Abstract: In this dissertation,we mainly study the quantization dimension of random self-similar measureÎ¼supported on the random self-similar set K(Ï‰) and show two purely atomic cases of doubling measures on compact sets.In Chapter 1,we briefly review the fractal naissance and give some fundamental concepts and properties of the fractal theory which conclude Hausdorff dimension,IFS,measure dimension,Symbol space and the strong separation condition and the open set condition.In Chapter 2,we introduce some knowledge of probability theory firstly,then give some fundamental concepts and properties of the random fractal and especially introduce the random cantor set.In Chapter 3,we study the quantization dimension of random self-similar measureÎ¼supported on the random self-similar set K(Ï‰). We establish a relationship between the quantization dimension ofÎ¼and its distribution,then we give a simple example to show that how to use the formula of the quantization dimension.In Chapter 4,we consider the purely atomic cases of doubling measures on compact sets.We show that there are two compact sets X,Y whose dimensions are both in(0,1),and all doubling measures on X are purely atomic,while any doubling measure on Y isn't purely atomic.In Chapter 5,we propose the ideas of work in the next step…

Key words: Hausdorff dimension; random self-similar set; random measure; quantization dimension; purely atomic; doubling measure; cantor set

# Quantization Dimension of Random Measure and Purely Atomic Cases of Doubling Measure

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