View source for Ergodicity

''of a dynamical system''

{{MSC|37A25}}

[[Category:Ergodic theory]]

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,

<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>

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>

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).