Multifractal analysis of the Birkhoff sums of Saint-Petersburg potential

TL;DR

This paper investigates the multifractal structure of Birkhoff sums for the Saint-Petersburg potential under the doubling map, determining Hausdorff dimensions of various level sets and analyzing sums with different growth rates.

Contribution

It provides the first detailed Hausdorff dimension analysis of level sets for Birkhoff sums of the Saint-Petersburg potential with various normalization functions.

Findings

Hausdorff dimension of level sets for normalized Birkhoff sums is characterized.

Dimension results depend on the growth rate of the normalization function.

Analysis includes sums with super-logarithmic growth, such as $2^{n^eta}$.

Abstract

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.