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Suppose we have a functional question, such as the one here. Often a key step in these problems is to try to equate or chain results together (see my hint). From looking at these questions, it seems that there is often only a finite number of ways that we can attempt to join them together. Can this fact be examined more formally?

Note: I am not asking you to turn my answer from that question into something algebraic. It is that way purposely as I was only trying to give a hint, not a complete solution. What I am asking, is whether there are any results that show that the functions can only be "combined" or "equated" in a certain number of ways on some reasonably general class of problems.

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1 Answer 1

One example is that there's an easy way to get uniqueness of functional equations in certain cases. Consider the "cohomological" equation $f(Tx) - f(x) = g(x)$ (which is important in dynamical systems, e.g. it comes up when you are trying to find an invariant $n$-form). The goal, of course, is to prove that there exists $f$ given $g$. If $T: X \to X$ is topologically transitive, then any two solutions differ by a constant. This is because if $x_0$ has a dense orbit, then the value of $f(x_0)$ determines $f(Tx_0), f(T^2x_0)$, and so on.

More generally, there is the same uniqueness for the "twisted" cohomological equation $f(Tx) - \alpha f(x) = g(x)$, where $\alpha$ is a nonzero real number.

(This is not, I think, relevant to the question, but I'll mention that one can get existence of a Hölder solution in the untwisted case for transitive Anosov diffeomorphisms of compact manifolds if the sum of $g$ over every finite orbit is zero, though it requires a clever trick. This is a result of Livsic.)

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This involves a lot of maths I don't know - will take me a while before I can understand this answer –  Casebash Aug 10 '10 at 23:28

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