Multifractal analysis of the Birkhoff sums of Saint-Petersburg potential

Abstract

Let $((0,1], T)$ be the doubling map in the unit interval and $φ$ be the Saint-Petersburg potential, defined by $φ(x)=2^n$ if $x\in (2^{-n-1}, 2^{-n}]$ for all $n\geq 0$. We consider the asymptotic properties of the Birkhoff sum $S\_n(x)=φ(x)+\cdots+φ(T^{n-1}(x))$. With respect to the Lebesgue measure, the Saint-Petersburg potential is not integrable and it is known that $\frac{1}{n\log n}S\_n(x)$ converges to $\frac{1}{\log 2}$ in probability. We determine the Hausdorff dimension of the level set $\{x: \lim\_{n\to\infty}S\_n(x)/n=α\} \ (α>0)$, as well as that of the set $\{x: \lim\_{n\to\infty}S\_n(x)/Ψ(n)=α\} \ (α>0)$, when $Ψ(n)=n\log n, n^a $ or $2^{n^γ}$ for $a>1$, $γ>0$. The fast increasing Birkhoff sum of the potential function $x\mapsto 1/x$ is also studied.