 Time average

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 $n^{-1}{\sum }_{j=0}^{n-1}h(T^{j}x)$.

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.