Some Studies Of Cantor Expansions And Quantization Theory

This thesis consists of five chapters. The first chapter is the introduction. In this chapter,we first briefly review the development of fractal geometry and its present studies. Then weintroduce the development and present studies of Cantor expansion and quantization theory.After that, the main research results of this thesis are presented.In the second chapter, we briefly present some preliminary knowledge that is used inthis thesis. First, we introduce the relevant concepts and qualities of measure to prepare usfor further study on fractal geometry. Then we discuss the Hausdorff measure and Hausdorffdimension, as well as the relevant concepts and qualities of box dimension. Next, weintroduce the self-similar set, which is the most important study subject of fractal geometry,and give the dimensions of it. Meanwhile, we also introduce the qualities of self-similarmeasure, and afterwards, we give the dimensions of homogeneous Moran set. At last, weintroduce the basic concepts of quantization theory and make comparisons between quantizationdimension and Hausdorff dimension, box measure.The following three chapters are the main parts of the thesis. In chapter three, weconsider a class of fractals generated by the Cantor series expansions. By constructing somehomogeneous Moran subsets nicely, we prove that these sets have full dimension.In chapter four, let {f_i}_i=1^N be a family of contracting similitudes with contraction rations{c_i}_i=1^N. Letμ_s be the self-similar measure associated with {f_i}_i=1^N and the probabilityvector (c₁^s,…, c_N^s). Under the strong separation condition, we determine the point densitymeasure for probability measures v<<μ_s.In the last chapter, we will extend the above result to the F-conformal measureμon R which is associated with a finite conformal iterated function system. LetF={f⁽ⁱ⁾: 1≤i≤N} be a strongly H(o|¨)lder family of continuous functions on Rand let {φ_i: 1≤i≤N} be a conformal iterated function system. We determine theupper(lower)quantization dimension (of order zero) of the F-conformal measure withrespect to the geometric mean error.