Behaviour of Birkhoff sums generated by rotations of the circle

Author:

Antonevich Anatolij Borisovich1,Kochergin Andrey Vasilyevich2,Shukur Ali Abdulhussein31

Affiliation:

1. Belarusian State University, Minsk, Belarus

2. Faculty of Economics, Lomonosov Moscow State University, Moscow, Russia

3. Faculty of Computer Science and Mathematics, University of Kufa, Kufa, Iraq

Abstract

For continuous functions $f$ with zero mean on the circle we consider the Birkhoff sums $f(n,x,h)$ generated by the rotations by $2\pi h$, where $h$ is an irrational number. The main result asserts that the growth rate of the sequence $\max_x f(n,x,h)$ as $n \to \infty$ depends only on the uniform convergence to zero of the Birkhoff means $\frac{1}{n}f(n,x,h)$. Namely, we show that for any sequence $\sigma_k \to 0$ and any irrational $h$ there exists a function $f$ such that the sequence $\max_x f(n,x,h)$ increases faster than $n\sigma_n$. We also show that for any function $f$ that is not a trigonometric polynomial there exist irrational $h$ for which some subsequence $\max_x f(n_k,x,h)$ increases faster than the corresponding subsequence $n_k\sigma_{n_k}$. We present applications to weighted shift operators generated by irrational rotations and to their resolvents. Namely, we show that the resolvent of such an operator can increase arbitrarily fast in approaching the spectrum. Bibliography: 46 titles.

Publisher

Steklov Mathematical Institute

Subject

Algebra and Number Theory

Reference69 articles.

1. Ergodic Theory

2. Introduction to the Modern Theory of Dynamical Systems

3. Sur les courbes définies par les équations différentielles;H. Poincaré;C. R. Acad. Sci. Paris,1882, 1884

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