%Received: from localhost (arnold@localhost) by pi.ceremade.dauphine.fr with SMTP (8.7.6/8.7.3) id SAA26467 for <duzhin@botik.ru>; Mon, 30 Mar 1998 18:12:33 +0200 (METDST)
%Date: Mon, 30 Mar 1998 18:12:33 +0200 (METDST)
%From: Arnold Vladimir <arnold@pi.ceremade.dauphine.fr>
%To: duzhin@botik.ru
%---------- Forwarded message ----------
%Date: Thu, 19 Feb 1998 16:07:14 +0300
%From: "Boris A. Amosov" <boris.amosov@relcom.ru>
%To: Arnold Vladimir <arnold@pi.ceremade.dauphine.fr>
%Subject: New paper

%Dear Vladimir Igorevich,
%
%I am sending you the english version of your new article
%as is (without editing). I hope you kept sty-file faa_eng.sty
%for compiling this amstex-file. Unfortunately, I cannot send
%ps-file from home (the file is too long for my bad telephone line).
%I will send it from steklovka.
%
%Russian version of this paper will be edited by Tsukerman
%and send you next week (I hope).
%
%Best regards. Boris Amosov

% below follows the FAA style edited by S.Duzhin to fit the printout
% on normal human paper

\magnification\magstep1

\expandafter\ifx\csname faa_eng.sty\endcsname\relax
  \expandafter\edef\csname faa_eng.sty\endcsname{%
    \catcode`\noexpand\@=\the\catcode`\@\relax}
\else \message{... already loaded}\endinput\fi
\catcode`\@=11
\def\W@#1{}\let\@message\message \def\message#1{}
\input amsppt.sty \relax
%% commented out by SVD
%\pagewidth{29pc}
\let\message\@message \def\W@{\immediate\write\sixt@@n}
\def\styversion{2.0} \def\styname{FAA_ENG}
\W@{\styname.STY - Version \styversion}\W@{}
\let\proclaim\relax
\outer\def\proclaim{%
  \let\savedef@\proclaim \let\proclaim\relax
  \add@missing\endroster \add@missing\enddefinition
  \add@missing\endproclaim \envir@stack\endproclaim
 \def\proclaim##1{\restoredef@\proclaim
   \penaltyandskip@{-100}\medskipamount\varindent@
   \def\usualspace{{\proclaimheadfont@\enspace}}\proclaimheadfont@
   \ignorespaces##1\unskip\frills@{.\enspace}%
  \it\ignorespaces}%
 \nofrillscheck\proclaim}
\outer\def\zh#1\endzh{\csname specialhead\endcsname#1\endspecialhead}
\outer\def\ah#1\endah{\csname head\endcsname#1\endhead}
\def\shortah{\rightheadtext}
\outer\def\bh#1\endbh{\csname subhead\endcsname#1\endsubhead}
\outer\def\ch#1\endch{\csname subsubhead\endcsname#1\endsubsubhead}
\outer\def\tm{\csname proclaim\endcsname}
\def\endtm{\endproclaim}
\def\pf{\csname definition\endcsname}
\def\endpf{\enddefinition}
\def\de{\csname definition\endcsname}
\def\endde{\enddefinition}
\def\rk{\csname definition\endcsname}
\def\endrk{\enddefinition}
\def\ex{\csname definition\endcsname}
\def\endex{\enddefinition}
\def\ext{\block\ninepoint}
\def\endext{\endblock}
\Monograph
\monograph@false
\let\headmark\eat@
\def\fg{\topinsert}
\def\endfg{\endinsert}
\def\fm{\midinsert}
\def\endfm{\endinsert}
\def\fp{\pageinsert}
\def\endfp{\endinsert}
\def\tc#1#2\endtc{\caption{#1}#2\endcaption}
\def\fc#1#2\endfc{\botcaption{#1}#2\endcaption}
\def\cn{\chapter}
\def\shortct{\leftheadtext}
\def\RM{\rom}
\def\scr{\Cal}
\begingroup
\let\head\relax \let\subhead\relax \let\subsubhead\relax
\let\ah\relax \let\bh\relax \let\ch\relax
\gdef\widestnumber{\begingroup
  \let\ah\relax \let\bh\relax \let\ch\relax
  \expandafter\endgroup\setwidest@}
%
\gdef\setwidest@#1#2{%
   \ifx#1\ah\setbox\tocheadbox@\hbox{#2.\enspace}%
   \else\ifx#1\bh\setbox\tocsubheadbox@\hbox{#2.\enspace}%
   \else\ifx#1\ch\setbox\tocsubheadbox@\hbox{#2.\enspace}%
   \else\ifx#1\key\refstyle A%
       \setboxz@h{\refsfont@\keyformat{#2}}%
       \refindentwd\wd\z@
   \else\ifx#1\no\refstyle C%
       \setboxz@h{\refsfont@\keyformat{#2}}%
       \refindentwd\wd\z@
   \else\ifx#1\page\setbox\z@\hbox{\quad\bf#2}%
       \pagenumwd\wd\z@
   \else\ifx#1\item
       \edef\next@{\the\revert@\rosteritemwd\the\rosteritemwd\relax
              \revert@{\the\revert@}}%
       \revert@\expandafter{\next@}%
       \setboxz@h{(#2)}\rosteritemwd\wdz@
   \else\message{\string\widestnumber\space not defined for this
      option (\string#1)}%
\fi\fi\fi\fi\fi\fi\fi}
\endgroup
\mathsurround=1.2pt
\def\<{\leavevmode\kern-\mathsurround}
\def\punctcheck@{\futurelet\nextchar@\removemathsurround@}
\let\everymath@\everymath
\let\everydisplay@\everydisplay
\newtoks\everymath
\everymath@{\the\everymath}
\newtoks\everydisplay
\everydisplay@{\the\everydisplay}
\everymath{\aftergroup\punctcheck@}
\def\removemathsurround@{%
  {\edef\next{\noexpand\nextchar@}%
  \ifcat.\noexpand\nextchar@
    \expandafter\ifx\next,\else\expandafter\ifx\next.\else
      \expandafter\ifx\next;\else\kern-\mathsurround\fi\fi\fi
  \fi
  \let\nextchar@\relax
  }%
}
\let\@@eqno=\eqno
\let\@@leqno=\leqno
\def\eqno{\everymath{}\@@eqno}
\def\leqno{\everymath{}\@@leqno}
\def\mathcomma@{\mathpunct{\mathchoice{\mkern1.2mu\mathchar"013B\mkern1.2mu}
%

