Let $((0,1], T)$ be the doubling map in the unit interval and $\varphi$ be the Saint-Petersburg potential, defined by $\varphi(x)=2^n$ if $x\in (2^{ -n-1}, 2^{ -n}]$ for all $n\geq 0$. We consider the asymptotic properties of the Birkhoff sum $S\_n(x)=\varphi(x)+\cdots+\varphi(T^{ n-1}(x))$. 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.
Categories: math.DS