Dynamical observable

A dynamical observable is a function $\phi :X\to \mathbb{R}$ or $\mathbb{C}$ that assigns a measurable quantity to each state of a dynamical system. If the dynamics is $T:X\to X$, the observed sequence from state $x$ is $\phi (x),\phi (Tx),\phi (T^{2}x),\dots$. The state space may be high dimensional even when this record is scalar.

For a harmonic oscillator with state $(q,p)$, position $\phi (q,p)=q$ is one dynamical observable; energy is another. Position generally oscillates, while energy is constant along an ideal conservative trajectory. Composition $\phi \mapsto \phi \circ T$ describes evolution of the measured quantity instead of evolution of states. Its time average, $n^{-1}{\sum }_{j=0}^{n-1}\phi (T^{j}x)$, is the object compared with a space integral in ergodic theory.