{\mkern1.2mu\mathchar"013B\mkern1.2mu}%

{\mkern0.2mu\mathchar"013B\mkern1.2mu}%

{\mkern0.2mu\mathchar"013B\mkern1.2mu}}}%
\def\mathsemicolon@{\mathpunct{\mkern3mu\mathchar"003B\mkern3mu}}%
\def\mathcolon@{\mathrel{\mkern\thickmuskip\nonscript\mkern-\thickmuskip
  \mathchar"003A\mkern\thickmuskip\nonscript\mkern-\thickmuskip}}
\def\coloneq{\mathrel{\mathchar"003A\mathchar"003D}}
\def\eqcolon{\mathrel{\mathchar"003D\mathchar"003A}}
\mathcode`\,="8000 \mathcode`\;="8000 \mathcode`\:="8000
\begingroup
\catcode`\,=\active \catcode`\;=\active \catcode`\:=\active
\global\everymath@{%
  \let,\mathcomma@ \let;\mathsemicolon@ \let:\mathcolon@
  \the\everymath}
\global\everydisplay@{%
  \let,\mathcomma@ \let;\mathsemicolon@ \let:\mathcolon@
  \def\.{\thinspace.}%
  \the\everydisplay}
\endgroup
\def\cprime{\begingroup\everymath{}\m@th$'$\endgroup}
\let\mz\cprime % ``myagkii znak'' synonym for \cprime
\def\index{\openindexfile
 \def\@sf{}\ifhmode\edef\@sf{\spacefactor\the\spacefactor\relax}\fi
 \unskip
 \begingroup\def\do##1{\catcode`##112 }%
 \iffalse{\fi \let\\=\relax \do\\\iffalse}\fi
 \do\&\do\#\do\~\do\@\do\%\writetoindexfile}
\def\writetoindexfile#1{\write\indexfile{\noexpand\indexentry
 {#1}{\the\pageno}}\endgroup\@sf}
\def\indexpageno{\ifnum\pageno<10 0\fi\ifnum\pageno<100 0\fi\the\pageno}
\newwrite\indexfile
\newif\ifindexing@
\def\openindexfile{%
  \openout\indexfile=\jobname.idx\gdef\openindexfile{}\global\indexing@true}
\font@\ninerm=cmr9
\font@\ninei=cmmi9    \skewchar\ninei='177
\font@\ninesy=cmsy9   \skewchar\ninesy='60
\font@\nineex=cmex9
\font@\ninebf=cmbx9
\font@\nineit=cmti9
\font@\ninesmc=cmcsc9
\font@\ninemsa=msam9
\font@\ninemsb=msbm9
\font@\nineeufm=eufm9
\newtoks\ninepoint@
\def\ninepoint{\normalbaselineskip11\p@
 %\abovedisplayskip10\p@ plus2.4\p@ minus7.2\p@
 \abovedisplayskip10\p@ plus2.4\p@ minus4.2\p@
 \belowdisplayskip\abovedisplayskip
 \abovedisplayshortskip\z@ plus2.4\p@
 \belowdisplayshortskip5.6\p@ plus2.4\p@ minus3.2\p@
 \textonlyfont@\rm\ninerm \textonlyfont@\it\nineit
 \textonlyfont@\bf\ninebf \textonlyfont@\smc\ninesmc
 \ifsyntax@\def\big##1{{\hbox{$\left##1\right.$}}}%
  \let\Big\big \let\bigg\big \let\Bigg\big
 \else
  \textfont\z@\ninerm \scriptfont\z@\sevenrm
       \scriptscriptfont\z@\fiverm
  \textfont\@ne\ninei \scriptfont\@ne\seveni
       \scriptscriptfont\@ne\fivei
  \textfont\tw@\ninesy \scriptfont\tw@\sevensy
       \scriptscriptfont\tw@\fivesy
  \textfont\thr@@\nineex \scriptfont\thr@@\sevenex
   \scriptscriptfont\thr@@\sevenex
  \textfont\itfam\nineit \scriptfont\itfam\sevenit
   \scriptscriptfont\itfam\sevenit
  \textfont\bffam\ninebf \scriptfont\bffam\sevenbf
   \scriptscriptfont\bffam\fivebf
  \textfont\msafam\ninemsa \scriptfont\msafam\sevenmsa
   \scriptscriptfont\msafam\fivemsa
  \textfont\msbfam\ninemsb \scriptfont\msbfam\sevenmsb
   \scriptscriptfont\msbfam\fivemsb
  \textfont\eufmfam\nineeufm \scriptfont\eufmfam\seveneufm
   \scriptscriptfont\eufmfam\fiveeufm
 \setbox\strutbox\hbox{\vrule height7.5\p@ depth3.5\p@ width\z@}%
 \setbox\strutbox@\hbox{\raise.5\normallineskiplimit\vbox{%
   \kern-\normallineskiplimit\copy\strutbox}}%
 \setbox\z@\vbox{\hbox{$($}\kern\z@}\bigsize@1.2\ht\z@
 \fi
 \normalbaselines\ninerm\dotsspace@1.5mu\ex@.2326ex\jot3\ex@
 \the\ninepoint@}
\font@\twelverm=cmr10 scaled\magstep1
\font@\twelvei=cmmi10 scaled\magstep1    \skewchar\twelvei='177
\font@\twelvesy=cmsy10 scaled\magstep1   \skewchar\twelvesy='60
\font@\twelveex=cmex10 scaled\magstep1
\font@\twelvebf=cmbx10 scaled\magstep1
\font@\twelveit=cmti10 scaled\magstep1
\font@\twelvemsa=msam10 scaled\magstep1
\font@\twelvemsb=msbm10 scaled\magstep1
\font@\twelveeufm=eufm10 scaled\magstep1
\newtoks\twelvepoint@
\def\twelvepoint{\normalbaselineskip15\p@
 \abovedisplayskip10\p@ plus2.4\p@ minus7.2\p@
 \belowdisplayskip\abovedisplayskip
 \abovedisplayshortskip\z@ plus2.4\p@
 \belowdisplayshortskip5.6\p@ plus2.4\p@ minus3.2\p@
 \textonlyfont@\rm\twelverm \textonlyfont@\bf\twelvebf
 \textonlyfont@\it\twelveit
 \ifsyntax@\def\big##1{{\hbox{$\left##1\right.$}}}%
  \let\Big\big \let\bigg\big \let\Bigg\big
 \else
  \textfont\z@\twelverm \scriptfont\z@\tenrm
       \scriptscriptfont\z@\sevenrm
  \textfont\@ne\twelvei \scriptfont\@ne\teni
       \scriptscriptfont\@ne\seveni
