Geodesic
An Additive Cohomological Equation and Typical Behavior of Birkhoff Sums over a Translation of the Multidimensional Torus
Informatics and Automation, Dynamical systems and optimization, Tome 256 (2007), pp. 278-289 
A. V. Rozhdestvenskii. An Additive Cohomological Equation and Typical Behavior of Birkhoff Sums over a Translation of the Multidimensional Torus. Informatics and Automation, Dynamical systems and optimization, Tome 256 (2007), pp. 278-289. http://geodesic.mathdoc.fr/item/TRSPY_2007_256_a14/
@article{TRSPY_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 = {Informatics and Automation},
     pages = {278--289},
     year = {2007},
     volume = {256},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2007_256_a14/}
}
TY  - JOUR
AU  - A. V. Rozhdestvenskii
TI  - An Additive Cohomological Equation and Typical Behavior of Birkhoff Sums over a Translation of the Multidimensional Torus
JO  - Informatics and Automation
PY  - 2007
SP  - 278
EP  - 289
VL  - 256
UR  - http://geodesic.mathdoc.fr/item/TRSPY_2007_256_a14/
LA  - ru
ID  - TRSPY_2007_256_a14
ER  - 
%0 Journal Article
%A A. V. Rozhdestvenskii
%T An Additive Cohomological Equation and Typical Behavior of Birkhoff Sums over a Translation of the Multidimensional Torus
%J Informatics and Automation
%D 2007
%P 278-289
%V 256
%U http://geodesic.mathdoc.fr/item/TRSPY_2007_256_a14/
%G ru
%F TRSPY_2007_256_a14

Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Kolmogorov A.N., “O dinamicheskikh sistemakh s integralnym invariantom na tore”, DAN SSSR, 93 (1953), 763–766 | MR | Zbl

[2] Gottschalk W.H., Hedlund G.A., Topological dynamics, AMS Colloq. Publ., 36, Amer. Math. Soc., Providence (RI), 1955 | MR | Zbl

[3] Anosov D.V., “Ob additivnom funktsionalnom gomologicheskom uravnenii, svyazannom s ergodicheskim povorotom okruzhnosti”, Izv. AN SSSR. Ser. mat., 37:6 (1973), 1259–1274 | MR | Zbl

[4] Katok A., Robinson E.A., Jr., “Cocycles, cohomology and combinatorial constructions in ergodic theory”, Smooth ergodic theory and its applications (Seattle (WA), 1999), Amer. Math. Soc., Providence (RI), 2001, 107–173 | MR | Zbl

[5] Moore C.C., Schmidt K., “Coboundaries and homomorphisms for nonsingular actions and a problem of H. Helson”, Proc. London Math. Soc. Ser. 3, 40 (1980), 443–475 | DOI | MR | Zbl

[6] Kozlov V.V., “Ob odnoi zadache Puankare”, PMM, 40:2 (1976), 352–355 | MR | Zbl

[7] Lemanczyk M., Parreau F., Volný D., “Ergodic properties of real cocycles and pseudo-homogeneous Banach spaces”, Trans. Amer. Math. Soc., 348:12 (1996), 4919–4938 | DOI | MR | Zbl

[8] Shmidt V., Diofantovy priblizheniya, Mir, M., 1983

[9] Groshev A.V., “Teorema o sisteme lineinykh form”, DAN SSSR, 19 (1938), 151–152 | Zbl

[10] Sprindzhuk V.G., Metricheskaya teoriya diofantovykh priblizhenii, Nauka, M., 1977 | MR

[11] Moschevitin N.G., “Raspredelenie znachenii lineinykh funktsii i asimptoticheskoe povedenie traektorii nekotorykh dinamicheskikh sistem”, Mat. zametki, 58:3 (1995), 394–410 | MR | Zbl

[12] Zigmund A., Trigonometricheskie ryady, t. 2, Mir, M., 1965 | MR

[13] Keipers L., Niderreiter G., Ravnomernoe raspredelenie posledovatelnostei, Nauka, M., 1985

[14] Schmidt K., Cocycles on ergodic transformation groups, Macmillan Lect. Math., 1, Macmillan, Delhi etc., 1977 | MR | Zbl