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\begin{document}

\begin{center}
{\Huge Thermodynamic Formalism, MA 793}
\end{center}

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\section*{Preface}

Thermodynamic formalism is the theory of Gibbs Measures. Technically,
those are defined on abstract spaces of symbolic sequences called
subshifts of finite type (the elements of those spaces are infinite
sequences of some symbols). In that context, it is hard to see the real
meaning of Gibbs measures and their relevance for dynamical systems and
classical physics. It is also difficult to learn Gibbs measures
starting with that abstract context.

We adopt a different approach. First we describe two classes of
dynamical systems: {\em Anosov diffeomorphisms} and {\em Axiom~A
diffeomorphisms}. Then we construct {\em Markov partitions}, which will
naturally lead to a symbolic representation (coding) by {\em subshifts
of finite type}. Gibbs measures will then correspond to invariant
measures for Anosov and Axiom~A diffeomorphisms.

\section*{Notation}

\begin{itemize}
\item $M$ denotes a smooth (of class $C^{\infty}$) compact
connected manifold with a Riemannian metric. \item $d$ denotes the
dimension of $M$, i.e.\ dim$\,M=d$. \item $\cT_xM$ denotes the
tangent space to $M$ at $x\in M$, and $\cT M=\cup_x\cT_xM$ the
tangent bundle of $M$. \item $\|v\|$ denotes the norm of $v\in\cT
M$. \item $\angle(u,v)$ denotes the angle between $u,v\in\cT_x M$.
\item $T:M\to M$ denotes a diffeomorphisms of class $C^p$, $p\geq
1$. \item $D_xT: \cT_xM \to \cT_{T(x)}M$ denotes the derivative of
$T$ at $x\in M$.
\end{itemize}

\newpage

\section{Anosov Diffeomorphisms}

\noindent {\bf 1.1 Definition (Anosov Diffeomorphism)}\\

Suppose that each tangent space $\cT_xM$ with $x\in M$ is a direct sum
$$
      \cT_xM = E_x^u \oplus E_x^s
$$
of subspaces so that
\begin{itemize}
\item[(a)] $D_xT(E_x^u) = E_{Tx}^u$ and $D_xT(E_x^s) = E_{Tx}^s$
\item[(b)] there exist constants $C>0$ and $\lambda\in (0,1)$ such that
$$
        \|D_xT^n(v)\| \leq C\lambda^n \|v\|\quad
        {\rm when}\quad v\in E_x^s,\quad n\geq 0
$$
$$
        \|D_xT^{-n}(v)\| \leq C\lambda^n \|v\|\quad
        {\rm when}\quad v\in E_x^u,\quad n\geq 0
$$
\item[(c)] The spaces $E^u_x$ and $E^s_x$ vary continuously with $x$,
and the angle between $E^u_x$ and $E^s_x$ is uniformly bounded away
from zero.
\end{itemize}
%\vspace*{5mm}
Note: the dimensions $d^u=\,{\rm dim}\,E^u_x$ and $d^s=\,{\rm
dim}\,E^s_x$ are constant on $M$.

\vspace*{10mm} \centerline{\epsffile{TF-1.eps}}

\begin{center}
Figure 1: The action of $D_xT$ on the subspaces $E^u_x$ and $E^s_x$.
\end{center} \vspace*{5mm}

\noindent{\bf 1.2 Exercise}

\begin{itemize}
\item[(a)] Show that (a) and (b) of 1.1 imply $\|D_xT^{-n}(v)\| \geq
C\lambda^{-n}\|v\|$ for all $v\in E_x^s$ and $n\geq 0$ and $\|D_xT^n(v)\|
\geq C\lambda^{-n}\|v\|$ for all $v\in E_x^u$ and $n\geq 0$. \item
[(b)] Prove that $T^{-1}:M\to M$ is an Anosov diffeomorphism.
\item[(c)] Show that if $v\in\cT_xM$ is such that $v\notin E_x^u$ and
$v\notin E_x^s$, then there is a $C_v>0$ such that $\|D_xT^n(v)\| \geq
C_v\lambda^{-|n|}$ for all $n\in\ZZ$, i.e.\ the images of $v$ grow
exponentially both in the future and in the past.\item[(d)] Prove that
$T:M\to M$ has at least one invariant measure $\mu$. \item[(e)] Let
$\mu$ be a $T$-invariant measure on $M$. Prove that at $\mu$-almost
every point $x\in M$ Lyapunov exponents exist (recall the Oseledec
Theorem). Then show that there are exactly $d^u$ positive Lyapunov
exponents (all $\geq \log\lambda^{-1}$) and exactly $d^s$ negative
Lyapunov exponents (all $\leq\log\lambda$).
\end{itemize}\vspace*{5mm}

