3) for $L=L_0$, $\exists M_2(L_0)$ such that $\forall \b \in \o_{M_2(L_0)}$ and $\forall M\ge M_2(L_0)$ % $$ \sup_{i} {2\over\sqrt{\r}} \|\a_i^{(\b E)} \|_{\io} + \sup_{A\in \G^A_p} \sum_{D\supset A} 2 \| W_D(\aa_D) \|_{\io} \le {\bar\k \over 3} \; . $$ % Therefore, if $\b\in \o_{M_0}$, with $M_0=\max\{M_1(L_0),M_2(L_0)\}$, there exist $\k\=\k(\b,L_0,M_0)\le \bar\k$ such that $\CC_D\le\k$. If $\bar\k$ is so chosen that $\k$ $\le$ $\ln[\sqrt{\tilde\r}$ $(2-\sqrt{\tilde\r})]^{-1}$, one can apply again [CO], Lemma 1: then there is a constant $G(\tilde\r,\k)$ such that % $$ \lim_{p\to\io}{1\over|\L_p|} \ln Z_p(H+\b E) \le G(\tilde\r,\k) + {1\over L} \ln \Big| \sum_{\aa_i} e^{\bar\a^{(H)}(\aa_i)} \Big| \; . $$ % The uniformity of the bound and the existence of the second limit uniformly in $L$ (for real $H$, the proof of such a result is standard, [R2]) allow us to apply Vitali's convergence theorem, [T], \S 5.21, and complete the proof of Lemma A3.1. \qed \vskip1.truecm %\pagina \centerline{\titolo References} \* \halign{\hbox to 1.2truecm {[#]\hss} & \vtop{\advance\hsize by -1.25 truecm \\#}\cr A& {D.V. Anosov: {\sl Geodesic flows on closed Riemann manifolds with negative curvature}, Proc. Steklov Inst. Math. {\bf 90} (1967), American Mathematical Journal, Providence, Rhode Island, 1969. }\cr % %AS& {D.V. Anosov, Ya.G. Sinai: %Some smooth ergodic systems, {\it Russ. Math. Surveys} {\bf 22}, %103--167 (1967). }\cr % Bb& {N. Bourbaki: {\sl El\'ements de Math\'ematique, Topologie G\'en\'erale}, Hermann, Paris, 1963. }\cr % B1& {R. Bowen: One-dimensional hyperbolic sets for flows, {\it J. Differential Equations} {\bf 12}, 173--179 (1972). }\cr % B2& {R. Bowen: Symbolic dynamics for hyperbolic flows. {\it Amer. J. Math.} {\bf 95}, 429--459 (1973). }\cr % B3& {R. Bowen: {\sl Equilibrium states and the ergodic theory of Anosov diffeomorphisms}, Lectures Notes on Math. {\bf 470}, Springer, Berlin, 1975. }\cr % B4& {R. Bowen: {\sl On Axiom $A$ diffeomorphisms}, Regional Conference Series in Mathematics {\bf 35}, American Mathematical Society, Providence, Rhode Island, 1978. }\cr % BR& {R. Bowen, D. Ruelle: The ergodic theory of Axiom $A$ flows. {\it Inventiones Math.} {\bf 29}, 181--202 (1975). }\cr % BW& {R. Bowen, P. Walters: Expansive one-parameter flows, {\it J. Differential Equations} {\bf 12}, 180--193 (1972). }\cr % Ch& {N.I. Chernov: Markov approximations and decay of correlations for Anosov flows, Preprint (1995). }\cr % CG1& {E.G.D. Cohen, G. Gallavotti: Dynamical ensembles in stationary states, {\it J. Stat. Phys.} {\bf 80}, 931--970 (1995). }\cr % CG2& {E.G.D. Cohen, G. Gallavotti: Chaoticity hypothesis and Onsager re\-ci\-pro\-ci\-ty, Pre\-print (1995). }\cr % CO& {M. Cassandro, E. Olivieri: Renormalization group and analyticity in one dimension. A proof of Dobrushin's theorem, {\it Comm. Math. Phys.} {\bf 80}, 255--269 (1981). }\cr % ER& {J.-P. Eckmann, D. Ruelle: Ergodic theory of chaos and strange attractors, {\it Rev. Mod. Phys} {\bf 57}, 617--656 (1985). }\cr % %Fe& {W. Feller: %{\sl An introduction to probability theory and its applications}, %Vol. I, John Wiley and Sons, New York, 1950, third edition, 1968. }\cr % FW& {J. Franks, R. Williams: Anomalous Anosov flows, in {\sl Global theory of dynamical systems}, Proceedings, Evanston (1979), Lecture notes in Math. {\bf 819}, 158--174, Ed. Z. Nitecki \& C. Robinson, Springer, Berlin, 1980. }\cr % G1& {G. Gallavotti: {\sl Aspetti della teoria ergodica, qualitativa e sta\-ti\-sti\-ca del mo\-to}, Pi\-ta\-go\-ra, Bo\-lo\-gna, 1981. }\cr % G2& {G. Gallavotti: Topics on chaotic dynamics, in {\sl Third Granada Lectures in Computational Physics. Proceedings, Granada, Spain, 1994}, Lectures Notes on Phys. {\bf 448}, Ed. P.L. Garrido \& J. Marro, Springer, 1995. }\cr % G3& {G. Gallavotti: Reversible Anosov diffeomorphisms and large deviations, {\it Math. Phys. Electronic J.} {\bf 1}, 1--12 (1995). }\cr % GL& {G. Gallavotti, T.F. Lin: One dimensional lattice gases with rapidly decreasing interactions, {\it Arch. Rational Mech. Anal.