# functions whose average along orbits is zero or a constant

Is there some name in ergodic theory or integrable systems theory for a function whose average value on every orbit is zero? (Of course when I say "every orbit" in the context of ergodic theory I mean "modulo a set of measure zero".)

The space of $L^2$ functions with ergodic average zero is the orthocomplement of the space of invariant $L^2$ functions, under the dynamical inner product $\langle f,g \rangle = \lim_{n \rightarrow \infty} (1/n) \sum_{k=0}^{n-1} f(T^k x)\overline{g}(x)$, so it seems like a natural space for ergodic theorists to consider. And the space of functions that average to zero along orbits seems even more natural in the setting of integrable systems, where evolution laws and conserved quantities can switch places.

I'm also interested in knowing if there's a name for functions whose average value on every orbit is some orbit-independent constant. I don't want to invent terminology for such functions if satisfactory terminology already exists, and I suspect it does, though I haven't been able to find it on the web. I looked at an introductory article on cocycles and coboundaries in ergodic theory, but didn't find what I was looking for.

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I presume you know this already in the L2 setting: a nice proof of the Birkhoff ergodic theorem works by showing that the orthogonal complement of the coboundaries is the collection of invariant functions. So the orthocomplement of the invariant functions is just the closure of the set of coboundaries. –  Anthony Quas Apr 22 '12 at 15:15
An indication that the answer to your question may be negative: in the paper of my colleague, Ian Putnam, Orbit equivalence of Cantor minimal systems: a survey and a new proof'. In the paper he first considers coboundaries and then considers what he calls infinitesimals': functions whose integral with respect to any invariant measure is 0. These presumably are the functions that you want, but the infinitesimal terminology seems to come more from dimension groups/operator algebras than dynamics and is surely not universal. –  Anthony Quas Apr 24 '12 at 17:55

Second part: if your system is nontrivial, and function $f$ is "smooth", than condition of the form $$\lim_{n\to\infty} \frac 1n\sum_{i=0}^{n-1} f(T^i x) =c$$ for all $\ x$, should imply that $$f(x)= g(x)-g(Tx)+c$$ for some function $g$. In this case, one says that $f$ is cohomologous to a constant.
the most famous example of a result of this nature is the so-called Livshic lemma: suppose $X$ is a mixing subshift, $T:X\to X$ is a left shift, and $f$ is a Holder-continuous function such that $$\sum_{i=0}^{p-1} f(T^ix) =0$$ for every periodic $x$: $x=T^px$. Then $f=g-g\circ T$. There are many generalizations of this result. –  user12345678 Apr 23 '12 at 13:31