dynamical systems theory Dynamical systems theory is an area of mathematics used to describe the behavior of complex systems, complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theo ...

, a subset Λ of a

smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One m ...

''M'' is said to have a hyperbolic structure with respect to a

smooth map In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...

''f'' if its

tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and ...

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 subbundl ...

s, one of which is contracting and the other is expanding under ''f'', with respect to some

Riemannian metric In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent spac ...

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 ...

and has been much studied, cf. Axiom A.

Definition

Let ''M'' be a

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

, ''f'': ''M'' → ''M'' a

diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given tw ...

, 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 In mathematics, a vector bundle is a topology, topological construction that makes precise the idea of a family of vector spaces parameterized by another space (mathematics), space X (for example X could be a topological space, a manifold, or an ...

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

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 In the study of dynamical systems, a hyperbolic equilibrium point or hyperbolic fixed point is a fixed point that does not have any center manifolds. Near a hyperbolic point the orbits of a two-dimensional, non-dissipative system resemble hyperb ...

''p'' is a fixed point, or equilibrium point, of ''f'', such that (''Df'')_''p'' has no

eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denot ...

with absolute value 1. In this case, Λ = . * More generally, a

periodic orbit In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time. Iterated functions Given ...

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