= Hyperbolic invariant set
A compact <invariant set> $K$ of a smooth invertible map $f$ is uniformly hyperbolic when its tangent spaces split continuously as $E^s\oplus E^u$, the <derivative> preserves this splitting, and constants $C>0$, $0<\rho<1$ satisfy $\|Df^n v_s\|\le C\rho^n\|v_s\|$ and $\|Df^{-n}v_u\|\le C\rho^n\|v_u\|$ for every $n\ge0$. Thus the stable direction contracts forward and the unstable direction contracts backward. A <Smale horseshoe> is an example with nontrivial symbolic dynamics; such a set can contain infinitely many saddle <periodic orbits> without attracting an open neighborhood.
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