Matrix scheme - Revision history

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2014-12-14T22:56:32Z

Replaced 2 instances of "PS2" with "PS4", refering to a PlayStation 4 as reward.

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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.~~		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.<br><br>my web blog; [http://happywedding.lonnieheart.gethompy.com/zbxe/wc_apply/372747 over the counter std test]

	~~==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~~ is ~~a map~~
	~~''C'' : ''X×T'' → '''R'''<sup>''n×n''</sup> satisfying~~

	~~:<math>C(x,0)=I_n {\rm~for~all~} x\in X</math>~~

	~~:<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>~~

	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~~''<sub>''n''</sub> is the ''n''-dimensional unit matrix.~~
	~~The dimension ''n'' of the matrices ''C'' is not related to the phase space ''X''.~~

	~~=== Examples ===~~
	* 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''.
	* For any cocycle ''C'', the [[determinant]] det ''C''(''x'', ''t'') is a one-dimensional cocycle.

	~~== Statement of the theorem ==~~

	~~Let μ be an invariant measure on ''X'' and ''C'' a cocycle~~
	~~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~~
	~~with respect to μ. Then for μ-almost all ''x'' and~~ each ~~non-zero vector ''u''∈'''R'''<sup>''n''</sup> the limit~~

	~~:<math>\lambda=\lim_{t\to\infty}{1\over t} \log{\\|C(x,t)u\\| \over \\|u\\|}</math>~~

	~~exists and assumes, depending on ''u'' but not on ''x'', up to ''n'' different values.~~
	~~These are the Lyapunov exponents.~~

	~~Further, if λ<sub>1</sub> > ... > λ<sub>''m''</sub>~~
	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''.~~

	~~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.~~

	~~== Additive versus multiplicative ergodic theorems ==~~

	~~Verbally, ergodicity means that time and space averages are equal, formally:~~
	~~:<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>~~

	~~where the integrals~~ and ~~the limit exist.~~
	~~Space average (right hand side, μ is an ergodic measure on ''X'')~~
	~~is the accumulation of ''f''(''x'') values weighted by μ(''dx'').~~
	~~Since addition is commutative, the accumulation of the~~ '~~'f''(''x'')μ(''dx'') values may~~ be ~~done in arbitrary order.~~
	~~In contrast, the time average (left hand side) suggests a specific ordering~~
	~~of the ''f''(''x''(''s'')) values~~ along ~~the trajectory.~~

	~~Since matrix multiplication is, in general, not commutative,~~
	~~accumulation of multiplied cocycle values (and limits thereof) according to~~
	~~''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>)~~
	~~— for ''t''<sub>''k''</sub> large and~~
	~~the steps ''t''<sub>''i''</sub> − ''t''~~<~~sub~~>~~''i''−1~~<~~/sub~~> ~~small &mdash~~; 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.

	~~==References==~~
	* V. I. Oseledets, "Multiplicative ergodic theorem: Characteristic Lyapunov exponents of dynamical systems", ''Trudy MMO'' '''19''' (1968), 179–210. ''(in Russian)''.
	* V. I. Oseledets, [http://~~www~~.~~scholarpedia~~.~~org~~/~~article~~/~~Oseledets_theorem ''Oseledets theorem''] at [[Scholarpedia]]~~
	* [[D. Ruelle]], "Ergodic theory of differentiable dynamic systems", ''IHES Publ. Math.'' '''50''' (1979), 27–58.

	~~[[Category:Ergodic theory]]~~
	~~[[Category:Theorems in dynamical systems]~~]

en>Ohconfucius: Script-assisted fixes per WP:TIES, MOS:NUM, MOS:CAPS, MOS:LINK

2014-01-06T03:50:34Z

Script-assisted fixes per WP:TIES, MOS:NUM, MOS:CAPS, MOS:LINK

← Older revision		Revision as of 05:50, 6 January 2014
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	~~I am Oscar~~ and ~~I totally dig that title~~. ~~Bookkeeping~~ is ~~her working day occupation now~~. ~~One~~ of the ~~issues she loves most~~ is ~~to read comics~~ and ~~she~~'~~ll be starting some thing else alongside~~ with it. ~~California~~ is ~~exactly where I~~'~~ve usually been living~~ and ~~I adore~~ each ~~working day residing right here~~.<br><br>~~My webpage~~: ~~[http~~://~~richlinked~~.~~com~~/~~index~~.~~php?do~~=/~~profile-32092~~/~~info~~/ http://~~richlinked~~.~~com~~]		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.

			==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 is a map
			''C'' : ''X×T'' → '''R'''<sup>''n×n''</sup> satisfying

			:<math>C(x,0)=I_n {\rm~for~all~} x\in X</math>

			:<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>

			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''<sub>''n''</sub> is the ''n''-dimensional unit matrix.
			The dimension ''n'' of the matrices ''C'' is not related to the phase space ''X''.

			=== Examples ===
			* 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''.
			* For any cocycle ''C'', the [[determinant]] det ''C''(''x'', ''t'') is a one-dimensional cocycle.

			== Statement of the theorem ==

			Let μ be an invariant measure on ''X'' and ''C'' a cocycle
			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
			with respect to μ. Then for μ-almost all ''x'' and each non-zero vector ''u''∈'''R'''<sup>''n''</sup> the limit

			:<math>\lambda=\lim_{t\to\infty}{1\over t} \log{\\|C(x,t)u\\| \over \\|u\\|}</math>

			exists and assumes, depending on ''u'' but not on ''x'', up to ''n'' different values.
			These are the Lyapunov exponents.

			Further, if λ<sub>1</sub> > ... > λ<sub>''m''</sub>
			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''.

			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.

			== Additive versus multiplicative ergodic theorems ==

			Verbally, ergodicity means that time and space averages are equal, formally:
			:<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>

			where the integrals and the limit exist.
			Space average (right hand side, μ is an ergodic measure on ''X'')
			is the accumulation of ''f''(''x'') values weighted by μ(''dx'').
			Since addition is commutative, the accumulation of the ''f''(''x'')μ(''dx'') values may be done in arbitrary order.
			In contrast, the time average (left hand side) suggests a specific ordering
			of the ''f''(''x''(''s'')) values along the trajectory.

			Since matrix multiplication is, in general, not commutative,
			accumulation of multiplied cocycle values (and limits thereof) according to
			''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>)
			— for ''t''<sub>''k''</sub> large and
			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.

			==References==
			* V. I. Oseledets, "Multiplicative ergodic theorem: Characteristic Lyapunov exponents of dynamical systems", ''Trudy MMO'' '''19''' (1968), 179–210. ''(in Russian)''.
			* V. I. Oseledets, [http://www.scholarpedia.org/article/Oseledets_theorem ''Oseledets theorem''] at [[Scholarpedia]]
			* [[D. Ruelle]], "Ergodic theory of differentiable dynamic systems", ''IHES Publ. Math.'' '''50''' (1979), 27–58.

			[[Category:Ergodic theory]]
			[[Category:Theorems in dynamical systems]]

en>Hyphz: /* Operation */

2012-05-30T16:44:20Z

Operation

New page

I am Oscar and I totally dig that title. Bookkeeping is her working day occupation now. One of the issues she loves most is to read comics and she'll be starting some thing else alongside with it. California is exactly where I've usually been living and I adore each working day residing right here.<br><br>My webpage: [http://richlinked.com/index.php?do=/profile-32092/info/ http://richlinked.com]