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 $x\in (2^{-n-1},2^{-n}]$ for all $n\ge 0$. We consider the asymptotic properties of the Birkhoff sum $S\_n(x)=\phi (x)+\cdots +\phi (T^{n-1}(x))$. 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 $\{x:\lim \_n\to \infty S\_n(x)/n=\alpha \}(\alpha >0)$, as well as that of the set $\{x:\lim \_n\to \infty S\_n(x)/\Psi (n)=\alpha \}(\alpha >0)$, 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 $x\mapsto 1/x$ is also studied.

17 pages

References in corpus (1)

• Subexponentially increasing sums of partial quotients in continued fraction expansions