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\title
A REMARK ON PAPERS BY PIXTON AND OLIVEIRA: GENERICITY OF SYMPLECTIC DIFFEOMORPHISMS OF
$\bold S^ {\bold 2}$ WITH POSITIVE TOPOLOGICAL ENTROPY
\endtitle
\author HOWARD WEISS \endauthor
\affil The Pennsylvania State University \endaffil
\address Department of Mathematics, University Park, PA 16802, USA \endaddress
\email weiss\@math.psu.edu \endemail
\thanks This work was
partially supported by a National Science Foundation Postdoctoral Research
Fellowship and National Science Foundation Brazil Science and Technology
Research Fellowship. The author would like to thank IMPA for their immense
hospitality.\endthanks
\keywords{genericity, chaos, topological entropy} \endkeywords
\abstract We prove the existence of an open and dense subset of maps $ f \in Diff^{\infty}_{\omega}(S^2)$ which have positive topological entropy. It
follows that these maps have infinitely many hyperbolic periodic
points and an exponential growth rate of hyperbolic periodic points.
The proof is an application of Pixton's theorem.
\endabstract
\endtopmatter
\document
\normalbaselineskip=18pt
\normalbaselines
Topological entropy characterizes the total exponential orbit complexity
of a map with a single number (see \cite{KH} for definitions and properties).
Topological entropy, especially in low
dimensional cases, provides a wealth of qualitative structural information
about the system including the growth rate of the number of periodic
orbits \cite{K1}, existence of {\it large} horseshoes \cite{K2}, and the growth
rate of the
volume of cells of various dimensions \cite{Y}. Any map which possesses a
horseshoe, i.e., some power of the map is topologically conjugate to a
Bernoulli shift, has positive topological entropy and Katok \cite{K1} has
shown that the converse is true for surface diffeomorphisms. Hence
surface diffeomorphisms having positive topological entropy exhibit
very stochastic behavior {\it on some subset} of the surface - possibly
a set of Lebesgue measure zero. Thus, the stochastic behavior of
a surface diffeomorphism with positive topological entropy may not be
{\it physically observable}.
In this note, we observe that there exists an open and dense subset of $C^{\infty}$
symplectic (area-preserving) diffeomorphisms (symplectomorphisms) on $S^2$ having positive topological entropy,
and we observe a related result for symplectomorphisms of
the 2-torus. The symplectic hypothesis is essential,
for the Morse-Smale diffeomorphisms of $S^2$ form an open subset of
diffeomorphisms with zero topological entropy \cite{Pa}.
The proofs are easy applications of Pixton's Theorem \cite{P} and Olivera's
Theorem \cite{O}.
Let $M^2$ denote a $C^{\infty}$ compact surface. A symplectic form (or area form) $\omega$ on $M^2$ is a smooth positive differential two form. Denote by $Diff^{\infty}_{\omega}(M^2)$ the set
of symplectomorphisms of $M^2$ equipped with the Whitney topology, i.e., the
diffeomorphisms $f$ of $M^2$ which preserve the symplectic form $\omega$, i.e.,
$f^* \omega = \omega$.
We now state our main result:
\proclaim {Theorem 1}
There exists an open and dense subset of maps $ f \in Diff^{\infty}_{\omega}(S^2)$ which have positive topological entropy.
\endproclaim
\medskip
Theorem 1 is also true for $Diff^{2}_{\omega}(S^2)$ or even $Diff^{1 + \alpha}_{\omega}(S^2)$.
Applying Katok's result \cite{K1} that a surface diffeomorphism with positive
topological entropy contains horseshoes (which carry {\it most} of the entropy) to Theorem 1 yields the following corollary:
\proclaim {Corollary 1}
There exists an open and dense subset of maps $f \in Diff^{\infty}_{\omega}(S^2)$
which have infinitely many hyperbolic
periodic points and an exponential growth rate of (hyperbolic) periodic points.
\endproclaim
\medskip
\demo{Proof of Theorem 1} \enddemo
The heart of the proof is the following theorem of Pixton:
\proclaim {Theorem (Pixton [P])}
A residual subset of $Diff^{\infty}_{\omega}(S^2)$ has the property that the stable and unstable manifolds of every hyperbolic
periodic point intersect transversely.
\endproclaim
Note that Pixton's theorem makes no claim about the existence of hyperbolic periodic points. The openness statement in Theorem 1 immediately follows from the
$C^1$ structural stability of hyperbolic sets \cite{R}.
To prove the density statement, Let $f: S^2 \rightarrow S^2$ be an
symplectomorphism having the generic property in Pixton's
Theorem, i.e., that every hyperbolic periodic point has transverse homoclinic
points. Choose a neighborhood $U \subset Diff^{\infty}_{\omega}(S^2) $ of $f$.
