\magnification \magstep1
%
%The original version of these macros is due to J.P. Eckmann
%
%\magnification \magstep1
\vsize=22 truecm
\hsize=16 truecm
\hoffset=0.8 truecm
\normalbaselineskip=5.25mm
\baselineskip=5.25mm
\parskip=10pt
\immediate\openout1=key
\font\titlefont=cmbx10 scaled\magstep1
\font\authorfont=cmcsc10
\font\footfont=cmr7
\font\sectionfont=cmbx10 scaled\magstep1
\font\subsectionfont=cmbx10
\font\small=cmr7
\font\smaller=cmr5
%%%%%constant subscript positions%%%%%
\fontdimen16\tensy=2.7pt
\fontdimen17\tensy=2.7pt
\fontdimen14\tensy=2.7pt
%%%%%%%%%%%%%%%%%%%%%%
%%% macros %%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%
\def\dowrite #1{\immediate\write16 {#1} \immediate\write1 {#1} }
%\headline={\ifnum\pageno>1 {\hss\tenrm-\ \folio\ -\hss} \else {\hfill}\fi}
\newcount\EQNcount \EQNcount=1
\newcount\SECTIONcount \SECTIONcount=0
\newcount\APPENDIXcount \APPENDIXcount=0
\newcount\CLAIMcount \CLAIMcount=1
\newcount\SUBSECTIONcount \SUBSECTIONcount=1
\def\SECTIONHEAD{X}
\def\undertext#1{$\underline{\smash{\hbox{#1}}}$}
\def\QED{\hfill\smallskip
\line{\hfill\vrule height 1.8ex width 2ex depth +.2ex
\ \ \ \ \ \ }
\bigskip}
% These ones cannot be used in amstex
%
\def\real{{\bf R}}
\def\rational{{\bf Q}}
\def\natural{{\bf N}}
\def\complex{{\bf C}}
\def\integer{{\bf Z}}
\def\torus{{\bf T}}
%
% These ones can only be used in amstex
%
%\def\real{{\Bbb R}}
%\def\rational{{\Bbb Q}}
%\def\natural{{\Bbb N}}
%\def\complex{{\Bbb C}}
%\def\integer{{\Bbb Z}}
%\def\torus{{\Bbb T}}
%
%
%
\def\Re{{\rm Re\,}}
\def\Im{{\rm Im\,}}
\def\PROOF{\medskip\noindent{\bf Proof.\ }}
\def\REMARK{\medskip\noindent{\bf Remark.\ }}
\def\NOTATION{\medskip\noindent{\bf Notation.\ }}
\def\PRUEBA{\medskip\noindent{\bf Demostraci\'on.\ }}
\def\NOTA{\medskip\noindent{\bf Nota.\ }}
\def\NOTACION{\medskip\noindent{\bf Notaci\'on.\ }}
\def\ifundefined#1{\expandafter\ifx\csname#1\endcsname\relax}
\def\equ(#1){\ifundefined{e#1}$\spadesuit$#1 \dowrite{undefined equation #1}
\else\csname e#1\endcsname\fi}
\def\clm(#1){\ifundefined{c#1}$\clubsuit$#1 \dowrite{undefined claim #1}
\else\csname c#1\endcsname\fi}
\def\EQ(#1){\leqno\JPtag(#1)}
\def\NR(#1){&\JPtag(#1)\cr} %the same as &\tag(xx)\cr in eqalignno
\def\JPtag(#1){(\SECTIONHEAD.