} {\bf 37}, 181-191 (1970). }\cr % GMM& {G. Gallavotti, A. Martin-L\"of, S. Miracle-Sole: Some problems connected with the description of coexisting phases at low temperature in the Ising model, in {\sl Statistical mechanics and mathematical problems}, Battelle Seattle Rencontres (1971) Lectures Notes on Phys. {\bf 20}, 162--203, Ed. A. Lenard, Springer, Heidelberg, 1973. }\cr % KH1& {L.P. Kadanoff, A. Houghton: Numerical evaluations of the critical properties of the two--dimensional Ising model, {\it Phys. Rev} {\bf B11}, 377--386 (1975). }\cr % KH2& {L.P. Kadanoff, A. Houghton: %M.K. Grover in {\sl Renormalization group in critical phenomena and quantum field theory}, Proceedings of the Temple University Conference on critical phenomena, 1973, Ed. J.D. Gunton \& M.S. Green, Dept. of Physics, Temple University, Phyladelphia, 1973. }\cr % %KHY& {L.P. Kadanoff, A. Houghton, M.C. Yalabik: %{\it J. Stat. Phys.} {\bf 14}, 171-- (1976). }\cr % HP& {M. Hirsch, C.C. Plug: Stable manifolds and hyperbolic sets, in {\sl Global Analysis}, {\it Proc. Symp. in Pure Math.} {\bf 14}, 133--163 (1970). }\cr % Is& {R.B. Israel: High temperature analyticity in classical lattice systems, {\it Comm. Math. Phys.} {\bf 50}, 245--257 (1976). }\cr % La& {O. Lanford: Entropy and equilibrium states in classical statistical mechanics, in {\sl Statistical mechanics and mathematical problems}, Battelle Seattle Rencontres (1971) Lectures Notes on Phys. {\bf 20}, 1--113, Ed. A. Lenard, Springer, Heidelberg, 1973. }\cr % %Ma& {B. Marcus: %Unique ergodicity of some flows related to Axiom $A$ diffeomorphisms, %{\it Israel J. Math.} {\bf 21}, 111--132 (1975). }\cr % %Ma& {J. Mather: %Characterization of Anosov diffeomorphisms, %{\it Indag. Math.} {\bf 30}, 479--483 (1968). }\cr % Pl& {J. F. Plante: Anosov flows, {\it Amer. J. Math.} {\bf 94}, 729--754 (1972). }\cr % PS& {C.C. Plug, M. Shub: The $\O$-stability theorem for flows, {\it Inventiones Math.} {\bf 11}, 150--158 (1970). }\cr % Po& {M. Pollicott: On the rate of mixing of Axiom $A$ flows, {\it Invent. Math.} {\bf 81}, 413--426 (1985). }\cr % R1& {D. Ruelle: Statistical mechanics of a one dimensional lattice gas, {\it Comm. in Math. Phys.} {\bf 9}, 267--278 (1968). }\cr % R2& {D. Ruelle: {\sl Statistical Mechanics}, Benjamin, New York, 1969. }\cr % R3& {D. Ruelle: {\sl Thermodynamics Formalism}, Encyclopedia of Mathematics Vol. 5, Addison--Wesley, Reading, 1978. }\cr % R4& {D. Ruelle: Ergodic theory of differentiable dynamical systems, {\it Pu\-bli\-ca\-tions Ma\-th\'e\-ma\-ti\-ques de l'IHES} {\bf 50}, 275--306 (1979). }\cr % R5& {D. Ruelle: Measures describing a turbulent flow, {\it Ann. N.Y. Acad. Sc.} {\bf 357}, 1--9 (1980). }\cr % R6& {D. Ruelle: Flots qui ne m\'elangent pas exponentiellement, {\it C. R. Acad. Sc. Paris} {296}, 191--193 (1983). }\cr % Si1& {Ya. G. Sinai: Markov partitions and $C$-diffeomorphisms, {\it Func. Anal. and its Appl.} {\bf 2}, 61--82 (1968). }\cr % Si2& {Ya. G. Sinai: Construction of Markov partitions, {\it Func. Anal. and its Appl.} {\bf 2}, 245--253 (1968). }\cr % Si3& {Ya. G. Sinai: Gibbs measures in ergodic theory, {\it Russ. Math. Surveys} {\bf 27}, No. 4, 21--64 (1972). }\cr % Sm& {S. Smale: Differentiable dynamical systems, {\it Bull. A. M. S.} {\bf 73}, 747--817 (1967). }\cr % T& {E.C. Titchmarsch: {\sl The theory of functions}, Oxford University Press, London, 1933; second edition, 1939. }\cr } \ciao %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % FIGURES: BEGINNING % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %BEGINSFIGURE %%%% FIGURE f1.ps: SAVE IT AS f1.ps %! %%Creator: FeynDiagram 1.21 by Bill Dimm %%Adaptation: Sandro Ambrosanio %%BoundingBox: 177.3 325.8 540 651.06 %%LanguageLevel: 1 %%Pages: 1 %%EndComments %%BeginProlog % @(#) abbrev.ps 1.9@(#) /CP /charpath load def /CF /currentflat load def /CPT /currentpoint load def /C2 /curveto load def /FP /flattenpath load def /L2 /lineto load def /M2 /moveto load def /NP /newpath load def /PBX /pathbbox load def /RM2 /rmoveto load def /SD /setdash load def /SLC /setlinecap load def /SLW /setlinewidth load def /S /show load def /ST /stroke load def % @(#) vertex.ps 1.9@(#) /vtx_dict 20 dict def vtx_dict /vtx_mtrxstor matrix put /vtx_create { vtx_dict begin 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