Anosov diffeomorphism ($TM=E^s\oplus E^u,\Vert D{f}^n{|}_{E^s}\Vert ,\Vert D{f}^{-n}{|}_{E^u}\Vert \le C{e}^{-\lambda n}$)

An Anosov diffeomorphism is a continuously differentiable diffeomorphism of a compact smooth manifold with an invariant splitting of its tangent bundle into stable and unstable vector subbundles satisfying the displayed uniform exponential estimates for $n\ge 0$, with $C,\lambda >0$. Hyperbolic toral automorphisms give examples. An Anosov flow has a further one-dimensional invariant bundle along the flow direction; its definition is a distinct continuous-time version of uniform hyperbolicity.