\number\EQNcount)
\expandafter\xdef\csname
e#1\endcsname{(\SECTIONHEAD.\number\EQNcount)}
\dowrite{ EQ \equ(#1):#1 }
\global\advance\EQNcount by 1
}
\def\CLAIM #1(#2) #3\par{
\vskip.1in\medbreak\noindent
{\bf #1~\SECTIONHEAD.\number\CLAIMcount.} {\sl #3}\par
\expandafter\xdef\csname c#2\endcsname{#1\
\SECTIONHEAD.\number\CLAIMcount}
%\immediate \write16{ CLAIM #1 (\number\SECTIONcount.\number\CLAIMcount) :#2}
%\immediate \write1{ CLAIM #1 (\number\SECTIONcount.\number\CLAIMcount) :#2}
\dowrite{ CLAIM #1 (\SECTIONHEAD.\number\CLAIMcount) :#2}
\global\advance\CLAIMcount by 1
\ifdim\lastskip<\medskipamount
\removelastskip\penalty55\medskip\fi}
\def\CLAIMNONR #1(#2) #3\par{
\vskip.1in\medbreak\noindent
{\bf #1~#2} {\sl #3}\par
\global\advance\CLAIMcount by 1
\ifdim\lastskip<\medskipamount
\removelastskip\penalty55\medskip\fi}
\def\SECTION#1\par{\vskip0pt plus.3\vsize\penalty-75
\vskip0pt plus -.3\vsize\bigskip\bigskip
\global\advance\SECTIONcount by 1
\def\SECTIONHEAD{\number\SECTIONcount}
\immediate\dowrite{ SECTION \SECTIONHEAD:#1}\leftline
{{\sectionfont \SECTIONHEAD.}\ {\sectionfont #1} }
\EQNcount=1
\CLAIMcount=1
\SUBSECTIONcount=1
\nobreak\smallskip\noindent}
\def\APPENDIX#1\par{\vskip0pt plus.3\vsize\penalty-75
\vskip0pt plus -.3\vsize\bigskip\bigskip
\def\SECTIONHEAD{\ifcase \number\APPENDIXcount X\or A\or B\or C\or D\or E\or F \fi}
\global\advance\APPENDIXcount by 1
\vfill \eject
\immediate\dowrite{ APPENDIX \SECTIONHEAD:#1}\leftline
{\titlefont APPENDIX \SECTIONHEAD: }
{\sectionfont #1}
\EQNcount=1
\CLAIMcount=1
\SUBSECTIONcount=1
\nobreak\smallskip\noindent}
\def\SECTIONNONR#1\par{\vskip0pt plus.3\vsize\penalty-75
\vskip0pt plus -.3\vsize\bigskip\bigskip
\global\advance\SECTIONcount by 1
\immediate\dowrite{SECTION:#1}\leftline
{\sectionfont #1}
\EQNcount=1
\CLAIMcount=1
\SUBSECTIONcount=1
\nobreak\smallskip\noindent}
\def\SUBSECTION#1\par{\vskip0pt plus.2\vsize\penalty-75
\vskip0pt plus -.2\vsize\bigskip\bigskip
\def\SUBSECTIONHEAD{\number\SUBSECTIONcount}
\immediate\dowrite{ SUBSECTION \SECTIONHEAD.\SUBSECTIONHEAD :#1}\leftline
{\subsectionfont
\SECTIONHEAD.\number\SUBSECTIONcount.\ #1}
\global\advance\SUBSECTIONcount by 1
\nobreak\smallskip\noindent}
\def\SUBSECTIONNONR#1\par{\vskip0pt plus.2\vsize\penalty-75
\vskip0pt plus -.2\vsize\bigskip\bigskip
\immediate\dowrite{SUBSECTION:#1}\leftline{\subsectionfont
#1}
\nobreak\smallskip\noindent}
%%%%%%%%%%%%%TITLE PAGE%%%%%%%%%%%%%%%%%%%%
\let\endarg=\par
\def\finish{\def\endarg{\par\endgroup}}
\def\start{\endarg\begingroup}
\def\getNORMAL#1{{#1}}
\def\TITLE{\beginTITLE\getTITLE}
\def\beginTITLE{\start
\titlefont\baselineskip=1.728
\normalbaselineskip\rightskip=0pt plus1fil
\noindent
\def\endarg{\par\vskip.35in\endgroup}}
\def\getTITLE{\getNORMAL}
\def\AUTHOR{\beginAUTHOR\getAUTHOR}
\def\beginAUTHOR{\start
\vskip .25in\rm\noindent\finish}
\def\getAUTHOR{\getNORMAL}
\def\FROM{\beginFROM\getFROM}
\def\beginFROM{\start\baselineskip=3.0mm\normalbaselineskip=3.0mm
\obeylines\sl\finish}
\def\getFROM{\getNORMAL}
\def\ENDTITLE{\endarg}
\def\ABSTRACT#1\par{
\vskip 1in {\noindent\sectionfont Abstract.} #1 \par}
\def\ENDABSTRACT{\vfill\break}
\def\TODAY{\number\day~\ifcase\month\or January \or February \or March \or
April \or May \or June
\or July \or August \or September \or October \or November \or December \fi
\number\year}
\newcount\timecount
\timecount=\number\time
\divide\timecount by 60
\def\DRAFT{\font\footfont=cmti7
\footline={{\footfont \hfil File:\jobname, \TODAY, \number\timecount h}}
