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An additive cohomological equation and typical behavior of Birkhoff sums over a translation of the multidimensional torus. (English. Russian original) Zbl 1153.37304
Proc. Steklov Inst. Math. 256, 263-274 (2007); translation from Tr. Mat. Inst. Steklova 256, 278-289 (2007).
Summary: For a periodic function $$f$$ with a given decrease of the moduli of its Fourier coefficients, we analyze the solvability of the equation $$w(T_\alpha x) - w(x) = f(x) - \smallint \_{\mathbb{T}^d } f(t) dt$$ and the asymptotic behavior of the Birkhoff sums $$\Sigma _{s=0}^{n-1} f(T_{\alpha}^s x)$$ for almost every $$\alpha$$. The results obtained are applied to the study of ergodic properties of a cylindrical cascade and of a special flow on the torus.
##### MSC:
 37A05 Dynamical aspects of measure-preserving transformations 37A30 Ergodic theorems, spectral theory, Markov operators 42B05 Fourier series and coefficients in several variables
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