In dynamical systems theory, a subset Λ of a smooth manifold *M* is said to have a **hyperbolic structure** with respect to a smooth map *f* if its tangent bundle may be split into two invariant subbundles, one of which is contracting and the other is expanding under *f*, with respect to some Riemannian metric on *M*. An analogous definition applies to the case of flows.

In the special case when the entire manifold *M* is hyperbolic, the map *f* is called an Anosov diffeomorphism. The dynamics of *f* on a hyperbolic set, or **hyperbolic dynamics**, exhibits features of local structural stability and has been much studied, cf. Axiom A.

Let *M* be a compact smooth manifold, *f*: *M* → *M* a diffeomorphism, and *Df*: *TM* → *TM* the differential of *f*. An *f*-invariant subset Λ of *M* is said to be **hyperbolic**, or to have a **hyperbolic structure**, if the restriction to Λ of the tangent bundle of *M* admits a splitting into a Whitney sum of two *Df*-invariant subbundles, called the stable bundle and the unstable bundle and denoted *E*^{s} and *E*^{u}. With respect to some Riemannian metric on *M*, the restriction of *Df* to *E*^{s} must be a contraction and the restriction of *Df* to *E*^{u} must be an expansion. Thus, there exist constants 0<*λ*<1 and *c*>0 such that

and

- and for all

and

- for all and

and

- for all and .

If Λ is hyperbolic then there exists a Riemannian metric for which *c* = 1 — such a metric is called **adapted**.

- Hyperbolic equilibrium point
*p*is a fixed point, or equilibrium point, of*f*, such that (*Df*)_{p}has no eigenvalue with absolute value 1. In this case, Λ = {*p*}. - More generally, a periodic orbit of
*f*with period*n*is hyperbolic if and only if*Df*^{n}at any point of the orbit has no eigenvalue with absolute value 1, and it is enough to check this condition at a single point of the orbit.