\noindent {\bf 1.3 Remarks}\\

The subspace $E^u_x$ and its vectors are said to be {\em unstable}, the
subspace $E^s_x$ and its vectors are said to be {\em stable}. The
assumption (b) in Definition 1.1 implies a uniform contraction of
stable vectors and uniform expansion of unstable vectors (under the
future iterations). The set of assumptions (a)--(c) specifies what is
frequently called {\em uniform hyperbolicity}.

As Exercise 1.2 (e) shows, the assumptions (a)--(c) imply that there
are no zero Lyapunov exponents for the map $T$ (this property is often
referred to as {\em hyperbolicity}). Hence, uniform hyperbolicity
implies hyperbolicity.

The converse is not true, i.e.\ the hyperbolicity does not imply
uniform hyperbolicity. Indeed, the hyperbolicity is only an asymptotic
property meaning that the images of stable (unstable) vectors contract
(respectively, expand) {\em eventually}, in the limit, as time goes to
infinity. This does not prevent temporary opposite trends -- when
stable vectors expand and unstable
vectors contract during arbitrarily long time periods.\\

\noindent{\bf 1.4 Adapted Metric}\\

Technically, the definition 1.1 does not prevent temporary expansion of
stable vectors and contraction of unstable vectors (but the time
periods of such reverses are strictly bounded by $n$ such that
$C\lambda^n<1$). Interestingly, for every Anosov diffeomorphism there
exist a Riemannian metric in which Definition 1.1 (b) holds with $C=1$.
This metric is called {\em adapted metric} or {\em Lyapunov metric}.

The adapted metric makes many calculations simpler. For our
convenience, we will always use an adapted metric.
\\

\noindent{\bf 1.5 Proposition}\\

The assumption (c) in Definition 1.1 follows from (a) and (b).\\

\proof Suppose $x_n\to x$. By taking a subsequence, if necessary, we
can assume that $k=\,$dim$\,E^s_{x_n}$ is independent of $n$. Let
$e_1^{(n)},\ldots,e_k^{(n)}$ be an orthonormal basis in $E_{x_n}^s$. By
taking a subsequence, if necessary, we can enforce the convergence
$e_i^{(n)}\to e_i$ as $n\to\infty$, for each $i=1,\ldots,k$. Let
$E^1_x=\,$span$\{e_1,\ldots,e_k\}$. By continuity, $\|D_xT^n(v)\| \leq
\lambda^n \|v\|$ and $\|D_xT^{-n}(v)\| \geq \lambda^{-n} \|v\|$ for all
$v\in E_x^s$ and $n\geq 0$. Hence, $E^1_x\subset E^s_x$. A similar
argument shows that $E^u_{x_i}$ converge to a subspace $E^2_x\subset
E^u_x$. Since dim$\,E^1_x+\,$dim$\,E^2_x=d$, then we have $E^1_x=
E^s_x$ and $E^2_x= E^u_x$. $\Box$\\

\noindent{\bf 1.6 Proposition}\\

Let $\tilde{T}:M\to M$ be a small $C^1$ perturbation of an Anosov
diffeomorphism $T:M\to M$. That is, there is a small $\varepsilon>0$
such that dist$(T(x),\tilde{T}(x))<\varepsilon$ and
$\|D_xT-D_x\tilde{T}\|<\varepsilon$ for all $x\in M$. Then $\tilde{T}$
is also an Anosov diffeomorphism.
\\

\proof The main part is to construct the subspaces $\tilde{E}^u_x$ and
$\tilde{E}^s_x$ for the perturbed diffeomorphism $\tilde{T}$. We fix a
small $\alpha>0$ and take a cone $C^u_x$ around $E^u_x$ with opening
$\alpha$, i.e.\ $C^u_x=\{v\in\cT_xM:\ \angle(v,E^u_x)<\alpha\}$. It is
easy to check, by a perturbation argument, that
$D_x\tilde{T}(C^u_x)\subset C^u_{\tilde{T}x}$ and $\|D_x\tilde{T}(v)\|
\geq \lambda_1^{-1} \|v\|$ for all $v\in C^u_x$ and some constant
$\lambda_1\in (\lambda,1)$ that depends on $\alpha$. Then we define
$\tilde{E}^u_x=\cap_{n\geq 0}D_{\tilde{T}^{-n}x}\tilde{T}^n
(C^u_{\tilde{T}^{-n}x})$. Similarly, $\tilde{E}^s_x=\cap_{n\geq
0}D_{\tilde{T}^{n}x}\tilde{T}^{-n} (C^s_{\tilde{T}^{n}x})$.
\\

