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2014-03-05T07:45:17Z

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Transposition (logic)

2013-05-02T12:22:00Z

86.46.55.195: /* Formal notation */

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]]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 [[flow (mathematics)|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]].

== Definition ==
Let ''M'' be a [[compact space|compact]] [[smooth manifold]], ''f'': ''M'' → ''M'' a [[diffeomorphism]], and ''Df'': ''TM'' → ''TM'' the [[pushforward (differential)|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

:<math>T_\Lambda M = E^s\oplus E^u</math>

and

:<math>(Df)_x E^s_x = E^s_{f(x)}</math> and <math>(Df)_x E^u_x = E^u_{f(x)}</math> for all <math>x\in \Lambda</math>

and

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

and

:<math>\|Df^{-n}v\| \le c\lambda^n \|v\|</math> for all <math>v\in E^u</math> and <math>n>0</math>.

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 (mathematics)|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 ==
* Ralph Abraham and Jerrold E. Marsden, ''Foundations of Mechanics'', (1978) Benjamin/Cummings Publishing, Reading Mass. ISBN 0-8053-0102-X
* {{cite book | author1=Brin, Michael | author2=Garrett, Stuck | title=Introduction to Dynamical Systems | publisher=Cambridge University Press | year=2002 | isbn=0-521-80841-3}}

{{PlanetMath attribution|id=4338|title=Hyperbolic Set}}

[[Category:Dynamical systems]]
[[Category:Limit sets]]