On the Recurrence of Birkhoff Sums for a Circle Rotation and a Hölder Function

Key Points

• This research examines the recurrence behavior of Birkhoff sums for a zero-mean Hölder function on the circle under certain rotation conditions.
• Analyzed properties of zero-mean Hölder functions on the circle.
• Investigated conditions under which irrational rotations are approximated by rational ones.
• Studied the sequence of Birkhoff sums at various points on the circle.
• Established that under specific approximation conditions, Birkhoff sums remain finite across the circle.
• Identified that the nature of the rotation affects the finiteness of Birkhoff sums.