hyperbolic setHyperbolicSet2003-06-11 16:23:402006-06-08 19:02:34Definition% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\Z}{\mathbb{Z}}Let $M$ be a compact smooth manifold, and let $f:M\to M$ be a diffeomorphism.
An $f$-invariant subset $\Lambda$ of $M$ is said to be \emph{hyperbolic} (or to have an hyperbolic structure) if there is a splitting of the tangent bundle of $M$ restricted to $\Lambda$ into a (Whitney) sum of two $Df$-invariant subbundles, $E^s$ and $E^u$ such that the restriction of $Df|_{E^s}$ is a contraction and $Df|_{E^u}$ is an expansion. This means that there are constants $0<\lambda<1$ and $c>0$ such that
\begin{enumerate}
\item $T_\Lambda M = E^s\oplus E^u$;
\item $Df(x)E^s_x = E^s_{f(x)}$ and $Df(x)E^u_x = E^u_{f(x)}$ for each $x\in \Lambda$;
\item $\|Df^nv\| < c\lambda^n\|v\|$ for each $v\in E^s$ and $n> 0$;
\item $\|Df^{-n}v\| < c\lambda^n \|v\|$ for each $v\in E^u$ and $n>0$.
\end{enumerate}
using some Riemannian metric on $M$.
If $\Lambda$ is hyperbolic, then there exists an \emph{adapted} Riemannian metric, i.e. one such that $c=1$.
%\textbf{Remark}: Morse-Smale diffeomorphisms have very good behaviour, so the
%dynamics of a Morse-Smale system (i.e. the behaviour of the orbits of
%elements of $M$ when $f$ is applied to them) is reasonably easy to describe.
%Since the nonwandering set of such a diffeomorphism consists only of periodic
%points (and finitely many of them), every other orbit converges to one of
%those periodic orbits. The notion of hyperbolic sets allow us to (somehow)
%generalize these ideas for a much wider class of diffeomorphisms (namely,
%those satisfying Axiom A) by means Smale's spectral decomposition theorem.
% Furthermore: the stable manifold theorem for hyperbolic sets says that
% the local stable and unstable manifold of a point in a hyperbolic set are
% always $\Cdiff^r$-embedded disks, and the (global) stable and unstable
% manifolds are (injectively) $C^k$-immersed disks in $M$.
%Also, if $\Lambda$
% has local product structure (or, equivalently, if it is locally maximal), the
% stable (likewise unstable) manifolds form a foliation of $M$.
% Some other notions from the fixed point theory can be generalized as well.
%The standard reference for this topic is \cite{Smale}; but it is covered in
%many differentiable dynamical systems books, for example \cite{Shub}.
%\begin{thebibliography}{9}
%\bibitem{Shub}
%Shub, M.
%\emph{Global Stability of Dynamical Systems}, New York, Springer-Verlag, 1987.
%\bibitem{Smale}
%Smale, S.
%\emph{Differnetiable dynamical systems},
%Bull. Amer. Math. Soc. \textbf{73} (1967), 747-817.
%\end{thebibliography}