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| In [[mathematics]], the '''multiplicative ergodic theorem''', or '''Oseledets theorem''' provides the theoretical background for computation of [[Lyapunov exponent]]s of a [[nonlinear]] [[dynamical system]]. It was proved by [[Valery Oseledets]] (also spelled "Oseledec") in 1965 and reported at the [[International Mathematical Congress]] in Moscow in 1966. A conceptually different proof of the multiplicative [[ergodic theorem]] was found by [[M. S. Raghunathan]]. The theorem has been extended to [[semisimple Lie group]]s by V. A. Kaimanovich and further generalized in the works of [[David Ruelle]], [[Gregory Margulis]], Anders Karlsson, and F. Ledrappier.
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| ==Cocycles==
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| The multiplicative ergodic theorem is stated in terms of matrix cocycles of a dynamical system. The theorem states conditions for the existence of the defining limits and describes the Lyapunov exponents. It does not address the rate of convergence.
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| A '''cocycle''' of an autonomous dynamical system is a map
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| ''C'' : ''X×T'' → '''R'''<sup>''n×n''</sup> satisfying
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| :<math>C(x,0)=I_n {\rm~for~all~} x\in X</math>
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| :<math>C(x,t+s)=C(x(t),s)\,C(x,t) {\rm~for~all~} x\in X {\rm~and~} t,s\in T</math>
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| where ''X'' and ''T'' (with ''T'' = '''Z''' or ''T'' = '''R''') are the phase space | |
| and the time range, respectively, of the dynamical system,
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| and ''I''<sub>''n''</sub> is the ''n''-dimensional unit matrix.
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| The dimension ''n'' of the matrices ''C'' is not related to the phase space ''X''.
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| === Examples ===
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| * A prominent example of a cocycle is given by the Matrix ''J''<sup>''t''</sup> in the theory of Lyapunov exponents. In this special case, the dimension ''n'' of the matrices is the same as the dimension of the manifold ''X''.
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| * For any cocycle ''C'', the [[determinant]] det ''C''(''x'', ''t'') is a one-dimensional cocycle.
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| == Statement of the theorem ==
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| Let μ be an invariant measure on ''X'' and ''C'' a cocycle
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| of the dynamical system such that for each ''t''∈T, the maps <math>x \rightarrow \log\|C(x,t)\|</math> and <math>x \rightarrow \log\|C(x,t)^{-1}\|</math> are ''L''<sup>1</sup>-integrable
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| with respect to μ. Then for μ-almost all ''x'' and each non-zero vector ''u''∈'''R'''<sup>''n''</sup> the limit
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| :<math>\lambda=\lim_{t\to\infty}{1\over t} \log{\|C(x,t)u\| \over \|u\|}</math>
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| exists and assumes, depending on ''u'' but not on ''x'', up to ''n'' different values.
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| These are the Lyapunov exponents.
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| Further, if λ<sub>1</sub> > ... > λ<sub>''m''</sub>
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| are the different limits then there are subspaces '''R'''<sup>''n''</sup> = ''R''<sub>1</sub> ⊃ ... ⊃ ''R''<sub>''m''</sub> ⊃ ''R''<sub>''m''+1</sub> = {0} such that the limit is λ<sub>''i''</sub> for ''u''∈ ''R''<sub>''i''</sub>\''R''<sub>''i''+1</sub> and ''i'' = 1, ..., ''m''.
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| The values of the Lyapunov exponents are invariant with respect to a wide range of coordinate transformations. Suppose that ''g'' : ''X'' → ''X'' is a one-to-one map such that <math>\partial g/\partial x</math>and its inverse exist then the values of the Lyapunov exponents do not change.
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| == Additive versus multiplicative ergodic theorems ==
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| Verbally, ergodicity means that time and space averages are equal, formally:
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| :<math>\lim_{t\to\infty}{1\over t} \int_0^t f(x(s))\,ds = {1\over \mu(X)} \int_X f(x)\,\mu(dx)</math>
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| where the integrals and the limit exist.
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| Space average (right hand side, μ is an ergodic measure on ''X'')
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| is the accumulation of ''f''(''x'') values weighted by μ(''dx'').
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| Since addition is commutative, the accumulation of the ''f''(''x'')μ(''dx'') values may be done in arbitrary order.
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| In contrast, the time average (left hand side) suggests a specific ordering
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| of the ''f''(''x''(''s'')) values along the trajectory.
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| Since matrix multiplication is, in general, not commutative,
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| accumulation of multiplied cocycle values (and limits thereof) according to
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| ''C''(''x''(''t''<sub>0</sub>),''t''<sub>''k''</sub>) = ''C''(''x''(''t''<sub>''k''−1</sub>),''t''<sub>''k''</sub> − ''t''<sub>''k''−1</sub>) ... ''C''(''x''(''t''<sub>0</sub>),''t''<sub>1</sub> − ''t''<sub>0</sub>)
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| — for ''t''<sub>''k''</sub> large and
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| the steps ''t''<sub>''i''</sub> − ''t''<sub>''i''−1</sub> small — makes sense only for a prescribed ordering. Thus, the time average may exist (and the theorem states that it actually exists), but there is no space average counterpart. In other words, the Oseledets theorem differs from additive ergodic theorems (such as [[G. D. Birkhoff]]'s and [[J. von Neumann]]'s) in that it guarantees the existence of the time average, but makes no claim about the space average.
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| ==References==
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| * V. I. Oseledets, "Multiplicative ergodic theorem: Characteristic Lyapunov exponents of dynamical systems", ''Trudy MMO'' '''19''' (1968), 179–210. ''(in Russian)''.
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| * V. I. Oseledets, [http://www.scholarpedia.org/article/Oseledets_theorem ''Oseledets theorem''] at [[Scholarpedia]]
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| * [[D. Ruelle]], "Ergodic theory of differentiable dynamic systems", ''IHES Publ. Math.'' '''50''' (1979), 27–58.
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| [[Category:Ergodic theory]]
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| [[Category:Theorems in dynamical systems]]
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Nice to satisfy you, I am Marvella Shryock. My day occupation is a meter reader. California is exactly where I've usually been living and I adore each day residing here. One of the things she loves most is to study comics and she'll be beginning something else along with it.
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