= Time average
{title2=$T^{-1}\int_0^T h(\Phi_t x)\,dt$}
A <time average> measures an <dynamical observable> along a single trajectory of a <dynamical system>. For a <flow> $\Phi_t$, define $A_T h(x)=T^{-1}\int_0^T h(\Phi_t x)\,dt$. If the limit as $T\to\infty$ exists, it is the long-time average at initial state $x$. For a discrete map $T$, the corresponding expression is $n^{-1}\sum_{j=0}^{n-1}h(T^jx)$.
If $h(\Phi_t x)=\cos t$, the average tends to zero although the observable never settles to a constant. The <Birkhoff ergodic theorem> supplies almost-everywhere existence for integrable observables under a probability-preserving map. Equality with a spatial average requires the appropriate <ergodicity> assumption; existence alone does not imply it.
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