  \textfont\tw@\twelvesy \scriptfont\tw@\tensy
       \scriptscriptfont\tw@\sevensy
  \textfont\thr@@\twelveex \scriptfont\thr@@\tenex
   \scriptscriptfont\thr@@\sevenex
  \textfont\itfam\twelveit \scriptfont\itfam\tenit
   \scriptscriptfont\itfam\sevenit
  \textfont\bffam\twelvebf \scriptfont\bffam\tenbf
   \scriptscriptfont\bffam\sevenbf
  \textfont\msafam\twelvemsa \scriptfont\msafam\tenmsa
   \scriptscriptfont\msafam\sevenmsa
  \textfont\msbfam\twelvemsb \scriptfont\msbfam\tenmsb
   \scriptscriptfont\msbfam\sevenmsb
  \textfont\eufmfam\twelveeufm \scriptfont\eufmfam\teneufm
   \scriptscriptfont\eufmfam\seveneufm
 \setbox\strutbox\hbox{\vrule height12\p@ depth5\p@ width\z@}%
 \setbox\strutbox@\hbox{\raise.5\normallineskiplimit\vbox{%
   \kern-\normallineskiplimit\copy\strutbox}}%
 \setbox\z@\vbox{\hbox{$($}\kern\z@}\bigsize@1.2\ht\z@
 \fi
 \normalbaselines\twelverm\dotsspace@1.5mu\ex@.2326ex\jot3\ex@
 \the\twelvepoint@}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newbox\plenumcopyright
\newbox\editor
\output={\faaoutput}
\def\faaoutput{\shipout\vbox{\pagebody\makefootline}%
  \advancepageno
  \ifnum\outputpenalty>-\@MM \else\dosupereject\fi}
\def\pagebody{\advance\vsize-0.6\ht\plenumcopyright
\vbox to\vsize{\boxmaxdepth\maxdepth \pagecontents}}
\def\makeheadline{\vbox to\z@{\vskip-22.5\p@
  \line{\vbox to8.5\p@{}\the\headline}\vss}\nointerlineskip}
\def\makefootline{\baselineskip24\p@\line{\the\footline}}
\def\dosupereject{\ifnum\insertpenalties>\z@ % something is being held over
  \line{}\kern-\topskip\nobreak\vfill\supereject\fi}
\def\pagecontents{\ifvoid\topins\else\unvbox\topins\fi
  \dimen@=\dp\@cclv \unvbox\@cclv % open up \box255
  \ifvoid\footins\else % footnote info is present
    \vskip\skip\footins
    \footnoterule
    \unvbox\footins\fi
      \ifr@ggedbottom \kern-\dimen@ \vfil \fi}
\def\footnoterule{\vskip2pt plus 1pt
  \hrule width 2in \kern 2.6\p@} % the \hrule is .4pt high
\footline{\ifvoid\plenumcopyright\sevenrm\ifodd\pageno
  \hss\the\pageno\else\the\pageno\hss\fi\else\box\plenumcopyright\fi}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\outer\def\enddocument{\par% \par will do a runaway check for \endref
  \add@missing\endRefs
  \add@missing\endroster \add@missing\endproclaim
  \add@missing\enddefinition
  \add@missing\enddemo \add@missing\endremark \add@missing\endexample
 \ifmonograph@ % do nothing
 \else
 \nobreak
 \thetranslator@
 \count@\z@ \loop\ifnum\count@<\addresscount@\advance\count@\@ne
 \csname address\number\count@\endcsname
 \csname email\number\count@\endcsname
 \repeat
\fi
\ifindexing@\closeout\indexfile
   \W@{Index information written to \jobname.idx}\fi
 \vfill\supereject\end}
%    Restore the previous catcode of @ character:
\csname faa_eng.sty\endcsname
%\pagewidth{38.33pc} \pageheight{52.5pc}
%\hoffset=-1.25cm
%\voffset=-27pt
% changed by SVD
\pagewidth{160truemm} \pageheight{230truemm}
\voffset=-5truemm
\baselineskip=12.2pt plus 0.2pt minus 0.2pt
\abovedisplayskip=7pt plus 2pt minus 2pt
\belowdisplayskip=7pt plus 2pt minus 2pt
\abovedisplayshortskip=5pt plus 1pt minus 3pt
\belowdisplayshortskip=5pt plus 1pt minus 3pt
\def\title#1#2#3#4{\flushpar\eightit Functional Analysis and Its
Applications, Vol.~#1, No.~#2,~19#3
   \vglue 2.5pc \flushpar{\twelvepoint \bf #4}\rm}
\def\titleb#1#2#3#4{\flushpar\eightit Functional Analysis and Its
Applications, Vol.~#1, No.~#2,~19#3
   \vglue 2.5pc
\flushpar{\twelvepoint\bf BRIEF COMMUNICATIONS}
   \vglue 2.5pc
\flushpar{\twelvepoint \bf #4}\rm}
\def\extratitleline#1{\smallskip\flushpar{\twelvepoint\bf #1}\rm}
\def\authorinfo#1#2#3#4#5#6#7{\medskip\medskip \bf#1
\hfill\rm UDC\ #2\medskip\medskip
\setbox\plenumcopyright=\vbox{\vskip12pt\hrule width \hsize \kern 3pt
\baselineskip9.5pt\eightrm #3. Translated from Funktsio\-nal\smash{\mz}nyi
Ana\-liz i Ego Pri\-lo\-zhe\-niya, Vol.~#4, No.~#5, pp.~#6,
{\ifcase#5 ?????\or January--March\or April--June\or July--September\or
October--December\else ?????\fi}, \count111=#4\advance\count111 by 66
19\the\count111. Original article submitted #7.\newline
\centerline{\vrule height12pt
width0pt\ifodd\pageno{}\else{\sevenrm\the\pageno}\fi
\hfill 0016--2663/\the\count111/#40#5--{\ifnum \pageno<10
000\the\pageno\else%
