We start by considering a very simple problem. Let {X} be a set, {T: X \rightarrow X} be a bijection function, and {g: X \rightarrow  \mathbb{R}} a function. We want to know when the cohomological equation

\displaystyle  g = f \circ T -  f

can be solved for some {f: X \rightarrow  \mathbb{R}}.

It turns out that this very simple question has an equally simple answer. The answer is that the equation can be solved if and only if for every finite (i.e., periodic) orbit {O \subset  X}, we have {\sum_{x \in O} g(x) =  0}. The necessity of this is evident, because if we have such a solution, then

\displaystyle  \sum_O  g(x) = \sum_0 f \circ T(x) - f(x) = \sum_O f(x) - \sum_O f(x) =  0

because {T} induces a bijection of {O} with itself. This condition is called the vanishing of the periodic obstruction.

(more…)