\vspace*{10mm} \centerline{\epsffile{TF-2.eps}}

\begin{center}
Figure 2: Stable and unstable cones.
\end{center} \vspace*{5mm}


\noindent {\bf 1.7 Definition (local stable and unstable manifolds)}\\

Let $x\in M$ and $\varepsilon>0$. A {\em local unstable manifold}
of $x$ is
$$
     W^u_{\varepsilon}(x)=\{y\in M:\
    {\rm dist}(T^{-n}x, T^{-n}y)<\varepsilon
    \quad\forall n\geq 0\}
$$
Similarly, a {\em local stable manifold} of $x$ is
$$
     W^s_{\varepsilon}(x)=\{y\in M:\
    {\rm dist}(T^nx, T^ny)<\varepsilon
    \quad\forall n\geq 0\}
$$
\\

\noindent {\bf 1.8 Remark}\\

There is a natural duality between stable and unstable objects in this
theory. These objects get interchanged if one replaces $T:M\to M$ with
$T^{-1}:M\to M$. For brevity, we will sometimes mention just one of
them and suppress the other.
\\

\noindent{\bf 1.9 Theorem}\\

Let $T:M\to M$ be a $C^p$ Anosov diffeomorphism. There is a small
$\varepsilon>0$ such that for any point $x\in M$
\begin{itemize}\item[(a)] $W^u_{\varepsilon}(x)$ and
$W^s_{\varepsilon}(x)$ are embedded $C^p$ disks in $M$ with
$\cT_xW^{u,s}_{\varepsilon}(x)=E^{u,s}_x$.
\item[(b)] We have
$$
   {\rm dist}(T^{-n}x, T^{-n}y)\leq \lambda^n\,
   {\rm dist}(x,y)\quad\forall y\in W^u_{\varepsilon}(x),\ \
   n\geq 0
$$
and
$$
   {\rm dist}(T^{n}x, T^{n}y)\leq \lambda^n\,
   {\rm dist}(x,y)\quad\forall y\in W^s_{\varepsilon}(x),\ \
   n\geq 0
$$
\item[(c)] We have
$$
     \cT_yW^{u,s}_{\varepsilon}(x) = E^{u,s}_y\quad
       \forall y\in W^{u,s}_{\varepsilon}(x)
$$
\item[(d)] The disks $W^{u,s}(x)$ vary continuously with $x$.
\end{itemize}

\proof A complete proof of this theorem is quite involved, and its
details are not much relevant to our further studies. We only
sketch a proof here. The idea of the proof resembles that of
Proposition~1.6. Define a cone $C^u_x$ with a small opening
$\alpha>0$. Then fix a small $\varepsilon>0$. At any point $x\in
M$ the $\varepsilon$-ball in the tangent space $\cT_xM$ can be
identified with a small neighborhood $\cU(x)$ of $x$ via the
exponential map. Now consider all local manifolds of dimension
$d^u$ in $\cU(x)$ passing through $x$ whose tangent vectors lie
within the cone $C^u_x$.

Under the map $T$, each of these manifolds stretches out in all
directions by a factor $\lambda^{-1}$. It is easy to see that the
image of each such manifold covers one of the manifolds
constructed similarly at the point $T(x)$. Removing the excesses
(the parts of the images stretching beyond $\cU(Tx)$), we obtain a
transformation of manifolds in $\cU(x)$ onto those in $\cU(Tx)$.
Now it is easy to check that the images get closer together: their
variation in the direction parallel to $E^s_{Tx}$ shrinks by a
factor $\sim\lambda^2$. Also, the variation of their tangent
planes shrinks by a factor of  $\sim\lambda^2$. Now we take all
those manifolds constructed at the point $T^{-n}(x)$ and map them
under $T^n$. Their images make a very thin ``pancake''. It is easy
to see that in the limit $n\to\infty$ they converge to a single
$C^1$ manifold, call it $W^u_{\varepsilon}(x)$. It is also easy to
check that $\cT_xW^u_{\varepsilon}(x)=E^u_x$ (the ``pancake''
always contains some manifolds whose tangent plane at $x$ is
$E^u_x$).