         \ifnum\pageno<100 00\the\pageno
         \else \ifnum\pageno<1000 0\the\pageno\fi\fi\fi}\,\$15.00\quad
\ifnum#5>2\advance\count111 by 1\fi\copyright19\the\count111\ Plenum
Publishing Corporation\hfill\ifodd\pageno{\sevenrm\the\pageno}\else{}\fi}}
}
\def\transl#1{\vskip12pt\rightline{Translated by #1}
\ifvoid\editor\else\line{\the\day.\the\month.\the\year\hss Edited by
\box\editor}\fi}
\def\bS{\pmb{\S}}
\define\wt{\widetilde}
\define\wh{\widehat}
\def\({\hskip1pt(}
\def\){)\hskip1pt}
\def\>{\hskip1pt}
\def\ov{\overline}
\def\pa{\partial}
\def\op{\operatorname}
\def\a{\alpha}
\def\be{\beta}
\def\ga{\gamma}
\def\dt{\delta}
\def\e{\varepsilon}
\def\th{\theta}
\def\ka{\varkappa}
\def\la{\lambda}
\def\si{\sigma}
\def\ph{\varphi}
\def\om{\omega}
\def\Ga{\Gamma}
\def\De{\Delta}
\def\Th{\Theta}
\def\Si{\Sigma}
\def\Om{\Omega}
\def\La{\Lambda}
\def\R{{\Bbb R}}
%\def\C{{\Bbb C}}
\def\C{\Bbb{C}\kern.7pt}
\TagsOnRight
\def\breakpage{\vadjust{\eject}\ }
\redefine\Refs{\bgroup\vskip10pt%
     \centerline{\bf References}\vskip10pt\leftskip5pt\frenchspacing}
\redefine\endRefs{\egroup}
\def\ft#1{\footnote"{${}^\star$}"{#1}}
\def\ftt#1{\footnote"{${}^\ast$}"{#1}}
\def\pre#1{\baselineskip16pt plus 0.5pt minus 0.5pt\setbox\editor=\hbox{#1}}
\def\pred#1{\baselineskip16pt plus 0.5pt minus 0.5pt}
\define\ovl#1#2#3{\rlap{\<$#3$\<}\kern#1pt\ov{\phantom{#3}%
                                 \kern-#1pt\kern-#2pt}\kern#2pt}
\define\scirc{\kern2pt\lower-0.5pt\hbox{\<$\ssize\circ$\<}\kern2pt}
\define\lan{\langle}
\define\ran{\rangle}
\define\rost{\par\leftskip12pt}
\define\rostrost{\par\leftskip24pt}
\define\endrost{\par\leftskip0pt}
\define\whs#1{\mathchoice%
         {\kern2pt\skew2\wh{\hbox spread-4pt{\hss\<$#1$\<\hss}}\kern2pt}%
         {\kern2pt\skew2\wh{\hbox spread-4pt{\hss\<$#1$\<\hss}}\kern2pt}%
  {\kern1pt\skew2\wh{\hbox spread-2pt{\hss\<$\scriptstyle
#1$\<\hss}}\kern1pt}%
  {\kern1pt\skew2\wh{\hbox spread-2pt{\hss\<$\scriptscriptstyle
#1$\<\hss}}\kern1pt}}
\define\wts#1{\mathchoice%
         {\kern2pt\skew2\wt{\hbox spread-4pt{\hss\<$#1$\<\hss}}\kern2pt}%
         {\kern2pt\skew2\wt{\hbox spread-4pt{\hss\<$#1$\<\hss}}\kern2pt}%
         {\kern1pt\skew2\wt{\hbox spread-2pt{\hss\<$\scriptstyle
#1$\<\hss}}\kern1pt}%
         {\kern1pt\skew2\wt{\hbox spread-2pt{\hss\<$\scriptscriptstyle
#1$\<\hss}}\kern1pt}}
\redefine\otimes{\mathchoice%
                 {\hskip2pt\mathchar"220A\hskip2pt}%
                 {\hskip2pt\mathchar"220A\hskip2pt}
                 {\mathchar"220A}
                 {\mathchar"220A}}
\redefine\oplus{\mathchoice%
                 {\hskip2pt\mathchar"2208\hskip2pt}%
                 {\hskip2pt\mathchar"2208\hskip2pt}
                 {\mathchar"2208}
                 {\mathchar"2208}}
\redefine\otimes{{\hskip2pt\mathchar"220A\hskip2pt}}
\redefine\oplus{{\hskip2pt\mathchar"2208\hskip2pt}}
%\let\cal=\Cal
\font\tenscr=eusm10.pk scaled 1000
\font\sevenscr=eusm7.pk scaled 1000
\font\fivescr=eusm5.pk scaled 1000
\define\cal#1{\mathchoice{\hbox{\tenscr #1}}{\hbox{\tenscr #1}}%
              {\hbox{\sevenscr #1}}{\hbox{\fivescr #1}}}
\font\tensscr=rsfs10.pk scaled 1000
\font\sevensscr=rsfs7.pk scaled 1000
\font\fivesscr=rsfs6.pk scaled 1000
\redefine\Cal#1{\mathchoice{\hbox{\tensscr #1}}{\hbox{\tensscr #1}}%
              {\hbox{\sevensscr #1}}{\hbox{\fivesscr #1}}}
\define\picture#1#2#3{\rlap{\kern #1\smash{\lower #2\hbox{\special{em:graph
#3}}}}}
\define\capt#1#2#3{\rlap{\kern #1\smash{\lower #2\hbox{#3}}}}
\newdimen\theight
\define\lab#1{\vadjust{\setbox0=\hbox{\tenpoint%
            \quad${}^{\dsize\checkmark}{}^{\text{#1}}$}%
            \theight=\ht0
            \advance\theight by\dp0\advance\theight by \lineskip
            \kern-\theight\vbox to\theight{\rightline{\rlap{\box0}}%
            \vss}%
            }}%
\loadbold
\define\bcdot{\,\boldsymbol\cdot\,}
\define\inn{\,{\in}\,}
\define\gee{\,{\ge}\,}
\define\lee{\,{\le}\,}
\define\={\,{=}\,}
\define\itemi{\par\parindent13.8pt{\rm(i)}\kern4pt}
\define\itemii{\par\parindent11pt{\rm(ii)}\kern4pt}
\define\itemiii{\par\parindent8pt{\rm(iii)}\kern4pt}
\define\itemiv{\par\parindent8.5pt{\rm(iv)}\kern4pt}
\define\itemv{\par\parindent11pt{\rm(v)}\kern4pt}
\define\itemvi{\par\parindent8.5pt{\rm(vi)}\kern4pt}
\define\noitem{\parindent12pt}

