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Fast and slow points of Birkhoff sums
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 40 / Issue 12 / December 2020
- Published online by Cambridge University Press:
- 11 July 2019, pp. 3236-3256
- Print publication:
- December 2020
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- Article
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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.
A POLYNOMIAL APPROACH TO COCYCLES OVER ELEMENTARY ABELIAN GROUPS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 85 / Issue 2 / October 2008
- Published online by Cambridge University Press:
- 01 October 2008, pp. 177-190
- Print publication:
- October 2008
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- Article
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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.
-dimensional GF(2