An extra technical work (we skip it) is required to show that
$W^u_{\varepsilon}(x)$ is $C^p$ smooth, i.e.\ the stable and
unstable manifolds are as smooth as the Anosov diffeomorphism $T$
itself. Thus we obtain (a). The claim (b) can be verified
directly. The claim (c) holds since the same construction can be
applied to the point $y$. The claim (d) follows from (c) and the
continuity of the spaces $E^{u,s}_x$.
\\

\vspace*{10mm} \centerline{\epsffile{TF-3.eps}}

\begin{center}
Figure 3: Stable and unstable manifolds.
\end{center} \vspace*{5mm}

\noindent {\bf 1.10 Remark}\\

We will denote local stable and unstable manifolds by $W^u(x)$ and
$W^s(x)$ suppressing their size $\varepsilon$. Note that local
stable and unstable manifolds are thus not unique. However they
are essentially unique. In particular, if $z\in W^u(x)\cap
W^u(y)$, then the intersection $W^u(x)\cap W^u(y)$ contains an
open neighborhood of $z$ in both manifolds (a similar statement
holds for stable manifolds).
\\

\noindent {\bf 1.11 Definition (global stable and unstable manifolds)}\\

{\em Global stable and unstable manifolds} are defined by
$$
   \cW^s(x)=\cup_{n\geq 0}T^{-n}W^s(T^{n}x)
     \quad {\rm and}\quad
   \cW^u(x)=\cup_{n\geq 0}T^nW^u(T^{-n}x)
$$

\noindent {\bf 1.12 Remarks}\\

We have
$$
     \cW^u(x)=\{y\in M:\
    {\rm dist}(T^{-n}x, T^{-n}y)\to 0\ \
    {\rm as}\ \  n\to\infty\}
$$
(a similar property holds for $\cW^s(x)$).

Also, $T^n(\cW^u(x))=\cW^u(T^nx)$ and $T^n(\cW^s(x))=\cW^s(T^nx)$
for all $n\in\ZZ$. Global stable (and unstable) manifolds are
unique. In particular, if $\cW^u(x)\cap \cW^u(y)\neq\emptyset$,
then $\cW^u(x) =\cW^u(y)$ (a similar statement holds for stable
manifolds).

The global stable and unstable manifolds are ``infinitely large'', so
that they never terminate. However, since the entire manifold $M$ is
compact, the global stable and unstable manifold have to wrap around
$M$ and, typically, become dense in $M$.
\\

\noindent {\bf 1.13 Examples}\\

In the course of Dynamical Systems, MA 760, we have studied hyperbolic
toral automorphisms. Those are Anosov diffeomorphisms. Their stable and
unstable manifolds are straight lines parallel to the eigenvectors of
the corresponding matrix. (Recall that we proved that each such line is
dense on the torus.)

By Proposition 1.6, small $C^1$ perturbations of hyperbolic toral
automorphisms are Anosov diffeomorphisms, too. It is rather strange
that there are no more simple examples...
\\

\vspace*{10mm} \centerline{\epsffile{TF-4.eps}}

\begin{center}
Figure 4: Canonical coordinates.
\end{center} \vspace*{5mm}

\noindent {\bf 1.14 Proposition (Canonical coordinates)}\\

There exists a small $\varepsilon_0>0$ such that if
dist$(x,y)<\varepsilon_0$, then the intersection
$$
       [x,y]:=W^s(x)\cap W^u(y)
$$
consists exactly of one point. Furthermore, the point $[x,y]$ depends
continuously on $x$ and $y$.

As a result, one can represent every point $y$ in the
$\varepsilon_0$-neighborhood of $x$ by two points
$$
    y'=[x,y]\in W^s(x)
    \quad{\rm and}\quad
    y''=[y,x]\in W^u(x)
$$
The points $y',y''$ play the role of coordinates of the point $y$, and
the manifolds $W^s(x)$ and $W^u(x)$ are coordinate axes.
\\

\proof The intersection $W^s(x)\cap W^u(x)$ is transversal, and such
intersections are preserved under small perturbations.
\\

\noindent {\bf 1.15 Corollary}\\

The map $T:M\to M$ is expansive, i.e.\ for any $x\neq y$ there is an
$n\in\ZZ$ such that dist$(T^nx,T^ny)>\varepsilon_0$.
\\

\proof Otherwise $y\in W^u(x)\cap W^s(x)$ by 1.10. However, this
is impossible by 1.14.

