The Research Of Hausdorff Measures And Dimensions Of Generalized Sierpinski Gaskets

In fractal geometry, estimation and calculation of dimensions and measures are important but also difficult problems. So far, except for few special fractals (such as Cantor set [1]), dimensions and measures of most fractals are still problems we need to solve. Even for the Sierpinski-gasket, one of three classical self-similar sets, calculation of its dimension is so easy, but the measure is very difficult. So far, we just have an upper and lower bound of it.However, when changing compression ratio c, the fractal graph are transformed from disconnected to overlapping structure when c increases from0to1, even with full overlapping structure for some special c.(In this paper, Sierpinski-gasket simple denoted as S while the author has a generalized Sierpinski gasket simple denoted as S*). Obviously, the calculation methods of S and S*are completely different. The literature [14] had discussed Hausdorff measure of S for and got an conclusion:(while In this paper, the author discussed the measure and dimension of Sierpinski gasket for other ranges:-and For different c, this paper calculated them with different methods.(In particular, while S*has overlapping structure at this time, we can see the discussion of S*is less for the present. The author discussed their Hausdorff dimensions and an upper bound of the Hausdorff measures).In the introduction, the writer briefly introduced conception of fractal, research status of this problem we discuss and main working in this paper.The second chapter described Hausdorff measure and dimension in details, including their concepts and properties.In the third chapter, the author introduced the structure of self-similar sets, dimension of self-similar sets, open sets condition and self-similar sets with overlapped.The forth and fifth chapters, the author discussed Hausdorff measure and dimension of some special Sierpinski gaskets with different compression ratio c. Then this paper has the following conclusion:while through the projection and structural mass distribution, the calculations of Cantor set measure and Sierpinski gasket are combined, then this paper has this conclusion:Hs (S)=1.The fifth chapter got a Sierpinski-dust S*using the same compression maps with S while attested that S*is a self-similar set. The author also discussed structure of S*while it is overlapped at this time. Then the author has a compact C={c1,c2…cn}(c1is the solution of the equation c1+1+c1+c-1=0) And while c∈C, S*have completely overlapping structure and dim B S*<s. While c∈C, S*satisfies open set condition, dimH...