Geodesic
An Additive Cohomological Equation and Typical Behavior of Birkhoff Sums over a~Translation of the Multidimensional Torus
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Dynamical systems and optimization, Tome 256 (2007), pp. 278-289

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For a periodic function $f$ with a given decrease of the moduli of its Fourier coefficients, we analyze the solvability of the equation $w(T_\alpha x)-w(x)=f(x)-\int_{\mathbb T^d}f(t)\,dt$ and the asymptotic behavior of the Birkhoff sums $\sum _{s=0}^{n-1} f(T^s_\alpha x)$ 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. 
@article{TM_2007_256_a14,
     author = {A. V. Rozhdestvenskii},
     title = {An {Additive} {Cohomological} {Equation} and {Typical} {Behavior} of {Birkhoff} {Sums} over {a~Translation} of the {Multidimensional} {Torus}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {278--289},
     publisher = {mathdoc},
     volume = {256},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2007_256_a14/}
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A. V. Rozhdestvenskii. An Additive Cohomological Equation and Typical Behavior of Birkhoff Sums over a~Translation of the Multidimensional Torus. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Dynamical systems and optimization, Tome 256 (2007), pp. 278-289. http://geodesic.mathdoc.fr/item/TM_2007_256_a14/