\newpage

\section{Axiom A Diffeomorphisms}

\noindent {\bf 2.1 Definition (Hyperbolic Set)}\\

A closed subset $\Lambda\subset M$ is {\em hyperbolic} if $T(\Lambda)
=\Lambda$ and each tangent space $\cT_xM$ with $x\in \Lambda$ is a
direct sum
$$
      \cT_xM = E_x^u \oplus E_x^s
$$
of subspaces so that
\begin{itemize}
\item[(a)] $D_xT(E_x^u) = E_{Tx}^u$ and $D_xT(E_x^s) = E_{Tx}^s$
\item[(b)] there exist constants $C>0$ and $\lambda\in (0,1)$ such that
$$
        \|D_xT^n(v)\| \leq C\lambda^n \|v\|\quad
        {\rm when}\quad v\in E_x^s,\quad n\geq 0
$$
$$
        \|D_xT^{-n}(v)\| \leq C\lambda^n \|v\|\quad
        {\rm when}\quad v\in E_x^u,\quad n\geq 0
$$
\item[(c)] The spaces $E^u_x$ and $E^s_x$ vary continuously with $x$,
and the angle between $E^u_x$ and $E^s_x$ is uniformly bounded away
from zero.
\end{itemize}
Note that the conditions (a)--(c) are identical to those in Definition
1.1, except now we require them only for $x\in\Lambda$.

Observe that if $T:M\to M$ is an Anosov diffeomorphism, then the entire
manifold $M$ is a hyperbolic set.
\\

\noindent {\bf 2.2 Definition (Nonwandering Points)}\\

A point $x\in M$ is nonwandering if
$$
        \cU \cap \cup_{n>0} T^n(\cU) \neq \emptyset
$$
for every open neighborhood $\cU$ of $x$. The set of all nonwandering
points $x\in M$ is denoted by $\Omega(T)$.
\\

\noindent{\bf 2.3 Exercise}\\

Show that $\Omega(T)$ is nonempty, closed and $T$-invariant. Show
that $\Omega(T)$ contains all periodic points of $T$.
\\

\noindent {\bf 2.4 Definition (Axiom A Diffeomorphism)}\\

$T:M\to M$ is an {\em Axiom A diffeomorphism} if $\Omega(T)$ is
hyperbolic and periodic points are dense in $\Omega(T)$.
\\

\noindent {\bf 2.5 Example: Smale's horseshoe}
\\

Let $R$ be a rectangle and $T:R\to \IR^2$ be a map that stretches $R$
in the horizontal direction, bends it into the shape of a ``horseshoe''
(see Figure 5) so that $R\cap T(R)$ consists of two narrow horizontal
rectangles. We assume that $T$ restricted to $R\cap T^{-1}(R)$ is
linear and its derivative is
$$
     DT = \left (\begin{array}{cc} \lambda_1 & 0 \\
     0 & \lambda_2 \end{array} \right )
$$
for some $\lambda_1>2$ and $\lambda_2<1/2$.

The maximal invariant set
$$
           \Omega(T)=\cap_{n=-\infty}^{\infty} T^n(R)
$$
is a Cantor set with a natural product structure (it is the product of
a Cantor set on the $x$ axis and a Cantor set on the $y$ axis).


\vspace*{5mm} \centerline{\epsffile{TF-5.eps}}

\begin{center}
Figure 5: Smale's horseshoe.
\end{center} \vspace*{5mm}

Note: the map $T$ on $R$ is not exactly a diffeomorphism of a compact
manifold. It can be extended to such, but this is rather difficult, and
the details of such an extension would be completely irrelevant. We
will simply pretend that we deal with a diffeomorphism of a compact
manifold, and in fact all our considerations will be restricted to $R$.
\\

\noindent{\bf 2.6 Exercise}\\

(a) Show that the conditions (a)--(c) of Definition 2.1 are satisfied
for $\Lambda=\Omega(T)$ in the horseshoe example. Describe the spaces
$E^u_x$ and $E^s_x$, find the constants $C$ and $\lambda$. Deduce that
$\Omega(T)$ is a hyperbolic set.

(b) [This is optional, it might be difficult!] Show that periodic
points are dense in $\Omega(T)$. Conclude that $T$ is an Axiom~A
diffeomorphism. [Note: this fact will actually follow from some
statements that we prove below.]
\\

\noindent{\bf 2.7 Remark}\\

Smale's horseshoe is a canonical example of an Axiom~A diffeomorphism,
which is essentially different from an Anosov diffeomorphism. One can
easily extend the horseshoe construction to any dimension.

