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In mathematics, the multiplicative ergodic theorem, or Oseledets theorem provides the theoretical background for computation of Lyapunov exponents 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 groups by V. A. Kaimanovich and further generalized in the works of David Ruelle, Gregory Margulis, Anders Karlsson, and F. Ledrappier. ==Cocycles== 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. A cocycle of an autonomous dynamical system ''X'' is a map ''C'' : ''X×T'' → R''n×n'' satisfying : : where ''X'' and ''T'' (with ''T'' = Z⁺ or ''T'' = R⁺) are the phase space and the time range, respectively, of the dynamical system, and ''I''''n'' is the ''n''-dimensional unit matrix. The dimension ''n'' of the matrices ''C'' is not related to the phase space ''X''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Oseledets theorem」の詳細全文を読む スポンサード リンク
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