Selfsimilarity and growth in Birkhoff sums for the golden rotation

TL;DR

This paper investigates the behavior of Birkhoff sums for the golden rotation, revealing a connection between their growth, a limiting function, and a functional equation related to critical KAM phenomena.

Contribution

It introduces a new limit function for Birkhoff sums at the golden rotation and links its properties to a functional equation, advancing understanding of critical KAM phenomena.

Findings

Existence of a limit function f(x) for Birkhoff sums

f(x) satisfies a specific functional equation

Limit of Birkhoff sums expressed via f(x)

Abstract

We study Birkhoff sums S(k,a) = g(a)+g(2a)+...+g(ka) at the golden mean rotation number a with periodic continued fraction approximations p(n)/q(n), where g(x) = log(2-2 cos(2 pi x). The summation of such quantities with logarithmic singularity is motivated by critical KAM phenomena. We relate the boundedness of log- averaged Birkhoff sums S(k,a)/log(k) and the convergence of S(q(n),a) with the existence of an experimentally established limit function f(x) = lim S([x q(n)])(p(n+1)/q(n+1))-S([x q(n)])(p(n)/q(n)) for n to infinity on the interval [0,1]. The function f satisfies a functional equation f(ax) + (1-a) f(x)= b(x) with a monotone function b. The limit lim S(q(n),a) for n going to infinity can be expressed in terms of the function f.