While the map $f$ need not have a fixed point, it follows from the Lefschetz fixed point
theorem that $f$ has a periodic point $x$ \cite{F}. If $x$ is hyperbolic,
by hypothesis the stable and unstable manifolds of $x$
intersect transversely and thus $f$ has positive topological entropy \cite{S}.
Suppose $x$ is elliptic (or parabolic). By studying the Birkhoff normal
form of $f$ near $x$ and working with generating functions (to effect
symplectic
perturbations), Moser \cite{Mo} showed
that one can find a symplectomorphism $g \in U$
having a {\it hyperbolic} periodic point $y$ near $x$. Moser's result
was actually stated for real analytic maps but several authors (see for
instance \cite{P}) have observed that a simple modification of his argument
yields $C^r$ versions of the result for all $r$. Since hyperbolic period
points are $C^1$ structurally stable, Pixton's theorem yields a
symplectomorphism $h \in U$ with a hyperbolic fixed point
$z$ near $y$ such that the stable and unstable manifolds of $z$ intersect
transversely. Hence $h$ has positive topological entropy. \quad \qed
\medskip
Oliveira [O] generalized Pixton's Theorem to symplectomorphisms
of the 2-torus $\Bbb T^2$. The extension to surfaces of higher genus is unknown. The same argument as in Theorem 1 would prove the
existence of an open and dense subset of symplectomorphisms
of $\Bbb T^2$ which have positive topological entropy, {\it provided} one
knew that a dense set of symplectomorphisms of $\Bbb T^2$ has a
periodic point. This is a weak form of the $C^{\infty}$ closing
lemma on $\Bbb T^2$ and has not yet been proved.
The irrational translation on $\Bbb T^2$ is an example of a
symplectomorphism of $\Bbb T^2$ without periodic points.
However, it follows from the Lefschetz fixed point theorem \cite{V} that any
diffeomorphism $f:\Bbb T^2 \rightarrow \Bbb T^2$ for which $f^*:H^1(\Bbb T^2)
\rightarrow H^1(\Bbb T^2) $ does not have $+1$ as an eigenvalue
has a fixed point, where $ H^1(\Bbb T^2)$ denotes the first cohomology group
of $ \Bbb T^2$. The set of symplectomorphisms with
this property clearly form a {\it large} open set of symplectomorphisms.
This proves the following theorem:
\proclaim {Theorem 2}
In the open set of diffeomorphisms $f:\Bbb T^2 \rightarrow
\Bbb T^2$ for which $f^*:H^1(\Bbb T^2)
\rightarrow H^1(\Bbb T^2) $ does not have $+1$ as an eigenvalue,
there exists an open and dense subset of symplectic diffeomorphisms
which have positive topological entropy.
\endproclaim
\medskip
It follows that in every connected component of $Diff_{\omega}^{\infty}$ with
one exception, there exists an open and dense subset of symplectic diffeomorphisms
which have positive topological entropy.
It should be pointed out that in {\it many}
cases, there already exists
a result stronger than Theorem 2. Let $A$ be an element of $SL(2, \Bbb Z)$ and
$f$ diffeomorphism of $\Bbb T^2$ inducing $A$ in homology. Then one has a short
list of possibilities:
\roster
\item Both eigenvalues of $A$ are real and of modulus different from $1$. In this case
it easily follows from Manning's {\it entropy inequality} \cite{Ma} between entropy and the log of the
spectral radius of $A$ that $f$ {\it always} has positive topological entropy.
\item Eigenvalues of $A$ are roots of unity of order 1, 2, 3, 4, or 6. If, as in
Theorem 2 we exclude $1$ as an eigenvalue, we are left with only finitely many
conjugacy classes in $SL(2, \Bbb Z)$ for which Theorem 2 is non-trivial. These
classes are just conjugacy classes of elements of finite order in $SL(2, \Bbb Z)$.
In other words, Theorem 2 yields new information for finitely many connected components of $Diff_{\omega}(\Bbb T^2)$.
\endroster
\medskip
We leave the reader with two intriguing open questions:
\roster
\item {\rm Is it true that
an open and dense set of symplectomorphisms on {\it every} surface,
or more generally, every smooth compact symplectic manifold has positive
topological entropy? }
\item {\rm A more refined invariant measuring the complexity of the
orbit structure for a symplectic map is the measure theoretic entropy (with
respect to the Lebesgue measure induced by the symplectic form.) Metric entropy
gives the exponential growth rate of the statistically significant orbits.
If a map has positive metric entropy, then it exhibits very stochastic
behavior of a set of positive Lebesgue measure. Recall that a map
with positive topological entropy may exhibit stochastic behavior on a set of Lebesgue measure zero. Is it true that
an open and dense set of symplectic diffeomorphisms on every surface,
or more generally, every smooth compact symplectic manifold has positive
metric entropy? The KAM theorem implies that the set of ergodic
symplectomorphisms is not dense.}
\endroster
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\enddocument