We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We investigate the growth rate of the Birkhoff sums 
$S_{n,\unicode[STIX]{x1D6FC}}f(x)=\sum _{k=0}^{n-1}f(x+k\unicode[STIX]{x1D6FC})$, where 
$f$ is a continuous function with zero mean defined on the unit circle 
$\mathbb{T}$ and 
$(\unicode[STIX]{x1D6FC},x)$ is a ‘typical’ element of 
$\mathbb{T}^{2}$. The answer depends on the meaning given to the word ‘typical’. Part of the work will be done in a more general context.
We derive bivariate polynomial formulae for cocycles and coboundaries in Z2(ℤpn,ℤpn), and a basis for the (pn−1−n)-dimensional GF(pn)-space of coboundaries. When p=2 we determine a basis for the 
-dimensional GF(2n)-space of cocycles and show that each cocycle has a unique decomposition as a direct sum of a coboundary and a multiplicative cocycle of restricted form.
