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An additive cocycle over a dynamical system records a quantity accumulated along a flow. It satisfies $c(0,x)=0$ and $c(s+t,x)=c(t,x)+c(s,{\Phi }_{t}x)$: accumulation over two intervals equals accumulation over the first interval plus accumulation over the second interval from the evolved state.

For an dynamical observable $h$, the integral $c(t,x)={\int }_0^{t}h({\Phi }_{u}x)du$ is an example, as splitting the integral at $t$ proves the identity. A coboundary has the special form $c(t,x)=u({\Phi }_{t}x)-u(x)$. Matrix-valued multiplicative versions replace addition by an ordered product; the order matters when the matrices do not commute.

For a map $T:X\to X$ and a given dynamical observable $f$, the additive cohomological equation asks for a function $u$ satisfyingWhen such $u$ exists, $f$ is a difference accumulated between consecutive states. Summing along a trajectory makes the intermediate terms cancel:Thus bounded $u$ forces these sums to remain bounded, and any periodic orbit of period $n$ must have sum zero. These are necessary conditions, not a universal existence theorem.

For the circle rotation $T(x)=x+\alpha (\mathrm{mod}1)$, try $f(x)=e^{2\pi ikx}$ and $u(x)=c{e}^{2\pi ikx}$. Then $c=(e^{2\pi ik\alpha }-1{)}^{-1}$ when the denominator is nonzero. If it vanishes, this mode is an obstruction; if it is very small, the coefficient is large. This explains how a simple equation leads to delicate regularity questions in dynamical systems.

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.

A periodic orbit obstruction is a necessary compatibility condition for solving a cohomological equation. If an dynamical observable is a flow coboundary, $h=Xu$, where $X$ differentiates along the flow, then on a periodic orbit of period $T$,A nonzero integral therefore rules out a single-valued transfer function $u$. Under suitable transitive Anosov flow and Hölder regularity hypotheses, the Livšic theorem provides a converse. Without those dynamical and regularity assumptions, vanishing periodic integrals alone is not a general existence theorem.^[ref]

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.