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

subbundle In mathematics, a subbundle U of a vector bundle V on a topological space X is a collection of linear subspaces U_xof the fibers V_x of V at x in X, that make up a vector bundle in their own right. In connection with foliation theory, a subbundle ...

s, 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 In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold ''M'' is a certain type of mapping, from ''M'' to itself, with rather clearly marked local directions of "expansion" and "cont ...

. The dynamics of ''f'' on a hyperbolic set, or hyperbolic dynamics, exhibits features of local

structural stability In mathematics, structural stability is a fundamental property of a dynamical system which means that the qualitative behavior of the trajectories is unaffected by small perturbations (to be exact ''C''1-small perturbations). Examples of such q ...

and has been much studied, cf.

Axiom A In mathematics, Smale's axiom A defines a class of dynamical systems which have been extensively studied and whose dynamics is relatively well understood. A prominent example is the Smale horseshoe map. The term "axiom A" originates with Stephen Sm ...

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 In mathematics, a stable vector bundle is a (holomorphic or algebraic) vector bundle that is stable in the sense of geometric invariant theory. Any holomorphic vector bundle may be built from stable ones using Harder–Narasimhan filtration. Stable ...

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 :

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

and :

(Df)_x E^s_x = E^s_

and

(Df)_x E^u_x = E^u_

for all

x\in \Lambda

and :

\, Df^nv\,  \le c\lambda^n\, v\,

for all

v\in E^s

and

n> 0

and :

\, Df^v\,  \le c\lambda^n \, v\,

for all

v\in E^u

and

n>0

. 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 In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...

1. In this case, Λ = . * 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

* * {{PlanetMath attribution, id=4338, title=Hyperbolic Set Dynamical systems Limit sets