On The Level Sets Related To The Run-length Function

In this thesis.we study the Hausdorff dimension of the level sets in fractal geometry related t0 the run-length function.For any x∈[0,1),let x=∈1(x)/2∈2(x)/22+…+∈n(x)/2n+… be the dyadic expansion of x.We shall write x=(∈1,…,∈n…)for brevity.For each n>1 and x∈[0,1),the run-length function rn(x)is defined to be the length of the longest run of 1’s in(∈1,…,∈n),that is rn(x)=max{j≥1:∈i+1=…∈i+j,0≤i≤n-j}. In order to ensure uniqueness of dyadic expansion,we only write terminating expansions in the form with infinitely many 0’s.Because 0 and 1 are symmetrical.we define ln(x)=max{l≥1:∈i+1=…=∈i+1=0,0≤i≤n-l). In this thesis,we consider the Hausdorff dimensions of the following sets: where{δn)n=1∞1,{θn}n=1∞ are nondecreasing integer sequences with δnâ†'∞,θnâ†'∞ as nâ†'∞.We obtain the following results:(2)If δn≤n,θn≤n.Then where α-lim infnâ†'∞ δn/n,β=lim infnâ†'∞ θn/n.This thesis is organized as followos:In Chapter 1,the history and the development of fractal geometry and run-length function are presented.In Chapter 2,we introduce the concepts and properties of frractal dimensions,symbol space,run-length function and Moran sets.Chapter 3 and 4,the main part of this thesis,include the main results of my research.In Chapter 3,we construct a Moran set and prove that it has Hausdorff dimension 1.Moreover,we prove that the Moran set is a subset of the level set E({δn}n=1∞,{θn}n=1∞), then we get the Hausdorff dimension of E({δn)n=1∞1,{θn}n=1∞).In Chapter 4,we consider the Hausdorff dimension of the set F({δn)n=1∞,{θn)n=1∞) by the method similar to chapter 3.