}
%%%%%%%%%%%%%%%%BIBLIOGRAPHY%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\period{\unskip.\spacefactor3000 { }}
%
% ...invisible stuff
%
\newbox\noboxJPE
\newbox\byboxJPE
\newbox\paperboxJPE
\newbox\yrboxJPE
\newbox\jourboxJPE
\newbox\pagesboxJPE
\newbox\volboxJPE
\newbox\preprintboxJPE
\newbox\toappearboxJPE
\newbox\bookboxJPE
\newbox\bybookboxJPE
\newbox\publisherboxJPE
\def\refclearJPE{
\setbox\noboxJPE=\null \gdef\isnoJPE{F}
\setbox\byboxJPE=\null \gdef\isbyJPE{F}
\setbox\paperboxJPE=\null \gdef\ispaperJPE{F}
\setbox\yrboxJPE=\null \gdef\isyrJPE{F}
\setbox\jourboxJPE=\null \gdef\isjourJPE{F}
\setbox\pagesboxJPE=\null \gdef\ispagesJPE{F}
\setbox\volboxJPE=\null \gdef\isvolJPE{F}
\setbox\preprintboxJPE=\null \gdef\ispreprintJPE{F}
\setbox\toappearboxJPE=\null \gdef\istoappearJPE{F}
\setbox\bookboxJPE=\null \gdef\isbookJPE{F} \gdef\isinbookJPE{F}
\setbox\bybookboxJPE=\null \gdef\isbybookJPE{F}
\setbox\publisherboxJPE=\null \gdef\ispublisherJPE{F}
}
\def\ref{\refclearJPE\bgroup}
\def\no {\egroup\gdef\isnoJPE{T}\setbox\noboxJPE=\hbox\bgroup}
\def\by {\egroup\gdef\isbyJPE{T}\setbox\byboxJPE=\hbox\bgroup}
\def\paper{\egroup\gdef\ispaperJPE{T}\setbox\paperboxJPE=\hbox\bgroup}
\def\yr{\egroup\gdef\isyrJPE{T}\setbox\yrboxJPE=\hbox\bgroup}
\def\jour{\egroup\gdef\isjourJPE{T}\setbox\jourboxJPE=\hbox\bgroup}
\def\pages{\egroup\gdef\ispagesJPE{T}\setbox\pagesboxJPE=\hbox\bgroup}
\def\vol{\egroup\gdef\isvolJPE{T}\setbox\volboxJPE=\hbox\bgroup\bf}
\def\preprint{\egroup\gdef
\ispreprintJPE{T}\setbox\preprintboxJPE=\hbox\bgroup}
\def\toappear{\egroup\gdef
\istoappearJPE{T}\setbox\toappearboxJPE=\hbox\bgroup}
\def\book{\egroup\gdef\isbookJPE{T}\setbox\bookboxJPE=\hbox\bgroup\it}
\def\publisher{\egroup\gdef
\ispublisherJPE{T}\setbox\publisherboxJPE=\hbox\bgroup}
\def\inbook{\egroup\gdef\isinbookJPE{T}\setbox\bookboxJPE=\hbox\bgroup\it}
\def\bybook{\egroup\gdef\isbybookJPE{T}\setbox\bybookboxJPE=\hbox\bgroup}
\def\endref{\egroup \sfcode`.=1000
\if T\isnoJPE \item{[\unhbox\noboxJPE\unskip]}
\else \item{} \fi
\if T\isbyJPE \unhbox\byboxJPE\unskip: \fi
\if T\ispaperJPE \unhbox\paperboxJPE\unskip\period \fi
\if T\isbookJPE ``\unhbox\bookboxJPE\unskip''\if T\ispublisherJPE, \else.
\fi\fi
\if T\isinbookJPE In ``\unhbox\bookboxJPE\unskip''\if T\isbybookJPE,
\else\period \fi\fi
\if T\isbybookJPE (\unhbox\bybookboxJPE\unskip)\period \fi
\if T\ispublisherJPE \unhbox\publisherboxJPE\unskip \if T\isjourJPE, \else\if
T\isyrJPE \ \else\period \fi\fi\fi
\if T\istoappearJPE (To appear)\period \fi
\if T\ispreprintJPE Preprint\period \fi
\if T\isjourJPE \unhbox\jourboxJPE\unskip\ \fi
\if T\isvolJPE \unhbox\volboxJPE\unskip, \fi
\if T\ispagesJPE \unhbox\pagesboxJPE\unskip\ \fi
\if T\isyrJPE (\unhbox\yrboxJPE\unskip)\period \fi
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% SOME SMALL TRICKS%%%%%%%%%%
\def \breakline{\vskip 0em}
\def \script{\bf}
\def \norm{\vert \vert}
\def \endnorm{\vert \vert}
\def\bfx{{\bf x}}
\def\bfn{{\bf n}}
\def\bfe{{\bf e}}
\def\bfu{{\bf u}}
\def\SS{{\cal S}}
\def\RR{{\cal R}}
\def\OO{{\cal O}}
\def\BB{{\cal B}}
\def\CC{{\cal C}}
\def\Tau{{\cal T}}
\def\spec{{\mathop {\rm spec}}}
\def\FF{{\cal F}}
\def\cite#1{{[#1]}}
\def\natural{{\rm I\kern-.18em N}}
\def\integer{{\rm Z\kern-.32em Z}}
\def\real{{\rm I\kern-.2em R}}
\def\complex{\kern.1em{\raise.47ex\hbox{
$\scriptscriptstyle |$}}\kern-.40em{\rm C}}
\TITLE HYPERBOLIC DYNAMICAL SYSTEMS AND
GENERATION OF MAGNETIC FIELDS BY PERFECTLY CONDUCTING FLUIDS.