In a sense, all Axiom~A diffeomorphisms are either Anosov or horseshoes
or combinations thereof.
\\

\noindent{\bf 2.8 Extension of Anosov theory}\\

Nearly all the results we obtained in Section 1 for Anosov
diffeomorphisms extend to Axiom~A diffeomorphisms with obvious
modifications:
\begin{itemize} \item The claims in Exercise~1.2, the existence of an
adapted metric 1.4, and Proposition 1.5 apply word for word. \item
Definition 1, Remark 1.8, Theorem 1.9, and Remark 1.10 apply to
all the points $x\in\Omega(T)$. Note, however: stable and unstable
manifolds may not completely belong to $\Omega(T)$ (recall the
horseshoe, for example!). \item Theorem 1.9 applies to all points
$x\in \Omega(T)$ and $y\in W^{u,s}(x)\cap\Omega(T)$. \item Remark
1.10, Definition 1.11, and Remark 1.12 remain unchanged. \item
Proposition 1.14 needs an important addition: if $x,y\in\Omega(T)$
and dist$(x,y)<\varepsilon_0$, then $[x,y]\in\Omega(T)$ (this
follows from Lemma 2.9 below). \item Corollary 1.15 applies to the
map $T:\Omega(T)\to\Omega(T)$.
\end{itemize}

\noindent{\bf 2.9 Lemma}\\

Let $T$ be an Axiom~A diffeomorphism and $x,y\in\Omega(T)$. Assume
that $\exists z\in\cW^u(x)\cap\cW^s(y)$ and $\exists
w\in\cW^s(x)\cap\cW^u(y)$ and both intersections are transversal
at the points $z$ and $w$. Then $z,w\in\Omega(T)$.
\\

\proof Let $z\in\cW^u(x)\cap\cW^s(y)$ and $w\in\cW^s(x)\cap\cW^u(y)$.
We need to show that $z$ is nonwandering. Let $\cU$ be an open
neighborhood of $z$. Since periodic points are dense in $\Omega(T)$,
then $x$ and $y$ can be approximated by periodic points, say $p$ and
$q$, with periods $m$ and $n$, respectively, so that
$\cL:=\cW^u(p)\cap\cU\neq\emptyset$ and $\cL\cap\cW^s(q)\neq\emptyset$.
We will show that $T^k(\cL)\cap\cU\neq\emptyset$ for some large $k$.

Note that $p$ and $q$ are fixed points for the map $S=T^{mn}$.
Observe that the images $S^k(\cL)$ accumulate on $W^u(q)$ as $k$
grows, i.e.\ $W^u(q)\subset\lim_{k\to\infty}S^k({\cal L})$. Next,
for large $k$ those images stretch out along $\cW^u(q)$ and cross
$\cW^s(p)$ near the point $w$. Hence, for even larger $k$ they
will come back to $p$ along $\cW^s(p)$, and therefore will
accumulate on $\cW^u(p)$. For still larger $k$, they will stretch
out along $\cW^u(p)$ and cross $\cU$.
\\

\vspace*{10mm} \centerline{\epsffile{TF-6.eps}}

\begin{center}
Figure 6: Proof of Lemma 2.9.
\end{center} \vspace*{5mm}

%There are three fundamental theorems on Axiom~A diffeomorphisms that we
%will need.
% For their proofs we refer to Bowen's book (these proofs can
% hardly be written better than in his book!).

\noindent{\bf 2.10 Definition (topological transitivity and mixing)}\\

A map $T:X\to X$ of a topological space $X$ is {\em topologically
transitive} if for any open subsets $U,V\subset X$ there is an $n\geq
0$ such that $U\cap T^n(V)\neq\emptyset$.

$T$ is {\em topologically mixing} if for any open subsets $U,V\subset
X$ there is an $n_0\geq 0$ such that $U\cap T^n(V)\neq\emptyset$ for
all $n>n_0$.
\\

From now on $T:M\to M$ will always be an Axiom~A diffeomorphism.
\\

\noindent{\bf 2.11 Theorem (Spectral Decomposition)}\\

(a) We have $\Omega(T)=\Omega_1\cup\Omega_2\cup\cdots\cup\Omega_s$, a
disjoint union of closed sets, such that $T(\Omega_i)=\Omega_i$ and
$T|_{\Omega_i}$ is topologically transitive for each $i$.

