functions whose average along orbits is zero or a constant

2012-04-22T06:12:38Z

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.

Answer by waler for functions whose average along orbits is zero or a constant

2012-04-22T09:23:50Z

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.

functions whose average along orbits is zero or a constant - MathOverflow

functions whose average along orbits is zero or a constant

Answer by waler for functions whose average along orbits is zero or a constant