\footnote{${}^{\rm (1)}$ } {\rm \baselineskip = 12 pt
This preprint is available from the
math-physics electronic preprints archive.
Send e-mail to
{\tt mp\_arc@math.utexas.edu} for instructions.}
\AUTHOR Rafael de la Llave
\footnote{${}^2$}{Supported in part by National Science Foundation Grants.}
\footnote{${}^3$}{ e-mail address {\tt llave@math.utexas.edu}}
\FROM Mathematics Department
The University of Texas at Austin
Austin TX\ \ 78712
\ENDTITLE
\ABSTRACT
We study the spectrum of the operator describing
the growth of magnetic fields in an infinitely conducting,
incompressible fluid.
We show that, in many circumstances, it consists of
an anulus and also give characterization
of this spectrum by the Lyapunov
exponents of periodic orbits of the Lagrangian flow.
\ENDABSTRACT
\SECTION Introduction
In this paper we will describe some recent results about the
infinitely conducting dynamo problem.
That is, we assume that we are given a divergence free
velocity field $u$ and
want to study the time evolution it induces on magnetic fields.
Using Maxwell's equations, Ohm's law and incompressibility of
the fluid, we obtain that the magnetic field evolves
according to the equation:
$$
{\partial B\over\partial t} + u\cdot \nabla B = (B\cdot\nabla) u
+ {1\over R_m} \nabla^2 B
\EQ(dynamo)
$$
In this paper, we will only consider the case
when ${1 / R_m} = 0$ which corresponds to the case that
the fluid is perfectly conducting.
It is well known that in when $1/R_m = 0$,
equation \equ(dynamo) has a purely geometric meaning and
a purely geometric solution that can be written as
$$
B_t = {f_t}_* B_0
$$
where $f_t$ is the Lagrangian flow generated by
$u$ according to $ {d \over dt} f_t = u \circ f_t$
and the push-forward ${f_t}_*$ is defined
by $\left( {f_t}_* B\right)(x) \equiv Df_t( f^{-1}_t(x) ) B( f^{-1}_t(x))$.
We notice that this solution indicates that the magnetic field
will grow when the term $Df$ is large, that is, when the
trajectories of particles suspended in the fluid
exhibit a very sensitive dependence to initial conditions
(See \cite{BC} \cite{FO}.)
Since a trajectory could expand for a while and then
start contracting, the substained rates of growth is
a global problem that can be best formulated as
studying $\lim_{t \to \infty }{1 \over t} \log || {f_t}_* ||$
This reduces to studying the spectral properties of
$f_1$.
The spectral properties of such
push-forward operators have been studied in \cite{Ma}.
We recall some results from this paper:
\CLAIM Theorem (Mather)
Let $f$ be a $C^1$ diffeomorphism of a manifold
\item{A)} If aperiodic points of $f$ are dense, then the
spectrum of $f_*$ acting on the complexification of
continuous vector fields with the supremum norm is
invariant under rotations in the complex plane
around the origin.(That is, if a complex number belongs to
the spectrum, all the complex numbers of the same
modulus also do.)
\item{B)} In that case, the spectrum consists of
a finite number of disjoint anuli. The number of
such anuli is not larger than the dimension of the
manifold.
\item{C)} $f$ is an Anosov diffeomorphism if and
only if, $1$ does not belong to this spectrum.
We recall that we say that $f$ is an Anosov diffeomorphism
if we can find an splitting of the tangent bundle
$T_xM = E^s_x \bigoplus E^u_x$ and constants $C > 0$,
$\lambda < 1$ such that:
$$| Df^n(x) u| \le C \lambda^n |u| n \ge 0 \iff u \in E^s_x$$
$$| Df^n(x) u| \le C \lambda^{-n} |u| n \le 0 \iff u \in E^u_x.$$
That is, a system is Anosov if and only if all the
infinitesimal displacements get exponentially stretched in the
future or in the past.
We will refer to the spectrum of $f_*$ on the complexification
of continuous vector fields equipped with the
sup norm as the {\sl Mather spectrum of $f$}
and denote it by $\Sigma_M(f)$. When there is no
risk of confusion, we will suppress the $f$.
Actually, in \cite{Ma}, we can find a generalization of
C) in \clm(Mather), namely:
\CLAIM Theorem (Mather2)
Assume that $\Sigma_M(f)$ is disjoint from
$\{ z \in \complex \big| |z| = \rho \}$.