% ------- Style file finished. Paper begins ---------

\title{32}{?}{98}{Towards the Legendrian Sturm Theory of Space
Curves\ftt{Partially supported by the Russian Foundation for Basic
Research, project 96-01-01104.}}

\authorinfo{V. I. Arnold}{517.9} {V.~A.~Steklov Mathematical
Institute and CEREMADE, Universit\'e Paris-Dauphine}
{32}{2}{??--??}{December 29, 1997}

\ah\bS1.\enspace Introduction\endah
Sturm theory extends the Morse inequality (minorating the number of
critical points of functions on a circle) to the higher derivatives.
The Legendrian Morse theory (created by Yu.~V.~Chekanov in 1986)
provides the Morse inequality for the multivalued functions
(corresponding to the unknoted Legendrian submanifolds of the space
of $1$-jets of functions). It is a generalization of the Lagrangian
intersection theory due to Conley, Zehnder, Chaperon, Floer, Sikorav,
Laudenbach, Hofer, Gromov, and others.
Below an attempt to extend the Legendrian Morse theory to higher
derivatives is presented. It extends the Legendrian Morse theory in
the same sense in which the ordinary Sturm theory extends Morse
theory.
Sturm theory implies the existence of at least four flattening points
on every curve in projective $3$-space having a convex projection.
The present paper results imply, for instance, the preservation of at
least four flattening points under (not necessary small) deformations
of the standard curve of Sturm theory (\<$x=\cos t$, $y=\sin t$,
$z=\cos2t$),
provided that the curve has no inflection (zero curvature) points at
any moment and that the Legendrian knot type remains unchanged under
the deformation. The Legendrian knot is a singular surface in the
five-dimensional manifold of the contact elements of the space,
associated to the curve.
In terms of the front of the curve (which is a surface in the dual
projective space) the restrictions on the deformation forbid the
selftangency of the front, the selfintersections of its cuspidal
edge and the tangencies of the cuspidal edge to the front, whose
cuspidal edge should remain irreducible during the deformation.
The main idea of the present paper is the attempt to isolate in the
topological study of the evolution of the fronts of the space curve
the part which is independent on the fact that the front is related to
a curve, making it applicable to the more general fronts of arbitrary
Legendrian varieties. In a similar way, the theory by A.~V.~Alexeev
and V.~A.~Vassiliev of the earlier 80ths (described for instance
in \cite5 and \cite7) already contained in a more general form many
results of \cite6.
\ah\bS2.\enspace The Legendrian Hulls of Curves in Projective
Space $\pmb{P^N}$ and Their Fronts\endah
I first recall some basic facts of the contact geometry of curve
$\ga$ in the $N$-dimensional projective space $P^N$ (due to
O.~P.~Shcherbak \cite1). We shall deduce below global topological
corollaries of this local theory.
The {\it Legendrian hull} $L$ of curve $\ga$ is the manifold formed
by all the contact elements of $P^N$\!, tangent to~$\ga$. It is a
smooth Legendrian submanifold of the manifold
$E=PT^*P^N=PT^*(P^N)^{\vee}$ of contact elements on $P^N$\!. A point of
$E$ is a pair $(a\text{,}A)$ 
where $a$ is a point of $P^N$ and $A$ is a
hyperplane containing this point. The manifold $E$ of contact
elements of $P^N$ is the total space of two Legendrian fibrations
(over the initial and over the dual projective spaces):
$$
\rho\:E\to P^N\text{,}\quad\rho^{\vee}\:E\to(P^N)^{\vee}\qquad
(\text{where $\rho(\text{,}A)=a$, $\rho^{\vee}(\text{,}A)=A$})\.
$$
The {\it front} $F$ of curve $\ga$ is the projection
$F=\rho^{\vee}L$ of the Legendrian hull of the curve into the dual
projective space. The dimension of the front equals $N-1$.
The front of a generic curve has only the standard Legendrian
singularities of type $A_k$ (in the three space they are just the
semicubical cuspidal edges $A_2$ and the swallowtails $A_3$). The
front is the ``{\it tangents developable}'' (the union of the
$(N-2)$-dimensional osculating spaces) of the dual curve $\ga^{\vee}
\subset(P^N)^{\vee}$ (in the three-space-the union of its tangent
lines). The {\it dual curve} $\ga^{\vee}$ is formed by the osculating
hyperplanes of the original curve.
The singularities of the curve, dual to a generic curve of the
three-space, are the semicubical cusps (corresponding to the flattening
points of the original curve, to the swallowtail vertices
of the front and to the folded umbrellas vertices on the ``tangents
developable'' surface of the original curve).
In a neighborhood of such a point the original curve admits the
parametric equations (\<$x=t$, $y=t^2+\dots$, $z=t^4+\dots$) (where the
points mean higher order terms) in suitable affine coordinates
$(x\text{,}y\text{,}z)$. The dual curve is described by the equations
(\<$p=t^2+\dots$, $q=t^3+\dots$, $r=t^4+\dots$) in suitable local affine
coordinates of the dual space.
In the generic one-parameter families of curves in $P^3$ there may
occur (for some isolated values of the parameter) two more events:
the {\it biflattening\/} of the curve (the degrees $(1\text{,}2\text{,}5)$ replacing
$(1\text{,}2\text{,}4)$ in the original space and the degrees $(2\text{,}3\text{,}5)$ replacing
$(2\text{,}3\text{,}4)$ in the dual space) and the {\it inflections} of the curve
(degrees $(1\text{,}3\text{,}4)$ in both spaces).
The corresponding perestroikas of the fronts are the standard
Legendrian perestroikas of type ``butterfly'' $(A_4)$ in the first
case and ``bec to bec'' $(A_3)$ in the second.
We shall give below the minoration of the number of the topologically
necessary flattening points on a curve, obtained from a fixed curve
of the projective space by a generic regular homotopy (in terms of
the selfintersection lines of the front of the original curve or of
the selfintersection lines of its ``tangent developable'' and of
their perestroikas under the homotopy).
\ah\bS3.\enspace The Sturmianity of a Curve of Projective Space\endah
Orient the
projective space $P^N$\!, $N=2n+1$. In a neighborhood of a flattening
curve a generic parametrized curve is defined by the equations
(\<$x_1=t$, $x_2=t^2+\dots$, $x_{N-1}=t^{N-1}+ \dots$,
$x_N=t^{N+1}+\dots$) in
an affine coordinate system whose origin is the flattening point.
\de{Definition} A flattening point is {\it positive}, if this
coordinate system coorient the space $P^N$ positively (and {\it
negative\/} in the opposite case).
\endde
The signs of the consecutive flattening points on a generic curve
alternate.
\rk{Remark} To define the sign one might also use the dual curve. The
reversal of the curve direction multiplies the signs of all its
flattening points by $(-1)^n$.
\endrk
The number of the flattening points of a closed curve in an
odd-dimensional projective space is even, since the sign of the
