Behaviour of Birkhoff sums
generated by rotations of the circle
الباحث الأول:
A. B. Antonevich
الباحثين الآخرين:
A. V. Kochergin and A. A. Shukur
المجلة:
Sbornik: Mathematics
تاريخ النشر:
None
مختصر البحث:
For continuous functions f with zero mean on the circle we
consider the Birkhoff sums f(n, x, h) generated by the rotations by 2πh,
where h is an irrational number. The main result asserts that the growth
rate of the sequence maxx f(n, x, h) as n…
For continuous functions f with zero mean on the circle we
consider the Birkhoff sums f(n, x, h) generated by the rotations by 2πh,
where h is an irrational number. The main result asserts that the growth
rate of the sequence maxx f(n, x, h) as n → ∞ depends only on the uniform
convergence to zero of the Birkhoff means 1
n f(n, x, h). Namely, we show
that for any sequence σk → 0 and any irrational h there exists a function f
such that the sequence maxx f(n, x, h) increases faster than nσn. We also
show that for any function f that is not a trigonometric polynomial there
exist irrational h for which some subsequence maxx f(nk, x, h) increases
faster than the corresponding subsequence nkσnk .
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.