Then, we can find an splitting of the
tangent bundle $T_xM = E^<_x \bigoplus E^>_x$
and $C>0$, $\epsilon >0$ such that:
$$| Df^n(x) u| \le C (\rho + \epsilon)^n |u| n \ge 0 \iff u \in E^<_x$$
$$| Df^n(x) u| \le C (\rho - \epsilon)^{-n} |u| n \le 0 \iff u \in E^>_x$$
Moreover, if we denote by $\Pi^>$, $\Pi^<$ the spectral projections of
$f_*$ associated to the two components of the complement of
the circle of radius $\rho$,
and by $\Pi^<_x$, $\Pi^>_x$ the
projections in the tangent bundle associated to the
splitting we have:
$$
\eqalign{ \left( \Pi^< v\right)(x) &= \Pi^<_x v(x) \cr
\left( \Pi^> v\right)(x) &= \Pi^>_x v(x)}
$$
Notice that \clm(Mather2) implies immediately the consequence that
there cannot be more anuli in the spectrum than the dimension of the
manifold.
The possible applications of these theorems to the dynamo problem
were suggested in \cite{Vi},(in which, moreover, there
is a thorough discussion of
the problem of the asymptotics of the spectrum as
${ 1 \over R_m}$ approaches zero, which we will not discuss here).1
Unfortunately, there are some reasons why the previous results
\clm(Mather), \clm(Mather2) may not be directly applicable to dynamo
theory. Let us just mention:
1)Since the $f_t$ is generated by the vector field $u$,
we always have ${f_t}_* u = u$ so that it is not an Anosov map.
2) In realistic situations, one does not expect stretching in the
whole manifold but only in a subset of it.
3) The vector field $B$ satisfies the non-trivial constraint of
having divergence zero. It is conceivable that
considering spaces of vector fields
with such constraint
changes the spectral properties.
4) For applications in hydrodynamics, one would like to
consider the spectrum in other spaces of more differentiable functions.
We just point that 1) and 2) are not very difficult objections.
If we consider a time periodic vector field in an appropriate
manifold, we could have an Anosov diffeomorphisms as the
time-T map. Also, Mather's arguments only needs small
adaptations to work for the cases of
Anosov flows or for cases in which a
closed invariant set takes the place of the manifold.
We will not discuss such minor problems here.
Objection 3), however is rather substantial, and, as
we will see later, it indeed changes the spectrum. For 4),
we just present some bounds on how the spectrum could change
when we consider spaces of differentiable vector fields.
Since there is considerable interest in computing the
spectrum of the dynamo operator, we also present some characterization
of $\Sigma_M$ and of the spectrum of the dynamo operator in terms of
periodic orbits. We just point out that periodic orbits have
played an important role in the study of the dynamo problem in
\cite{Vi}, \cite{VF} \cite{AG}. Of course, in the dynamical systems
literature, there are many results relating properties of
a dynamical system to properties of its periodic orbits.
We also point out that the characterization
we propose is quite easy to implement in the computer and
we have found it rather reliable.
\SECTION Statement of results
The arguments of \cite{Ma} work in the case that
spaces considered are closed under $\sup$ norms.
So, our first task
is to formulate precisely the Banach spaces
under $\sup$ norm that may have divergence zero.
Here are several reasonable possibilities
of spaces. Notice that if
$f$ is a volume preserving diffeomorphism, it preserves
all of them.
\item{$\chi^0$}: The $C^0$ closure of all the
$C^1$ vector fields with zero divergence.
\item{$\chi^1$}: The set of continuous vector fields with zero divergence in the
sense of distributions.
\item{$\chi^2$}: The set of $C^0$ vector fields which have zero flux over any
$C^1$ boundaryless surface.
\item{$\chi^{1,b}$}: The set of bounded vector fields
with zero divergence in the sense of distributions.
\item{$\chi^{2,b}$}: The set of bounded vector fields with zero
flux over ant $C^1$ boundaryless surface.
There are some obvious relations between those spaces.
For example $\chi^0 \subset \chi^1$, $\chi^0 \subset \chi^2$
$\chi^1 \subset \chi^{1,b}$,
$\chi^1 \ne \chi^{1,b}$,
$\chi^2 \subset \chi^{2,b}$,
$\chi^2 \ne \chi^{2,b}$.
In the torus, it is easy to show by using convolutions
that $\chi^0 = \chi^1 = \chi^2$ but I do not know how to prove
the same on an arbitrary manifold. Even if it would be interesting to
clarify their inclusions in general manifolds.
for the concerns of these paper, all the results can
be proved by the same methods for all the spaces.
We will start by stating some generalizations of characterizations
of Mather spectrum that we will find useful later.
\CLAIM Theorem (NewMather)
Let $f$ be a $C^1$ diffeomorphism on $M$ such that the
set of aperiodic orbits is dense.
