Oseledets theorem

E695940

mathematical theorem result in dynamical systems result in ergodic theory

Oseledets theorem is a fundamental result in dynamical systems and ergodic theory that guarantees the existence of Lyapunov exponents and an invariant splitting of the tangent space for almost every point under suitable conditions.

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Label	Occurrences
Oseledets theorem canonical	1

Statements (48)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in dynamical systems ⓘ result in ergodic theory ⓘ
appliesTo	flows and diffeomorphisms via derivative cocycle ⓘ invertible measure-preserving transformations ⓘ linear cocycles over dynamical systems ⓘ measure-preserving dynamical systems ⓘ
assumes	ergodicity of the underlying measure ⓘ integrability conditions on the cocycle ⓘ
concerns	asymptotic behavior of products of random matrices ⓘ growth rates of vectors under iteration ⓘ
concludes	almost sure existence of Lyapunov exponents ⓘ filtration by Oseledets subspaces ⓘ measurable invariant splitting of the space ⓘ
describes	decomposition into Oseledets subspaces ⓘ exponential growth rates along invariant subspaces ⓘ
domain	finite-dimensional vector spaces ⓘ
field	dynamical systems ⓘ ergodic theory ⓘ linear cocycle theory ⓘ probability theory ⓘ
guarantees	existence of Lyapunov exponents ⓘ existence of invariant splitting of tangent space ⓘ
hasAlternativeName	multiplicative ergodic theorem NERFINISHED ⓘ
hasConsequence	characterization of typical orbits by Lyapunov spectrum ⓘ existence of stable and unstable directions in nonuniformly hyperbolic systems ⓘ
hasGeneralization	infinite-dimensional multiplicative ergodic theorems ⓘ
implies	Lyapunov exponents are invariant under the dynamics ⓘ Lyapunov exponents are well-defined almost everywhere ⓘ
namedAfter	Vladimir Oseledets NERFINISHED ⓘ
originalLanguage	Russian ⓘ
originalPublication	Trudy Moskovskogo Matematicheskogo Obshchestva NERFINISHED ⓘ
relatedTo	Birkhoff ergodic theorem NERFINISHED ⓘ Furstenberg–Kesten theorem NERFINISHED ⓘ Kingman subadditive ergodic theorem NERFINISHED ⓘ Lyapunov exponent NERFINISHED ⓘ
requires	integrability of logarithm of operator norm ⓘ measurable linear cocycle ⓘ
typeOf	ergodic theorem for matrix products ⓘ limit theorem ⓘ
usedIn	Pesin theory NERFINISHED ⓘ nonuniformly hyperbolic dynamics ⓘ random dynamical systems ⓘ smooth ergodic theory ⓘ stability analysis of differential equations ⓘ statistical properties of dynamical systems ⓘ theory of random matrix products ⓘ
yearProved	1965 ⓘ

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Lyapunov exponents → relatedTo → Oseledets theorem ⓘ