osculating frame changes under each transition through a flattening
point.
There is an involution with no fixed points acting on the set of the
flattening points of a generic closed curve in the odd-dimensional
projective space: each flattening point has its {\it joint point}.
The joining is due to the fact, that each swallowtail vertices is an
endpoint of a selfintersection line of the front, starting at some
other swallowtail vertices.
\de{Definition} A {\it pair of joint vertices\/} of swallowtails of a
generic front in a three-manifold is a pair of end points of a segment
of a front selfintersection line (this line may contain cusps and
intersection points of two nontangent branches of the
selfintersection line).
\endde
\rk{Remark} Similarly, one might define the pairs of the joint
swallowtails vertices on the fronts in \text{$(2n+1)$-manifolds}, having
just $A_k$-singularities.
\endrk
In a neighborhood of the vertices of a ``swallowtail $A_{2n+1}$'' the
front is diffeomorphic to the variety of the polynomials
$x^{2n+2}+\la_1x^{2n}+\dots+\la_{2n+1}$ having a multiple real root.
The vertices (\<$\la=0$) is the endpoint of the curve of the polynomials
$2A_n$ of the type $(x^2-a^2)^{n+1}$.
\de{Definition} Two flattening points of a generic curve in an
odd-dimensional projective space form a {\it joint pair},
if the corresponding vertices of the front swallowtails
from a joint pair.
\endde
\rk{Remark} In the original-three space the flattening points forming
a joint pair are joint by a segment of the selfintersection line of
the ``tangent developable'' curve. This follows from the fact, that the
intersection point $q\in P^3$ of the tangent lines $l_a$ and $l_b$ of
the original curve $\ga$ at $a$ and at $b$ belongs to the plane~$Q$,
generated by $l_a$ and $l_b$. This plane is a point
$Q\in(P^3)^{\vee}$ of the dual space, contained in the lines $L_A$
and $L_B$ of the dual space, tangent to the dual curve $\ga^{\vee}$
at the points $A$ and $B$ corresponding to $a$ and~$b$. Namely, the
line $L_A$ consists of those planes of the original space, which are
tangent to $\ga$ at $a$, and $L_B$ at $B$.
\endrk
Thus the point $q$ of the selfintersection of the ``tangent
developable'' of the curve $\ga$ (which is the front surface of the
dual curve $\ga^{\vee}$) corresponds to a selfintersection point $Q$
of the front $F$ of the curve $\ga$ (which is the
selfintersection point of the ``tangent developable'' of the curve
$\ga^{\vee}$ dual to $\ga$).
The point $Q$ sweeps the segment of the selfintersection line of the
front $F$ of the curve $\ga$ joining two front's swallowtail
vertices while $q$ sweeps the segment of the selfintersection line of
the ``tangent developable'' of the
original curve $\ga$ joining two flattening points of $\ga$.
\de{Main definition} The {\it sturmianity} $s$ of a generic curve in an
odd-dimensional projective space is the number of joined pairs of
flattening points of the curve having equal signs.
\endde
\rk{Remark} The sturmianity of a closed curve is even, since the
number of positive pairs is equal to the number of negative ones.
\endrk
\ex{Example} The sturmianity of the standard curve of the Sturm
theory in $P^{2n+1}$
$$
x_1=\cos t\text{,}\quad y_1=\sin t\text{,}\quad\dots\text{,}\quad
x_n=\cos nt\text{,}\quad y_n=\sin nt\text{,}\quad z=\cos(n+1\)t\text{,}
$$
where $n$ is odd, equals $s=n+1$ (the joined points being opposite).
\endex
The sturmianity of the curve (\<$x=\cos t$, $y=\sin t$, $z=\cos2nt$) in the
$3$-space is $s=2n$ (the joined points being opposite).
\ah\bS4.\enspace The Minoration of the Number off Flattening Points\endah
Consider a smooth closed generic curve $\ga$, embedded into the
projective $3$-space (genericity means that we suppose that the curve
torsion is vanishing at most at finite set of first order roots).
\de{Definition} A regular homotopy of curve $\ga$ is called {\it
admissible}, if it is a generic regular homotopy (containing a finite
set of transversal biflattening but no inflections) whose front
experiences no (i) selftangency moments; (ii) moments of
selfintersection of the cuspidal edge (iii) moments of tangency of
the cuspidal edge of the front with the front.
\endde
\rk{Remark} A selftangency of the front means a selfintersection of
the original curve. A selfintersection of the cuspidal edge of the
front means a selfintersection of the dual curve (and hence a
selftangency of the ``tangent developable'' of the original
curve). A tangency of the cuspidal edge of the front with the
front means a tangency of the original curve with one of its
osculating planes (and hence a tangency of the original curve with
its ``tangent developable'' surface).
\endrk
At the inflection moment the cuspidal edge would be reducible (would
contain a straight line).
\tm{Theorem} The sturmianity of a curve remains constant under the
admissible homotopies.
\endtm
\tm{Corollary} The number of flattening points of a curve
experiencing an admissible homotopy remains bounded from below by the
doubled sturmianity of the original curve.
\endtm
\rk{Remark} The above lower bound of the number of the flattening
points is a multiple of $4$.
\endrk
\pf{Proof} The list of the perestroikas in the typical one-parameter
families of the fronts (in the spaces of not two big dimension) is
discovered in \cite2. O.~P.~Shcherbak has proved in \cite1 that the
perestroikas of the fronts of typical one-parameter families of
curves in the projective $3$-space form the following part of this
list:
\endpf
I. {\it The perestroikas changing the number of the swallowtails}.
1. $A_4$
(the Thom's ``butterfly''): the birth or death of a joined pair of
two close vertices of the swallowtails on the cuspidal edge of the
front (connected by a short segment of the front selfintersection
line, containing two cusps).
2. $A_3$ (the Thom's ``bec to bec''): the birth or the death of two
close vertices of the swallowtails on the cuspidal edge of the front
(which are separated by a long segment of the swallowtail).
II. {\it The perestroikas preserving the swallowtails number}.
3. The front selftangency (the Morse perestroika of the trace, formed
by the intersection with a different smooth branch of the front on a
given branch).
4. The tangency of the cuspidal edge of the front with a smooth branch
of the front (accompanied with the birth of the death of two cusps on
the front selfintersection line).
5. The selfintersection of the cuspidal edge of the front (moving two
cusps of the front selfintersection line from one branch of the
cuspidal edge to another).
6. The passing of a smooth branch of the front through the vertex of
the swallowtail (accompanied with the birth or the death of two cusps
on the front selfintersection line).
7. The transition of the cuspidal edge of the front through the
intersection line of two smooth branches of the front (accompanied
