= Cohomological equation
For a map $T:X\to X$ and a given <dynamical observable> $f$, the additive <cohomological equation> asks for a <function> $u$ satisfying
$$
u\circ T-u=f.
$$
When such $u$ exists, $f$ is a difference accumulated between consecutive states. Summing along a <trajectory> makes the intermediate terms cancel:
$$
\sum_{j=0}^{n-1}f(T^jx)=u(T^nx)-u(x).
$$
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 $T(x)=x+\alpha\pmod1$, 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>.
Back to article page
