The Hausdorff Dimension Of The Graph Of The Generalized Takagi Function

Fractal sets are the basic object on studying fractal geometry. Fractal dimension, on the other hand, plays an important role in the research of fractal geometry theory. When people do theoretical or practical research related to time, fractals often occur. For example, the stock function in finance and the wind velocity function in physics, are all well-defined fractals. Thus, it is important to study fractals and the properties they left behind.This thesis studies the Hausdorff dimension of the graph of the generalized Takagi function. Wc consider a generalization of classical Takagi function and definite it as the generalized Takagi function, that is where 1< s< 2, x ∈ I=[0,1], aj>1,φ(x)= min{x-n|:n ∈N}. This function is continuous but nowhere differentiable, its graph is a kind of typical fractal set.Inspired in Daochun Sun and Zhiying Wen’s work, we give{aj} a suitable gap condition, that is αj+1/αj(?)∞,jâ†'∞. In this condition, we can determine the Hausdorff dimension of the graph of the generalized Takagi function by Natural Cover and Quality Distribution Principle, the result isThe result can get the following corollarys:(1) when logan+i/loganâ†' 1, nâ†'∞, we have dimH G(K,I)= s (The result has proved by Besicovitch and Ursellt using other method);(2) when logan+1/loganâ†'∞,nâ†'∞, we have dimH G(K, I)=1.Finally, we consider two types of generalized Takagi functions containing phase "θ" and use similar methods of proving theorem 4.8 and theorem 4.9, we have that three types of functions’dimensions remain the same.