MULTIFRACTAL ANALYSIS OF THE BIRKHOFF SUMS OF SAINT-PETERSBURG POTENTIAL

Abstract

Let ((0, 1], T) be the doubling map in the unit interval and phi be the Saint-Petersburg potential, defined by phi(x) = 2(n) if x is an element of (2(-n-1),2(-n)] for all n >= 0. We consider asymptotic properties of the Birkhoff sum S-n(x) = phi(x) + . . . + phi(Tn-1 (x)). With respect to the Lebesgue measure, the Saint-Petersburg potential is not integrable and it is known that 1/n log n S-n(x) converges to 1/log 2 in probability. We determine the Hausdorff dimension of the level set {x : lim(n ->infinity)S(n)(x)/n = alpha} (alpha > 0), as well as that of the set {x: lim(n ->infinity) S-n(x)/Psi(n) = alpha} (alpha > 0), when Psi(n) = n log n, n(a) or 2(n gamma) for a > 1, gamma > 0. The fast increasing Birkhoff sum of the potential function x bar right arrow 1/x is also studied.