Dissertation > Mathematical sciences and chemical > Physics > Theoretical Physics > Nonlinear physics > Chaos Theory

Generalized Besicovitch-Eggleston Function and Self-similar Structure of Cartesian Product of Cantor Sets

Author YeFuGeNi·PaTeLiKeYeFu
Tutor LiWenXia
School East China Normal University
Course Basic mathematics
Keywords Hausdorff dimension Self-similarity Besicovitch-Eggleston function Generalised Cantorsets
CLC O415.5
Type Master's thesis
Year 2013
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In the present paper, we study two kinds of fractal sets. The first one is the graph of the Besicovitch-Eggleston vector function. This function is defined by the frequencies of digits, occurring in the N-adic expansion of the variable and its graph is a subset of [0,1]N with complicated geometric structure and zero-value topological dimension. We explore its properties and show it has Hausdorff dimension N. The second one is related to the Cartesian product of the middle-(1-2β) Cantor sets. Such sets are a natural generalization of the classical middle-third Cantor sets. Compared with the latter they are constructed by removing middle-(1-2β) intervals on each step, instead of one third. We obtain the sufficient and necessary conditions for the intersection of this product with its translation to be a self-similar set.

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