(b) Next, $\Omega_i=\Omega_{i,1}\cup\cdots\cup\Omega_{i,s_i}$, a
disjoint union of closed sets, such that
$T(\Omega_{i,j})=\Omega_{i,j+1}$ (with identification
$\Omega_{i,s_i+1}=\Omega_{i,1}$), and $T^{s_i}|_{\Omega_{i,j}}$ is
topologically mixing for each $j$.
\\

The sets $\Omega_i$ are called {\em basic sets}.
\\

\proof We will define an equivalence relation on $\Omega(T)$ and
obtain $\Omega_{ij}$ as equivalence classes. Let $x\sim y$ iff
$\cW^u(x)\cap\cW^s(y)\neq\emptyset$ and
$\cW^s(x)\cap\cW^u(y)\neq\emptyset$. (Note: the reflexivity and
symmetry are trivial, and transitivity follows from Lemma 2.9).
The analogue of 1.14 implies that each equivalence class is open
in $\Omega(T)$. By compactness, there are finitely many
equivalence classes. If $X$ is one of them, then for each
$n\in\ZZ$ the set $T^n(X)$ is an equivalence class, too. Thus $T$
permutes equivalence classes.

The topological transitivity in (a) follows from the topological mixing
in (b). To prove the latter, let $U,V\subset\Omega_{ij}$ be open sets
and pick two periodic points $p\in U$ and $q\in V$. Let $T_0=T^{s_i}$
be the minimal iteration of $T$ that leaves $\Omega_{ij}$ invariant.
Denote by $n$ and $m$ the periods of $p$ and $q$ with respect to the
map $T_0$ and set $S=T_0^{mn}$. An argument similar to the proof of
Lemma~2.9 shows that $U\cap S^k(V)\neq\emptyset$ for all large enough
$k$. Lastly, the topological mixing follows from the same argument
applied to each pair of points $T_0^i(p)$, $1\leq i\leq n$, and
$T_0^j(q)$, $1\leq j\leq m$.
\\

\noindent{\bf 2.12 Definition (Shadowing)}\\

A sequence of points $\{x_n\}_{n=a}^b$ ($a=-\infty$ or $b=+\infty$
is permitted) in $M$ is an $\delta$-{\em pseudo-orbit} if
$$
      {\rm dist}(Tx_i,x_{i+1})<\delta
      \quad {\rm for}\ \ {\rm all}\ \ i\in[a,b)
$$
A point $x\in \Omega(T)$ $\varepsilon$-{\em shadows}
$\{x_n\}_{n=a}^b$ if
$$
      {\rm dist}(T^i(x),x_{i})<\varepsilon
      \quad {\rm for}\ \ {\rm all}\ \ i\in[a,b]
$$

\noindent{\bf 2.13 Lemma (Shadowing Lemma)}\\

For any $\varepsilon>0$ there is a $\delta>0$ such that every
$\delta$-{\em pseudo-orbit} in $\Omega(T)$ (i.e., every
$x_i\in\Omega(T)$) is $\varepsilon$-shadowed by a points $x\in M$.
\\

\proof For each $n\in[a,b]$ consider a neighborhood
$\cU_n\subset\Omega(T)$ of the point $x_n$ (in the set
$\Omega(T))$ obtained by taking a direct product, in the sense of
1.14, of the closed $\varepsilon$-balls on $W^u(x_n)\cap\Omega(T)$
and $W^s(x_n)\cap\Omega(T)$. By reducing $\varepsilon$ if
necessary we can make all those neighborhoods very close to
regular cylindrical sets, i.e.\ make their distortions arbitrarily
small (since our manifold is compact, this can be done uniformly
over all $n$'s).

Next we assume that $\delta\ll\varepsilon$. It is easy to verify that
for all $y\in \cU_n\cap T^{-1}\cU_{n+1}$
$$
    T^{-1}(W^s(Ty)\cap\cU_{n+1})\supset W^s(y)\cap\cU_n
$$
and
$$
    T^{-1}(W^u(Ty)\cap\cU_{n+1})\subset W^u(y)\cap\cU_{n}
$$
Hence, $\cU_n\cap T^{-1}\cU_{n+1}$ is a direct product of two closed
sets: one is $W^s(x_n)\cap\cU_n$ and the other is $W^u(x_n)\cap
T^{-1}\cU_{n+1}$, which is a subset of $W^u(x_n)\cap \cU_{n}$. By
induction on $k$, it follows that for each $n+k\leq b$ the set
$\cap_{i\in[n,n+k]}T^{n-i}\cU_i$ is a direct product of
$W^s(x_n)\cap\cU_n$ and a closed subset of $W^u(x_n)\cap \cU_{n}$,
which decreases as $k$ grows.

