Abstract
Even Poincaré had constructed an example which implies the existence of an irrational rotation of the circle and a function continuous on it with zero mean for which the Birkhoff sums at some points tend to infinity as the number of iterations grows. The strict ergodicity in this case is a natural constraint on the growth rate of Birkhoff sums: the sequence of Birkhoff means uniformly tends to zero on the circle. The paper shows that any prescribed admissible rate of growth of Birkhoff sums within the ergodic theorem can be realized, and the set of points at which the sums grow with a given speed is massive: it has the Hausdorff dimension one.
Similar content being viewed by others
Notes
The existence of topologically transitive cylindrical mappings in a more general situation was studied in [5].
References
H. Poincaré, “Mémoire sur les courbes définies par une équation différentielle, (I)–(II),” Journal de Mathématiques Pures et Appliquées 7, 375–422 (1881); 8 251 (1882); “Sur les courbes définies par les équations différentielles (III–IV) ,” Journal de Mathématiques Pures et Appliquées (Ser. 4) 1, 167–244 (1885); “Sur les courbes définies par les équations différentielles (quatriéme partie) ,” (1886), Vol. 2, pp. 151–217.
H. Poincaré, “Sur les series trigonometriques,” Comptes rendus 101 (2), 1131–1134 (1885).
V. V. Kozlov, Methods of Qualitative Analysis in the Dynamics of a Rigid Body (Nauchno-Izdatel’skii Tsentr “Regulyarnaya i Khaoticheskaya Dinamika”, Izhevsk, 2000) [in Russian].
W. H. Gottschalk and G. A. Hedlund, Topological Dynamics, in Amer. Math. Soc. Colloq. Publ. (Amer. Math. Soc., Providence, RI, 1955), Vol. 36.
E. A. Sidorov, “Topological transitivity of cylindrical cascades,” Math. Notes 14 (3), 810–816 (1973).
A. V. Kochergin, “On the absence of mixing in special flows over the rotation of a circle and in flows on a two-dimensional torus,” Soviet Mathematics - Doklady 13 (4), 949–952 (1972).
V. V. Kozlov, “On a problem of Poincaré,” J. Appl. Math. Mech. 40 (2), 326–329 (1976).
A. B. Krygin, “\(\omega\)-Limit sets of smooth cylindrical cascades,” Math. Notes 23 (6), 479–485 (1978).
E. A. Sidorov, “Conditions for uniform Poisson stability of cylindrical systems,” Russian Math. Surveys 34 (6), 220–224 (1979).
A. S. Besicovitch, “A problem on topological transformations of the plane. II,” Proc. Cambridge Philos. Soc. 47, 38–45 (1951).
K. Fraczek and M. Lemańczyk, “On the Hausdorff dimension of the set of closed orbits for a cylindrical transformation,” Nonlinearity 23 (10), 2393–2422 (2010).
A. V. Kochergin, “Besicovitch cylindrical transformation with a Hölder function,” Math. Notes 99 (3), 382–389 (2016).
A. V. Kochergin, “New examples of Besicovitch transitive cylindrical cascades,” Sb. Math. 209 (9), 1257–1272 (2018).
E. Dymek, Transitive Cylinder Flows Whose Set of Discrete Points Is of Full Hausdorff Dimension, arXiv: math. DS/1303.3099v1.
N. G. Moshchevitin, “Distribution of values of linear functions and asymptotic behavior of trajectories of some dynamical systems,” Math. Notes, 58 (3), 948–959 (1995).
A. B. Antonevich and Ali A. Shukur, “Estimations of the norm of the powers of the operator generated by irrational rotation,” Dokl. Nats. Akad. Nauk Belarusi 61 (1), 30–35 (2017).
A. B. Antonevich, A. V. Kochergin, and A. A. Shukur, “Behaviour of Birkhoff sums generated by rotations of the circle,” Sb. Math. 213 (7), 891–924 (2022).
A. Ya. Khinchin, Continued Fractions (University of Chicago Press, Chicago, 1964).
K. Falconer, Fractal Geometry. Mathematical Foundations and Applications (Wiley, Hoboken, NJ, 2003).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Matematicheskie Zametki, 2023, Vol. 113, pp. 836–848 https://doi.org/10.4213/mzm13654.
Rights and permissions
About this article
Cite this article
Kochergin, A.V. On the Growth of Birkhoff Sums over a Rotation of the Circle. Math Notes 113, 784–793 (2023). https://doi.org/10.1134/S0001434623050206
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434623050206