Hyperbolic set

Definition

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 00 such that

:T_\Lambda M = E^s\oplus E^u

and

:(Df)x E^s_x = E^s{f(x)} and (Df)x E^u_x = E^u{f(x)} for all x\in \Lambda

and

:|Df^nv| \le c\lambda^n|v| for all v\in E^s and n 0

and

:|Df^{-n}v| \le c\lambda^{n} |v| for all v\in E^u and n0.

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

Examples

• 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.

References