\vspace*{10mm} \centerline{\epsffile{TF-7.eps}}

\begin{center}
Figure 7: Proof of Lemma 2.13.
\end{center} \vspace*{5mm}

Similarly, for each $n-k\geq a$ the set
$\cap_{i\in[n-k,n]}T^{n-i}\cU_i$ is a direct product of
$W^u(x_n)\cap\cU_n$ and a closed subset of $W^s(x_n)\cap \cU_{n}$,
which decreases as $k$ grows.

Now we fix $n\in [a,b]$ and define
$$
   K=\cap_{i\in[a,b]}T^{n-i}\cU_i
$$
This is a direct product of two closed subsets of $W^s(x_n)\cap\cU_n$
and $W^u(x_n)\cap \cU_{n}$. Any point $x\in T^{-n}(K)$ will now shadow
our pseudo-orbit.

Note: if $a=-\infty$ and $b=+\infty$, then $K$ is a singleton, and the
point $x$ is unique.
\\

\noindent{\bf 2.14 Corollary (Anosov's Closing Lemma)}\\

For any $\varepsilon>0$ there is a $\delta>0$ such that if
$x\in\Omega(T)$ and ${\rm dist}(T^nx,x)<\delta$, then there is an
$x'\in\Omega(T)$ with $T^n(x')=x'$ and
$$
   {\rm dist}(T^ix,T^ix')<\varepsilon
$$
for all $i\in [0,n]$.
\\

\proof Let $x_i=T^k(x)$ for $i\equiv k$ (mod $n$) and $0\leq k<n$.
Then $\{x_i\}_{i=-\infty}^{\infty}$ is a $\delta$-pseudo-orbit. By
2.13, it is $\varepsilon$-shadowed by some point $x'\in\Omega(T)$.
Next,
$$
   {\rm dist}(T^ix',T^{i+n}x')<
   {\rm dist}(T^ix',x_i)+
   {\rm dist}(x_i,T^{i+n}x')<2\varepsilon
$$
for all $i\in\ZZ$. We can assume that $\varepsilon$ is small
enough and obtain $x'=T^nx'$ by expansivity, cf.\ 1.15.
\\

\noindent{\bf 2.15 Corollary}\\

Every Anosov diffeomorphism is an Axiom A diffeomorphism.
\\

\proof Indeed, periodic points are dense in $\Omega(T)$ by 2.14.
\\


\noindent{\bf 2.16 Remark}\\

If $T:M\to M$ is an Anosov diffeomorphism and $\Omega(T)=M$, then 2.11
implies that $T$ is topologically transitive and mixing. Conversely, if
$T:M\to M$ is a topologically transitive Anosov diffeomorphism, then it
is easily seen that $\Omega(T)=M$.
\\

\noindent Open Problem: It is unknown if $\Omega(T)=M$ for every Anosov
diffeomorphism.
\\


\end{document}
\end

\noindent{\bf 2.17 Proposition (Fundamental Neighborhood)}\\

There is an open neighborhood $\cU$ of $\Omega(T)$ such that
$$
       \cap_{n=-\infty}^{\infty} T^n(U) = \Omega(T)
$$

\proof Let $\varepsilon$ be small and $\delta$ be as in 2.13. Pick
$\gamma<\delta/2$ so that
$$
    {\rm dist}(x,y)<\gamma\quad\Rightarrow\quad
    {\rm dist}(Tx,Ty)<\delta/2
$$
Let $\cU=\{x:\ {\rm dist}(x,\Omega(T))<\gamma\}$. We need to prove
that if $T^n(y)\in\cU$ for all $n\in\ZZ$, then $y\in\Omega(T)$.
For every $n\in\ZZ$ select a point $x_n\in\Omega(T)$ such that
dist$(T^n(y),x_n)<\gamma$. Then $\{x_n\}$ is a
$\delta$-pseudo-orbit, which is $\varepsilon$-shadowed by a point
$x\in\Omega(T)$, hence dist$(T^ny,T^nx)<\varepsilon+\gamma$ for
all $n\in\ZZ$. This implies $y=x$.
