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Difference between revisions of "Ergodicity"

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''of a dynamical system''
 
''of a dynamical system''
  
A property considered in [[Ergodic theory|ergodic theory]]. Originally it was defined for a [[Cascade|cascade]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036170/e0361701.png" /> or a [[Flow (continuous-time dynamical system)|flow (continuous-time dynamical system)]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036170/e0361702.png" /> with a finite [[Invariant measure|invariant measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036170/e0361703.png" /> in the following way: If a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036170/e0361704.png" /> is given on the phase space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036170/e0361705.png" />, then for almost-every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036170/e0361706.png" /> the time average along the trajectory of this point exists, that is,
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{{MSC|37A25}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036170/e0361707.png" /></td> </tr></table>
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[[Category:Ergodic theory]]
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A property considered in [[Ergodic theory|ergodic theory]]. Originally it was defined for a [[Cascade|cascade]] $\{T^k\}$ or a [[Flow (continuous-time dynamical system)|flow (continuous-time dynamical system)]] $\{T_t\}$ with a finite [[Invariant measure|invariant measure]] $\mu$ in the following way: If a function $f\in L_1(W,\mu)$ is given on the phase space $W$, then for almost-every point $w$ the time average along the trajectory of this point exists, that is,
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$$\lim_{n\to\infty}\frac1n\sum_{k=0}^{n-1}f(T^kw),$$
  
 
or
 
or
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036170/e0361708.png" /></td> </tr></table>
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$$\lim_{T\to\infty}\frac1T\int\limits_0^Tf(T_tw)dt,$$
  
exists, and coincides with the space average (that is, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036170/e0361709.png" />). In this case one also speaks of ergodicity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036170/e03617010.png" />. In particular, for any measurable set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036170/e03617011.png" /> the average time of a trajectory staying in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036170/e03617012.png" /> is proportional to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036170/e03617013.png" /> for almost-every point (in fact, this property is equivalent to ergodicity). When the [[Birkhoff ergodic theorem|Birkhoff ergodic theorem]] had been proved it became clear that ergodicity is equivalent to [[Metric transitivity|metric transitivity]]. Therefore one spoke of ergodicity, meaning metric transitivity, in the more general situation when it was no longer suitable to talk of the equality of time and space averages (systems with an infinite invariant or quasi-invariant measure, not only flows and cascades, but also more general transformation groups and semi-groups).
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exists, and coincides with the space average (that is, with $(\int fd\mu)/\mu(W)$). In this case one also speaks of ergodicity of $\mu$. In particular, for any measurable set $A\subset W$ the average time of a trajectory staying in $A$ is proportional to $\mu(A)$ for almost-every point (in fact, this property is equivalent to ergodicity). When the [[Birkhoff ergodic theorem|Birkhoff ergodic theorem]] had been proved it became clear that ergodicity is equivalent to [[Metric transitivity|metric transitivity]]. Therefore one spoke of ergodicity, meaning metric transitivity, in the more general situation when it was no longer suitable to talk of the equality of time and space averages (systems with an infinite invariant or quasi-invariant measure, not only flows and cascades, but also more general transformation groups and semi-groups).

Latest revision as of 21:18, 8 November 2014

of a dynamical system

2010 Mathematics Subject Classification: Primary: 37A25 [MSN][ZBL]

A property considered in ergodic theory. Originally it was defined for a cascade $\{T^k\}$ or a flow (continuous-time dynamical system) $\{T_t\}$ with a finite invariant measure $\mu$ in the following way: If a function $f\in L_1(W,\mu)$ is given on the phase space $W$, then for almost-every point $w$ the time average along the trajectory of this point exists, that is,

$$\lim_{n\to\infty}\frac1n\sum_{k=0}^{n-1}f(T^kw),$$

or

$$\lim_{T\to\infty}\frac1T\int\limits_0^Tf(T_tw)dt,$$

exists, and coincides with the space average (that is, with $(\int fd\mu)/\mu(W)$). In this case one also speaks of ergodicity of $\mu$. In particular, for any measurable set $A\subset W$ the average time of a trajectory staying in $A$ is proportional to $\mu(A)$ for almost-every point (in fact, this property is equivalent to ergodicity). When the Birkhoff ergodic theorem had been proved it became clear that ergodicity is equivalent to metric transitivity. Therefore one spoke of ergodicity, meaning metric transitivity, in the more general situation when it was no longer suitable to talk of the equality of time and space averages (systems with an infinite invariant or quasi-invariant measure, not only flows and cascades, but also more general transformation groups and semi-groups).

How to Cite This Entry:
Ergodicity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ergodicity&oldid=17465
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article