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| 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]].
| | Andrew Simcox is the name his mothers and fathers gave him and he totally loves this title. He is an info officer. What I love doing is football but I don't have the time lately. I've always loved residing in Mississippi.<br><br>Here is my blog post; online psychics [[http://test.jeka-nn.ru/node/129 jeka-nn.ru]] |
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| 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]].
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| == Definition ==
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| 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''<sup>''s''</sup> and ''E''<sup>''u''</sup>. With respect to some [[Riemannian metric]] on ''M'', the restriction of ''Df'' to ''E''<sup>''s''</sup> must be a contraction and the restriction of ''Df'' to ''E''<sup>''u''</sup> must be an expansion. Thus, there exist constants 0<''λ''<1 and ''c''>0 such that
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| :<math>T_\Lambda M = E^s\oplus E^u</math>
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| and
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| :<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>
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| and
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| :<math>\|Df^nv\| \le c\lambda^n\|v\|</math> for all <math>v\in E^s</math> and <math>n> 0</math>
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| and
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| :<math>\|Df^{-n}v\| \le c\lambda^n \|v\|</math> for all <math>v\in E^u</math> and <math>n>0</math>.
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| If ''Λ'' is hyperbolic then there exists a Riemannian metric for which ''c''=1 — such a metric is called '''adapted'''.
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| == Examples ==
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| * [[Hyperbolic equilibrium point]] ''p'' is a [[fixed point (mathematics)|fixed point]], or equilibrium point, of ''f'', such that (''Df'')<sub>''p''</sub> has no eigenvalue with [[absolute value]] 1. In this case, ''Λ'' = {''p''}.
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| * More generally, a [[periodic orbit]] of ''f'' with period ''n'' is hyperbolic if and only if ''Df''<sup>''n''</sup> 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.
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| == References ==
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| * Ralph Abraham and Jerrold E. Marsden, ''Foundations of Mechanics'', (1978) Benjamin/Cummings Publishing, Reading Mass. ISBN 0-8053-0102-X
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| * {{cite book | author1=Brin, Michael | author2=Garrett, Stuck | title=Introduction to Dynamical Systems | publisher=Cambridge University Press | year=2002 | isbn=0-521-80841-3}}
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| {{PlanetMath attribution|id=4338|title=Hyperbolic Set}}
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| [[Category:Dynamical systems]]
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| [[Category:Limit sets]]
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Andrew Simcox is the name his mothers and fathers gave him and he totally loves this title. He is an info officer. What I love doing is football but I don't have the time lately. I've always loved residing in Mississippi.
Here is my blog post; online psychics [jeka-nn.ru]