Multifractal Analysis On The Fast Increasing Birkhoff Sums Of Saint-Petersburg Potential

Multifractal analysis is a vital tool to study the dimension theory in dynamical systems.In this paper,we are mainly concerned with multifractal analysis of fast increasing Birkhoff sums of Saint-Petersburg potential.Let((0,1],T)be the doubling map in the unit interval and ? be the Saint-Petersburg potential,defined by ?(x)=2n if x? 2-n-1,2-n for all n?0.Thus,Saint-Petersburg potential is locally constant,not continued and unbound.We consider the properties of Birkhoff sum Sn?(x)=?(x)+?(T(x))+…+?(Tn-1(x)).With respect to the Lebesgue measure,the Saint-Petersburg potential is not integrable and Sn?(x)/nlogn converges to 1/log2 in probability.Subsequently,we determine the Hausdorff dimension of the following set (?) when ?(n)=na(logn)b,a?1,b?1 or ?(n)=2(logn)?,?>0 and ??1.Furthermore,we also study the fast increasing Birkhoff sums of the potential function x(?)1/x,and show the Hausdorff dimension of its corresponding sets.In Chapter 1,we present overall development of fractal geometry and multifractal,and show their research backgrounds,related concepts and main research contents.In Chapter 2,we give the preliminaries.In Chapter 3,the origin of the name of Saint-Petersburg potential is well explained,and the related concepts and conclusions of Saint-Petersburg potential are elaborated.In Chapter 4,we determine the Hausdorff dimension of the set{x?(0,1]:limn??Sn?(x)/?(x)=?}(??0),when ?(n)=na(log n)b,a?1,b?1 or?(n)=2(logn)?,?>0 and ??1.The fast increasing Birkhoff sums of the potential function x(?)1/x is also studied.In the last chapter,we sum up the innovations of this paper and show some prospects for further study.