Oseledets theorem
E695940
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.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Oseledets multiplicative ergodic theorem | 2 |
| multiplicative ergodic theorem | 2 |
| Oseledec multiplicative ergodic theorem | 1 |
| Oseledec theorem | 1 |
| Oseledets theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7833158 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Oseledets theorem Context triple: [Lyapunov exponents, relatedTo, Oseledets theorem]
-
A.
Sylvester’s law of inertia
Sylvester’s law of inertia is a theorem in linear algebra stating that the numbers of positive, negative, and zero eigenvalues (the inertia) of a real symmetric matrix are invariant under change of basis.
-
B.
May–Wigner stability theorem
The May–Wigner stability theorem is a result in theoretical ecology and random matrix theory showing that large, complex systems with many random interactions are generically unstable beyond a critical level of complexity.
-
C.
Lyapunov exponents
Lyapunov exponents are quantitative measures in dynamical systems theory that characterize the rates at which nearby trajectories diverge or converge, indicating the presence and strength of chaos.
-
D.
Khinchin–Kolmogorov theorem
The Khinchin–Kolmogorov theorem is a fundamental result in probability theory that provides conditions under which series of independent random variables converge almost surely.
-
E.
Subspace theorem
The Subspace theorem is a fundamental result in Diophantine approximation that describes how solutions to certain inequalities involving linear forms over algebraic numbers must lie in a finite union of proper subspaces.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Oseledets theorem Target entity description: 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.
-
A.
Sylvester’s law of inertia
Sylvester’s law of inertia is a theorem in linear algebra stating that the numbers of positive, negative, and zero eigenvalues (the inertia) of a real symmetric matrix are invariant under change of basis.
-
B.
May–Wigner stability theorem
The May–Wigner stability theorem is a result in theoretical ecology and random matrix theory showing that large, complex systems with many random interactions are generically unstable beyond a critical level of complexity.
-
C.
Lyapunov exponents
Lyapunov exponents are quantitative measures in dynamical systems theory that characterize the rates at which nearby trajectories diverge or converge, indicating the presence and strength of chaos.
-
D.
Khinchin–Kolmogorov theorem
The Khinchin–Kolmogorov theorem is a fundamental result in probability theory that provides conditions under which series of independent random variables converge almost surely.
-
E.
Subspace theorem
The Subspace theorem is a fundamental result in Diophantine approximation that describes how solutions to certain inequalities involving linear forms over algebraic numbers must lie in a finite union of proper subspaces.
- F. None of above. chosen
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
ⓘ
linked to:
Oseledets theorem
|
| 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 ⓘ |
| originalLanguage | Russian ⓘ |
| originalPublication |
Trudy Moskovskogo Matematicheskogo Obshchestva
ⓘ
linked to:
Moscow Mathematical Society
|
| relatedTo |
Birkhoff ergodic theorem
ⓘ
linked to:
ergodic theorem
Furstenberg–Kesten theorem ⓘ
linked to:
subadditive ergodic theorem
Kingman subadditive ergodic theorem ⓘ
linked to:
subadditive ergodic theorem
Lyapunov exponent ⓘ
linked to:
Lyapunov exponents
|
| requires |
integrability of logarithm of operator norm
ⓘ
measurable linear cocycle ⓘ |
| typeOf |
ergodic theorem for matrix products
ⓘ
limit theorem ⓘ |
| usedIn |
Pesin theory
ⓘ
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 ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Oseledets theorem Description of subject: 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.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Oseledets theorem
linked to: Oseledets theorem
linked to: Oseledets theorem
linked to: Oseledets theorem
linked to: Oseledets theorem
linked to: Oseledets theorem