with the transition of this intersection line through a cusp of
another branch of the selfintersection line).
8. The transition of four smooth branches of the front through the
same point (all the six lines of the pairs of branches intersection
moving smoothly under the perestroika).
9. The tangency of the front selfintersection line with the smooth
part of the front (all the three lines of the pairs of branches
intersection moving smoothly under the perestroika).
We should prove that the sturmianity remains unchanged under the
admissible perestroikas. This is obvious for the perestroikas
$6\text{,}7\text{,}8\text{,}9$ which are not changing neither the joined pairs nor the
signs of the swallowtails vertices.
The perestroikas $3$, $4$, $5$ are forbidden by the admissible deformations
definition.
The perestroika $1$ happens at the biflattening moment. This is the
moment of the birth or of the death of a joined pair of neighboring
flattening points of the curve. They have opposite signs and they are
joined by a selfintersection segment on the front (or on the
``tangent developable'' surface). Hence this birth or death does not
change the sturmianity.
The perestroika $2$ happens only at the inflection moment. Since
admissible homotopies have no inflection moments, the theorem is
proven.
The proof shows also what is happening to the sturmianity under the
nonadmissible perestroikas $2$, $3$, $4$, $5$. Thus we might evaluate the
sturmianity (and hence the number of the flattening points) of the
deformed curve in terms of the sturmianity of the original curve and
of the number of the nonadmissible perestroikas.
\rk{Remark} At the transition of an inflection moment two close
flattening points are born or killed on the curve. These points have
opposite signs, but they are not necessarily joint. The two points,
joint with them, might have the same signs as the joint vanishing
flattening points respectively. In this case the sturmianity of the
curve is decreased by $2$ at the death of the vanishing flattenings.
\endrk
If at least one of the joint flattening points has a sign different
from the sign of the vanishing flattening point joint to it, the
sturmianity of the curve does not change under the inflection
transition. The transition through such inflection perestroikas might
be included into the list of the admissible perestroikas.
The same way, one might admit the ``elliptical'' perestroikas $3$, $4$,
and $5$ (accompanied with the birth or the death of small front
selfintersection lines). They do not change the curve sturmianity.
One also might get better minorations distinguishing the joining
segments of the selfintersection line of the ``tangent developable''
surface which are contractible to $\ga$ and which are not.
\ah\bS5.\enspace Examples\endah
\ex{Example 1} Consider the space curve described by the equations
(\<$r=1-a\cos\ph$, $z=\sin2\ph$) in cylindrical coordinates. While the
parameter $a$ is growing from $0$ to $1/2$, this curve has four
flattening points. Two of them vanish (at the inflection point
$\ph=0$) for $a=1/2$.
\endex
While $a$ is smaller than $1/2$, the curve has a quadratically convex
projection and two alternating joint pairs of flattening points of
signs $(++)$ and $(--)$. The sturmianity of this curve is hence equal
to $2$.
At the transition of the parameter $a$ through the value $1/2$ the
perestroika $2$ happens. The sturmianity is diminished by $2$ and the
resulting curve has only $2$ flattening points. For instance, the
resulting curve has no convex projection (from any point of the
projective $3$-space), since a curve having a convex projection has
at least $4$ flattening points (see \cite{3, 4}).
This example shows that one should not accept the inflections
transitions, at least in the case where the inflections joint to the
vanishing ones have the same signs as the corresponding vanishing
inflections.
\ex{Example 2} Consider the space curve, defined by the equations
(\<$x=\cos\ph$, $y=\sin\ph$, $z=\cos3\ph$). This curve has $6$ flattening
points. The selfintersection line of the ``tangent developable''
surface consists of $3$ straight lines segments joining the opposite
flattening points intersecting the infinite line and of the infinite line
$z=0$. At the intersection points of the radial lines with the
infinite line both smooth branches of the ``tangent developable''
surface are Morse tangent to each other.
\endex
A small deformation of the original curve (\<$z=\cos3\ph+\e f(\ph)$)
modifies the selfintersection line into a smooth curve, and the joint
pairs may be $(+-) (+-)(+-)$ along the curve (say, for $f=\sin6t$).
The sturmianity of the deformed curve is then equal to zero. For the
deformations of this curve our theorem does not, guarantee any
flattening points, while the curve has a convex projection.
The ``tangent developable'' surface of a curve having a convex
projection may be considered as a two-valued section of a
one-dimensional line bundle over the M\"obius band (which is the
complement in the projective plane to the convex domain, bounded by
the projection of the curve).
The projections of the selfintersection lines of the ``tangent
developable'' surface divide the M\"obius band into connected parts.
These parts may be colored into two colors (the color being changed
at each generic crossing of the projection of the intersection line).
If the parts are orientable, the sturmianity of the original curve
cannot be smaller than $2$. In this case the number of the flattening
points is at least $4$ (and remains at least equal to $4$ under the
admissible homotopies).
Example 2 shows that one of the parts may be nonorientable and the
sturmianity may vanish even for the curves having a convex projection.
\ex{Example 3} The immersed curve (\<$x=\sin2t$,
$y=\cos2t$, $z=\cos4t$) has $8$
flattening points. A small deformation, making it an embedded curve,
transforms this curve into a generic curve of sturmianity $4$. This
curve has no convex projections (since the planes containing the axis
intersecting the curve more than twice sweep all the projective
space).
\endex
\rk{Remark} One might probably extend the theorem to the more general
case of the fronts of the Legendrian varieties which are not related
to the curves. What is important is the distinction between the
swallowtails of different signs. The Legendrian variety associated to
the curve is the union of the Legendrian hulls of the original curve
and of its dual curve. Even the local normal forms of the
singularities of these varieties have not been studied as they should
be in contact geometry (as far as I know).
\endrk
\Refs
\item{1.} O. P. Shcherbak, ``Projectively dual space curves and Legendre
singularities,'' Proc. Tbilisi Univ., {\bf232/233}, 280--336 (1982);
English transl. in Selecta Math. Soviet., {\bf5}, No.~4, 391--421 (1986).

