Multifractal analysis of the Birkhoff sums of Saint-Petersburg potential

Fractal Geometry and Dynamics

07 September 14:00 - 14:50

Lingmin Liao - Université Paris-Est Créteil Val-de-Marne

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 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 $a>1$, $\gamma>0$. This is a joint work with D. H. Kim, M. Rams and B. W. Wang.
Kenneth J. Falconer
University of St Andrews
Maarit Järvenpää
University of Oulu
Antti Kupiainen
University of Helsinki
Francois Ledrappier
University of Notre Dame
Pertti Mattila
University of Helsinki


Maarit Järvenpää


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