\item{A)} The spectrum of $f_*$ acting on the complexification of
continuous vector fields is the same as the spectrum on the
complexification of bounded vector fields.
\item{B)} If periodic points of $f$ are dense, for any
$z \in \partial \Sigma_M(f)$, we can find a sequence $\{x_n\}_{n=0}^\infty$
of periodic orbits $f^{N_n}(x_n) = x_n$ such that for
some $\lambda_n \in \spec( Df^{N_n}(x_n))$
we have $|\lambda|_n^{1 / N_n} \to |z|$.
\item{C)} Assume that the splittings in the tangent bundle
induced by \clm(Mather2) are one-dimensional and that
$f$ is a transitive Anosov diffeomorphism.
Then,
$$
\Sigma_M(f) =\overline{ \left\{ \lambda e^{i \theta} \big|
\theta \in \real, \lambda^N \in \spec( Df^N(x)), f^N(x) = x \right\} }
$$
\CLAIM Theorem (new)
Let $f$ be a
volume preserving diffeomorphism with dense aperiodic orbits.
\item{A)} Then, the spectrum of $f_*$ acting on any of the
$\chi$ defined above is invariant under rotations.
(We will denote it by $\Sigma_{ND}(f))$
\item{B)} $\partial \Sigma_{ND} \subset \Sigma_M$,
$\partial \Sigma_M \subset \Sigma_{ND}$
The inner and outer boundaries of
$\Sigma_{ND}$ are the same as the inner and outer boundaries
of $\Sigma_M$.
\item{C)} If $f$ is a transitive Anosov
diffeomorphism with one-dimensional foliations,
then $\Sigma_{ND}$
consists of an anulus
whose inner and outer boundaries are the inner
and outer boundaries of $\Sigma_M(f)$.
Notice that a corollary is that the rate of growth of
the magnetic field evolving under \equ(dynamo) $( 1/R_m = 0 )$ can
be computed by taking the supremum of
the Lyapunov exponent at periodic orbits.
This is a rather simple algorithm to
implement numerically and preliminary
numerical calculations by the
author suggest it should be quite effective.
The relation of this exponent of growth of
the magnetic field and the behaviour at periodic orbits has
been studied numerically by \cite{AG} even for
the case when $1/R_m \ne 0$.
The work of \cite{Vi}, \cite{VF} also provides with some mathematically
rigorous results relating the behaviour at periodic orbits and
the behaviour of the exponent of growth
of \equ(dynamo).
\CLAIM Theorem (differentiable)
Let $f$ be a $C^{r+1}$ diffeomorphism, $r \in \natural$.
Assume that $\Sigma_M \subset \{ z \in \complex | \lambda_- \le |z|
\le \lambda_+\}$ then the spectrum of
$f_*$ acting on the Banach space of $C^r$ vector
fields is contained in
$ \{ z \in \complex | \lambda_- \lambda_+^r \le |z|
\le \lambda_+ \lambda_-^{-r}\}$
(If $f$ is volume preserving there are similar results)
\REMARK
We suspect that the conditions on the
spectrum on $C^r$ are optimal for most diffeomorphisms.
For H\"older regularities of sufficiently
small exponent, there are more detailed results
that we will not discuss here.
\SECTION Sketches of proofs.
In this section, we will just present sketches of proofs.
Full details as well as some extensions will appear
elsewhere.
As in the arguments in \cite{Ma} we will have to work simultaneously
on the level of the functional analysis properties of
the operator $f_*$ and on the level of the dynamics of $f$.
It is an standard result in functional analysis that
if $A$ is a bounded operator acting on a Banach space
${\cal X}$ and $z \in \complex$ is in the
boundary of the spectrum of $A$, we can find a
sequence $\{ u_n\}_{n=0}^\infty \subset {\cal X}$
such that $|| u_n|| = 1$, $|| Au_n - z u_n|| \to 0$.
Conversely, if such a sequence can be found, we may conclude that
$z$ is in the spectrum of $A$ ( it may not be in the boundary).
We well refer to the elements of such a sequence as
approximate eigenfunctions.
We also note that in general it is not true that all the
spectrum of a linear operator can be obtained through
approximate eigenfunctions, but all the spectrum
which is not residual can.
We will refer to the part of the spectrum that can
be characterized by the existence of
approximate eigenfunctions as the {\sl non-residual
spectrum}.
A large part of the argument will be to show that if
we have approximate eigenfunctions in one space,
we can use the dynamica properties of the system to
construct approximate eigenfunctions in another.
Let us start by discussing the proof of
\clm(NewMather).
Clearly, if we can find a sequence of approximate
eigenfunctions which are continuous, we can use it
as a sequence of bounded approximate eigenfunctions.
