Although each cocycle for a action of the integers is
specified by the sequence of Birkhoff sums of a function,
it is relatively difficult to specify cocycles for the actions of
multidimensional groups such as $\Bbb Z^2$.
We'll see that if $(X,T)$ is a transitive action of the finitely
generated (countable) group $\Gamma$ by homeomorphism of the polish space $X$,
and $\Bbb B$ is a separable Banach space, there is a cocycle
$F:\Gamma\times X \to\Bbb B$
with each $x\mapsto F(g,x)$ bounded and continuous
so that the skew product action $(X x \Bbb B,S)$ is transitive where
$S_g(x,b)=(T_gx,b+F(g,x))$.
Depending on time available, we'll also discuss measure theoretic analogues.
This result was shown for transitive actions of the integers by E.A.
Sidorov in 1973.
Joint work with Benjamin Weiss: arXiv:1712.05196