Cantor Set Structure And The Corresponding Fractal Dimension Calculated

The thesis is devoted to three kinds of Cantor sets that are changed and promoted based on the classics Cantor sets,then research on their dimensions and measure.Main new points:First,a class of general 3-part Cantor set is studied.Main conclunsions:(1)The formula of its box dimension is established;(2)The Hausdorff measure of the set is proved;(3)A new method to calculate the box dimension of separated and contacted fractals is obtained.Second,the homogeneous Cantor sets are studied.Main conclusions: (1)The arithmetic sum of two homogeneous Cantor sets X_ωand Y_ωis still a homogeneous Cantor set;(2)The relationship among X_ω,Y_ω,and X_ω+ Y_ωis obtained;(3)The relationship among X_ω,Y_ωand X_ω+ Y_ωis obtained.Third,overlap Cantor set is studied.The formula of Hausdorff dimension of incompletely overlap Cantor set is established.