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 by nature of the ergodic theory, ergodicity of dynamic systems. When differ ...

, 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 may ...

''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 (''differentiability class)'' it has over its domain. A function of class C^k is a function of smoothness at least ; t ...

''f'' if its

tangent bundle A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...

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 In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...

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 "contr ...

. 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.

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, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...

, ''f'': ''M'' → ''M'' a

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

, 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 topological construction that makes precise the idea of a Family of sets, family of vector spaces parameterized by another space (mathematics), space X (for example X could be a topological space, a manifold, ...

of two ''Df''-invariant subbundles, called the stable bundle 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^ \, 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 hyperbol ...

''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 is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...

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 x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...

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