Hence $\partial \Sigma_{C^0} \subset \Sigma_{\rm bounded}$.
If we have a bounded approximate eigenfunction, it is possible to
obtain
an approximate eigenfunction of the
form
$v = \sum_{j=0}^N v_j$,
where $v_j$ has support in $f^j(x)$. This can
be extended in a continuous
fashion to a small neighborhood of those points so that it
is still an approximate eigenfunction.
We conclude that $\partial \Sigma_{\rm bounded} \subset \Sigma_{C^0}$.
We can use the argument in \clm(Mather2) to conclude that the
gaps in the spectrum induce splittings of the tangent bundle
characterized by the rates of growth of individual vectors. So that
the circles have to agree.
If periodic points of $f$ are dense, given an approximate
eigenfunction of $f_*$, we can restrict it to a periodic orbit and
hence, find an orbit whose return maps make a vector grow
at an exponential rate close to the modulus of the point in the
spectrum we are considering. This implies that there is an
eigenvalue of, roughly, the same exponent.
For a transitive Anosov diffeomorphism, any pair of periodic orbits has
a homoclinic connections. By Smale's analysis of the horseshoe,
we can find periodic orbits that spend any proportion
(up to an arbitrarily small error) of their
time in any neighborhood of the two periodic orbits.
By the invariance of the splitting, and the fact that the
splittings are one-dimensional, we know that the eigenvectors
of the points in the {\sl intermediary}
periodic orbit will be
quite close to the eigenvectors of the original
periodic orbits. The multiplication factors will be
as close as we like to those of the original periodic
orbits for any proportion of the time that we want
(up to arbitrarily small errors).
>From that, it follows that the closure of the set
of Lyapunov exponents at periodic orbits is
a convex set in the real line. Since we showed that it
can get as close as desired to the edges of
$\Sigma_M$ this finishes the argument for
\clm(NewMather)
We notice that a consequence of part $C$ is that, under the
hypothesis of the theorem, all of $\Sigma_M$ is not residual,
since we can take as approximate eigenfunctions
vectors supported on the periodic orbit
in the case that we consider bounded vector fields
or supported in an small balls near the periodic orbit
for the case of continuous vector fields.
We will only prove \clm(new) in the case that
we consider the spectrum on $\chi^0$. the other
cases are very similar.
First of all, we observe that since
approximate eigenfunctions in $\chi^0$ are
also approximate eigenfunctions in
the space of continuous functions we
have $\partial \Sigma_{ND} \subset \Sigma_M$.
We also observe that if we have an approximate
eigenfunction in $\chi^0$ or in $C^0$ we can also produce
an approximate eigenfunction for $z$
$v = \sum_{j=0} v_j$ with $v_j$ has support on
$f^j(x)$.
We observe that $\sum_{j=0}^N v_j e^{ij\theta}$ is
an approximate eigenfunction for $z e^{i\theta}$.
As soon as we show that such finitely supported
approximate eigenfunctions can be turned into
approximate eigenfunctions in $\chi^0$ we would have shown that
$\partial \Sigma_{ND}$ is rotation invariant and also that
$\partial \Sigma_M \subset \Sigma_{ND}$.
Since the outer edges of the spectrum of
$f_*$ are
characterized by the rates of growth of $ || f_*^n||$ and this
quantity is larger is we consider larger spaces, we obtain that the
$\Sigma_{ND}$ is contained in the anulus bounded by the
inner and outer edges of $\Sigma_M$.
The basic idea of this construction
of divergence free approximate eigenfunctions
is that it is possible to
construct vector fields of zero divergence and arbitrarily
small support that, nevertheless, have well defined directions.
Of course, it is quite impossible to make divergence zero
vector fields that have small support and that their vector values
are well defined. Nevertheless, we can observe that
for the rates of growth of vectors, the orientation of the
vectors is irrelevant. If we pick a direction, we can imagine
two very thin pipes running along it joined
to form a closed circuit by much thinker pipes.
If we make a fluid go around this circuit, it will have appreciable
velocity only in the thin segments of pipe,
which have the direction we want.
In the segments that the velocity has the wrong direction, it
has so small modulus that it does not affect the property of
being an approximate eigenfunction.
This finishes the proof of part A) and B) of \clm(new).
Actually, we have proved something more. We have proved that
all the points in $\Sigma_M$ that belong to the non-residual
spectrum, are in the spectrum of $\Sigma_{ND}$.
For transitive Anosov diffeomorphisms with
one-dimensional splittings, we have shown that
all of $\Sigma_M$ is non-residual, so that,
for such systems, $\Sigma_M \subset \Sigma_{ND}$.
Under these circumstances, we see that
$\Sigma_{ND}$ should consist of anuli
that cover the anuli of $\Sigma_M$.
