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 $\Vert D{f}^n{v}_s\Vert \le C{\rho }^n\Vert v_s\Vert$ and $\Vert D{f}^{-n}{v}_u\Vert \le C{\rho }^n\Vert v_u\Vert$ for every $n\ge 0$. 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.