\item{2.} V. I. Arnold, ``Wave front evolution and equivariant Morse
lemma,'' Comm. Pure Appl. Math., {\bf29}, No.~6, 557--582 (1976).

\item{3.} V. I. Arnold. ``On the number of flattening points on
space curves,'' in: Ya.~G.~Sinai's Moscow Seminar on Dynamical System,
Am. Math. Soc. Transl., Ser.~2, Vol.~171, Providence, 1995, pp.~11--22.

\item{4.} M. Barner, ``\"Uber die Mindestanzahl station\"aren Schmeigebenen
bei geschlossenen streng konvexen Raumkurven,''
Abh. Math. Sem. Univ. Hamburg, {\bf20}, 196--215 (1956).

\item{5.} V. Arnold. Singularities of Caustics and Wave
Fronts, Kluwer, Dordrecht, 1990 (\S5.2, p.~100).

\item{6.} J. J. Nu\u no Ballesteros and M. C. Romero Fuster,
``Global bitangency properties of generic closed space
curves,'' Math. Proc. Cambridge Philos. Soc., {\bf112}, 519--526 (1992).

\item{7.} V. A. Vassiliev, ``Selfintersections of wavefronts and Legendre
(Lagrangian) characteristic numbers,'' Funkts. Anal. Prilozhen.,
{\bf16}, 131--133 (1982).

\endRefs
\transl{V. I. Arnold}
\end
