$\{\phi:\int \phi d\mu=0\}$ for a fixed shift invariant $\mu$ Given a shift invariant probability measure $\mu$ on a mixing subshift of finite type.
What are the Lipschitz functions with zero integral with respect to the measure $\mu?$
Clearly any $\phi\in\{-u+u\circ \sigma: u \mbox{ Lipschitz continuous}\}$ is solution. More general, any
$\phi\in\{\sum _{i=1}^m (-1)^iu\circ \sigma^{a_i}: u \mbox{ Lipschitz continuous}, m \mbox{ even},(a_i)_{i=1}^m\in \mathbb{Z}^m\}$ is solution.
Are there other solutions? How to solve this, is there a general technique?
 A: Even for integer-valued continuous $f: X \to $ Z, there are generically lots of solutions which are not coboundaries (of either integer or real-valued ct functions). For example, suppose that $\mu$ is ergodic, and is supported on a uniquely ergodic minimal subset of the mSFT. Then the GPS (Giordano, Putnam, Skau) results apply, and the dimension group invariant gives plenty of examples with nontrivial infinitesimals. Every nontrivial infinitesimal gives rise to an integer-valued continuous function $f$ which vanishes under integration at $\mu$. On the other hand, if the minimal subset has more than one ergodic measure, then there must be non-coboundaries that integrate to zero (although there need not be any integer-valued examples of these).
I suppose there are some that must be Lipschitz continuous ...
A: I am not quite sure what kind of answer you are looking for, but perhaps this helps ...
Of course every function of the form $g-g\circ\sigma$ (a coboundary) has integral $0$ with respect to every shift-invariant measure.  So does every function in the closed linear span of coboundaries.  Let us call this close linear span $B$.  Are there functions outside $B$ with integral $0$ with respect to $\mu$?
As David mentions above, it can be shown that the property of having $0$ integral with respect to all shift-invariant measures characterizes $B$:


*

*a continuous function $f$ is in $B$ if and only if $\pi(f)=0$ for all
shift-invariant measures $\pi$.


On the other hand, if you have a continuous function $f$, then $f'(\cdot)\triangleq f(\cdot)-\mu(f)$ is a function with integral $0$ with respect to $\mu$.
So, if you are interested in continuous (of Lipschitz) functions with $0$ integral with respect to $\mu$ that are not in $B$, you can look for any continuous (or Lipschitz) function $f$ such that $\pi_1(f)\neq\pi_2(f)$ for two (arbitrary) shift-invariant measures $\pi_1$ and $\pi_2$, and form $f'(\cdot)\triangleq f(\cdot)-\mu(f)$.  This would give you all the possible solutions.
In fact, since you are working with a mixing shift of finite type, you can use the following simpler characterization of Lipschitz functions in $B$:


*

*$f$ is in $B$ if and only if $\overline{f}(x)=0$ for every periodic
point $x$,


where $\overline{f}(x)$ denotes the average of $f$ over one period of $x$.  So, you can look for any function $f$ that has different averages over two periodic points.