The only possible gaps in $\Sigma_{ND}$
are the gaps in
$\Sigma_M$.
We want to derive a contradiction with the fact that there actually
some gaps. If some of these gaps existed,
we could find spectral projections on $\chi^0$ associated
with them. These spectral projections would
be characterized by rates of growth, of $f^n_* u$.
Hence, they should be the restrictions to the
space of vector fields of zero divergence of the
spectral projections of continuous vector fields.
By \clm(Mather2), we know that such projections can
only be projections on invariant subbundles, which is
easy to see that never preserve divergence free
vector fields. Hence, we conclude that
the existence of such
projections is impossible. Hence, there are no gaps in
$\Sigma_{ND}$.
Notice that, part B) alone does not allow us to
reach the same conclusion we reached here. If B)
was the only restriction, it could happen that
$\Sigma_{ND}$ had gaps but they could be
contained in one of anuli of $\Sigma_M$.
This looks rather implausible, but we do not
know at the moment how to exclude it nor how to present
examples.
This finishes the arguments for
\clm(new).
To prove \clm(differentiable),
it suffices to show that if $|z| > \lambda_+\lambda_-^{-r}$
then $-\sum_{i=0}^\infty z^{-i-1}{f_*}^i$
converges uniformly as operators in $C^r$ and that
if $|z| < \lambda_-\lambda_+^{-r}$
so does $\sum_{i=0}^\infty z^i f^{-i-1}_*$.
Once we prove uniform convergence,
one can justify formal rearrangements
so that those sums
are the resolvent of the operator in $C^r$.
To prove the desired uniform convergence for the
first series
we consider $L_1,\cdots,L_k$, $k \le r$, fist order
differential operators with $C^\infty$
coefficients and show that
$$
||L_1 \cdots L_k \left( f_*^i v\right)||_{C^0} \le
C (\lambda_+ \lambda_-^{-k} )^i i^{k} || v||_{C^k}
$$
Such estimates can be obtained by
applying the chain rule and estimating the number of terms.
Similar calculations are done in \cite{LMM}.
The reason to conjecture that these estimates are
optimal is that it is easy to check that they are
optimal for linear maps since then,
a simple calculation can get the
rate of groth of $C^r$ norm in terms of the
eigenvalues. One could imagine localizing
examples in a neighborhood of a periodic points that
has eigenvalues
whose moduli are close to the edges of the spectrum simultaneously.
It is not clear that all diffeomorphisms have periodic
orbits whose Lyapunov exponents are simultaneously close
to both edges -- we have proved that there are orbits
whose Lyapunov exponents are close to each of the
edges --, but using the homoclinic connections, it
seems plausible that a generic one does.
For $\alpha$ small enough, it is possible
to find regions arbitrarily close to
$\Sigma_M$ that enclose the
spectrum on $C^\alpha$ vector fields.
\SECTION Acknowledgements
I got interested in dynamo theory trough discussions
with B. Bailey. I have profited and enjoyed the
explanations
and encouragement of M. Vishik.
\SECTION References
\ref
\no{AG}
\by{E. Aurell, A. Gilbert}
\paper{Fast dynamos and determinants of singular integral operators}
\jour{Preprint}
\yr{1992}
\endref
\ref
\no{BC}
\by{B.J. Bayley. S. Childress}
\paper{Unsteady dynamo effects at large magnetic Reynolds numbers}
\jour{Geophys. Atrophys. Fluid. Dyn.}
\vol{49}
\pages{23--43}
\yr{1989}
\endref
\ref
\no{FO}
\by{J.M. Finn, E. Ott}
\paper{Chaotic flows and fast magnetic dynamos}
\jour{Phys. Fluids}
\vol{31}
\pages{2992}
\yr{1988}
\endref
\ref
\no{LMM}
\by{R. de la Llave, J.M. Marco, R. Moriy\'on}
\paper{Canonical perturbation theories of Anosov
diffeomorphisms and regularity results for the Livsic cohomology equation}
\jour{Ann. of Math.}
\vol{123}
\pages{537--611}
\yr{1986}
\endref
\ref
\no{Ma}
\by{J. Mather}
\paper{Characterization of Anosov diffeomorphisms}
\jour{Indag. Mat.}
\vol{30}
\pages{479-483}
\yr{1968}
\endref
\ref
\no{Vi}
\by{M. Vishik}
\paper{Magnetic field generation by the motion of a highly conducting fluid}
\jour{Geophys. Astrophys. Fluid. Mech.}
\vol{48}
\pages{151--167}
\yr{1989}
\endref
\ref
\no{VF}
\by{M. Vishik, S. Friedlander}
\paper{Dynamo theory methods for hydrodynamic instability}
\jour{Jour. Anal. Math. }
\yr{to appear}
\endref
\end