= Hyperbolic invariant set
Let $f$ be a smooth <diffeomorphism> and $K$ a compact <invariant set>. It is uniformly hyperbolic if its tangent spaces have a continuous splitting $T_xM=E_x^s\oplus E_x^u$ preserved by the <derivative>, with constants $C>0$ and $0<\rho<1$ such that
$$
\|Df_x^n v_s\|\leq C\rho^n\|v_s\|,\qquad
\|Df_x^{-n}v_u\|\leq C\rho^n\|v_u\|,\qquad n\geq0.
$$
The same constants must work for every $x\in K$. Thus the stable directions contract forward and the unstable directions contract backward, uniformly over the set. A sign observed in a finite-time numerical <Lyapunov exponent> calculation does not establish these uniform estimates.
For the <cat map> on a two-dimensional <torus>, the constant matrix $\begin{pmatrix}2&1\\1&1\end{pmatrix}$ has <eigenvalues> $(3\pm\sqrt5)/2$. Its two <eigenvectors> give fixed stable and unstable directions; the smaller <eigenvalue> is the reciprocal of the larger. With the flat Euclidean metric, $C=1$ and $\rho=(3-\sqrt5)/2$ give the required estimates everywhere. The whole <torus> is therefore hyperbolic.
A <Smale horseshoe> supplies a different example: its hyperbolic <invariant set> contains infinitely many saddle <periodic orbits>, yet need not attract a neighborhood. Hyperbolicity describes stretching and contraction, while being an <attractor> concerns nearby trajectories. For a nonsingular <flow map>, the analogous splitting also has the neutral direction along the <vector field>; it should not be omitted by copying the map definition unchanged.
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