= Dynamical observable
A <dynamical observable> is a <function> $\varphi:X\to\mathbb R$ 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 $\varphi(x),\varphi(Tx),\varphi(T^2x),\ldots$. The <state space> may be high dimensional even when this record is scalar.
For a <harmonic oscillator> with state $(q,p)$, position $\varphi(q,p)=q$ is one <dynamical observable>; energy is another. Position generally oscillates, while energy is constant along an ideal conservative <trajectory>. Composition $\varphi\mapsto\varphi\circ T$ describes evolution of the measured quantity instead of evolution of states. Its <time average>, $n^{-1}\sum_{j=0}^{n-1}\varphi(T^jx)$, is the object compared with a space <integral> in <ergodic theory>.
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